[{"data":1,"prerenderedAt":152585},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F0012-shu-xue-yu-hua-jia":3,"i-ci:tag":-1,"i-mingcute:time-line":-1},{"post":4,"nextPost":203,"prevPost":20148},{"id":5,"title":6,"body":7,"description":189,"draft":190,"enableCodeHighlight":190,"enableCodeHightlight":190,"enableComment":191,"enableD2":190,"enableKatex":190,"extension":192,"image":186,"important":190,"location":193,"meta":194,"navigation":191,"ogImage":193,"path":195,"seo":196,"stem":197,"summary":198,"tags":199,"time":201,"__hash__":202},"blog\u002Fblog\u002F2026\u002F0012-shu-xue-yu-hua-jia.md","数学与画家",{"type":8,"value":9,"toc":185},"minimark",[10,23,54,57,60,63,66,69,72,75,78,81,84,87,90,93,96,99,102,105,108,111,114,117,120,123,126,129,132,135,138,141,144,147,150,153,156,159,162,165,168,171,174,177,180],[11,12,13,14,22],"p",{},"前段时间，OpenAI 声称，他们用内部尚未正式发布的模型产出了七百多篇论文，「AI 接替数学家」的叙事于是再一次被推到台前。市场是狂热还是炒作，我已经疲于争论。那七百篇论文是否名副其实，也先放在一边。真正让我想写点东西的，",[15,16,21],"a",{"href":17,"rel":18,"target":20},"https:\u002F\u002Fx.com\u002FNoe1017_Monad\u002Fstatus\u002F2112008540592410951",[19],"nofollow","_blank","是 Twitter 上一条获得不少认同的帖子","。它把数学结构说成人类为了迁就自己那点脑容量而被迫发明的拐杖，把 AI 说成不需要几何直观、可以直接站在高维展开式里往下看的怪物。最后一句尤其漂亮：我们终于拿到了上帝账本里的答案，却发现上帝算账根本不用微积分，他老人家是直接拿计算器硬按出来的。",[24,25,26],"blockquote",{},[11,27,28,29,32,33,35,36,38,39,41,42,44,45,47,48,50,51,53],{},"所谓优雅的数学结构，本质上是人类为了迁就自己那点可怜的脑容量，被迫发明出来的拐杖。  ",[30,31],"br",{},"\n一个顶尖人类数学家的工作内存，撑死也就同时追踪五六个变量的交互。  ",[30,34],{},"\n当你在算一个偏微分方程或者组合不等式时，一旦展开到三阶以上，几十个交叉项涌出来，人脑就直接宕机了。  ",[30,37],{},"\n所以人类数学家极其厌恶展开高阶项，一旦看到高阶项发散，人类的本能反应是此路不通，赶紧退回来找对称性、找不变量、搞同调代数或者几何直观，试图不展开式子就把问题绕过去。  ",[30,40],{},"\n但 AI 没有这个生理缺陷。  ",[30,43],{},"\n对于上万个并发的 Agent 来说，同时追踪一万个五阶交叉项的系数变化，跟我们算 1+1 一样轻松。  ",[30,46],{},"\n在它的视野里，根本不需要什么几何直观，它直接站在高维的代数展开式里往下看，人类觉得像乱麻一样的百项高阶余项，在它眼里就是一张明明白白的线性方程组。  ",[30,49],{},"\n既然把式子硬展开到第七阶，再凑几个待定系数就能让所有讨厌的发散项精准归零，它为什么要费劲去帮你发明一个人类能看懂的新概念？  ",[30,52],{},"\n这 700 个结果背后最无奈的黑色幽默：我们终于拿到了上帝账本里的答案，却发现上帝算账根本不用微积分，他老人家是直接拿计算器硬按出来的。",[11,55,56],{},"这套论证值得认真对待，恰恰因为它错得很有代表性：它把人类的工作记忆、数学的审美偏好、AI 的暴力计算串成了一条看似严密的因果链。但这条链中间有一个关键的偷换——它把「人类因为认知限制而偏爱结构」偷换成了「结构只是认知限制的产物」。前者是发生学上的事实，后者是一个关于数学本体的判断。两者之间隔着一整个哲学鸿沟。",[11,58,59],{},"画家这个比喻，也许比「拐杖」更接近真相。画家画不出世界上的每一个光子，所以他发明了透视、明暗和构图。但这些都长在真实的结构上：透视是射影几何，明暗是光的传播，构图是视觉心理。一个能逐像素渲染世界的渲染器，仍然跑在同一套射影几何上。暴力计算并没有绕开结构。它把结构执行到了人眼跟不上的规模。数学里的「硬展开」也一样。",[11,61,62],{},"它对人类工作记忆的描述大体成立，对数学的判断则基本错了。结构就是数学本身，拐杖之说站不住。AI 同样依赖结构，只不过那种结构人读不懂。问题在于「能不能翻译」，美丑反而次要。而翻译，恰恰是当前 AI 数学产出链条上最缺失的一环。",[11,64,65],{},"先承认其中合理的部分。人类的工作记忆确实窄得可怜。Miller 在 1956 年给出著名的 7±2，后来 Cowan 又把它修正到 4±1 左右。数字有争议，但「窄」这个结论没有被推翻。数学家偏爱对称性、不变量、同调，厌恶把式子硬展开，很大程度上确实是认知经济学的产物：找不变量，本质上是把高维信息压缩成一两个人类握得住的符号。AI 在大规模符号展开和并行系数追踪上的优势也是真实的。AlphaTensor 找矩阵乘法算法、各种组合搜索里那些看似暴力、实则依赖结构化搜索空间的路线，确实绕开了人类偏好的那条美学路径。",[11,67,68],{},"问题出在下一步。从「人类因为认知限制而偏爱结构」推到「结构只是认知限制的产物」，这是一个发生学谬误。一个东西因为什么被发明出来，并不等于它是什么。望远镜是因为人眼太弱才被造出来的，但这不意味着望远镜里的木星卫星是人类的幻觉。十进制是因为人有十根手指才流行的，但这也不意味着素数分布依赖于手指的数量。数学结构也一样：它可能是因为人脑太窄才被优先发现的，但一旦被发现，它说的就是对象本身的事，与人类脑容量无关。",[11,70,71],{},"把 Galois 群说成「人类算不动五次方程，所以编个借口说它不可解」，是一种误解。它本身是一个定理：一个方程能用根式解，当且仅当它的 Galois 群可解。五次一般方程对应的对称群 S5 不是可解群，所以没有根式解。这个结论与人类的工作记忆无关。就算人类有无限的工作记忆，S5 依然不是可解群，这个定理依然成立。同样，Noether 定理把连续对称性和守恒律绑在一起。把它说成「人类看不懂，所以画个对称性的图安慰自己」，也是一种误解。它说的是物理世界的一条结构事实：如果世界没有这种对称性，相应的守恒律就不存在。同调群也常被误解成「空间太复杂，人类算不动，所以发明一个符号糊弄过去」。它探测的是空间的洞，而它作为不变量能被计算，本身依赖链复形这一整套结构。",[11,73,74],{},"此外，它的说服力有很大一部分来自修辞。「可怜的脑容量」「拐杖」「硬按」——这些词先把你放在一个需要被怜悯的位置，再让你接受结论。把修辞剥掉，论证的核心其实只有一句：人脑窄，AI 不窄，所以结构是拐杖。而这句话恰恰是跳步的。",[11,76,77],{},"退一步说，就算 AI 真的把式子硬展开到第七阶，把所有发散项精准归零，它拿到的只是一张针对这个特例的系数表，换一个情形就未必管用。它能告诉你「这里可以归零」，但不能告诉你「为什么这里必然可以归零」。前者是答案，后者才是数学。打个比方：给你七百个点，你总可以找到一个不超过 699 次的多项式，让它精确穿过每一个点。但这只是插值，够不上理解。它在七百个点之外没有任何预测力。硬展开加待定系数，本质上就是高维版本的插值——你用一个足够大的展开式，把眼前的项全部按住，然后宣布问题解决了。可数学要的是「为什么会这样，以及换一个情形会怎样」。「按住」远远不够。",[11,79,80],{},"照这个逻辑，AI 不需要几何直观，可以直接站在高维的代数展开式里往下看。这个画面假设存在一个「没有直观」的视角。但任何计算都发生在某种表示里，而表示本身就是一种结构。Transformer 的注意力是内容相关的路由，等变网络把对称性硬编码进架构，神经算子学的是函数空间之间的映射——这些做法都把结构写进了模型，只是藏得比较深。没有免费午餐定理早就说过，不存在对所有问题都最优的通用学习器。每一个能学的模型，都带着自己的归纳偏置。所谓「不需要直观的 AI」，只是把直观藏进了架构里，让人看不见而已。",[11,82,83],{},"AlphaTensor 常被拿来当例子，但它恰恰站在这个论证的反面。DeepMind 并没有枚举所有可能的算法。它在张量分解的结构化空间里做强化学习搜索。它能找到 Strassen 式的算法，正是因为搜索空间被结构约束过。把 AlphaTensor 说成「暴力硬算」的证据，是对它的误读。",[11,85,86],{},"更有意思的是，AI 自己的前沿正在往结构走，纯暴力那条路反而越走越窄。等变网络、神经算子、状态空间模型、注意力机制，都是在把某种先验写进模型。为什么？因为纯暴力在高维会指数爆炸。这个领域最聪明的一批人，正在用工程手段重新发明数学家一百年前就熟悉的那些东西：对称性、局部性、可分解性。AI 的研究者自己正在沿着那条路径追赶，「绕开审美」的预言并没有发生。",[11,88,89],{},"技术细节上也站不住。「高阶项发散」常被当作「项太多、人脑装不下」的同义词。但在很多场合，发散属于表示问题，跟项数多少关系不大：这个展开本身是一个渐近级数，并不收敛。渐近级数在前几项可以给出极好的近似，但继续展开不会逼近真值。你需要的是重求和、解析延拓，或者换一个表示。数学家看到发散就后退，是因为他们知道「继续展开」不会自动收敛。这时候该做的是换一种表示，把项数堆得更高没有用。AI 就算把发散项凑没了，也可能只找到一种形式上的消去，并未得到收敛的表示。",[11,91,92],{},"即便并行 Agent 真的能同时追踪一万个五阶交叉项，代价也不是零。算力、内存、通信都会随着项数组合爆炸。「并行」被当成了免费的午餐。更关键的是，这个天花板背后还有复杂性理论：在普遍接受的复杂性假设下，有些问题是内在难解的——问题本身的复杂度摆在那里，跟聪明与否关系不大。「人脑太小」和「问题太难」被混为一谈了。就算给一个无限工作记忆的理想智能体，它也会撞上指数爆炸。人类偏爱结构，既因为脑子小，也因为问题本身硬。",[11,94,95],{},"所以，AI 可以使用一种人读不懂的结构。硬展开同样有结构，只不过这种结构被执行到了人跟不上的规模。一个逐像素渲染的渲染器，算得比任何画家都细，却始终跑在射影几何上，把透视算到了人眼看不见的精度。AI 的硬展开也一样：它把代数结构展开到了人脑装不下的长度，却始终在代数结构之内。",[11,97,98],{},"「上帝拿计算器硬按」假设答案先于结构存在。它背后是一个很朴素的柏拉图式画面：答案早就躺在那里，结构只是人类够不着时用的梯子。但数学的实践不是这样。很多答案是在结构被发明出来之后才成为「答案」的。",[11,100,101],{},"一个猜想在 n=5 成立、在 n=6 崩掉，硬算给你两个数。结构告诉你边界在哪里，以及为什么边界在那里。两个数可以和无穷多个故事相容，结构帮你挑出正确的那个。更一般地，硬算只能处理有限多个情形。即使你在计算机上验证费马大定理到 n 等于十亿，也不构成对所有的 n 的证明。Wiles 的证明之所以成立，靠的是模形式这座桥，把问题接到了另一个世界。计算量的大小并非关键。",[11,103,104],{},"知道一个命题为真，和知道它为什么为真，不是一回事。前者是证明，后者是理解。硬算可以给你前者，未必给你后者。而且，硬算能证明的通常只是有限多个特例。一旦命题是全称的，你就需要结构、归纳或者某种不变量，才能跨过「有限」到「所有」的鸿沟。七百个结果里可能有一些本身就是关于有限对象的定理，本身就有价值。但如果它们大多是硬展开出来的特例，那么它们更像是一批精确的数值事实，未必是可迁移的定理。另外，「七百篇论文」和「七百个结果」不是一回事。前者可能包含大量增量工作，后者才是数学意义上的成果。这个区分本身就值得追问，但先放在一边。",[11,106,107],{},"数学史上早有「先有黑箱，后有理解」的先例。四色定理 1976 年由机器辅助证明，当时吵得很厉害。后来人类花了很多年去简化、去压缩，形式化验证也完成了，但「机器证明算不算理解」这个问题并没有完全消失。Kepler 猜想被 Hales 用大规模计算机辅助证明，后来又有 Flyspeck 项目做形式化验证。模式很清楚：先拿到一个可验证但不可读的答案，再花一代人把它翻译成人类握得住的结构。翻译可能失败，也可能要几代人，但「现在翻译不出来」从来不足以让人停下来。",[11,109,110],{},"而且，人类数学自己也早就遇到过不可读的问题。有限单群分类定理的证明上万页，散落在几百篇文章里，据说至今没有人从头到尾读完。这是人类自己造出来的黑箱。区别只在于规模：人类数学在二十世纪就已经开始生产超出个体阅读能力的结果，AI 只是把这个趋势推得更远。",[11,112,113],{},"黑箱当然有价值。它可以被引用、被使用、被当作下一步的踏脚石。但它的价值有一个天花板：验证成本无法摊销。一个你理解了的定理，可以在一段话里教给别人，可以在新问题里直接调用。一个你不理解的黑箱，每用一次都要重新跑一遍、重新信一遍。理解之所以重要，是因为它把验证成本压缩成了可复用的结构。没有理解，知识就不能以低成本累积。",[11,115,116],{},"把这种思路推到极端：也许黑箱就是数学的新常态。也许未来的数学单位会从定理变成证书。数学家追求的东西，也会从「看懂」转向「可验证、可组合、可规模化」。这是一个严肃的可能性，我不打算轻易否定它。但它比「AI 算得更快」是一个大得多的主张：它改变的是数学知识的定义，速度倒在其次。而且，就算走到那一步，信任基础设施本身仍然是结构：证书格式、证明助理、可复现性标准、社区共识。黑箱的未来并未取消结构，只不过把结构挪到了另一个地方。",[11,118,119],{},"至于审美，AI 正在用一个丑但有效的答案挑战「美是数学的先验判据」这个前提。这个前提站不住。",[11,121,122],{},"数学史上「丑的真理」屡见不鲜。三次方程的求根公式在 16 世纪就撞上了 casus irreducibilis：三个实根，却必须经过复数才能算出来。当时的数学家觉得这近乎荒谬。非欧几何刚出来的时候，高斯不敢公开发表。Cantor 的无穷集合遭到 Kronecker 的激烈反对。Weierstrass 造出处处连续、处处不可导的函数，被当时的许多人视为病态。Banach-Tarski 分球悖论更是像在说「数学不该长这样」。后来的处理方式，是扩写美的定义，把这些东西留下来。复数变成了复平面上的几何，非欧几何变成了曲率，病态函数催生了测度论，也逼着拓扑学重新审视连续与收敛。分球悖论则逼着人们去追问选择公理到底意味着什么。丑的东西待得够久，人类就会给它找一套它配得上的语言。语言找到了，它就不丑了。",[11,124,125],{},"所以，美更像事后的评价。数学家说一个证明美，通常是在说：它把原本分散的东西统一了，没有多余的假设，并且揭示了机制。这些都是理解完成之后的判词，很难充当发现之前的门槛。历史上真正被审美直觉挡在门外的结果——比如 Galois 生前没能发表的手稿——事后看，那是审美的失败，数学本身并没有失败。",[11,127,128],{},"但也不要走到另一个极端，把审美说成纯粹的非理性。审美其实是一个关于「可压缩性」的启发式：一个漂亮的证明，往往更容易推广、更容易迁移、更容易在别的问题里复用。它当然会犯错，但命中率并不低。数学家对丑的警惕，除了对陌生的恐惧，还包含一种合理怀疑：这东西也许无法一般化。把审美当成纯粹的偏见，这太便宜了。",[11,130,131],{},"所以，「能不能翻译」比「美不美」更值得问。丑是可以忍受的，只要它能在可接受的成本内被压缩成一套可迁移的结构。丑而无法批量翻译，才是真正的问题。对 AI 那七百个结果，关键在翻译成本：把它们变成人类可读的结构要花多大代价，这个代价能不能被规模化地摊薄。如果每个结果都需要一位数学家花十年去反推，那七百个结果的边际价值就取决于翻译的规模化程度——而这恰恰是 AI 可能帮上忙的地方：让 AI 去搜「能压缩这七百个结果的统一框架」。丑不可怕，可怕的是丑得无法批量处理。",[11,133,134],{},"「地球灭亡之前，人类没法理解的事情肯定存在。」这句话我同意。Gödel 的不完备定理保证了真而不可证的东西存在，复杂性理论则告诉我们，一大批问题在可接受的资源内算不动。但这句话不能推出「所以现在别理解了」。",[11,136,137],{},"从「存在无法理解的东西」推不出「眼前这七百个就是无法理解的」。这是一个很基本的逻辑错误：存在量词不能这样实例化。而且，可理解域的边界是未知的，历史上每一次都被推得比预期远得多。Hilbert 纲领失败了，但失败本身催生了证明论和可计算性理论，也深刻改变了模型论的问题意识。数学的边界更像一片雾。你往前走，雾就退一点。",[11,139,140],{},"理解有许多层次。我们理解热力学，不需要追踪每一个分子。同样，我们理解一个函数，不需要它的封闭形式。七百个结果可能在细粒度上不可读，但在粗粒度上仍然可以被理解：它们可能共享某种对称性、某种生成机制、某种还没被命名的结构。放弃理解需要「已经证明这里推不动」。「现在看着像一团乱麻」还不够。前者是数学，后者是情绪。",[11,142,143],{},"另外，AI 的方法太有针对性，这是数学界最实在的批评之一。以前一个理论出来，不仅解决目标问题，还会带来一整套工具箱，影响其他领域。现在的 AI 方法往往只解决目标问题，然后就结束了。",[11,145,146],{},"这条批评有它的道理。人类数学的溢出效应是结构性的。Galois 理论本来是为了解五次方程，产出的群论和域扩张后来成了整个代数的主干。Riemann 为了理解素数分布把复分析引进数论，他的几何工作又养活了微分几何和拓扑。同调代数从拓扑里长出来，最后变成了交换代数、代数几何、表示论的通用语法。目标问题是钩子，工具箱才是本体。数学界评价一个工作，隐含的标准是「它留下了什么可复用的东西」。「它证明了什么」只是起点。",[11,148,149],{},"AI 目前的产出恰好踩在这个标准的反面。AlphaTensor 找矩阵乘法算法、组合搜索、符号回归凑公式，共同特征是：针对特定实例、依赖大规模搜索、结果的可迁移性有限。它优化的是「这个问题有没有解、解是什么」。「这类问题的结构长什么样」则被放在一边。一个只针对单点的工具，边际价值确实低，尤其当它消耗的算力还能拿去做别的事。",[11,151,152],{},"但针对性太强，是 AI 的局限，还是当前阶段的局限？历史上「针对性太强」的批评也落在过人类自己头上。数值分析早期被纯数学家看不起，觉得是工程凑数，后来它长出了稳定性、收敛阶、条件数这一整套理论。概率论被当成赌博工具几百年，Kolmogorov 公理化之后才翻身。针对性强的工具，如果被反复使用，往往会倒逼出一般理论。差别在于：人类工具的一般化是被使用者的需求驱动的，慢但稳。AI 工具的一般化目前没有人系统性地做，因为发论文的激励在「又刷了一个纪录」，不在「把这套方法抽象成可复用的框架」。",[11,154,155],{},"所以，真正缺的，是有人负责把针对性的方法一般化。「AI 方法针对性太强」只是表象。人类数学家有同行评议和长期声誉在逼他们想「这东西能不能推广」。AI 数学的产出链条里，这个环节目前是断的。如果哪天真有团队认真做「把七百个硬展开结果反推成一个统一框架」，而且做成了，那这条批评就自动消解了。但这件事的难度不比发现那七百个结果低，甚至更高：归纳一个框架，靠的是猜对抽象层级，算力反而次要。而这恰恰是目前最说不清楚 AI 能不能做好的部分。",[11,157,158],{},"翻译是一项制度功能，很难靠某位数学家的英雄行为完成。人类数学之所以能把计算变成结构，靠的是一整套同行评议、教科书、研讨会和长期声誉的机器，在逼着每一代人去问「这东西能不能推广」。个别天才只是其中一环。AI 数学现在缺的，正是这台机器，算力和聪明的模型都还在其次。真正的瓶颈在分工，认知带宽不是主要问题。",[11,160,161],{},"这套论证的力量来自三个漂亮的偷换，把动机当成定义，把事后的判词当成先验的门槛，把答案当成理解。",[11,163,164],{},"但挑战是真的，只是位置放错了：我们把计算翻译成结构的制度，没有跟上计算的速度。「AI 比人算得快」只是表面。",[11,166,167],{},"画家画不出世界上的每一个光子，所以他发明透视。但透视是射影几何，跟「眼睛不行」没有关系。",[11,169,170],{},"今天最强的渲染器可以逐像素地硬算，但它仍然跑在透视的数学上。硬算并没有取消结构。它把结构算到了人眼跟不上的地方。",[11,172,173],{},"上帝或许真的在用计算器。但账本不是给计算器看的。人类接下来要做的，是翻译账本，让计算器的结果变得可读。",[11,175,176],{},"所以说，AI 在未来会彻底取代数学研究的劳动吗？",[11,178,179],{},"我认为不会，而且在未来很长一段时间里，都不会，至少在 AGI 诞生之前永远不会。",[24,181,182],{},[11,183,184],{},"砸掉计算器解决不了问题。数学的下一份工作，也许要从「发现」转向「翻译」。",{"title":186,"searchDepth":187,"depth":187,"links":188},"",2,[],"前段时间，OpenAI 声称，他们用内部尚未正式发布的模型产出了七百多篇论文，「AI 接替数学家」的叙事于是再一次被推到台前。市场是狂热还是炒作，我已经疲于争论。那七百篇论文是否名副其实，也先放在一边。真正让我想写点东西的，是 Twitter 上一条获得不少认同的帖子。它把数学结构说成人类为了迁就自己那点脑容量而被迫发明的拐杖，把 AI 说成不需要几何直观、可以直接站在高维展开式里往下看的怪物。最后一句尤其漂亮：我们终于拿到了上帝账本里的答案，却发现上帝算账根本不用微积分，他老人家是直接拿计算器硬按出来的。",false,true,"md",null,{},"\u002Fblog\u002F2026\u002F0012-shu-xue-yu-hua-jia",{"title":6,"description":189},"blog\u002F2026\u002F0012-shu-xue-yu-hua-jia","结构本身就是定理，AI 也有结构，只是那种结构人读不懂。关键在于可翻译性，以及把计算翻译成结构的制度是否跟得上。硬算常给出有限情形的答案而非「为什么」，而把针对性的结果一般化、翻译成可复用结构的制度，在 AI 数学的产出链条里是断的。",[200],"thoughts","2026-10-25T16:52:45+08:00","MTXFn_wz9eNH537cFyhWDAYMtvfSTEdfulMeMndlEyU",{"id":204,"title":205,"body":206,"description":210,"draft":190,"enableCodeHighlight":191,"enableCodeHightlight":190,"enableComment":191,"enableD2":191,"enableKatex":191,"extension":192,"image":186,"important":190,"location":193,"meta":20139,"navigation":191,"ogImage":193,"path":20140,"seo":20141,"stem":20142,"summary":20143,"tags":20144,"time":20146,"__hash__":20147},"blog\u002Fblog\u002F2026\u002F0013-satisfiability-modulo-theories.md","可满足性模理论 (Satisfiability Modulo Theories)",{"type":8,"value":207,"toc":20100},[208,211,214,366,381,384,568,571,579,601,607,612,617,655,658,661,810,874,1045,1049,1052,1383,1946,1949,2673,3041,3100,3111,3118,3122,3125,3338,3344,3376,3379,3382,3386,3390,3397,3443,3573,4205,4623,4626,4629,4657,4765,4984,4987,4991,4994,4997,5192,5195,5581,5772,5775,5778,5782,5785,5788,5959,5962,6390,6651,6787,6794,6798,6804,6807,6844,6847,7463,7466,7469,7473,7476,7490,7496,7500,7504,7511,7514,7613,7648,7667,7670,7674,7680,7864,7867,8037,8040,8480,8580,8587,8719,8723,8729,8736,8739,8909,9354,9357,9360,9545,9646,9652,9842,9846,9852,9931,9937,9941,9948,9951,9964,9967,9971,9975,9986,9993,9996,10017,10020,10024,10027,10030,10246,10249,10503,10506,10602,10809,10816,10972,11052,11361,11365,11368,11376,11379,11399,11402,11406,11442,11445,11450,11453,11594,11597,11823,12044,12047,12050,12055,12191,12193,12390,12515,12920,12923,12925,12930,12933,13132,13135,13142,13406,13549,13551,13559,13570,13572,13873,14093,14096,14102,14106,14113,14116,14271,14274,14503,14506,14617,14678,14681,14685,14688,14903,15358,15486,15642,15737,15860,15863,15870,15873,15910,15914,15920,15948,15951,15954,15958,15962,15965,15970,15973,16403,16489,16498,16504,16508,16511,16600,16617,16628,16759,16762,16954,16957,16989,16992,16998,17134,17137,17177,17180,17373,17379,17393,17431,17448,17452,17545,17623,17626,17734,17737,18128,18193,18196,18238,18241,18247,18250,18258,18454,18540,18550,18554,18557,18622,18633,18724,18730,18877,18883,18886,18952,19042,19099,19109,19116,19120,19194,19197,19306,19316,19319,19359,19362,19525,19530,19541,19861,19952,19956,19959,19966,19972,19979,19983,20086,20096],[11,209,210],{},"SMT 的出现，本质上是为了解决一个核心矛盾：很多实际问题用纯命题逻辑（SAT）表达时，要么根本无法表达，要么表达后规模爆炸；而用一般的一阶逻辑或高阶逻辑表达虽然自然，却无法自动高效求解。",[11,212,213],{},"看一段再普通不过的代码：",[215,216,220],"pre",{"className":217,"code":218,"language":219,"meta":186,"style":186},"language-c shiki shiki-themes github-light github-light github-dark","void f(int x, int y) {\n    int a[100] = {0};\n    a[0] = 5;\n    if (x + y > 10 && a[0] == 5) {\n        \u002F\u002F 目标分支\n    }\n}\n","c",[221,222,223,258,288,308,347,354,360],"code",{"__ignoreMap":186},[224,225,228,232,236,240,243,247,250,252,255],"span",{"class":226,"line":227},"line",1,[224,229,231],{"class":230},"s8jYJ","void",[224,233,235],{"class":234},"snPdu"," f",[224,237,239],{"class":238},"sxrX7","(",[224,241,242],{"class":230},"int",[224,244,246],{"class":245},"sP4rz"," x",[224,248,249],{"class":238},", ",[224,251,242],{"class":230},[224,253,254],{"class":245}," y",[224,256,257],{"class":238},") {\n",[224,259,260,263,266,269,273,276,279,282,285],{"class":226,"line":187},[224,261,262],{"class":230},"    int",[224,264,265],{"class":245}," a",[224,267,268],{"class":238},"[",[224,270,272],{"class":271},"sBjJW","100",[224,274,275],{"class":238},"] ",[224,277,278],{"class":230},"=",[224,280,281],{"class":238}," {",[224,283,284],{"class":271},"0",[224,286,287],{"class":238},"};\n",[224,289,291,294,296,298,300,302,305],{"class":226,"line":290},3,[224,292,293],{"class":245},"    a",[224,295,268],{"class":238},[224,297,284],{"class":271},[224,299,275],{"class":238},[224,301,278],{"class":230},[224,303,304],{"class":271}," 5",[224,306,307],{"class":238},";\n",[224,309,311,314,317,320,323,326,329,332,334,336,338,340,343,345],{"class":226,"line":310},4,[224,312,313],{"class":230},"    if",[224,315,316],{"class":238}," (x ",[224,318,319],{"class":230},"+",[224,321,322],{"class":238}," y ",[224,324,325],{"class":230},">",[224,327,328],{"class":271}," 10",[224,330,331],{"class":230}," &&",[224,333,265],{"class":245},[224,335,268],{"class":238},[224,337,284],{"class":271},[224,339,275],{"class":238},[224,341,342],{"class":230},"==",[224,344,304],{"class":271},[224,346,257],{"class":238},[224,348,350],{"class":226,"line":349},5,[224,351,353],{"class":352},"sCsY4","        \u002F\u002F 目标分支\n",[224,355,357],{"class":226,"line":356},6,[224,358,359],{"class":238},"    }\n",[224,361,363],{"class":226,"line":362},7,[224,364,365],{"class":238},"}\n",[11,367,368,369,372,373,376,377],{},"假设 ",[221,370,371],{},"x","、",[221,374,375],{},"y"," 是来自外部的整数。我们想知道：",[378,379,380],"strong",{},"有没有一组输入，能走进这个分支？",[11,382,383],{},"沿着这条路径，把进入分支需要满足的条件收集起来：",[224,385,389],{"className":386,"translate":388},[387],"katex-display","no",[224,390,393,451],{"className":391,"translate":388},[392],"katex",[224,394,397],{"className":395},[396],"katex-mathml",[398,399,402],"math",{"xmlns":400,"display":401},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML","block",[403,404,405,446],"semantics",{},[406,407,408,411,414,416,418,422,426,429,431,433,436,438,441,443],"mrow",{},[409,410,371],"mi",{},[412,413,319],"mo",{},[409,415,375],{},[412,417,325],{},[419,420,421],"mn",{},"10",[423,424,425],"mtext",{},"  ",[412,427,428],{},"∧",[423,430,425],{},[409,432,15],{},[412,434,268],{"stretchy":435},"false",[419,437,284],{},[412,439,440],{"stretchy":435},"]",[412,442,278],{},[419,444,445],{},"5",[447,448,450],"annotation",{"encoding":449},"application\u002Fx-tex","x + y > 10 \\;\\wedge\\; a[0] = 5",[224,452,456,482,504,529,559],{"className":453,"ariaHidden":455},[454],"katex-html","true",[224,457,460,465,470,475,479],{"className":458},[459],"base",[224,461],{"className":462,"style":464},[463],"strut","height:0.6667em;vertical-align:-0.0833em;",[224,466,371],{"className":467},[468,469],"mord","mathnormal",[224,471],{"className":472,"style":474},[473],"mspace","margin-right:0.2222em;",[224,476,319],{"className":477},[478],"mbin",[224,480],{"className":481,"style":474},[473],[224,483,485,489,493,497,501],{"className":484},[459],[224,486],{"className":487,"style":488},[463],"height:0.7335em;vertical-align:-0.1944em;",[224,490,375],{"className":491,"style":492},[468,469],"margin-right:0.0359em;",[224,494],{"className":495,"style":496},[473],"margin-right:0.2778em;",[224,498,325],{"className":499},[500],"mrel",[224,502],{"className":503,"style":496},[473],[224,505,507,511,514,517,520,523,526],{"className":506},[459],[224,508],{"className":509,"style":510},[463],"height:0.6444em;",[224,512,421],{"className":513},[468],[224,515],{"className":516,"style":496},[473],[224,518],{"className":519,"style":474},[473],[224,521,428],{"className":522},[478],[224,524],{"className":525,"style":496},[473],[224,527],{"className":528,"style":474},[473],[224,530,532,536,539,543,546,550,553,556],{"className":531},[459],[224,533],{"className":534,"style":535},[463],"height:1em;vertical-align:-0.25em;",[224,537,15],{"className":538},[468,469],[224,540,268],{"className":541},[542],"mopen",[224,544,284],{"className":545},[468],[224,547,440],{"className":548},[549],"mclose",[224,551],{"className":552,"style":496},[473],[224,554,278],{"className":555},[500],[224,557],{"className":558,"style":496},[473],[224,560,562,565],{"className":561},[459],[224,563],{"className":564,"style":510},[463],[224,566,445],{"className":567},[468],[11,569,570],{},"问题于是变成：这组约束有没有解？",[11,572,573,574,372,576,578],{},"纯 SAT 求解器也能做，但你得先把 ",[221,575,371],{},[221,577,375],{}," 各拆成 32 个布尔变量，把加法展开成加法器电路，把数组展开成 100 个元素，最后交给它几千条子句。它只认识真假，不认识整数，也不认识数组。",[11,580,581,582,585,586,589,590,593,594,596,597,600],{},"SMT 求解器则直接读原式：",[221,583,584],{},"x + y > 10"," 交给算术，",[221,587,588],{},"a[0] = 5"," 交给数组。算术说取 ",[221,591,592],{},"x = 6, y = 5"," 就行，数组说 ",[221,595,588],{}," 本来就成立。两边一对，返回 ",[378,598,599],{},"sat","，并给出这组解。",[11,602,603,604],{},"同一个问题，一个要把语义拆成门电路，一个能直接利用语义。",[378,605,606],{},"SMT 要做的，就是后者。",[608,609,611],"h2",{"id":610},"一smt-是什么","一、SMT 是什么",[613,614,616],"h3",{"id":615},"_11-定义","1.1 定义",[11,618,619,620],{},"把上面的过程抽象一下。",[378,621,622,623,654],{},"SMT 是「SAT + 背景理论」：在某个理论 ",[224,624,626,640],{"className":625,"translate":388},[392],[224,627,629],{"className":628},[396],[398,630,631],{"xmlns":400},[403,632,633,638],{},[406,634,635],{},[409,636,637],{},"T",[447,639,637],{"encoding":449},[224,641,643],{"className":642,"ariaHidden":455},[454],[224,644,646,650],{"className":645},[459],[224,647],{"className":648,"style":649},[463],"height:0.6833em;",[224,651,637],{"className":652,"style":653},[468,469],"margin-right:0.1389em;"," 规定的语义下，判定一个一阶逻辑公式是否存在满足它的赋值。",[11,656,657],{},"这里的关键词是「模理论」（modulo theories）。「模」可以理解为「相对于」：我们不再问公式在全部数学结构里是否成立，而是问它在某个特定理论——比如整数算术、位向量、数组——的解释下是否成立。",[11,659,660],{},"形式化一点，SMT 要回答的是：",[224,662,664],{"className":663,"translate":388},[387],[224,665,667,708],{"className":666,"translate":388},[392],[224,668,670],{"className":669},[396],[398,671,672],{"xmlns":400,"display":401},[403,673,674,705],{},[406,675,676,680,683,686,689,692,694,702],{},[409,677,679],{"mathvariant":678},"normal","∃",[423,681,682],{}," ",[409,684,685],{},"σ",[409,687,688],{"mathvariant":678},".",[473,690],{"width":691},"1em",[409,693,685],{},[695,696,697,700],"msub",{},[412,698,699],{},"⊨",[409,701,637],{},[409,703,704],{},"φ",[447,706,707],{"encoding":449},"\\exists\\, \\sigma.\\quad \\sigma \\models_T \\varphi",[224,709,711,800],{"className":710,"ariaHidden":455},[454],[224,712,714,718,721,725,728,731,735,738,741,797],{"className":713},[459],[224,715],{"className":716,"style":717},[463],"height:0.999em;vertical-align:-0.249em;",[224,719,679],{"className":720},[468],[224,722],{"className":723,"style":724},[473],"margin-right:0.1667em;",[224,726,685],{"className":727,"style":492},[468,469],[224,729,688],{"className":730},[468],[224,732],{"className":733,"style":734},[473],"margin-right:1em;",[224,736,685],{"className":737,"style":492},[468,469],[224,739],{"className":740,"style":496},[473],[224,742,744,747],{"className":743},[500],[224,745,699],{"className":746},[500],[224,748,751],{"className":749},[750],"msupsub",[224,752,756,788],{"className":753},[754,755],"vlist-t","vlist-t2",[224,757,760,783],{"className":758},[759],"vlist-r",[224,761,765],{"className":762,"style":764},[763],"vlist","height:0.3283em;",[224,766,768,773],{"style":767},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[224,769],{"className":770,"style":772},[771],"pstrut","height:2.7em;",[224,774,780],{"className":775},[776,777,778,779],"sizing","reset-size6","size3","mtight",[224,781,637],{"className":782,"style":653},[468,469,779],[224,784,787],{"className":785},[786],"vlist-s","​",[224,789,791],{"className":790},[759],[224,792,795],{"className":793,"style":794},[763],"height:0.15em;",[224,796],{},[224,798],{"className":799,"style":496},[473],[224,801,803,807],{"className":802},[459],[224,804],{"className":805,"style":806},[463],"height:0.625em;vertical-align:-0.1944em;",[224,808,704],{"className":809},[468,469],[11,811,812,813,843,844,873],{},"也就是：是否存在一个赋值 ",[224,814,816,830],{"className":815,"translate":388},[392],[224,817,819],{"className":818},[396],[398,820,821],{"xmlns":400},[403,822,823,827],{},[406,824,825],{},[409,826,685],{},[447,828,829],{"encoding":449},"\\sigma",[224,831,833],{"className":832,"ariaHidden":455},[454],[224,834,836,840],{"className":835},[459],[224,837],{"className":838,"style":839},[463],"height:0.4306em;",[224,841,685],{"className":842,"style":492},[468,469],"，把每个变量映射到理论域里的值，使得 ",[224,845,847,861],{"className":846,"translate":388},[392],[224,848,850],{"className":849},[396],[398,851,852],{"xmlns":400},[403,853,854,858],{},[406,855,856],{},[409,857,704],{},[447,859,860],{"encoding":449},"\\varphi",[224,862,864],{"className":863,"ariaHidden":455},[454],[224,865,867,870],{"className":866},[459],[224,868],{"className":869,"style":806},[463],[224,871,704],{"className":872},[468,469]," 在这个理论的所有公理下为真。",[11,875,876,877,949,950,1015,1016,1044],{},"回到开头的例子：",[224,878,880,901],{"className":879,"translate":388},[392],[224,881,883],{"className":882},[396],[398,884,885],{"xmlns":400},[403,886,887,899],{},[406,888,889,891,893,895,897],{},[409,890,371],{},[412,892,319],{},[409,894,375],{},[412,896,325],{},[419,898,421],{},[447,900,584],{"encoding":449},[224,902,904,922,940],{"className":903,"ariaHidden":455},[454],[224,905,907,910,913,916,919],{"className":906},[459],[224,908],{"className":909,"style":464},[463],[224,911,371],{"className":912},[468,469],[224,914],{"className":915,"style":474},[473],[224,917,319],{"className":918},[478],[224,920],{"className":921,"style":474},[473],[224,923,925,928,931,934,937],{"className":924},[459],[224,926],{"className":927,"style":488},[463],[224,929,375],{"className":930,"style":492},[468,469],[224,932],{"className":933,"style":496},[473],[224,935,325],{"className":936},[500],[224,938],{"className":939,"style":496},[473],[224,941,943,946],{"className":942},[459],[224,944],{"className":945,"style":510},[463],[224,947,421],{"className":948},[468]," 是算术原子，",[224,951,953,976],{"className":952,"translate":388},[392],[224,954,956],{"className":955},[396],[398,957,958],{"xmlns":400},[403,959,960,974],{},[406,961,962,964,966,968,970,972],{},[409,963,15],{},[412,965,268],{"stretchy":435},[419,967,284],{},[412,969,440],{"stretchy":435},[412,971,278],{},[419,973,445],{},[447,975,588],{"encoding":449},[224,977,979,1006],{"className":978,"ariaHidden":455},[454],[224,980,982,985,988,991,994,997,1000,1003],{"className":981},[459],[224,983],{"className":984,"style":535},[463],[224,986,15],{"className":987},[468,469],[224,989,268],{"className":990},[542],[224,992,284],{"className":993},[468],[224,995,440],{"className":996},[549],[224,998],{"className":999,"style":496},[473],[224,1001,278],{"className":1002},[500],[224,1004],{"className":1005,"style":496},[473],[224,1007,1009,1012],{"className":1008},[459],[224,1010],{"className":1011,"style":510},[463],[224,1013,445],{"className":1014},[468]," 是数组原子，理论 ",[224,1017,1019,1032],{"className":1018,"translate":388},[392],[224,1020,1022],{"className":1021},[396],[398,1023,1024],{"xmlns":400},[403,1025,1026,1030],{},[406,1027,1028],{},[409,1029,637],{},[447,1031,637],{"encoding":449},[224,1033,1035],{"className":1034,"ariaHidden":455},[454],[224,1036,1038,1041],{"className":1037},[459],[224,1039],{"className":1040,"style":649},[463],[224,1042,637],{"className":1043,"style":653},[468,469]," 就是整数算术与数组的组合。SMT 要回答的，正是这组原子能不能同时为真。",[613,1046,1048],{"id":1047},"_12-公式的样子","1.2 公式的样子",[11,1050,1051],{},"SMT 公式由布尔结构和理论原子组成：",[224,1053,1055],{"className":1054,"translate":388},[387],[224,1056,1058,1150],{"className":1057,"translate":388},[392],[224,1059,1061],{"className":1060},[396],[398,1062,1063],{"xmlns":400,"display":401},[403,1064,1065,1147],{},[406,1066,1067,1069,1072,1074,1076,1079,1081,1084,1086,1089,1091,1093,1095,1098,1100,1102,1104,1107,1109,1111,1113,1115,1117,1119,1121,1123,1125,1127,1129,1132,1134,1136,1138,1140,1142,1145],{},[409,1068,704],{},[412,1070,1071],{},":",[412,1073,1071],{},[412,1075,278],{},[409,1077,1078],{"mathvariant":678},"⊤",[423,1080,425],{},[412,1082,1083],{},"∣",[423,1085,425],{},[409,1087,1088],{"mathvariant":678},"⊥",[423,1090,425],{},[412,1092,1083],{},[423,1094,425],{},[409,1096,1097],{},"A",[423,1099,425],{},[412,1101,1083],{},[423,1103,425],{},[409,1105,1106],{"mathvariant":678},"¬",[409,1108,704],{},[423,1110,425],{},[412,1112,1083],{},[423,1114,425],{},[409,1116,704],{},[412,1118,428],{},[409,1120,704],{},[423,1122,425],{},[412,1124,1083],{},[423,1126,425],{},[409,1128,704],{},[412,1130,1131],{},"∨",[409,1133,704],{},[423,1135,425],{},[412,1137,1083],{},[423,1139,425],{},[409,1141,704],{},[412,1143,1144],{},"→",[409,1146,704],{},[447,1148,1149],{"encoding":449},"\\varphi ::= \\top \\;\\mid\\; \\bot \\;\\mid\\; A \\;\\mid\\; \\neg\\varphi \\;\\mid\\; \\varphi \\wedge \\varphi \\;\\mid\\; \\varphi \\vee \\varphi \\;\\mid\\; \\varphi \\rightarrow \\varphi",[224,1151,1153,1172,1196,1220,1244,1271,1290,1314,1332,1356,1374],{"className":1152,"ariaHidden":455},[454],[224,1154,1156,1159,1162,1165,1169],{"className":1155},[459],[224,1157],{"className":1158,"style":806},[463],[224,1160,704],{"className":1161},[468,469],[224,1163],{"className":1164,"style":496},[473],[224,1166,1168],{"className":1167},[500],"::=",[224,1170],{"className":1171,"style":496},[473],[224,1173,1175,1178,1181,1184,1187,1190,1193],{"className":1174},[459],[224,1176],{"className":1177,"style":535},[463],[224,1179,1078],{"className":1180},[468],[224,1182],{"className":1183,"style":496},[473],[224,1185],{"className":1186,"style":496},[473],[224,1188,1083],{"className":1189},[500],[224,1191],{"className":1192,"style":496},[473],[224,1194],{"className":1195,"style":496},[473],[224,1197,1199,1202,1205,1208,1211,1214,1217],{"className":1198},[459],[224,1200],{"className":1201,"style":535},[463],[224,1203,1088],{"className":1204},[468],[224,1206],{"className":1207,"style":496},[473],[224,1209],{"className":1210,"style":496},[473],[224,1212,1083],{"className":1213},[500],[224,1215],{"className":1216,"style":496},[473],[224,1218],{"className":1219,"style":496},[473],[224,1221,1223,1226,1229,1232,1235,1238,1241],{"className":1222},[459],[224,1224],{"className":1225,"style":535},[463],[224,1227,1097],{"className":1228},[468,469],[224,1230],{"className":1231,"style":496},[473],[224,1233],{"className":1234,"style":496},[473],[224,1236,1083],{"className":1237},[500],[224,1239],{"className":1240,"style":496},[473],[224,1242],{"className":1243,"style":496},[473],[224,1245,1247,1250,1253,1256,1259,1262,1265,1268],{"className":1246},[459],[224,1248],{"className":1249,"style":535},[463],[224,1251,1106],{"className":1252},[468],[224,1254,704],{"className":1255},[468,469],[224,1257],{"className":1258,"style":496},[473],[224,1260],{"className":1261,"style":496},[473],[224,1263,1083],{"className":1264},[500],[224,1266],{"className":1267,"style":496},[473],[224,1269],{"className":1270,"style":496},[473],[224,1272,1274,1278,1281,1284,1287],{"className":1273},[459],[224,1275],{"className":1276,"style":1277},[463],"height:0.75em;vertical-align:-0.1944em;",[224,1279,704],{"className":1280},[468,469],[224,1282],{"className":1283,"style":474},[473],[224,1285,428],{"className":1286},[478],[224,1288],{"className":1289,"style":474},[473],[224,1291,1293,1296,1299,1302,1305,1308,1311],{"className":1292},[459],[224,1294],{"className":1295,"style":535},[463],[224,1297,704],{"className":1298},[468,469],[224,1300],{"className":1301,"style":496},[473],[224,1303],{"className":1304,"style":496},[473],[224,1306,1083],{"className":1307},[500],[224,1309],{"className":1310,"style":496},[473],[224,1312],{"className":1313,"style":496},[473],[224,1315,1317,1320,1323,1326,1329],{"className":1316},[459],[224,1318],{"className":1319,"style":1277},[463],[224,1321,704],{"className":1322},[468,469],[224,1324],{"className":1325,"style":474},[473],[224,1327,1131],{"className":1328},[478],[224,1330],{"className":1331,"style":474},[473],[224,1333,1335,1338,1341,1344,1347,1350,1353],{"className":1334},[459],[224,1336],{"className":1337,"style":535},[463],[224,1339,704],{"className":1340},[468,469],[224,1342],{"className":1343,"style":496},[473],[224,1345],{"className":1346,"style":496},[473],[224,1348,1083],{"className":1349},[500],[224,1351],{"className":1352,"style":496},[473],[224,1354],{"className":1355,"style":496},[473],[224,1357,1359,1362,1365,1368,1371],{"className":1358},[459],[224,1360],{"className":1361,"style":806},[463],[224,1363,704],{"className":1364},[468,469],[224,1366],{"className":1367,"style":496},[473],[224,1369,1144],{"className":1370},[500],[224,1372],{"className":1373,"style":496},[473],[224,1375,1377,1380],{"className":1376},[459],[224,1378],{"className":1379,"style":806},[463],[224,1381,704],{"className":1382},[468,469],[11,1384,1385,1386,1414,1415,1418,1419,372,1557,372,1691,372,1842,1945],{},"其中 ",[224,1387,1389,1402],{"className":1388,"translate":388},[392],[224,1390,1392],{"className":1391},[396],[398,1393,1394],{"xmlns":400},[403,1395,1396,1400],{},[406,1397,1398],{},[409,1399,1097],{},[447,1401,1097],{"encoding":449},[224,1403,1405],{"className":1404,"ariaHidden":455},[454],[224,1406,1408,1411],{"className":1407},[459],[224,1409],{"className":1410,"style":649},[463],[224,1412,1097],{"className":1413},[468,469]," 是",[378,1416,1417],{},"理论原子","，形如 ",[224,1420,1422,1451],{"className":1421,"translate":388},[392],[224,1423,1425],{"className":1424},[396],[398,1426,1427],{"xmlns":400},[403,1428,1429,1448],{},[406,1430,1431,1439,1441],{},[695,1432,1433,1436],{},[409,1434,1435],{},"t",[419,1437,1438],{},"1",[412,1440,278],{},[695,1442,1443,1445],{},[409,1444,1435],{},[419,1446,1447],{},"2",[447,1449,1450],{"encoding":449},"t_1 = 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属于",[378,2753,2754],{},"线性整数算术","：它知道 ",[224,2757,2759,2772],{"className":2758,"translate":388},[392],[224,2760,2762],{"className":2761},[396],[398,2763,2764],{"xmlns":400},[403,2765,2766,2770],{},[406,2767,2768],{},[412,2769,319],{},[447,2771,319],{"encoding":449},[224,2773,2775],{"className":2774,"ariaHidden":455},[454],[224,2776,2778,2781],{"className":2777},[459],[224,2779],{"className":2780,"style":464},[463],[224,2782,319],{"className":2783},[468]," 是加法、",[224,2786,2788,2801],{"className":2787,"translate":388},[392],[224,2789,2791],{"className":2790},[396],[398,2792,2793],{"xmlns":400},[403,2794,2795,2799],{},[406,2796,2797],{},[412,2798,325],{},[447,2800,325],{"encoding":449},[224,2802,2804],{"className":2803,"ariaHidden":455},[454],[224,2805,2807,2811],{"className":2806},[459],[224,2808],{"className":2809,"style":2810},[463],"height:0.5782em;vertical-align:-0.0391em;",[224,2812,325],{"className":2813},[500]," 是全序；",[2677,2816,2817,2751,2882,2755,2885,2887,2888,2891],{},[224,2818,2820,2843],{"className":2819,"translate":388},[392],[224,2821,2823],{"className":2822},[396],[398,2824,2825],{"xmlns":400},[403,2826,2827,2841],{},[406,2828,2829,2831,2833,2835,2837,2839],{},[409,2830,15],{},[412,2832,268],{"stretchy":435},[419,2834,284],{},[412,2836,440],{"stretchy":435},[412,2838,278],{},[419,2840,445],{},[447,2842,588],{"encoding":449},[224,2844,2846,2873],{"className":2845,"ariaHidden":455},[454],[224,2847,2849,2852,2855,2858,2861,2864,2867,2870],{"className":2848},[459],[224,2850],{"className":2851,"style":535},[463],[224,2853,15],{"className":2854},[468,469],[224,2856,268],{"className":2857},[542],[224,2859,284],{"className":2860},[468],[224,2862,440],{"className":2863},[549],[224,2865],{"className":2866,"style":496},[473],[224,2868,278],{"className":2869},[500],[224,2871],{"className":2872,"style":496},[473],[224,2874,2876,2879],{"className":2875},[459],[224,2877],{"className":2878,"style":510},[463],[224,2880,445],{"className":2881},[468],[378,2883,2884],{},"数组理论",[221,2886,1907],{}," 和 ",[221,2889,2890],{},"store"," 的读写语义；",[2677,2893,2894,2751,3008,3011,3012,3040],{},[224,2895,2897,2927],{"className":2896,"translate":388},[392],[224,2898,2900],{"className":2899},[396],[398,2901,2902],{"xmlns":400},[403,2903,2904,2924],{},[406,2905,2906,2908,2910,2912,2914,2916,2918,2920,2922],{},[409,2907,1705],{},[412,2909,239],{"stretchy":435},[409,2911,371],{},[412,2913,1716],{"stretchy":435},[412,2915,2048],{"mathvariant":678},[409,2917,1705],{},[412,2919,239],{"stretchy":435},[409,2921,375],{},[412,2923,1716],{"stretchy":435},[447,2925,2926],{"encoding":449},"f(x) \\ne f(y)",[224,2928,2930,2990],{"className":2929,"ariaHidden":455},[454],[224,2931,2933,2936,2939,2942,2945,2948,2951,2987],{"className":2932},[459],[224,2934],{"className":2935,"style":535},[463],[224,2937,1705],{"className":2938,"style":1740},[468,469],[224,2940,239],{"className":2941},[542],[224,2943,371],{"className":2944},[468,469],[224,2946,1716],{"className":2947},[549],[224,2949],{"className":2950,"style":496},[473],[224,2952,2954,2981,2984],{"className":2953},[500],[224,2955,2957],{"className":2956},[500],[224,2958,2960],{"className":2959},[468,2553],[224,2961,2963],{"className":2962},[2557],[224,2964,2966,2969,2978],{"className":2965},[2561],[224,2967],{"className":2968,"style":2565},[463],[224,2970,2972],{"className":2971},[2569],[224,2973,2975],{"className":2974},[468],[224,2976,2576],{"className":2977},[500],[224,2979],{"className":2980},[2580],[224,2982],{"className":2983},[473,2584],[224,2985,278],{"className":2986},[500],[224,2988],{"className":2989,"style":496},[473],[224,2991,2993,2996,2999,3002,3005],{"className":2992},[459],[224,2994],{"className":2995,"style":535},[463],[224,2997,1705],{"className":2998,"style":1740},[468,469],[224,3000,239],{"className":3001},[542],[224,3003,375],{"className":3004,"style":492},[468,469],[224,3006,1716],{"className":3007},[549],[378,3009,3010],{},"未解释函数","：它不知道 ",[224,3013,3015,3028],{"className":3014,"translate":388},[392],[224,3016,3018],{"className":3017},[396],[398,3019,3020],{"xmlns":400},[403,3021,3022,3026],{},[406,3023,3024],{},[409,3025,1705],{},[447,3027,1705],{"encoding":449},[224,3029,3031],{"className":3030,"ariaHidden":455},[454],[224,3032,3034,3037],{"className":3033},[459],[224,3035],{"className":3036,"style":2565},[463],[224,3038,1705],{"className":3039,"style":1740},[468,469]," 具体算什么，只知道「同一个输入必须得到同一个输出」。",[11,3042,3043,3044,3099],{},"对 SAT 来说，这三个原子只是三个裸的布尔变量 ",[224,3045,3047,3069],{"className":3046,"translate":388},[392],[224,3048,3050],{"className":3049},[396],[398,3051,3052],{"xmlns":400},[403,3053,3054,3066],{},[406,3055,3056,3058,3060,3062,3064],{},[409,3057,11],{},[412,3059,1877],{"separator":455},[409,3061,2023],{},[412,3063,1877],{"separator":455},[409,3065,2061],{},[447,3067,3068],{"encoding":449},"p, q, r",[224,3070,3072],{"className":3071,"ariaHidden":455},[454],[224,3073,3075,3078,3081,3084,3087,3090,3093,3096],{"className":3074},[459],[224,3076],{"className":3077,"style":806},[463],[224,3079,11],{"className":3080},[468,469],[224,3082,1877],{"className":3083},[1917],[224,3085],{"className":3086,"style":724},[473],[224,3088,2023],{"className":3089,"style":492},[468,469],[224,3091,1877],{"className":3092},[1917],[224,3094],{"className":3095,"style":724},[473],[224,3097,2061],{"className":3098,"style":2473},[468,469],"。对 SMT 来说，它们各自带着一整套内部结构。",[3101,3102,3103],"div",{},[3104,3105],"img",{"height":3106,"width":3107,"style":3108,"data-d2-theme":3109,"alt":186,"src":3110},614,552,"max-width:100%;height:auto","default","data:image\u002Fsvg+xml;base64,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\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",[11,3112,3113,3114,3117],{},"这张图就是 SMT 的骨架：",[378,3115,3116],{},"布尔层负责「怎么组合」，理论层负责「每个原子是什么意思」","。",[613,3119,3121],{"id":3120},"_13-背景理论","1.3 背景理论",[11,3123,3124],{},"SMT 里常用的理论有这么几类：",[2674,3126,3127,3137,3214,3226,3299,3316,3325],{},[2677,3128,3129,3136],{},[378,3130,3131,3132,3135],{},"EUF（",[221,3133,3134],{},"QF_UF","）","：只处理相等和未解释函数，适合描述对象相等、指针别名。",[2677,3138,3139,3149,3150,372,3178,3208,3209,3211,3212,3117],{},[378,3140,3141,3142,3145,3146,3135],{},"线性算术（",[221,3143,3144],{},"QF_LRA"," \u002F ",[221,3147,3148],{},"QF_LIA","：处理 ",[224,3151,3153,3166],{"className":3152,"translate":388},[392],[224,3154,3156],{"className":3155},[396],[398,3157,3158],{"xmlns":400},[403,3159,3160,3164],{},[406,3161,3162],{},[412,3163,319],{},[447,3165,319],{"encoding":449},[224,3167,3169],{"className":3168,"ariaHidden":455},[454],[224,3170,3172,3175],{"className":3171},[459],[224,3173],{"className":3174,"style":464},[463],[224,3176,319],{"className":3177},[468],[224,3179,3181,3195],{"className":3180,"translate":388},[392],[224,3182,3184],{"className":3183},[396],[398,3185,3186],{"xmlns":400},[403,3187,3188,3192],{},[406,3189,3190],{},[412,3191,2070],{},[447,3193,3194],{"encoding":449},"\\le",[224,3196,3198],{"className":3197,"ariaHidden":455},[454],[224,3199,3201,3205],{"className":3200},[459],[224,3202],{"className":3203,"style":3204},[463],"height:0.7719em;vertical-align:-0.136em;",[224,3206,2070],{"className":3207},[500]," 和常数；实数版是 ",[221,3210,3144],{},"，整数版是 ",[221,3213,3148],{},[2677,3215,3216,3225],{},[378,3217,3218,3219,3145,3222,3135],{},"非线性算术（",[221,3220,3221],{},"QF_NRA",[221,3223,3224],{},"QF_NIA","：允许乘法；实数版可判定，整数版不可判定。",[2677,3227,3228,3234,3235,3298],{},[378,3229,3230,3231,3135],{},"位向量（",[221,3232,3233],{},"QF_BV","：描述机器整数和硬件电路，加法是模 ",[224,3236,3238,3258],{"className":3237,"translate":388},[392],[224,3239,3241],{"className":3240},[396],[398,3242,3243],{"xmlns":400},[403,3244,3245,3255],{},[406,3246,3247],{},[3248,3249,3250,3252],"msup",{},[419,3251,1447],{},[409,3253,3254],{},"n",[447,3256,3257],{"encoding":449},"2^n",[224,3259,3261],{"className":3260,"ariaHidden":455},[454],[224,3262,3264,3268],{"className":3263},[459],[224,3265],{"className":3266,"style":3267},[463],"height:0.6644em;",[224,3269,3271,3274],{"className":3270},[468],[224,3272,1447],{"className":3273},[468],[224,3275,3277],{"className":3276},[750],[224,3278,3280],{"className":3279},[754],[224,3281,3283],{"className":3282},[759],[224,3284,3286],{"className":3285,"style":3267},[763],[224,3287,3289,3292],{"style":3288},"top:-3.063em;margin-right:0.05em;",[224,3290],{"className":3291,"style":772},[771],[224,3293,3295],{"className":3294},[776,777,778,779],[224,3296,3254],{"className":3297},[468,469,779]," 的。",[2677,3300,3301,3310,3311,2887,3313,3315],{},[378,3302,3303,3304,3145,3307,3135],{},"数组（",[221,3305,3306],{},"QF_AX",[221,3308,3309],{},"QF_AUFBV","：用 ",[221,3312,1907],{},[221,3314,2890],{}," 描述内存读写。",[2677,3317,3318,3324],{},[378,3319,3320,3321,3135],{},"浮点（",[221,3322,3323],{},"QF_FP","：描述 IEEE 754 浮点运算。",[2677,3326,3327,3330,3331,372,3334,3337],{},[378,3328,3329],{},"字符串","：描述 ",[221,3332,3333],{},"concat",[221,3335,3336],{},"contains"," 等操作，工业实现常常不完整。",[11,3339,3340,3343],{},[221,3341,3342],{},"QF"," 是 quantifier-free 的缩写，意思是「无量词」。绝大多数 SMT 求解器面对的都是这个片段。",[11,3345,3346,3347,3375],{},"可以把 SAT 看成 SMT 的一个特例：当理论 ",[224,3348,3350,3363],{"className":3349,"translate":388},[392],[224,3351,3353],{"className":3352},[396],[398,3354,3355],{"xmlns":400},[403,3356,3357,3361],{},[406,3358,3359],{},[409,3360,637],{},[447,3362,637],{"encoding":449},[224,3364,3366],{"className":3365,"ariaHidden":455},[454],[224,3367,3369,3372],{"className":3368},[459],[224,3370],{"className":3371,"style":649},[463],[224,3373,637],{"className":3374,"style":653},[468,469]," 为空、所有原子都是裸命题时，SMT 就退化成 SAT。",[11,3377,3378],{},"反过来，SMT 也可以看成 SAT 的一次「语义升级」：布尔骨架不变，但每个命题不再是黑盒，而是有内部结构、可以被理论求解器检查和反驳的对象。",[11,3380,3381],{},"这个升级带来的好处，要放到具体的历史困境里才看得清楚。",[608,3383,3385],{"id":3384},"二smt-之前","二、SMT 之前",[613,3387,3389],{"id":3388},"_21-位爆破","2.1 位爆破",[11,3391,3392,3393,3396],{},"SAT 只认布尔变量。要用它处理整数，标准做法叫",[378,3394,3395],{},"位爆破","（bit-blasting）：把整数拆成一个个布尔变量，再把算术运算展开成逻辑门电路，最后用 Tseitin 变换把电路转成 CNF 子句。",[11,3398,3399,3400,3442],{},"看一个具体的小例子。取两个 4 位无符号整数 ",[224,3401,3403,3421],{"className":3402,"translate":388},[392],[224,3404,3406],{"className":3405},[396],[398,3407,3408],{"xmlns":400},[403,3409,3410,3418],{},[406,3411,3412,3414,3416],{},[409,3413,371],{},[412,3415,1877],{"separator":455},[409,3417,375],{},[447,3419,3420],{"encoding":449},"x, y",[224,3422,3424],{"className":3423,"ariaHidden":455},[454],[224,3425,3427,3430,3433,3436,3439],{"className":3426},[459],[224,3428],{"className":3429,"style":806},[463],[224,3431,371],{"className":3432},[468,469],[224,3434,1877],{"className":3435},[1917],[224,3437],{"className":3438,"style":724},[473],[224,3440,375],{"className":3441,"style":492},[468,469],"，约束是：",[224,3444,3446],{"className":3445,"translate":388},[387],[224,3447,3449,3483],{"className":3448,"translate":388},[392],[224,3450,3452],{"className":3451},[396],[398,3453,3454],{"xmlns":400,"display":401},[403,3455,3456,3480],{},[406,3457,3458,3460,3462,3464,3466,3468,3470,3472,3474,3476,3478],{},[409,3459,371],{},[412,3461,319],{},[409,3463,375],{},[412,3465,278],{},[419,3467,421],{},[423,3469,425],{},[412,3471,428],{},[423,3473,425],{},[409,3475,371],{},[412,3477,325],{},[409,3479,375],{},[447,3481,3482],{"encoding":449},"x + y = 10 \\;\\wedge\\; x > y",[224,3484,3486,3504,3522,3546,3564],{"className":3485,"ariaHidden":455},[454],[224,3487,3489,3492,3495,3498,3501],{"className":3488},[459],[224,3490],{"className":3491,"style":464},[463],[224,3493,371],{"className":3494},[468,469],[224,3496],{"className":3497,"style":474},[473],[224,3499,319],{"className":3500},[478],[224,3502],{"className":3503,"style":474},[473],[224,3505,3507,3510,3513,3516,3519],{"className":3506},[459],[224,3508],{"className":3509,"style":806},[463],[224,3511,375],{"className":3512,"style":492},[468,469],[224,3514],{"className":3515,"style":496},[473],[224,3517,278],{"className":3518},[500],[224,3520],{"className":3521,"style":496},[473],[224,3523,3525,3528,3531,3534,3537,3540,3543],{"className":3524},[459],[224,3526],{"className":3527,"style":510},[463],[224,3529,421],{"className":3530},[468],[224,3532],{"className":3533,"style":496},[473],[224,3535],{"className":3536,"style":474},[473],[224,3538,428],{"className":3539},[478],[224,3541],{"className":3542,"style":496},[473],[224,3544],{"className":3545,"style":474},[473],[224,3547,3549,3552,3555,3558,3561],{"className":3548},[459],[224,3550],{"className":3551,"style":2810},[463],[224,3553,371],{"className":3554},[468,469],[224,3556],{"className":3557,"style":496},[473],[224,3559,325],{"className":3560},[500],[224,3562],{"className":3563,"style":496},[473],[224,3565,3567,3570],{"className":3566},[459],[224,3568],{"className":3569,"style":806},[463],[224,3571,375],{"className":3572,"style":492},[468,469],[11,3574,3575,3576,2887,3786,3995,3996,4102,4103,4204],{},"SAT 看到的是 8 个布尔变量 ",[224,3577,3579,3616],{"className":3578,"translate":388},[392],[224,3580,3582],{"className":3581},[396],[398,3583,3584],{"xmlns":400},[403,3585,3586,3613],{},[406,3587,3588,3595,3601,3607],{},[695,3589,3590,3592],{},[409,3591,371],{},[419,3593,3594],{},"3",[695,3596,3597,3599],{},[409,3598,371],{},[419,3600,1447],{},[695,3602,3603,3605],{},[409,3604,371],{},[419,3606,1438],{},[695,3608,3609,3611],{},[409,3610,371],{},[419,3612,284],{},[447,3614,3615],{"encoding":449},"x_3 x_2 x_1 x_0",[224,3617,3619],{"className":3618,"ariaHidden":455},[454],[224,3620,3622,3626,3666,3706,3746],{"className":3621},[459],[224,3623],{"className":3624,"style":3625},[463],"height:0.5806em;vertical-align:-0.15em;",[224,3627,3629,3632],{"className":3628},[468],[224,3630,371],{"className":3631},[468,469],[224,3633,3635],{"className":3634},[750],[224,3636,3638,3658],{"className":3637},[754,755],[224,3639,3641,3655],{"className":3640},[759],[224,3642,3644],{"className":3643,"style":1479},[763],[224,3645,3646,3649],{"style":767},[224,3647],{"className":3648,"style":772},[771],[224,3650,3652],{"className":3651},[776,777,778,779],[224,3653,3594],{"className":3654},[468,779],[224,3656,787],{"className":3657},[786],[224,3659,3661],{"className":3660},[759],[224,3662,3664],{"className":3663,"style":794},[763],[224,3665],{},[224,3667,3669,3672],{"className":3668},[468],[224,3670,371],{"className":3671},[468,469],[224,3673,3675],{"className":3674},[750],[224,3676,3678,3698],{"className":3677},[754,755],[224,3679,3681,3695],{"className":3680},[759],[224,3682,3684],{"className":3683,"style":1479},[763],[224,3685,3686,3689],{"style":767},[224,3687],{"className":3688,"style":772},[771],[224,3690,3692],{"className":3691},[776,777,778,779],[224,3693,1447],{"className":3694},[468,779],[224,3696,787],{"className":3697},[786],[224,3699,3701],{"className":3700},[759],[224,3702,3704],{"className":3703,"style":794},[763],[224,3705],{},[224,3707,3709,3712],{"className":3708},[468],[224,3710,371],{"className":3711},[468,469],[224,3713,3715],{"className":3714},[750],[224,3716,3718,3738],{"className":3717},[754,755],[224,3719,3721,3735],{"className":3720},[759],[224,3722,3724],{"className":3723,"style":1479},[763],[224,3725,3726,3729],{"style":767},[224,3727],{"className":3728,"style":772},[771],[224,3730,3732],{"className":3731},[776,777,778,779],[224,3733,1438],{"className":3734},[468,779],[224,3736,787],{"className":3737},[786],[224,3739,3741],{"className":3740},[759],[224,3742,3744],{"className":3743,"style":794},[763],[224,3745],{},[224,3747,3749,3752],{"className":3748},[468],[224,3750,371],{"className":3751},[468,469],[224,3753,3755],{"className":3754},[750],[224,3756,3758,3778],{"className":3757},[754,755],[224,3759,3761,3775],{"className":3760},[759],[224,3762,3764],{"className":3763,"style":1479},[763],[224,3765,3766,3769],{"style":767},[224,3767],{"className":3768,"style":772},[771],[224,3770,3772],{"className":3771},[776,777,778,779],[224,3773,284],{"className":3774},[468,779],[224,3776,787],{"className":3777},[786],[224,3779,3781],{"className":3780},[759],[224,3782,3784],{"className":3783,"style":794},[763],[224,3785],{},[224,3787,3789,3825],{"className":3788,"translate":388},[392],[224,3790,3792],{"className":3791},[396],[398,3793,3794],{"xmlns":400},[403,3795,3796,3822],{},[406,3797,3798,3804,3810,3816],{},[695,3799,3800,3802],{},[409,3801,375],{},[419,3803,3594],{},[695,3805,3806,3808],{},[409,3807,375],{},[419,3809,1447],{},[695,3811,3812,3814],{},[409,3813,375],{},[419,3815,1438],{},[695,3817,3818,3820],{},[409,3819,375],{},[419,3821,284],{},[447,3823,3824],{"encoding":449},"y_3 y_2 y_1 y_0",[224,3826,3828],{"className":3827,"ariaHidden":455},[454],[224,3829,3831,3834,3875,3915,3955],{"className":3830},[459],[224,3832],{"className":3833,"style":806},[463],[224,3835,3837,3840],{"className":3836},[468],[224,3838,375],{"className":3839,"style":492},[468,469],[224,3841,3843],{"className":3842},[750],[224,3844,3846,3867],{"className":3845},[754,755],[224,3847,3849,3864],{"className":3848},[759],[224,3850,3852],{"className":3851,"style":1479},[763],[224,3853,3855,3858],{"style":3854},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[224,3856],{"className":3857,"style":772},[771],[224,3859,3861],{"className":3860},[776,777,778,779],[224,3862,3594],{"className":3863},[468,779],[224,3865,787],{"className":3866},[786],[224,3868,3870],{"className":3869},[759],[224,3871,3873],{"className":3872,"style":794},[763],[224,3874],{},[224,3876,3878,3881],{"className":3877},[468],[224,3879,375],{"className":3880,"style":492},[468,469],[224,3882,3884],{"className":3883},[750],[224,3885,3887,3907],{"className":3886},[754,755],[224,3888,3890,3904],{"className":3889},[759],[224,3891,3893],{"className":3892,"style":1479},[763],[224,3894,3895,3898],{"style":3854},[224,3896],{"className":3897,"style":772},[771],[224,3899,3901],{"className":3900},[776,777,778,779],[224,3902,1447],{"className":3903},[468,779],[224,3905,787],{"className":3906},[786],[224,3908,3910],{"className":3909},[759],[224,3911,3913],{"className":3912,"style":794},[763],[224,3914],{},[224,3916,3918,3921],{"className":3917},[468],[224,3919,375],{"className":3920,"style":492},[468,469],[224,3922,3924],{"className":3923},[750],[224,3925,3927,3947],{"className":3926},[754,755],[224,3928,3930,3944],{"className":3929},[759],[224,3931,3933],{"className":3932,"style":1479},[763],[224,3934,3935,3938],{"style":3854},[224,3936],{"className":3937,"style":772},[771],[224,3939,3941],{"className":3940},[776,777,778,779],[224,3942,1438],{"className":3943},[468,779],[224,3945,787],{"className":3946},[786],[224,3948,3950],{"className":3949},[759],[224,3951,3953],{"className":3952,"style":794},[763],[224,3954],{},[224,3956,3958,3961],{"className":3957},[468],[224,3959,375],{"className":3960,"style":492},[468,469],[224,3962,3964],{"className":3963},[750],[224,3965,3967,3987],{"className":3966},[754,755],[224,3968,3970,3984],{"className":3969},[759],[224,3971,3973],{"className":3972,"style":1479},[763],[224,3974,3975,3978],{"style":3854},[224,3976],{"className":3977,"style":772},[771],[224,3979,3981],{"className":3980},[776,777,778,779],[224,3982,284],{"className":3983},[468,779],[224,3985,787],{"className":3986},[786],[224,3988,3990],{"className":3989},[759],[224,3991,3993],{"className":3992,"style":794},[763],[224,3994],{},"。为了表达加法，需要 4 个全加器。每个全加器有三个输入 ",[224,3997,3999,4030],{"className":3998,"translate":388},[392],[224,4000,4002],{"className":4001},[396],[398,4003,4004],{"xmlns":400},[403,4005,4006,4027],{},[406,4007,4008,4010,4012,4015,4017],{},[409,4009,15],{},[412,4011,1877],{"separator":455},[409,4013,4014],{},"b",[412,4016,1877],{"separator":455},[695,4018,4019,4021],{},[409,4020,219],{},[406,4022,4023,4025],{},[409,4024,1880],{},[409,4026,3254],{},[447,4028,4029],{"encoding":449},"a, b, c_{in}",[224,4031,4033],{"className":4032,"ariaHidden":455},[454],[224,4034,4036,4039,4042,4045,4048,4051,4054,4057],{"className":4035},[459],[224,4037],{"className":4038,"style":2565},[463],[224,4040,15],{"className":4041},[468,469],[224,4043,1877],{"className":4044},[1917],[224,4046],{"className":4047,"style":724},[473],[224,4049,4014],{"className":4050},[468,469],[224,4052,1877],{"className":4053},[1917],[224,4055],{"className":4056,"style":724},[473],[224,4058,4060,4063],{"className":4059},[468],[224,4061,219],{"className":4062},[468,469],[224,4064,4066],{"className":4065},[750],[224,4067,4069,4094],{"className":4068},[754,755],[224,4070,4072,4091],{"className":4071},[759],[224,4073,4076],{"className":4074,"style":4075},[763],"height:0.3117em;",[224,4077,4078,4081],{"style":767},[224,4079],{"className":4080,"style":772},[771],[224,4082,4084],{"className":4083},[776,777,778,779],[224,4085,4087],{"className":4086},[468,779],[224,4088,4090],{"className":4089},[468,469,779],"in",[224,4092,787],{"className":4093},[786],[224,4095,4097],{"className":4096},[759],[224,4098,4100],{"className":4099,"style":794},[763],[224,4101],{},"、两个输出 ",[224,4104,4106,4136],{"className":4105,"translate":388},[392],[224,4107,4109],{"className":4108},[396],[398,4110,4111],{"xmlns":400},[403,4112,4113,4133],{},[406,4114,4115,4117,4119],{},[409,4116,1858],{},[412,4118,1877],{"separator":455},[695,4120,4121,4123],{},[409,4122,219],{},[406,4124,4125,4128,4131],{},[409,4126,4127],{},"o",[409,4129,4130],{},"u",[409,4132,1435],{},[447,4134,4135],{"encoding":449},"s, c_{out}",[224,4137,4139],{"className":4138,"ariaHidden":455},[454],[224,4140,4142,4145,4148,4151,4154],{"className":4141},[459],[224,4143],{"className":4144,"style":806},[463],[224,4146,1858],{"className":4147},[468,469],[224,4149,1877],{"className":4150},[1917],[224,4152],{"className":4153,"style":724},[473],[224,4155,4157,4160],{"className":4156},[468],[224,4158,219],{"className":4159},[468,469],[224,4161,4163],{"className":4162},[750],[224,4164,4166,4196],{"className":4165},[754,755],[224,4167,4169,4193],{"className":4168},[759],[224,4170,4173],{"className":4171,"style":4172},[763],"height:0.2806em;",[224,4174,4175,4178],{"style":767},[224,4176],{"className":4177,"style":772},[771],[224,4179,4181],{"className":4180},[776,777,778,779],[224,4182,4184,4187,4190],{"className":4183},[468,779],[224,4185,4127],{"className":4186},[468,469,779],[224,4188,4130],{"className":4189},[468,469,779],[224,4191,1435],{"className":4192},[468,469,779],[224,4194,787],{"className":4195},[786],[224,4197,4199],{"className":4198},[759],[224,4200,4202],{"className":4201,"style":794},[763],[224,4203],{},"，满足：",[224,4206,4208],{"className":4207,"translate":388},[387],[224,4209,4211,4303],{"className":4210,"translate":388},[392],[224,4212,4214],{"className":4213},[396],[398,4215,4216],{"xmlns":400,"display":401},[403,4217,4218,4300],{},[406,4219,4220,4222,4224,4226,4229,4231,4233,4243,4245,4248,4260,4262,4264,4266,4268,4270,4272,4274,4276,4286,4288,4290,4292,4294,4296,4298],{},[409,4221,1858],{},[412,4223,278],{},[409,4225,15],{},[412,4227,4228],{},"⊕",[409,4230,4014],{},[412,4232,4228],{},[695,4234,4235,4237],{},[409,4236,219],{},[406,4238,4239,4241],{},[409,4240,1880],{},[409,4242,3254],{},[412,4244,1877],{"separator":455},[473,4246],{"width":4247},"2em",[695,4249,4250,4252],{},[409,4251,219],{},[406,4253,4254,4256,4258],{},[409,4255,4127],{},[409,4257,4130],{},[409,4259,1435],{},[412,4261,278],{},[412,4263,239],{"stretchy":435},[409,4265,15],{},[412,4267,428],{},[409,4269,4014],{},[412,4271,1716],{"stretchy":435},[412,4273,1131],{},[412,4275,239],{"fence":455,"stretchy":455,"minsize":2004,"maxsize":2004},[695,4277,4278,4280],{},[409,4279,219],{},[406,4281,4282,4284],{},[409,4283,1880],{},[409,4285,3254],{},[412,4287,428],{},[412,4289,239],{"stretchy":435},[409,4291,15],{},[412,4293,4228],{},[409,4295,4014],{},[412,4297,1716],{"stretchy":435},[412,4299,1716],{"fence":455,"stretchy":455,"minsize":2004,"maxsize":2004},[447,4301,4302],{"encoding":449},"s = a \\oplus b \\oplus c_{in}, \\qquad c_{out} = (a \\wedge b) \\vee \\bigl(c_{in} \\wedge (a \\oplus b)\\bigr)",[224,4304,4306,4324,4342,4361,4478,4499,4520,4584,4605],{"className":4305,"ariaHidden":455},[454],[224,4307,4309,4312,4315,4318,4321],{"className":4308},[459],[224,4310],{"className":4311,"style":839},[463],[224,4313,1858],{"className":4314},[468,469],[224,4316],{"className":4317,"style":496},[473],[224,4319,278],{"className":4320},[500],[224,4322],{"className":4323,"style":496},[473],[224,4325,4327,4330,4333,4336,4339],{"className":4326},[459],[224,4328],{"className":4329,"style":464},[463],[224,4331,15],{"className":4332},[468,469],[224,4334],{"className":4335,"style":474},[473],[224,4337,4228],{"className":4338},[478],[224,4340],{"className":4341,"style":474},[473],[224,4343,4345,4349,4352,4355,4358],{"className":4344},[459],[224,4346],{"className":4347,"style":4348},[463],"height:0.7778em;vertical-align:-0.0833em;",[224,4350,4014],{"className":4351},[468,469],[224,4353],{"className":4354,"style":474},[473],[224,4356,4228],{"className":4357},[478],[224,4359],{"className":4360,"style":474},[473],[224,4362,4364,4367,4410,4413,4417,4420,4469,4472,4475],{"className":4363},[459],[224,4365],{"className":4366,"style":806},[463],[224,4368,4370,4373],{"className":4369},[468],[224,4371,219],{"className":4372},[468,469],[224,4374,4376],{"className":4375},[750],[224,4377,4379,4402],{"className":4378},[754,755],[224,4380,4382,4399],{"className":4381},[759],[224,4383,4385],{"className":4384,"style":4075},[763],[224,4386,4387,4390],{"style":767},[224,4388],{"className":4389,"style":772},[771],[224,4391,4393],{"className":4392},[776,777,778,779],[224,4394,4396],{"className":4395},[468,779],[224,4397,4090],{"className":4398},[468,469,779],[224,4400,787],{"className":4401},[786],[224,4403,4405],{"className":4404},[759],[224,4406,4408],{"className":4407,"style":794},[763],[224,4409],{},[224,4411,1877],{"className":4412},[1917],[224,4414],{"className":4415,"style":4416},[473],"margin-right:2em;",[224,4418],{"className":4419,"style":724},[473],[224,4421,4423,4426],{"className":4422},[468],[224,4424,219],{"className":4425},[468,469],[224,4427,4429],{"className":4428},[750],[224,4430,4432,4461],{"className":4431},[754,755],[224,4433,4435,4458],{"className":4434},[759],[224,4436,4438],{"className":4437,"style":4172},[763],[224,4439,4440,4443],{"style":767},[224,4441],{"className":4442,"style":772},[771],[224,4444,4446],{"className":4445},[776,777,778,779],[224,4447,4449,4452,4455],{"className":4448},[468,779],[224,4450,4127],{"className":4451},[468,469,779],[224,4453,4130],{"className":4454},[468,469,779],[224,4456,1435],{"className":4457},[468,469,779],[224,4459,787],{"className":4460},[786],[224,4462,4464],{"className":4463},[759],[224,4465,4467],{"className":4466,"style":794},[763],[224,4468],{},[224,4470],{"className":4471,"style":496},[473],[224,4473,278],{"className":4474},[500],[224,4476],{"className":4477,"style":496},[473],[224,4479,4481,4484,4487,4490,4493,4496],{"className":4480},[459],[224,4482],{"className":4483,"style":535},[463],[224,4485,239],{"className":4486},[542],[224,4488,15],{"className":4489},[468,469],[224,4491],{"className":4492,"style":474},[473],[224,4494,428],{"className":4495},[478],[224,4497],{"className":4498,"style":474},[473],[224,4500,4502,4505,4508,4511,4514,4517],{"className":4501},[459],[224,4503],{"className":4504,"style":535},[463],[224,4506,4014],{"className":4507},[468,469],[224,4509,1716],{"className":4510},[549],[224,4512],{"className":4513,"style":474},[473],[224,4515,1131],{"className":4516},[478],[224,4518],{"className":4519,"style":474},[473],[224,4521,4523,4526,4532,4575,4578,4581],{"className":4522},[459],[224,4524],{"className":4525,"style":2663},[463],[224,4527,4529],{"className":4528},[542],[224,4530,239],{"className":4531},[2363,2364],[224,4533,4535,4538],{"className":4534},[468],[224,4536,219],{"className":4537},[468,469],[224,4539,4541],{"className":4540},[750],[224,4542,4544,4567],{"className":4543},[754,755],[224,4545,4547,4564],{"className":4546},[759],[224,4548,4550],{"className":4549,"style":4075},[763],[224,4551,4552,4555],{"style":767},[224,4553],{"className":4554,"style":772},[771],[224,4556,4558],{"className":4557},[776,777,778,779],[224,4559,4561],{"className":4560},[468,779],[224,4562,4090],{"className":4563},[468,469,779],[224,4565,787],{"className":4566},[786],[224,4568,4570],{"className":4569},[759],[224,4571,4573],{"className":4572,"style":794},[763],[224,4574],{},[224,4576],{"className":4577,"style":474},[473],[224,4579,428],{"className":4580},[478],[224,4582],{"className":4583,"style":474},[473],[224,4585,4587,4590,4593,4596,4599,4602],{"className":4586},[459],[224,4588],{"className":4589,"style":535},[463],[224,4591,239],{"className":4592},[542],[224,4594,15],{"className":4595},[468,469],[224,4597],{"className":4598,"style":474},[473],[224,4600,4228],{"className":4601},[478],[224,4603],{"className":4604,"style":474},[473],[224,4606,4608,4611,4614,4617],{"className":4607},[459],[224,4609],{"className":4610,"style":2663},[463],[224,4612,4014],{"className":4613},[468,469],[224,4615,1716],{"className":4616},[549],[224,4618,4620],{"className":4619},[549],[224,4621,1716],{"className":4622},[2363,2364],[11,4624,4625],{},"把这两个等式转成 CNF，一个全加器大约 17 条子句。4 位加法器就是 4 个全加器，约 70 条子句。再加上「和等于 1010、进位为 0」的 5 条单元子句，以及一个 4 位比较器的几十条子句，整个约束变成一百多条子句、几十个辅助变量。",[11,4627,4628],{},"而原始约束只有一行：",[215,4630,4632],{"className":217,"code":4631,"language":219,"meta":186,"style":186},"x + y == 10 && x > y\n",[221,4633,4634],{"__ignoreMap":186},[224,4635,4636,4639,4641,4643,4645,4647,4649,4652,4654],{"class":226,"line":227},[224,4637,4638],{"class":238},"x ",[224,4640,319],{"class":230},[224,4642,322],{"class":238},[224,4644,342],{"class":230},[224,4646,328],{"class":271},[224,4648,331],{"class":230},[224,4650,4651],{"class":238}," x ",[224,4653,325],{"class":230},[224,4655,4656],{"class":238}," y\n",[11,4658,4659,4660,372,4712,4764],{},"求解器可能给出 ",[224,4661,4663,4682],{"className":4662,"translate":388},[392],[224,4664,4666],{"className":4665},[396],[398,4667,4668],{"xmlns":400},[403,4669,4670,4679],{},[406,4671,4672,4674,4676],{},[409,4673,371],{},[412,4675,278],{},[419,4677,4678],{},"0110",[447,4680,4681],{"encoding":449},"x = 0110",[224,4683,4685,4703],{"className":4684,"ariaHidden":455},[454],[224,4686,4688,4691,4694,4697,4700],{"className":4687},[459],[224,4689],{"className":4690,"style":839},[463],[224,4692,371],{"className":4693},[468,469],[224,4695],{"className":4696,"style":496},[473],[224,4698,278],{"className":4699},[500],[224,4701],{"className":4702,"style":496},[473],[224,4704,4706,4709],{"className":4705},[459],[224,4707],{"className":4708,"style":510},[463],[224,4710,4678],{"className":4711},[468],[224,4713,4715,4734],{"className":4714,"translate":388},[392],[224,4716,4718],{"className":4717},[396],[398,4719,4720],{"xmlns":400},[403,4721,4722,4731],{},[406,4723,4724,4726,4728],{},[409,4725,375],{},[412,4727,278],{},[419,4729,4730],{},"0100",[447,4732,4733],{"encoding":449},"y = 0100",[224,4735,4737,4755],{"className":4736,"ariaHidden":455},[454],[224,4738,4740,4743,4746,4749,4752],{"className":4739},[459],[224,4741],{"className":4742,"style":806},[463],[224,4744,375],{"className":4745,"style":492},[468,469],[224,4747],{"className":4748,"style":496},[473],[224,4750,278],{"className":4751},[500],[224,4753],{"className":4754,"style":496},[473],[224,4756,4758,4761],{"className":4757},[459],[224,4759],{"className":4760,"style":510},[463],[224,4762,4730],{"className":4763},[468],"，但它并不知道这是 6 和 4。它只知道这一组真假值恰好让所有子句为真。",[11,4766,4767,4768,4771,4772,4868,4869,4897,4898,4926,4927,4897,4955,4983],{},"规模还不是最麻烦的，",[378,4769,4770],{},"语义丢失","才是。假设我们想证明 ",[224,4773,4775,4801],{"className":4774,"translate":388},[392],[224,4776,4778],{"className":4777},[396],[398,4779,4780],{"xmlns":400},[403,4781,4782,4798],{},[406,4783,4784,4786,4788,4790,4792,4794,4796],{},[409,4785,371],{},[412,4787,319],{},[409,4789,375],{},[412,4791,278],{},[409,4793,375],{},[412,4795,319],{},[409,4797,371],{},[447,4799,4800],{"encoding":449},"x + y = y + x",[224,4802,4804,4822,4840,4859],{"className":4803,"ariaHidden":455},[454],[224,4805,4807,4810,4813,4816,4819],{"className":4806},[459],[224,4808],{"className":4809,"style":464},[463],[224,4811,371],{"className":4812},[468,469],[224,4814],{"className":4815,"style":474},[473],[224,4817,319],{"className":4818},[478],[224,4820],{"className":4821,"style":474},[473],[224,4823,4825,4828,4831,4834,4837],{"className":4824},[459],[224,4826],{"className":4827,"style":806},[463],[224,4829,375],{"className":4830,"style":492},[468,469],[224,4832],{"className":4833,"style":496},[473],[224,4835,278],{"className":4836},[500],[224,4838],{"className":4839,"style":496},[473],[224,4841,4843,4847,4850,4853,4856],{"className":4842},[459],[224,4844],{"className":4845,"style":4846},[463],"height:0.7778em;vertical-align:-0.1944em;",[224,4848,375],{"className":4849,"style":492},[468,469],[224,4851],{"className":4852,"style":474},[473],[224,4854,319],{"className":4855},[478],[224,4857],{"className":4858,"style":474},[473],[224,4860,4862,4865],{"className":4861},[459],[224,4863],{"className":4864,"style":839},[463],[224,4866,371],{"className":4867},[468,469],"。在算术里这是一条公理，一秒都不用。但在 SAT 里，左边是一个「先加 ",[224,4870,4872,4885],{"className":4871,"translate":388},[392],[224,4873,4875],{"className":4874},[396],[398,4876,4877],{"xmlns":400},[403,4878,4879,4883],{},[406,4880,4881],{},[409,4882,371],{},[447,4884,371],{"encoding":449},[224,4886,4888],{"className":4887,"ariaHidden":455},[454],[224,4889,4891,4894],{"className":4890},[459],[224,4892],{"className":4893,"style":839},[463],[224,4895,371],{"className":4896},[468,469]," 再加 ",[224,4899,4901,4914],{"className":4900,"translate":388},[392],[224,4902,4904],{"className":4903},[396],[398,4905,4906],{"xmlns":400},[403,4907,4908,4912],{},[406,4909,4910],{},[409,4911,375],{},[447,4913,375],{"encoding":449},[224,4915,4917],{"className":4916,"ariaHidden":455},[454],[224,4918,4920,4923],{"className":4919},[459],[224,4921],{"className":4922,"style":806},[463],[224,4924,375],{"className":4925,"style":492},[468,469],"」的电路，右边是一个「先加 ",[224,4928,4930,4943],{"className":4929,"translate":388},[392],[224,4931,4933],{"className":4932},[396],[398,4934,4935],{"xmlns":400},[403,4936,4937,4941],{},[406,4938,4939],{},[409,4940,375],{},[447,4942,375],{"encoding":449},[224,4944,4946],{"className":4945,"ariaHidden":455},[454],[224,4947,4949,4952],{"className":4948},[459],[224,4950],{"className":4951,"style":806},[463],[224,4953,375],{"className":4954,"style":492},[468,469],[224,4956,4958,4971],{"className":4957,"translate":388},[392],[224,4959,4961],{"className":4960},[396],[398,4962,4963],{"xmlns":400},[403,4964,4965,4969],{},[406,4966,4967],{},[409,4968,371],{},[447,4970,371],{"encoding":449},[224,4972,4974],{"className":4973,"ariaHidden":455},[454],[224,4975,4977,4980],{"className":4976},[459],[224,4978],{"className":4979,"style":839},[463],[224,4981,371],{"className":4982},[468,469],"」的电路，两个电路内部结构不同，SAT 只能把两边都展开，再逐位证明输出相等。它不知道「加法可交换」，只知道一堆门。",[11,4985,4986],{},"打个比方：这就像为了判断一首诗是否押韵，先把它拆成一个个字母，再让机器从字母的排列里重新发现什么是韵脚。信息没有丢，但结构没了。",[613,4988,4990],{"id":4989},"_22-不可判定","2.2 不可判定",[11,4992,4993],{},"那干脆用表达力最强的一阶逻辑，让自动定理证明器去证，不行吗？",[11,4995,4996],{},"不行。看一个具体方程：",[224,4998,5000],{"className":4999,"translate":388},[387],[224,5001,5003,5043],{"className":5002,"translate":388},[392],[224,5004,5006],{"className":5005},[396],[398,5007,5008],{"xmlns":400,"display":401},[403,5009,5010,5040],{},[406,5011,5012,5018,5020,5026,5028,5035,5037],{},[3248,5013,5014,5016],{},[409,5015,371],{},[419,5017,3594],{},[412,5019,319],{},[3248,5021,5022,5024],{},[409,5023,375],{},[419,5025,3594],{},[412,5027,319],{},[3248,5029,5030,5033],{},[409,5031,5032],{},"z",[419,5034,3594],{},[412,5036,278],{},[419,5038,5039],{},"33",[447,5041,5042],{"encoding":449},"x^3 + y^3 + z^3 = 33",[224,5044,5046,5093,5138,5183],{"className":5045,"ariaHidden":455},[454],[224,5047,5049,5053,5084,5087,5090],{"className":5048},[459],[224,5050],{"className":5051,"style":5052},[463],"height:0.9474em;vertical-align:-0.0833em;",[224,5054,5056,5059],{"className":5055},[468],[224,5057,371],{"className":5058},[468,469],[224,5060,5062],{"className":5061},[750],[224,5063,5065],{"className":5064},[754],[224,5066,5068],{"className":5067},[759],[224,5069,5072],{"className":5070,"style":5071},[763],"height:0.8641em;",[224,5073,5075,5078],{"style":5074},"top:-3.113em;margin-right:0.05em;",[224,5076],{"className":5077,"style":772},[771],[224,5079,5081],{"className":5080},[776,777,778,779],[224,5082,3594],{"className":5083},[468,779],[224,5085],{"className":5086,"style":474},[473],[224,5088,319],{"className":5089},[478],[224,5091],{"className":5092,"style":474},[473],[224,5094,5096,5100,5129,5132,5135],{"className":5095},[459],[224,5097],{"className":5098,"style":5099},[463],"height:1.0585em;vertical-align:-0.1944em;",[224,5101,5103,5106],{"className":5102},[468],[224,5104,375],{"className":5105,"style":492},[468,469],[224,5107,5109],{"className":5108},[750],[224,5110,5112],{"className":5111},[754],[224,5113,5115],{"className":5114},[759],[224,5116,5118],{"className":5117,"style":5071},[763],[224,5119,5120,5123],{"style":5074},[224,5121],{"className":5122,"style":772},[771],[224,5124,5126],{"className":5125},[776,777,778,779],[224,5127,3594],{"className":5128},[468,779],[224,5130],{"className":5131,"style":474},[473],[224,5133,319],{"className":5134},[478],[224,5136],{"className":5137,"style":474},[473],[224,5139,5141,5144,5174,5177,5180],{"className":5140},[459],[224,5142],{"className":5143,"style":5071},[463],[224,5145,5147,5151],{"className":5146},[468],[224,5148,5032],{"className":5149,"style":5150},[468,469],"margin-right:0.044em;",[224,5152,5154],{"className":5153},[750],[224,5155,5157],{"className":5156},[754],[224,5158,5160],{"className":5159},[759],[224,5161,5163],{"className":5162,"style":5071},[763],[224,5164,5165,5168],{"style":5074},[224,5166],{"className":5167,"style":772},[771],[224,5169,5171],{"className":5170},[776,777,778,779],[224,5172,3594],{"className":5173},[468,779],[224,5175],{"className":5176,"style":496},[473],[224,5178,278],{"className":5179},[500],[224,5181],{"className":5182,"style":496},[473],[224,5184,5186,5189],{"className":5185},[459],[224,5187],{"className":5188,"style":510},[463],[224,5190,5039],{"className":5191},[468],[11,5193,5194],{},"它看起来很简单。它有整数解，一个已知解是：",[224,5196,5198],{"className":5197,"translate":388},[387],[224,5199,5201,5312],{"className":5200,"translate":388},[392],[224,5202,5204],{"className":5203},[396],[398,5205,5206],{"xmlns":400,"display":401},[403,5207,5208,5309],{},[5209,5210,5214,5246,5278],"mtable",{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},"0.25em","right left","0em",[5215,5216,5217,5231],"mtr",{},[5218,5219,5220],"mtd",{},[5221,5222,5223],"mstyle",{"scriptlevel":284,"displaystyle":455},[406,5224,5225,5227,5229],{},[419,5226,5039],{},[412,5228,278],{},[406,5230],{},[5218,5232,5233],{},[5221,5234,5235],{"scriptlevel":284,"displaystyle":455},[406,5236,5237,5239],{},[406,5238],{},[3248,5240,5241,5244],{},[419,5242,5243],{},"8866128975287528",[419,5245,3594],{},[5215,5247,5248,5254],{},[5218,5249,5250],{},[5221,5251,5252],{"scriptlevel":284,"displaystyle":455},[406,5253],{},[5218,5255,5256],{},[5221,5257,5258],{"scriptlevel":284,"displaystyle":455},[406,5259,5260,5262,5264,5266,5269,5272],{},[406,5261],{},[412,5263,319],{},[412,5265,239],{"stretchy":435},[412,5267,5268],{},"−",[419,5270,5271],{},"8778405442862239",[3248,5273,5274,5276],{},[412,5275,1716],{"stretchy":435},[419,5277,3594],{},[5215,5279,5280,5286],{},[5218,5281,5282],{},[5221,5283,5284],{"scriptlevel":284,"displaystyle":455},[406,5285],{},[5218,5287,5288],{},[5221,5289,5290],{"scriptlevel":284,"displaystyle":455},[406,5291,5292,5294,5296,5298,5300,5303],{},[406,5293],{},[412,5295,319],{},[412,5297,239],{"stretchy":435},[412,5299,5268],{},[419,5301,5302],{},"2736111468807040",[3248,5304,5305,5307],{},[412,5306,1716],{"stretchy":435},[419,5308,3594],{},[447,5310,5311],{"encoding":449},"\\begin{aligned}\n33 ={}& 8866128975287528^3 \\\\\n      &+ (-8778405442862239)^3 \\\\\n      &+ (-2736111468807040)^3\n\\end{aligned}",[224,5313,5315],{"className":5314,"ariaHidden":455},[454],[224,5316,5318,5322],{"className":5317},[459],[224,5319],{"className":5320,"style":5321},[463],"height:4.2723em;vertical-align:-1.8862em;",[224,5323,5325],{"className":5324},[468],[224,5326,5328,5396],{"className":5327},[5209],[224,5329,5332],{"className":5330},[5331],"col-align-r",[224,5333,5335,5387],{"className":5334},[754,755],[224,5336,5338,5384],{"className":5337},[759],[224,5339,5342,5366,5375],{"className":5340,"style":5341},[763],"height:2.3862em;",[224,5343,5345,5348],{"style":5344},"top:-4.5221em;",[224,5346],{"className":5347,"style":2108},[771],[224,5349,5351,5354,5357,5360,5363],{"className":5350},[468],[224,5352,5039],{"className":5353},[468],[224,5355],{"className":5356,"style":496},[473],[224,5358,278],{"className":5359},[500],[224,5361],{"className":5362,"style":496},[473],[224,5364],{"className":5365},[468],[224,5367,5369,5372],{"style":5368},"top:-2.9979em;",[224,5370],{"className":5371,"style":2108},[771],[224,5373],{"className":5374},[468],[224,5376,5378,5381],{"style":5377},"top:-1.4738em;",[224,5379],{"className":5380,"style":2108},[771],[224,5382],{"className":5383},[468],[224,5385,787],{"className":5386},[786],[224,5388,5390],{"className":5389},[759],[224,5391,5394],{"className":5392,"style":5393},[763],"height:1.8862em;",[224,5395],{},[224,5397,5400],{"className":5398},[5399],"col-align-l",[224,5401,5403,5573],{"className":5402},[754,755],[224,5404,5406,5570],{"className":5405},[759],[224,5407,5409,5454,5512],{"className":5408,"style":5341},[763],[224,5410,5411,5414],{"style":5344},[224,5412],{"className":5413,"style":2108},[771],[224,5415,5417,5420,5424],{"className":5416},[468],[224,5418],{"className":5419},[468],[224,5421,5423],{"className":5422},[468],"886612897528752",[224,5425,5427,5431],{"className":5426},[468],[224,5428,5430],{"className":5429},[468],"8",[224,5432,5434],{"className":5433},[750],[224,5435,5437],{"className":5436},[754],[224,5438,5440],{"className":5439},[759],[224,5441,5443],{"className":5442,"style":5071},[763],[224,5444,5445,5448],{"style":5074},[224,5446],{"className":5447,"style":772},[771],[224,5449,5451],{"className":5450},[776,777,778,779],[224,5452,3594],{"className":5453},[468,779],[224,5455,5456,5459],{"style":5368},[224,5457],{"className":5458,"style":2108},[771],[224,5460,5462,5465,5468,5471,5474,5477,5480,5483],{"className":5461},[468],[224,5463],{"className":5464},[468],[224,5466],{"className":5467,"style":474},[473],[224,5469,319],{"className":5470},[478],[224,5472],{"className":5473,"style":474},[473],[224,5475,239],{"className":5476},[542],[224,5478,5268],{"className":5479},[468],[224,5481,5271],{"className":5482},[468],[224,5484,5486,5489],{"className":5485},[549],[224,5487,1716],{"className":5488},[549],[224,5490,5492],{"className":5491},[750],[224,5493,5495],{"className":5494},[754],[224,5496,5498],{"className":5497},[759],[224,5499,5501],{"className":5500,"style":5071},[763],[224,5502,5503,5506],{"style":5074},[224,5504],{"className":5505,"style":772},[771],[224,5507,5509],{"className":5508},[776,777,778,779],[224,5510,3594],{"className":5511},[468,779],[224,5513,5514,5517],{"style":5377},[224,5515],{"className":5516,"style":2108},[771],[224,5518,5520,5523,5526,5529,5532,5535,5538,5541],{"className":5519},[468],[224,5521],{"className":5522},[468],[224,5524],{"className":5525,"style":474},[473],[224,5527,319],{"className":5528},[478],[224,5530],{"className":5531,"style":474},[473],[224,5533,239],{"className":5534},[542],[224,5536,5268],{"className":5537},[468],[224,5539,5302],{"className":5540},[468],[224,5542,5544,5547],{"className":5543},[549],[224,5545,1716],{"className":5546},[549],[224,5548,5550],{"className":5549},[750],[224,5551,5553],{"className":5552},[754],[224,5554,5556],{"className":5555},[759],[224,5557,5559],{"className":5558,"style":5071},[763],[224,5560,5561,5564],{"style":5074},[224,5562],{"className":5563,"style":772},[771],[224,5565,5567],{"className":5566},[776,777,778,779],[224,5568,3594],{"className":5569},[468,779],[224,5571,787],{"className":5572},[786],[224,5574,5576],{"className":5575},[759],[224,5577,5579],{"className":5578,"style":5393},[763],[224,5580],{},[11,5582,5583,5584,5639,5640,5704,5705,5771],{},"这三个数都是 16 位数，直到 2019 年才被找到。如果把 ",[224,5585,5587,5609],{"className":5586,"translate":388},[392],[224,5588,5590],{"className":5589},[396],[398,5591,5592],{"xmlns":400},[403,5593,5594,5606],{},[406,5595,5596,5598,5600,5602,5604],{},[409,5597,371],{},[412,5599,1877],{"separator":455},[409,5601,375],{},[412,5603,1877],{"separator":455},[409,5605,5032],{},[447,5607,5608],{"encoding":449},"x, y, z",[224,5610,5612],{"className":5611,"ariaHidden":455},[454],[224,5613,5615,5618,5621,5624,5627,5630,5633,5636],{"className":5614},[459],[224,5616],{"className":5617,"style":806},[463],[224,5619,371],{"className":5620},[468,469],[224,5622,1877],{"className":5623},[1917],[224,5625],{"className":5626,"style":724},[473],[224,5628,375],{"className":5629,"style":492},[468,469],[224,5631,1877],{"className":5632},[1917],[224,5634],{"className":5635,"style":724},[473],[224,5637,5032],{"className":5638,"style":5150},[468,469]," 都编码成 64 位整数，搜索空间是 ",[224,5641,5643,5662],{"className":5642,"translate":388},[392],[224,5644,5646],{"className":5645},[396],[398,5647,5648],{"xmlns":400},[403,5649,5650,5659],{},[406,5651,5652],{},[3248,5653,5654,5656],{},[419,5655,1447],{},[419,5657,5658],{},"192",[447,5660,5661],{"encoding":449},"2^{192}",[224,5663,5665],{"className":5664,"ariaHidden":455},[454],[224,5666,5668,5672],{"className":5667},[459],[224,5669],{"className":5670,"style":5671},[463],"height:0.8141em;",[224,5673,5675,5678],{"className":5674},[468],[224,5676,1447],{"className":5677},[468],[224,5679,5681],{"className":5680},[750],[224,5682,5684],{"className":5683},[754],[224,5685,5687],{"className":5686},[759],[224,5688,5690],{"className":5689,"style":5671},[763],[224,5691,5692,5695],{"style":3288},[224,5693],{"className":5694,"style":772},[771],[224,5696,5698],{"className":5697},[776,777,778,779],[224,5699,5701],{"className":5700},[468,779],[224,5702,5658],{"className":5703},[468,779],"；按每秒试十亿个赋值来算，也要 ",[224,5706,5708,5727],{"className":5707,"translate":388},[392],[224,5709,5711],{"className":5710},[396],[398,5712,5713],{"xmlns":400},[403,5714,5715,5724],{},[406,5716,5717],{},[3248,5718,5719,5721],{},[419,5720,421],{},[419,5722,5723],{},"41",[447,5725,5726],{"encoding":449},"10^{41}",[224,5728,5730],{"className":5729,"ariaHidden":455},[454],[224,5731,5733,5736,5739],{"className":5732},[459],[224,5734],{"className":5735,"style":5671},[463],[224,5737,1438],{"className":5738},[468],[224,5740,5742,5745],{"className":5741},[468],[224,5743,284],{"className":5744},[468],[224,5746,5748],{"className":5747},[750],[224,5749,5751],{"className":5750},[754],[224,5752,5754],{"className":5753},[759],[224,5755,5757],{"className":5756,"style":5671},[763],[224,5758,5759,5762],{"style":3288},[224,5760],{"className":5761,"style":772},[771],[224,5763,5765],{"className":5764},[776,777,778,779],[224,5766,5768],{"className":5767},[468,779],[224,5769,5723],{"className":5770},[468,779]," 年量级。位爆破在这里基本不可行。",[11,5773,5774],{},"更根本的是，这不是「这一个方程特别难」的问题。希尔伯特第十问题问的是「是否存在算法判定整系数多项式方程有没有整数解」，这个问题在 1970 年被马蒂亚谢维奇证明为否。证明的思路很巧妙：任给一台图灵机和一段输入，都能构造出一个整系数多项式方程，使方程有解当且仅当图灵机停机。既然停机问题不可判定，一般的方程判定问题也不可判定。换句话说，想写一个能判定所有这类方程的通用算法，这件事本身就是不可能的。",[11,5776,5777],{},"工程上的后果很直接：自动定理证明器可能跑了一整天，最后告诉你「我不知道」。对于一个需要每秒判定成千上万条路径条件的程序分析工具来说，这是不可接受的。",[613,5779,5781],{"id":5780},"_23-难以组合","2.3 难以组合",[11,5783,5784],{},"另一条路是「一个问题配一个专用算法」：线性规划用单纯形法，只懂实数线性约束；并查集只懂「相等」；BDD 只懂布尔函数；自动机只懂正则语言和字符串匹配。",[11,5786,5787],{},"这些算法在各自领域都很好用，但真实的程序约束往往是混合的。看一个只有两个原子的例子：",[224,5789,5791],{"className":5790,"translate":388},[387],[224,5792,5794,5836],{"className":5793,"translate":388},[392],[224,5795,5797],{"className":5796},[396],[398,5798,5799],{"xmlns":400,"display":401},[403,5800,5801,5833],{},[406,5802,5803,5805,5807,5809,5811,5813,5815,5817,5819,5821,5823,5825,5827,5829,5831],{},[409,5804,371],{},[412,5806,278],{},[409,5808,375],{},[423,5810,425],{},[412,5812,428],{},[423,5814,425],{},[409,5816,1705],{},[412,5818,239],{"stretchy":435},[409,5820,371],{},[412,5822,1716],{"stretchy":435},[412,5824,2048],{"mathvariant":678},[409,5826,1705],{},[412,5828,239],{"stretchy":435},[409,5830,375],{},[412,5832,1716],{"stretchy":435},[447,5834,5835],{"encoding":449},"x = y \\;\\wedge\\; f(x) \\ne f(y)",[224,5837,5839,5857,5881,5941],{"className":5838,"ariaHidden":455},[454],[224,5840,5842,5845,5848,5851,5854],{"className":5841},[459],[224,5843],{"className":5844,"style":839},[463],[224,5846,371],{"className":5847},[468,469],[224,5849],{"className":5850,"style":496},[473],[224,5852,278],{"className":5853},[500],[224,5855],{"className":5856,"style":496},[473],[224,5858,5860,5863,5866,5869,5872,5875,5878],{"className":5859},[459],[224,5861],{"className":5862,"style":1277},[463],[224,5864,375],{"className":5865,"style":492},[468,469],[224,5867],{"className":5868,"style":496},[473],[224,5870],{"className":5871,"style":474},[473],[224,5873,428],{"className":5874},[478],[224,5876],{"className":5877,"style":496},[473],[224,5879],{"className":5880,"style":474},[473],[224,5882,5884,5887,5890,5893,5896,5899,5902,5938],{"className":5883},[459],[224,5885],{"className":5886,"style":535},[463],[224,5888,1705],{"className":5889,"style":1740},[468,469],[224,5891,239],{"className":5892},[542],[224,5894,371],{"className":5895},[468,469],[224,5897,1716],{"className":5898},[549],[224,5900],{"className":5901,"style":496},[473],[224,5903,5905,5932,5935],{"className":5904},[500],[224,5906,5908],{"className":5907},[500],[224,5909,5911],{"className":5910},[468,2553],[224,5912,5914],{"className":5913},[2557],[224,5915,5917,5920,5929],{"className":5916},[2561],[224,5918],{"className":5919,"style":2565},[463],[224,5921,5923],{"className":5922},[2569],[224,5924,5926],{"className":5925},[468],[224,5927,2576],{"className":5928},[500],[224,5930],{"className":5931},[2580],[224,5933],{"className":5934},[473,2584],[224,5936,278],{"className":5937},[500],[224,5939],{"className":5940,"style":496},[473],[224,5942,5944,5947,5950,5953,5956],{"className":5943},[459],[224,5945],{"className":5946,"style":535},[463],[224,5948,1705],{"className":5949,"style":1740},[468,469],[224,5951,239],{"className":5952},[542],[224,5954,375],{"className":5955,"style":492},[468,469],[224,5957,1716],{"className":5958},[549],[11,5960,5961],{},"假设我们分别去问两个求解器：",[2674,5963,5964,6096],{},[2677,5965,5966,5967,6018,6019,6021,6022,6095],{},"算术求解器只看 ",[224,5968,5970,5988],{"className":5969,"translate":388},[392],[224,5971,5973],{"className":5972},[396],[398,5974,5975],{"xmlns":400},[403,5976,5977,5985],{},[406,5978,5979,5981,5983],{},[409,5980,371],{},[412,5982,278],{},[409,5984,375],{},[447,5986,5987],{"encoding":449},"x = y",[224,5989,5991,6009],{"className":5990,"ariaHidden":455},[454],[224,5992,5994,5997,6000,6003,6006],{"className":5993},[459],[224,5995],{"className":5996,"style":839},[463],[224,5998,371],{"className":5999},[468,469],[224,6001],{"className":6002,"style":496},[473],[224,6004,278],{"className":6005},[500],[224,6007],{"className":6008,"style":496},[473],[224,6010,6012,6015],{"className":6011},[459],[224,6013],{"className":6014,"style":806},[463],[224,6016,375],{"className":6017,"style":492},[468,469],"，回答 ",[378,6020,599],{},"，比如 ",[224,6023,6025,6047],{"className":6024,"translate":388},[392],[224,6026,6028],{"className":6027},[396],[398,6029,6030],{"xmlns":400},[403,6031,6032,6044],{},[406,6033,6034,6036,6038,6040,6042],{},[409,6035,371],{},[412,6037,278],{},[409,6039,375],{},[412,6041,278],{},[419,6043,284],{},[447,6045,6046],{"encoding":449},"x = y = 0",[224,6048,6050,6068,6086],{"className":6049,"ariaHidden":455},[454],[224,6051,6053,6056,6059,6062,6065],{"className":6052},[459],[224,6054],{"className":6055,"style":839},[463],[224,6057,371],{"className":6058},[468,469],[224,6060],{"className":6061,"style":496},[473],[224,6063,278],{"className":6064},[500],[224,6066],{"className":6067,"style":496},[473],[224,6069,6071,6074,6077,6080,6083],{"className":6070},[459],[224,6072],{"className":6073,"style":806},[463],[224,6075,375],{"className":6076,"style":492},[468,469],[224,6078],{"className":6079,"style":496},[473],[224,6081,278],{"className":6082},[500],[224,6084],{"className":6085,"style":496},[473],[224,6087,6089,6092],{"className":6088},[459],[224,6090],{"className":6091,"style":510},[463],[224,6093,284],{"className":6094},[468],"；",[2677,6097,6098,6099,6212,6213,6215,6216,6257,6258,372,6324,3117],{},"EUF 求解器只看 ",[224,6100,6102,6131],{"className":6101,"translate":388},[392],[224,6103,6105],{"className":6104},[396],[398,6106,6107],{"xmlns":400},[403,6108,6109,6129],{},[406,6110,6111,6113,6115,6117,6119,6121,6123,6125,6127],{},[409,6112,1705],{},[412,6114,239],{"stretchy":435},[409,6116,371],{},[412,6118,1716],{"stretchy":435},[412,6120,2048],{"mathvariant":678},[409,6122,1705],{},[412,6124,239],{"stretchy":435},[409,6126,375],{},[412,6128,1716],{"stretchy":435},[447,6130,2926],{"encoding":449},[224,6132,6134,6194],{"className":6133,"ariaHidden":455},[454],[224,6135,6137,6140,6143,6146,6149,6152,6155,6191],{"className":6136},[459],[224,6138],{"className":6139,"style":535},[463],[224,6141,1705],{"className":6142,"style":1740},[468,469],[224,6144,239],{"className":6145},[542],[224,6147,371],{"className":6148},[468,469],[224,6150,1716],{"className":6151},[549],[224,6153],{"className":6154,"style":496},[473],[224,6156,6158,6185,6188],{"className":6157},[500],[224,6159,6161],{"className":6160},[500],[224,6162,6164],{"className":6163},[468,2553],[224,6165,6167],{"className":6166},[2557],[224,6168,6170,6173,6182],{"className":6169},[2561],[224,6171],{"className":6172,"style":2565},[463],[224,6174,6176],{"className":6175},[2569],[224,6177,6179],{"className":6178},[468],[224,6180,2576],{"className":6181},[500],[224,6183],{"className":6184},[2580],[224,6186],{"className":6187},[473,2584],[224,6189,278],{"className":6190},[500],[224,6192],{"className":6193,"style":496},[473],[224,6195,6197,6200,6203,6206,6209],{"className":6196},[459],[224,6198],{"className":6199,"style":535},[463],[224,6201,1705],{"className":6202,"style":1740},[468,469],[224,6204,239],{"className":6205},[542],[224,6207,375],{"className":6208,"style":492},[468,469],[224,6210,1716],{"className":6211},[549],"，也回答 ",[378,6214,599],{},"：它把 ",[224,6217,6219,6236],{"className":6218,"translate":388},[392],[224,6220,6222],{"className":6221},[396],[398,6223,6224],{"xmlns":400},[403,6225,6226,6234],{},[406,6227,6228,6230,6232],{},[409,6229,371],{},[412,6231,1877],{"separator":455},[409,6233,375],{},[447,6235,3420],{"encoding":449},[224,6237,6239],{"className":6238,"ariaHidden":455},[454],[224,6240,6242,6245,6248,6251,6254],{"className":6241},[459],[224,6243],{"className":6244,"style":806},[463],[224,6246,371],{"className":6247},[468,469],[224,6249,1877],{"className":6250},[1917],[224,6252],{"className":6253,"style":724},[473],[224,6255,375],{"className":6256,"style":492},[468,469]," 当成两个不同的常量，令 ",[224,6259,6261,6285],{"className":6260,"translate":388},[392],[224,6262,6264],{"className":6263},[396],[398,6265,6266],{"xmlns":400},[403,6267,6268,6282],{},[406,6269,6270,6272,6274,6276,6278,6280],{},[409,6271,1705],{},[412,6273,239],{"stretchy":435},[409,6275,371],{},[412,6277,1716],{"stretchy":435},[412,6279,278],{},[419,6281,284],{},[447,6283,6284],{"encoding":449},"f(x) = 0",[224,6286,6288,6315],{"className":6287,"ariaHidden":455},[454],[224,6289,6291,6294,6297,6300,6303,6306,6309,6312],{"className":6290},[459],[224,6292],{"className":6293,"style":535},[463],[224,6295,1705],{"className":6296,"style":1740},[468,469],[224,6298,239],{"className":6299},[542],[224,6301,371],{"className":6302},[468,469],[224,6304,1716],{"className":6305},[549],[224,6307],{"className":6308,"style":496},[473],[224,6310,278],{"className":6311},[500],[224,6313],{"className":6314,"style":496},[473],[224,6316,6318,6321],{"className":6317},[459],[224,6319],{"className":6320,"style":510},[463],[224,6322,284],{"className":6323},[468],[224,6325,6327,6351],{"className":6326,"translate":388},[392],[224,6328,6330],{"className":6329},[396],[398,6331,6332],{"xmlns":400},[403,6333,6334,6348],{},[406,6335,6336,6338,6340,6342,6344,6346],{},[409,6337,1705],{},[412,6339,239],{"stretchy":435},[409,6341,375],{},[412,6343,1716],{"stretchy":435},[412,6345,278],{},[419,6347,1438],{},[447,6349,6350],{"encoding":449},"f(y) = 1",[224,6352,6354,6381],{"className":6353,"ariaHidden":455},[454],[224,6355,6357,6360,6363,6366,6369,6372,6375,6378],{"className":6356},[459],[224,6358],{"className":6359,"style":535},[463],[224,6361,1705],{"className":6362,"style":1740},[468,469],[224,6364,239],{"className":6365},[542],[224,6367,375],{"className":6368,"style":492},[468,469],[224,6370,1716],{"className":6371},[549],[224,6373],{"className":6374,"style":496},[473],[224,6376,278],{"className":6377},[500],[224,6379],{"className":6380,"style":496},[473],[224,6382,6384,6387],{"className":6383},[459],[224,6385],{"className":6386,"style":510},[463],[224,6388,1438],{"className":6389},[468],[11,6391,6392,6393,6443,6444,6488,6489,6533,6534,6647,6648,3117],{},"如果把两个答案一拼，说「两边都可满足，所以整体可满足」，就错了。因为 ",[224,6394,6396,6413],{"className":6395,"translate":388},[392],[224,6397,6399],{"className":6398},[396],[398,6400,6401],{"xmlns":400},[403,6402,6403,6411],{},[406,6404,6405,6407,6409],{},[409,6406,371],{},[412,6408,278],{},[409,6410,375],{},[447,6412,5987],{"encoding":449},[224,6414,6416,6434],{"className":6415,"ariaHidden":455},[454],[224,6417,6419,6422,6425,6428,6431],{"className":6418},[459],[224,6420],{"className":6421,"style":839},[463],[224,6423,371],{"className":6424},[468,469],[224,6426],{"className":6427,"style":496},[473],[224,6429,278],{"className":6430},[500],[224,6432],{"className":6433,"style":496},[473],[224,6435,6437,6440],{"className":6436},[459],[224,6438],{"className":6439,"style":806},[463],[224,6441,375],{"className":6442,"style":492},[468,469]," 意味着 ",[224,6445,6447,6467],{"className":6446,"translate":388},[392],[224,6448,6450],{"className":6449},[396],[398,6451,6452],{"xmlns":400},[403,6453,6454,6464],{},[406,6455,6456,6458,6460,6462],{},[409,6457,1705],{},[412,6459,239],{"stretchy":435},[409,6461,371],{},[412,6463,1716],{"stretchy":435},[447,6465,6466],{"encoding":449},"f(x)",[224,6468,6470],{"className":6469,"ariaHidden":455},[454],[224,6471,6473,6476,6479,6482,6485],{"className":6472},[459],[224,6474],{"className":6475,"style":535},[463],[224,6477,1705],{"className":6478,"style":1740},[468,469],[224,6480,239],{"className":6481},[542],[224,6483,371],{"className":6484},[468,469],[224,6486,1716],{"className":6487},[549]," 必须等于 ",[224,6490,6492,6512],{"className":6491,"translate":388},[392],[224,6493,6495],{"className":6494},[396],[398,6496,6497],{"xmlns":400},[403,6498,6499,6509],{},[406,6500,6501,6503,6505,6507],{},[409,6502,1705],{},[412,6504,239],{"stretchy":435},[409,6506,375],{},[412,6508,1716],{"stretchy":435},[447,6510,6511],{"encoding":449},"f(y)",[224,6513,6515],{"className":6514,"ariaHidden":455},[454],[224,6516,6518,6521,6524,6527,6530],{"className":6517},[459],[224,6519],{"className":6520,"style":535},[463],[224,6522,1705],{"className":6523,"style":1740},[468,469],[224,6525,239],{"className":6526},[542],[224,6528,375],{"className":6529,"style":492},[468,469],[224,6531,1716],{"className":6532},[549],"，与 ",[224,6535,6537,6566],{"className":6536,"translate":388},[392],[224,6538,6540],{"className":6539},[396],[398,6541,6542],{"xmlns":400},[403,6543,6544,6564],{},[406,6545,6546,6548,6550,6552,6554,6556,6558,6560,6562],{},[409,6547,1705],{},[412,6549,239],{"stretchy":435},[409,6551,371],{},[412,6553,1716],{"stretchy":435},[412,6555,2048],{"mathvariant":678},[409,6557,1705],{},[412,6559,239],{"stretchy":435},[409,6561,375],{},[412,6563,1716],{"stretchy":435},[447,6565,2926],{"encoding":449},[224,6567,6569,6629],{"className":6568,"ariaHidden":455},[454],[224,6570,6572,6575,6578,6581,6584,6587,6590,6626],{"className":6571},[459],[224,6573],{"className":6574,"style":535},[463],[224,6576,1705],{"className":6577,"style":1740},[468,469],[224,6579,239],{"className":6580},[542],[224,6582,371],{"className":6583},[468,469],[224,6585,1716],{"className":6586},[549],[224,6588],{"className":6589,"style":496},[473],[224,6591,6593,6620,6623],{"className":6592},[500],[224,6594,6596],{"className":6595},[500],[224,6597,6599],{"className":6598},[468,2553],[224,6600,6602],{"className":6601},[2557],[224,6603,6605,6608,6617],{"className":6604},[2561],[224,6606],{"className":6607,"style":2565},[463],[224,6609,6611],{"className":6610},[2569],[224,6612,6614],{"className":6613},[468],[224,6615,2576],{"className":6616},[500],[224,6618],{"className":6619},[2580],[224,6621],{"className":6622},[473,2584],[224,6624,278],{"className":6625},[500],[224,6627],{"className":6628,"style":496},[473],[224,6630,6632,6635,6638,6641,6644],{"className":6631},[459],[224,6633],{"className":6634,"style":535},[463],[224,6636,1705],{"className":6637,"style":1740},[468,469],[224,6639,239],{"className":6640},[542],[224,6642,375],{"className":6643,"style":492},[468,469],[224,6645,1716],{"className":6646},[549]," 直接矛盾。整体其实是 ",[378,6649,6650],{},"unsat",[11,6652,6653,6654,6704,6705,6786],{},"要得到正确答案，两个求解器必须交换信息：算术把等式 ",[224,6655,6657,6674],{"className":6656,"translate":388},[392],[224,6658,6660],{"className":6659},[396],[398,6661,6662],{"xmlns":400},[403,6663,6664,6672],{},[406,6665,6666,6668,6670],{},[409,6667,371],{},[412,6669,278],{},[409,6671,375],{},[447,6673,5987],{"encoding":449},[224,6675,6677,6695],{"className":6676,"ariaHidden":455},[454],[224,6678,6680,6683,6686,6689,6692],{"className":6679},[459],[224,6681],{"className":6682,"style":839},[463],[224,6684,371],{"className":6685},[468,469],[224,6687],{"className":6688,"style":496},[473],[224,6690,278],{"className":6691},[500],[224,6693],{"className":6694,"style":496},[473],[224,6696,6698,6701],{"className":6697},[459],[224,6699],{"className":6700,"style":806},[463],[224,6702,375],{"className":6703,"style":492},[468,469]," 告诉 EUF，EUF 立刻推出 ",[224,6706,6708,6738],{"className":6707,"translate":388},[392],[224,6709,6711],{"className":6710},[396],[398,6712,6713],{"xmlns":400},[403,6714,6715,6735],{},[406,6716,6717,6719,6721,6723,6725,6727,6729,6731,6733],{},[409,6718,1705],{},[412,6720,239],{"stretchy":435},[409,6722,371],{},[412,6724,1716],{"stretchy":435},[412,6726,278],{},[409,6728,1705],{},[412,6730,239],{"stretchy":435},[409,6732,375],{},[412,6734,1716],{"stretchy":435},[447,6736,6737],{"encoding":449},"f(x) = f(y)",[224,6739,6741,6768],{"className":6740,"ariaHidden":455},[454],[224,6742,6744,6747,6750,6753,6756,6759,6762,6765],{"className":6743},[459],[224,6745],{"className":6746,"style":535},[463],[224,6748,1705],{"className":6749,"style":1740},[468,469],[224,6751,239],{"className":6752},[542],[224,6754,371],{"className":6755},[468,469],[224,6757,1716],{"className":6758},[549],[224,6760],{"className":6761,"style":496},[473],[224,6763,278],{"className":6764},[500],[224,6766],{"className":6767,"style":496},[473],[224,6769,6771,6774,6777,6780,6783],{"className":6770},[459],[224,6772],{"className":6773,"style":535},[463],[224,6775,1705],{"className":6776,"style":1740},[468,469],[224,6778,239],{"className":6779},[542],[224,6781,375],{"className":6782,"style":492},[468,469],[224,6784,1716],{"className":6785},[549],"，冲突。谁来负责交换？交换哪些事实？什么时候交换？在 SMT 之前，这些都是需要人工设计的事情。",[11,6788,6789,6790,6793],{},"数组和算术之间也有同样的坑。",[221,6791,6792],{},"i = j ∧ a[i] ≠ a[j]"," 单独看两边都可满足，合起来却不可满足，因为下标相等就意味着读到的值相等。没有任何一个专用算法能独自看出这一点。",[613,6795,6797],{"id":6796},"_24-门槛太高","2.4 门槛太高",[11,6799,6800,6801,3117],{},"还有一条路是 Coq、Isabelle、ACL2 这类交互式定理证明器。它们的表达力极强，能处理带量词、带归纳的复杂性质，但代价是",[378,6802,6803],{},"需要人写证明脚本",[11,6805,6806],{},"比如在 Coq 里证明「任何自然数加零都等于它自己」，要写：",[215,6808,6812],{"className":6809,"code":6810,"language":6811,"meta":186,"style":186},"language-coq shiki shiki-themes github-light github-light github-dark","Theorem plus_0_r : forall n : nat, n + 0 = n.\nProof.\n  induction n as [| n' IH].\n  - simpl. reflexivity.\n  - simpl. rewrite IH. reflexivity.\nQed.\n","coq",[221,6813,6814,6819,6824,6829,6834,6839],{"__ignoreMap":186},[224,6815,6816],{"class":226,"line":227},[224,6817,6818],{},"Theorem plus_0_r : forall n : nat, n + 0 = n.\n",[224,6820,6821],{"class":226,"line":187},[224,6822,6823],{},"Proof.\n",[224,6825,6826],{"class":226,"line":290},[224,6827,6828],{},"  induction n as [| n' IH].\n",[224,6830,6831],{"class":226,"line":310},[224,6832,6833],{},"  - simpl. reflexivity.\n",[224,6835,6836],{"class":226,"line":349},[224,6837,6838],{},"  - simpl. rewrite IH. reflexivity.\n",[224,6840,6841],{"class":226,"line":356},[224,6842,6843],{},"Qed.\n",[11,6845,6846],{},"这几行不是装饰：",[2674,6848,6849,7010,7103],{},[2677,6850,6851,6854,6855,2887,6906,6095],{},[221,6852,6853],{},"induction n"," 把目标拆成两种情况：",[224,6856,6858,6876],{"className":6857,"translate":388},[392],[224,6859,6861],{"className":6860},[396],[398,6862,6863],{"xmlns":400},[403,6864,6865,6873],{},[406,6866,6867,6869,6871],{},[409,6868,3254],{},[412,6870,278],{},[419,6872,284],{},[447,6874,6875],{"encoding":449},"n = 0",[224,6877,6879,6897],{"className":6878,"ariaHidden":455},[454],[224,6880,6882,6885,6888,6891,6894],{"className":6881},[459],[224,6883],{"className":6884,"style":839},[463],[224,6886,3254],{"className":6887},[468,469],[224,6889],{"className":6890,"style":496},[473],[224,6892,278],{"className":6893},[500],[224,6895],{"className":6896,"style":496},[473],[224,6898,6900,6903],{"className":6899},[459],[224,6901],{"className":6902,"style":510},[463],[224,6904,284],{"className":6905},[468],[224,6907,6909,6939],{"className":6908,"translate":388},[392],[224,6910,6912],{"className":6911},[396],[398,6913,6914],{"xmlns":400},[403,6915,6916,6936],{},[406,6917,6918,6920,6922,6925,6927,6934],{},[409,6919,3254],{},[412,6921,278],{},[409,6923,6924],{},"S",[412,6926,239],{"stretchy":435},[3248,6928,6929,6931],{},[409,6930,3254],{},[412,6932,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},"′",[412,6935,1716],{"stretchy":435},[447,6937,6938],{"encoding":449},"n = S(n')",[224,6940,6942,6960],{"className":6941,"ariaHidden":455},[454],[224,6943,6945,6948,6951,6954,6957],{"className":6944},[459],[224,6946],{"className":6947,"style":839},[463],[224,6949,3254],{"className":6950},[468,469],[224,6952],{"className":6953,"style":496},[473],[224,6955,278],{"className":6956},[500],[224,6958],{"className":6959,"style":496},[473],[224,6961,6963,6967,6971,6974,7007],{"className":6962},[459],[224,6964],{"className":6965,"style":6966},[463],"height:1.0019em;vertical-align:-0.25em;",[224,6968,6924],{"className":6969,"style":6970},[468,469],"margin-right:0.0576em;",[224,6972,239],{"className":6973},[542],[224,6975,6977,6980],{"className":6976},[468],[224,6978,3254],{"className":6979},[468,469],[224,6981,6983],{"className":6982},[750],[224,6984,6986],{"className":6985},[754],[224,6987,6989],{"className":6988},[759],[224,6990,6993],{"className":6991,"style":6992},[763],"height:0.7519em;",[224,6994,6995,6998],{"style":3288},[224,6996],{"className":6997,"style":772},[771],[224,6999,7001],{"className":7000},[776,777,778,779],[224,7002,7004],{"className":7003},[468,779],[224,7005,6933],{"className":7006},[468,779],[224,7008,1716],{"className":7009},[549],[2677,7011,7012,7013,7016,7017,7069,7070,7098,7099,7102],{},"第一种情况，",[221,7014,7015],{},"simpl"," 把 ",[224,7018,7020,7038],{"className":7019,"translate":388},[392],[224,7021,7023],{"className":7022},[396],[398,7024,7025],{"xmlns":400},[403,7026,7027,7035],{},[406,7028,7029,7031,7033],{},[419,7030,284],{},[412,7032,319],{},[419,7034,284],{},[447,7036,7037],{"encoding":449},"0 + 0",[224,7039,7041,7060],{"className":7040,"ariaHidden":455},[454],[224,7042,7044,7048,7051,7054,7057],{"className":7043},[459],[224,7045],{"className":7046,"style":7047},[463],"height:0.7278em;vertical-align:-0.0833em;",[224,7049,284],{"className":7050},[468],[224,7052],{"className":7053,"style":474},[473],[224,7055,319],{"className":7056},[478],[224,7058],{"className":7059,"style":474},[473],[224,7061,7063,7066],{"className":7062},[459],[224,7064],{"className":7065,"style":510},[463],[224,7067,284],{"className":7068},[468]," 化简成 ",[224,7071,7073,7086],{"className":7072,"translate":388},[392],[224,7074,7076],{"className":7075},[396],[398,7077,7078],{"xmlns":400},[403,7079,7080,7084],{},[406,7081,7082],{},[419,7083,284],{},[447,7085,284],{"encoding":449},[224,7087,7089],{"className":7088,"ariaHidden":455},[454],[224,7090,7092,7095],{"className":7091},[459],[224,7093],{"className":7094,"style":510},[463],[224,7096,284],{"className":7097},[468],"，",[221,7100,7101],{},"reflexivity"," 确认两边相等；",[2677,7104,7105,7106,7016,7108,7207,7208,7307,7308,7016,7311,7396,7397,7459,7460,7462],{},"第二种情况，",[221,7107,7015],{},[224,7109,7111,7139],{"className":7110,"translate":388},[392],[224,7112,7114],{"className":7113},[396],[398,7115,7116],{"xmlns":400},[403,7117,7118,7136],{},[406,7119,7120,7122,7124,7130,7132,7134],{},[409,7121,6924],{},[412,7123,239],{"stretchy":435},[3248,7125,7126,7128],{},[409,7127,3254],{},[412,7129,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,7131,1716],{"stretchy":435},[412,7133,319],{},[419,7135,284],{},[447,7137,7138],{"encoding":449},"S(n') + 0",[224,7140,7142,7198],{"className":7141,"ariaHidden":455},[454],[224,7143,7145,7148,7151,7154,7186,7189,7192,7195],{"className":7144},[459],[224,7146],{"className":7147,"style":6966},[463],[224,7149,6924],{"className":7150,"style":6970},[468,469],[224,7152,239],{"className":7153},[542],[224,7155,7157,7160],{"className":7156},[468],[224,7158,3254],{"className":7159},[468,469],[224,7161,7163],{"className":7162},[750],[224,7164,7166],{"className":7165},[754],[224,7167,7169],{"className":7168},[759],[224,7170,7172],{"className":7171,"style":6992},[763],[224,7173,7174,7177],{"style":3288},[224,7175],{"className":7176,"style":772},[771],[224,7178,7180],{"className":7179},[776,777,778,779],[224,7181,7183],{"className":7182},[468,779],[224,7184,6933],{"className":7185},[468,779],[224,7187,1716],{"className":7188},[549],[224,7190],{"className":7191,"style":474},[473],[224,7193,319],{"className":7194},[478],[224,7196],{"className":7197,"style":474},[473],[224,7199,7201,7204],{"className":7200},[459],[224,7202],{"className":7203,"style":510},[463],[224,7205,284],{"className":7206},[468]," 化成 ",[224,7209,7211,7239],{"className":7210,"translate":388},[392],[224,7212,7214],{"className":7213},[396],[398,7215,7216],{"xmlns":400},[403,7217,7218,7236],{},[406,7219,7220,7222,7224,7230,7232,7234],{},[409,7221,6924],{},[412,7223,239],{"stretchy":435},[3248,7225,7226,7228],{},[409,7227,3254],{},[412,7229,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,7231,319],{},[419,7233,284],{},[412,7235,1716],{"stretchy":435},[447,7237,7238],{"encoding":449},"S(n' + 0)",[224,7240,7242,7295],{"className":7241,"ariaHidden":455},[454],[224,7243,7245,7248,7251,7254,7286,7289,7292],{"className":7244},[459],[224,7246],{"className":7247,"style":6966},[463],[224,7249,6924],{"className":7250,"style":6970},[468,469],[224,7252,239],{"className":7253},[542],[224,7255,7257,7260],{"className":7256},[468],[224,7258,3254],{"className":7259},[468,469],[224,7261,7263],{"className":7262},[750],[224,7264,7266],{"className":7265},[754],[224,7267,7269],{"className":7268},[759],[224,7270,7272],{"className":7271,"style":6992},[763],[224,7273,7274,7277],{"style":3288},[224,7275],{"className":7276,"style":772},[771],[224,7278,7280],{"className":7279},[776,777,778,779],[224,7281,7283],{"className":7282},[468,779],[224,7284,6933],{"className":7285},[468,779],[224,7287],{"className":7288,"style":474},[473],[224,7290,319],{"className":7291},[478],[224,7293],{"className":7294,"style":474},[473],[224,7296,7298,7301,7304],{"className":7297},[459],[224,7299],{"className":7300,"style":535},[463],[224,7302,284],{"className":7303},[468],[224,7305,1716],{"className":7306},[549],"，再用归纳假设 ",[221,7309,7310],{},"IH",[224,7312,7314,7336],{"className":7313,"translate":388},[392],[224,7315,7317],{"className":7316},[396],[398,7318,7319],{"xmlns":400},[403,7320,7321,7333],{},[406,7322,7323,7329,7331],{},[3248,7324,7325,7327],{},[409,7326,3254],{},[412,7328,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,7330,319],{},[419,7332,284],{},[447,7334,7335],{"encoding":449},"n' + 0",[224,7337,7339,7387],{"className":7338,"ariaHidden":455},[454],[224,7340,7342,7346,7378,7381,7384],{"className":7341},[459],[224,7343],{"className":7344,"style":7345},[463],"height:0.8352em;vertical-align:-0.0833em;",[224,7347,7349,7352],{"className":7348},[468],[224,7350,3254],{"className":7351},[468,469],[224,7353,7355],{"className":7354},[750],[224,7356,7358],{"className":7357},[754],[224,7359,7361],{"className":7360},[759],[224,7362,7364],{"className":7363,"style":6992},[763],[224,7365,7366,7369],{"style":3288},[224,7367],{"className":7368,"style":772},[771],[224,7370,7372],{"className":7371},[776,777,778,779],[224,7373,7375],{"className":7374},[468,779],[224,7376,6933],{"className":7377},[468,779],[224,7379],{"className":7380,"style":474},[473],[224,7382,319],{"className":7383},[478],[224,7385],{"className":7386,"style":474},[473],[224,7388,7390,7393],{"className":7389},[459],[224,7391],{"className":7392,"style":510},[463],[224,7394,284],{"className":7395},[468]," 换成 ",[224,7398,7400,7418],{"className":7399,"translate":388},[392],[224,7401,7403],{"className":7402},[396],[398,7404,7405],{"xmlns":400},[403,7406,7407,7415],{},[406,7408,7409],{},[3248,7410,7411,7413],{},[409,7412,3254],{},[412,7414,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[447,7416,7417],{"encoding":449},"n'",[224,7419,7421],{"className":7420,"ariaHidden":455},[454],[224,7422,7424,7427],{"className":7423},[459],[224,7425],{"className":7426,"style":6992},[463],[224,7428,7430,7433],{"className":7429},[468],[224,7431,3254],{"className":7432},[468,469],[224,7434,7436],{"className":7435},[750],[224,7437,7439],{"className":7438},[754],[224,7440,7442],{"className":7441},[759],[224,7443,7445],{"className":7444,"style":6992},[763],[224,7446,7447,7450],{"style":3288},[224,7448],{"className":7449,"style":772},[771],[224,7451,7453],{"className":7452},[776,777,778,779],[224,7454,7456],{"className":7455},[468,779],[224,7457,6933],{"className":7458},[468,779],"，最后 ",[221,7461,7101],{}," 收尾。",[11,7464,7465],{},"每一步往哪走、用哪条归纳假设，都是人选的。这还只是自然数加零。真实的程序验证要证循环不变式、数组不变式、指针分离，脚本可能几百上千行。",[11,7467,7468],{},"程序分析场景里，工具每秒要生成并判定几千条路径条件。不可能每条都停下来等工程师写一段证明。自动化程度不够，就无法工业部署。",[613,7470,7472],{"id":7471},"_25-小结","2.5 小结",[11,7474,7475],{},"几条路，各有各的难处：",[2674,7477,7478,7481,7484,7487],{},[2677,7479,7480],{},"纯 SAT 快，但把高层语义压成门电路之后，规模爆炸、语义丢失；",[2677,7482,7483],{},"一阶逻辑表达力强，但一般情形不可判定，自动证明器可能永远给不出答案；",[2677,7485,7486],{},"专用算法在自己的理论里很好用，碰到混合约束就各自为政；",[2677,7488,7489],{},"交互式定理证明能处理复杂性质，但每条路径都等人写脚本，工业化无从谈起。",[11,7491,7492,7493],{},"SMT 要做的，就是在这几条路的夹缝里找一个中间地带：",[378,7494,7495],{},"表达力足够描述程序，判定过程保证自动，还能把多个理论组合起来。",[608,7497,7499],{"id":7498},"三smt-的解法","三、SMT 的解法",[613,7501,7503],{"id":7502},"_31-收缩片段","3.1 收缩片段",[11,7505,7506,7507,7510],{},"SMT 的第一个策略是",[378,7508,7509],{},"主动放弃一部分表达力","，只保留可判定的片段。",[11,7512,7513],{},"具体做法是：把主战场放在无量词片段，只允许特定理论的组合。这样既避开了 2.2 节里的不可判定性，又保留了描述程序行为所需的大部分能力。",[11,7515,7516,7517,7098,7546,7612],{},"为什么可以放弃量词？因为程序分析产生的验证条件，大多数天然就是无量词的。符号执行收集到的路径条件，是对「这一条路径上各个变量的具体取值」的约束，不需要「对所有 ",[224,7518,7520,7533],{"className":7519,"translate":388},[392],[224,7521,7523],{"className":7522},[396],[398,7524,7525],{"xmlns":400},[403,7526,7527,7531],{},[406,7528,7529],{},[409,7530,1880],{},[447,7532,1880],{"encoding":449},[224,7534,7536],{"className":7535,"ariaHidden":455},[454],[224,7537,7539,7543],{"className":7538},[459],[224,7540],{"className":7541,"style":7542},[463],"height:0.6595em;",[224,7544,1880],{"className":7545},[468,469],[224,7547,7549,7573],{"className":7548,"translate":388},[392],[224,7550,7552],{"className":7551},[396],[398,7553,7554],{"xmlns":400},[403,7555,7556,7570],{},[406,7557,7558,7560,7562,7564,7566,7568],{},[409,7559,15],{},[412,7561,268],{"stretchy":435},[409,7563,1880],{},[412,7565,440],{"stretchy":435},[412,7567,325],{},[419,7569,284],{},[447,7571,7572],{"encoding":449},"a[i] > 0",[224,7574,7576,7603],{"className":7575,"ariaHidden":455},[454],[224,7577,7579,7582,7585,7588,7591,7594,7597,7600],{"className":7578},[459],[224,7580],{"className":7581,"style":535},[463],[224,7583,15],{"className":7584},[468,469],[224,7586,268],{"className":7587},[542],[224,7589,1880],{"className":7590},[468,469],[224,7592,440],{"className":7593},[549],[224,7595],{"className":7596,"style":496},[473],[224,7598,325],{"className":7599},[500],[224,7601],{"className":7602,"style":496},[473],[224,7604,7606,7609],{"className":7605},[459],[224,7607],{"className":7608,"style":510},[463],[224,7610,284],{"className":7611},[468],"」这种全称陈述。",[11,7614,7615,7616,7647],{},"需要量词的地方——循环不变式、数组的整体性质、归纳证明——通常由上层工具负责：抽象解释负责生成不变式，",[224,7617,7619,7633],{"className":7618,"translate":388},[392],[224,7620,7622],{"className":7621},[396],[398,7623,7624],{"xmlns":400},[403,7625,7626,7631],{},[406,7627,7628],{},[409,7629,7630],{},"k",[447,7632,7630],{"encoding":449},[224,7634,7636],{"className":7635,"ariaHidden":455},[454],[224,7637,7639,7643],{"className":7638},[459],[224,7640],{"className":7641,"style":7642},[463],"height:0.6944em;",[224,7644,7630],{"className":7645,"style":7646},[468,469],"margin-right:0.0315em;","-induction 负责把归纳化归为有限次查询，SMT 只作为底层的判定引擎。这是一个分工，而不是缺陷。",[11,7649,7650,7651,372,7653,372,7655,7657,7658,7660,7661,7663,7664,7666],{},"当然，「可判定」不等于「容易」。",[221,7652,3134],{},[221,7654,3148],{},[221,7656,3233],{}," 都是 NP-complete；",[221,7659,3144],{}," 是多项式时间；",[221,7662,3221],{}," 可判定，但最坏情况是双指数；",[221,7665,3224],{}," 则直接不可判定。",[11,7668,7669],{},"SMT 在选定的片段内保证了「答案一定会来」，但不保证「答案来得快」。工程上的成败，很大程度上取决于启发式和问题编码。",[613,7671,7673],{"id":7672},"_32-直接表达","3.2 直接表达",[11,7675,7676,7677,3117],{},"第二个策略是",[378,7678,7679],{},"让公式直接说人话",[11,7681,7682,7683,7755,7756,7821,7822,7863],{},"同样是「",[224,7684,7686,7707],{"className":7685,"translate":388},[392],[224,7687,7689],{"className":7688},[396],[398,7690,7691],{"xmlns":400},[403,7692,7693,7705],{},[406,7694,7695,7697,7699,7701,7703],{},[409,7696,371],{},[412,7698,319],{},[409,7700,375],{},[412,7702,325],{},[419,7704,421],{},[447,7706,584],{"encoding":449},[224,7708,7710,7728,7746],{"className":7709,"ariaHidden":455},[454],[224,7711,7713,7716,7719,7722,7725],{"className":7712},[459],[224,7714],{"className":7715,"style":464},[463],[224,7717,371],{"className":7718},[468,469],[224,7720],{"className":7721,"style":474},[473],[224,7723,319],{"className":7724},[478],[224,7726],{"className":7727,"style":474},[473],[224,7729,7731,7734,7737,7740,7743],{"className":7730},[459],[224,7732],{"className":7733,"style":488},[463],[224,7735,375],{"className":7736,"style":492},[468,469],[224,7738],{"className":7739,"style":496},[473],[224,7741,325],{"className":7742},[500],[224,7744],{"className":7745,"style":496},[473],[224,7747,7749,7752],{"className":7748},[459],[224,7750],{"className":7751,"style":510},[463],[224,7753,421],{"className":7754},[468]," 且 ",[224,7757,7759,7782],{"className":7758,"translate":388},[392],[224,7760,7762],{"className":7761},[396],[398,7763,7764],{"xmlns":400},[403,7765,7766,7780],{},[406,7767,7768,7770,7772,7774,7776,7778],{},[409,7769,15],{},[412,7771,268],{"stretchy":435},[419,7773,284],{},[412,7775,440],{"stretchy":435},[412,7777,278],{},[419,7779,445],{},[447,7781,588],{"encoding":449},[224,7783,7785,7812],{"className":7784,"ariaHidden":455},[454],[224,7786,7788,7791,7794,7797,7800,7803,7806,7809],{"className":7787},[459],[224,7789],{"className":7790,"style":535},[463],[224,7792,15],{"className":7793},[468,469],[224,7795,268],{"className":7796},[542],[224,7798,284],{"className":7799},[468],[224,7801,440],{"className":7802},[549],[224,7804],{"className":7805,"style":496},[473],[224,7807,278],{"className":7808},[500],[224,7810],{"className":7811,"style":496},[473],[224,7813,7815,7818],{"className":7814},[459],[224,7816],{"className":7817,"style":510},[463],[224,7819,445],{"className":7820},[468],"」，SMT 不需要把 ",[224,7823,7825,7842],{"className":7824,"translate":388},[392],[224,7826,7828],{"className":7827},[396],[398,7829,7830],{"xmlns":400},[403,7831,7832,7840],{},[406,7833,7834,7836,7838],{},[409,7835,371],{},[412,7837,1877],{"separator":455},[409,7839,375],{},[447,7841,3420],{"encoding":449},[224,7843,7845],{"className":7844,"ariaHidden":455},[454],[224,7846,7848,7851,7854,7857,7860],{"className":7847},[459],[224,7849],{"className":7850,"style":806},[463],[224,7852,371],{"className":7853},[468,469],[224,7855,1877],{"className":7856},[1917],[224,7858],{"className":7859,"style":724},[473],[224,7861,375],{"className":7862,"style":492},[468,469]," 拆成 64 个布尔变量，也不需要把数组展开成 100 个元素。它把这两个条件原样保留为两个理论原子，交给对应的求解器。",[11,7865,7866],{},"关键在于「按需展开」：",[2674,7868,7869,7875,7881],{},[2677,7870,7871,7874],{},[378,7872,7873],{},"算术","：直接用单纯形法或分支定界，在整数\u002F实数域上推理，完全不经过布尔编码。",[2677,7876,7877,7880],{},[378,7878,7879],{},"数组","：用 read-over-write 公理，只在公式里真的出现某个下标时才实例化。",[2677,7882,7883,7886,7887,7890,7891,372,7963,8036],{},[378,7884,7885],{},"位向量","：这一条要诚实一点——大多数求解器内部",[378,7888,7889],{},"仍然会做位爆破","，因为位级编码对硬件电路太有效了。但这件事发生在求解器内部，只针对相关子表达式，而且会配合字级重写（比如 ",[224,7892,7894,7921],{"className":7893,"translate":388},[392],[224,7895,7897],{"className":7896},[396],[398,7898,7899],{"xmlns":400},[403,7900,7901,7918],{},[406,7902,7903,7905,7907,7910,7912,7914,7916],{},[409,7904,371],{},[423,7906,682],{},[409,7908,7909],{"mathvariant":678},"&",[423,7911,682],{},[419,7913,284],{},[412,7915,278],{},[419,7917,284],{},[447,7919,7920],{"encoding":449},"x \\,\\&\\, 0 = 0",[224,7922,7924,7954],{"className":7923,"ariaHidden":455},[454],[224,7925,7927,7930,7933,7936,7939,7942,7945,7948,7951],{"className":7926},[459],[224,7928],{"className":7929,"style":7642},[463],[224,7931,371],{"className":7932},[468,469],[224,7934],{"className":7935,"style":724},[473],[224,7937,7909],{"className":7938},[468],[224,7940],{"className":7941,"style":724},[473],[224,7943,284],{"className":7944},[468],[224,7946],{"className":7947,"style":496},[473],[224,7949,278],{"className":7950},[500],[224,7952],{"className":7953,"style":496},[473],[224,7955,7957,7960],{"className":7956},[459],[224,7958],{"className":7959,"style":510},[463],[224,7961,284],{"className":7962},[468],[224,7964,7966,7988],{"className":7965,"translate":388},[392],[224,7967,7969],{"className":7968},[396],[398,7970,7971],{"xmlns":400},[403,7972,7973,7985],{},[406,7974,7975,7977,7979,7981,7983],{},[409,7976,371],{},[412,7978,5268],{},[409,7980,371],{},[412,7982,278],{},[419,7984,284],{},[447,7986,7987],{"encoding":449},"x - x = 0",[224,7989,7991,8009,8027],{"className":7990,"ariaHidden":455},[454],[224,7992,7994,7997,8000,8003,8006],{"className":7993},[459],[224,7995],{"className":7996,"style":464},[463],[224,7998,371],{"className":7999},[468,469],[224,8001],{"className":8002,"style":474},[473],[224,8004,5268],{"className":8005},[478],[224,8007],{"className":8008,"style":474},[473],[224,8010,8012,8015,8018,8021,8024],{"className":8011},[459],[224,8013],{"className":8014,"style":839},[463],[224,8016,371],{"className":8017},[468,469],[224,8019],{"className":8020,"style":496},[473],[224,8022,278],{"className":8023},[500],[224,8025],{"className":8026,"style":496},[473],[224,8028,8030,8033],{"className":8029},[459],[224,8031],{"className":8032,"style":510},[463],[224,8034,284],{"className":8035},[468],"）和等价类合并。对使用者来说，它仍然是透明的。",[11,8038,8039],{},"数组的 read-over-write 公理长这样：",[224,8041,8043],{"className":8042,"translate":388},[387],[224,8044,8046,8188],{"className":8045,"translate":388},[392],[224,8047,8049],{"className":8048},[396],[398,8050,8051],{"xmlns":400,"display":401},[403,8052,8053,8185],{},[406,8054,8055,8069,8071,8083,8085,8087,8089,8091,8093,8095,8097,8099,8102,8104,8106],{},[406,8056,8057,8059,8061,8063,8065,8067],{},[409,8058,1858],{"mathvariant":678},[409,8060,1861],{"mathvariant":678},[409,8062,1864],{"mathvariant":678},[409,8064,1861],{"mathvariant":678},[409,8066,219],{"mathvariant":678},[409,8068,1435],{"mathvariant":678},[412,8070,239],{"fence":455,"stretchy":455,"minsize":2004,"maxsize":2004},[406,8072,8073,8075,8077,8079,8081],{},[409,8074,1858],{"mathvariant":678},[409,8076,1435],{"mathvariant":678},[409,8078,4127],{"mathvariant":678},[409,8080,2061],{"mathvariant":678},[409,8082,1861],{"mathvariant":678},[412,8084,239],{"stretchy":435},[409,8086,15],{},[412,8088,1877],{"separator":455},[409,8090,1880],{},[412,8092,1877],{"separator":455},[409,8094,1887],{},[412,8096,1716],{"stretchy":435},[412,8098,1877],{"separator":455},[409,8100,8101],{},"j",[412,8103,1716],{"fence":455,"stretchy":455,"minsize":2004,"maxsize":2004},[412,8105,278],{},[406,8107,8108,8111],{},[412,8109,8110],{"fence":455},"{",[5209,8112,8115,8139],{"rowspacing":8113,"columnalign":8114,"columnspacing":691},"0.36em","left left",[5215,8116,8117,8127],{},[5218,8118,8119],{},[5221,8120,8121],{"scriptlevel":284,"displaystyle":435},[406,8122,8123,8125],{},[409,8124,1887],{},[412,8126,1877],{"separator":455},[5218,8128,8129],{},[5221,8130,8131],{"scriptlevel":284,"displaystyle":435},[406,8132,8133,8135,8137],{},[409,8134,1880],{},[412,8136,278],{},[409,8138,8101],{},[5215,8140,8141,8173],{},[5218,8142,8143],{},[5221,8144,8145],{"scriptlevel":284,"displaystyle":435},[406,8146,8147,8161,8163,8165,8167,8169,8171],{},[406,8148,8149,8151,8153,8155,8157,8159],{},[409,8150,1858],{"mathvariant":678},[409,8152,1861],{"mathvariant":678},[409,8154,1864],{"mathvariant":678},[409,8156,1861],{"mathvariant":678},[409,8158,219],{"mathvariant":678},[409,8160,1435],{"mathvariant":678},[412,8162,239],{"stretchy":435},[409,8164,15],{},[412,8166,1877],{"separator":455},[409,8168,8101],{},[412,8170,1716],{"stretchy":435},[412,8172,1877],{"separator":455},[5218,8174,8175],{},[5221,8176,8177],{"scriptlevel":284,"displaystyle":435},[406,8178,8179,8181,8183],{},[409,8180,1880],{},[412,8182,2048],{"mathvariant":678},[409,8184,8101],{},[447,8186,8187],{"encoding":449},"\\mathrm{select}\\bigl(\\mathrm{store}(a, i, v), j\\bigr) =\n\\begin{cases}\nv, & i = j \\\\\n\\mathrm{select}(a, j), & i \\ne j\n\\end{cases}",[224,8189,8191,8267],{"className":8190,"ariaHidden":455},[454],[224,8192,8194,8197,8203,8209,8215,8218,8221,8224,8227,8230,8233,8236,8239,8242,8245,8248,8252,8258,8261,8264],{"className":8193},[459],[224,8195],{"className":8196,"style":2663},[463],[224,8198,8200],{"className":8199},[468],[224,8201,1907],{"className":8202},[468,1906],[224,8204,8206],{"className":8205},[542],[224,8207,239],{"className":8208},[2363,2364],[224,8210,8212],{"className":8211},[468],[224,8213,2890],{"className":8214},[468,1906],[224,8216,239],{"className":8217},[542],[224,8219,15],{"className":8220},[468,469],[224,8222,1877],{"className":8223},[1917],[224,8225],{"className":8226,"style":724},[473],[224,8228,1880],{"className":8229},[468,469],[224,8231,1877],{"className":8232},[1917],[224,8234],{"className":8235,"style":724},[473],[224,8237,1887],{"className":8238,"style":492},[468,469],[224,8240,1716],{"className":8241},[549],[224,8243,1877],{"className":8244},[1917],[224,8246],{"className":8247,"style":724},[473],[224,8249,8101],{"className":8250,"style":8251},[468,469],"margin-right:0.0572em;",[224,8253,8255],{"className":8254},[549],[224,8256,1716],{"className":8257},[2363,2364],[224,8259],{"className":8260,"style":496},[473],[224,8262,278],{"className":8263},[500],[224,8265],{"className":8266,"style":496},[473],[224,8268,8270,8274],{"className":8269},[459],[224,8271],{"className":8272,"style":8273},[463],"height:3em;vertical-align:-1.25em;",[224,8275,8277,8286,8476],{"className":8276},[2091],[224,8278,8282],{"className":8279,"style":8281},[542,8280],"delimcenter","top:0em;",[224,8283,8110],{"className":8284},[2363,8285],"size4",[224,8287,8289],{"className":8288},[468],[224,8290,8292,8369,8374],{"className":8291},[5209],[224,8293,8295],{"className":8294},[5399],[224,8296,8298,8360],{"className":8297},[754,755],[224,8299,8301,8357],{"className":8300},[759],[224,8302,8305,8321],{"className":8303,"style":8304},[763],"height:1.69em;",[224,8306,8308,8312],{"style":8307},"top:-3.69em;",[224,8309],{"className":8310,"style":8311},[771],"height:3.008em;",[224,8313,8315,8318],{"className":8314},[468],[224,8316,1887],{"className":8317,"style":492},[468,469],[224,8319,1877],{"className":8320},[1917],[224,8322,8324,8327],{"style":8323},"top:-2.25em;",[224,8325],{"className":8326,"style":8311},[771],[224,8328,8330,8336,8339,8342,8345,8348,8351,8354],{"className":8329},[468],[224,8331,8333],{"className":8332},[468],[224,8334,1907],{"className":8335},[468,1906],[224,8337,239],{"className":8338},[542],[224,8340,15],{"className":8341},[468,469],[224,8343,1877],{"className":8344},[1917],[224,8346],{"className":8347,"style":724},[473],[224,8349,8101],{"className":8350,"style":8251},[468,469],[224,8352,1716],{"className":8353},[549],[224,8355,1877],{"className":8356},[1917],[224,8358,787],{"className":8359},[786],[224,8361,8363],{"className":8362},[759],[224,8364,8367],{"className":8365,"style":8366},[763],"height:1.19em;",[224,8368],{},[224,8370],{"className":8371,"style":8373},[8372],"arraycolsep","width:1em;",[224,8375,8377],{"className":8376},[5399],[224,8378,8380,8468],{"className":8379},[754,755],[224,8381,8383,8465],{"className":8382},[759],[224,8384,8386,8409],{"className":8385,"style":8304},[763],[224,8387,8388,8391],{"style":8307},[224,8389],{"className":8390,"style":8311},[771],[224,8392,8394,8397,8400,8403,8406],{"className":8393},[468],[224,8395,1880],{"className":8396},[468,469],[224,8398],{"className":8399,"style":496},[473],[224,8401,278],{"className":8402},[500],[224,8404],{"className":8405,"style":496},[473],[224,8407,8101],{"className":8408,"style":8251},[468,469],[224,8410,8411,8414],{"style":8323},[224,8412],{"className":8413,"style":8311},[771],[224,8415,8417,8420,8423,8459,8462],{"className":8416},[468],[224,8418,1880],{"className":8419},[468,469],[224,8421],{"className":8422,"style":496},[473],[224,8424,8426,8453,8456],{"className":8425},[500],[224,8427,8429],{"className":8428},[500],[224,8430,8432],{"className":8431},[468,2553],[224,8433,8435],{"className":8434},[2557],[224,8436,8438,8441,8450],{"className":8437},[2561],[224,8439],{"className":8440,"style":2565},[463],[224,8442,8444],{"className":8443},[2569],[224,8445,8447],{"className":8446},[468],[224,8448,2576],{"className":8449},[500],[224,8451],{"className":8452},[2580],[224,8454],{"className":8455},[473,2584],[224,8457,278],{"className":8458},[500],[224,8460],{"className":8461,"style":496},[473],[224,8463,8101],{"className":8464,"style":8251},[468,469],[224,8466,787],{"className":8467},[786],[224,8469,8471],{"className":8470},[759],[224,8472,8474],{"className":8473,"style":8366},[763],[224,8475],{},[224,8477],{"className":8478},[549,8479],"nulldelimiter",[11,8481,8482,8483,8527,8528,8579],{},"如果公式里只关心 ",[224,8484,8486,8506],{"className":8485,"translate":388},[392],[224,8487,8489],{"className":8488},[396],[398,8490,8491],{"xmlns":400},[403,8492,8493,8503],{},[406,8494,8495,8497,8499,8501],{},[409,8496,15],{},[412,8498,268],{"stretchy":435},[419,8500,284],{},[412,8502,440],{"stretchy":435},[447,8504,8505],{"encoding":449},"a[0]",[224,8507,8509],{"className":8508,"ariaHidden":455},[454],[224,8510,8512,8515,8518,8521,8524],{"className":8511},[459],[224,8513],{"className":8514,"style":535},[463],[224,8516,15],{"className":8517},[468,469],[224,8519,268],{"className":8520},[542],[224,8522,284],{"className":8523},[468],[224,8525,440],{"className":8526},[549],"，求解器就只实例化 ",[224,8529,8531,8549],{"className":8530,"translate":388},[392],[224,8532,8534],{"className":8533},[396],[398,8535,8536],{"xmlns":400},[403,8537,8538,8546],{},[406,8539,8540,8542,8544],{},[409,8541,1880],{},[412,8543,278],{},[419,8545,284],{},[447,8547,8548],{"encoding":449},"i = 0",[224,8550,8552,8570],{"className":8551,"ariaHidden":455},[454],[224,8553,8555,8558,8561,8564,8567],{"className":8554},[459],[224,8556],{"className":8557,"style":7542},[463],[224,8559,1880],{"className":8560},[468,469],[224,8562],{"className":8563,"style":496},[473],[224,8565,278],{"className":8566},[500],[224,8568],{"className":8569,"style":496},[473],[224,8571,8573,8576],{"className":8572},[459],[224,8574],{"className":8575,"style":510},[463],[224,8577,284],{"className":8578},[468]," 这一条，剩下的 99 个元素根本不会出现。这是「按需」和「全局」的区别，也是 SMT 能在数组上跑赢位爆破的原因。",[11,8581,8582,8583,8586],{},"所以更准确的说法不是「SMT 消灭了底层编码」，而是：",[378,8584,8585],{},"SMT 把「怎么编码」的决定权从用户手里交给了求解器","。求解器可以根据理论特性选择最合适的内部表示，而不是让用户先做一次不可逆的全局展开。",[11,8588,8589,8590,8631,8632,3117],{},"用同一个约束对比一下。纯 SAT 的版本要生成数百到上千个布尔变量、上千条子句；SMT 的版本只保留两个理论原子。SAT 只能做布尔剪枝，SMT 能利用算术和数组的语义；SAT 的答案是一组布尔赋值，还要反解出 ",[224,8591,8593,8610],{"className":8592,"translate":388},[392],[224,8594,8596],{"className":8595},[396],[398,8597,8598],{"xmlns":400},[403,8599,8600,8608],{},[406,8601,8602,8604,8606],{},[409,8603,371],{},[412,8605,1877],{"separator":455},[409,8607,375],{},[447,8609,3420],{"encoding":449},[224,8611,8613],{"className":8612,"ariaHidden":455},[454],[224,8614,8616,8619,8622,8625,8628],{"className":8615},[459],[224,8617],{"className":8618,"style":806},[463],[224,8620,371],{"className":8621},[468,469],[224,8623,1877],{"className":8624},[1917],[224,8626],{"className":8627,"style":724},[473],[224,8629,375],{"className":8630,"style":492},[468,469],"，SMT 直接给出 ",[224,8633,8635,8661],{"className":8634,"translate":388},[392],[224,8636,8638],{"className":8637},[396],[398,8639,8640],{"xmlns":400},[403,8641,8642,8659],{},[406,8643,8644,8646,8648,8651,8653,8655,8657],{},[409,8645,371],{},[412,8647,278],{},[419,8649,8650],{},"6",[412,8652,1877],{"separator":455},[409,8654,375],{},[412,8656,278],{},[419,8658,445],{},[447,8660,592],{"encoding":449},[224,8662,8664,8682,8710],{"className":8663,"ariaHidden":455},[454],[224,8665,8667,8670,8673,8676,8679],{"className":8666},[459],[224,8668],{"className":8669,"style":839},[463],[224,8671,371],{"className":8672},[468,469],[224,8674],{"className":8675,"style":496},[473],[224,8677,278],{"className":8678},[500],[224,8680],{"className":8681,"style":496},[473],[224,8683,8685,8689,8692,8695,8698,8701,8704,8707],{"className":8684},[459],[224,8686],{"className":8687,"style":8688},[463],"height:0.8389em;vertical-align:-0.1944em;",[224,8690,8650],{"className":8691},[468],[224,8693,1877],{"className":8694},[1917],[224,8696],{"className":8697,"style":724},[473],[224,8699,375],{"className":8700,"style":492},[468,469],[224,8702],{"className":8703,"style":496},[473],[224,8705,278],{"className":8706},[500],[224,8708],{"className":8709,"style":496},[473],[224,8711,8713,8716],{"className":8712},[459],[224,8714],{"className":8715,"style":510},[463],[224,8717,445],{"className":8718},[468],[613,8720,8722],{"id":8721},"_33-组合理论","3.3 组合理论",[11,8724,8725,8726,3117],{},"第三个策略是",[378,8727,8728],{},"让不同理论通过共享等式来协作",[11,8730,8731,8732,8735],{},"最经典的方法是 Nelson–Oppen 组合。它的前提是：两个理论的签名不相交（各自管各自的符号），并且都满足「稳定无限」性质（任何可满足的公式都有无穷模型）。在这个前提下，两个理论只需要交换",[378,8733,8734],{},"变量之间的等式","，就能组合成一个完整的判定过程。",[11,8737,8738],{},"看一个例子：",[224,8740,8742],{"className":8741,"translate":388},[387],[224,8743,8745,8786],{"className":8744,"translate":388},[392],[224,8746,8748],{"className":8747},[396],[398,8749,8750],{"xmlns":400,"display":401},[403,8751,8752,8784],{},[406,8753,8754,8756,8758,8760,8762,8764,8766,8768,8770,8772,8774,8776,8778,8780,8782],{},[409,8755,371],{},[412,8757,278],{},[409,8759,375],{},[423,8761,425],{},[412,8763,428],{},[423,8765,425],{},[409,8767,1705],{},[412,8769,239],{"stretchy":435},[409,8771,371],{},[412,8773,1716],{"stretchy":435},[412,8775,2048],{"mathvariant":678},[409,8777,1705],{},[412,8779,239],{"stretchy":435},[409,8781,375],{},[412,8783,1716],{"stretchy":435},[447,8785,5835],{"encoding":449},[224,8787,8789,8807,8831,8891],{"className":8788,"ariaHidden":455},[454],[224,8790,8792,8795,8798,8801,8804],{"className":8791},[459],[224,8793],{"className":8794,"style":839},[463],[224,8796,371],{"className":8797},[468,469],[224,8799],{"className":8800,"style":496},[473],[224,8802,278],{"className":8803},[500],[224,8805],{"className":8806,"style":496},[473],[224,8808,8810,8813,8816,8819,8822,8825,8828],{"className":8809},[459],[224,8811],{"className":8812,"style":1277},[463],[224,8814,375],{"className":8815,"style":492},[468,469],[224,8817],{"className":8818,"style":496},[473],[224,8820],{"className":8821,"style":474},[473],[224,8823,428],{"className":8824},[478],[224,8826],{"className":8827,"style":496},[473],[224,8829],{"className":8830,"style":474},[473],[224,8832,8834,8837,8840,8843,8846,8849,8852,8888],{"className":8833},[459],[224,8835],{"className":8836,"style":535},[463],[224,8838,1705],{"className":8839,"style":1740},[468,469],[224,8841,239],{"className":8842},[542],[224,8844,371],{"className":8845},[468,469],[224,8847,1716],{"className":8848},[549],[224,8850],{"className":8851,"style":496},[473],[224,8853,8855,8882,8885],{"className":8854},[500],[224,8856,8858],{"className":8857},[500],[224,8859,8861],{"className":8860},[468,2553],[224,8862,8864],{"className":8863},[2557],[224,8865,8867,8870,8879],{"className":8866},[2561],[224,8868],{"className":8869,"style":2565},[463],[224,8871,8873],{"className":8872},[2569],[224,8874,8876],{"className":8875},[468],[224,8877,2576],{"className":8878},[500],[224,8880],{"className":8881},[2580],[224,8883],{"className":8884},[473,2584],[224,8886,278],{"className":8887},[500],[224,8889],{"className":8890,"style":496},[473],[224,8892,8894,8897,8900,8903,8906],{"className":8893},[459],[224,8895],{"className":8896,"style":535},[463],[224,8898,1705],{"className":8899,"style":1740},[468,469],[224,8901,239],{"className":8902},[542],[224,8904,375],{"className":8905,"style":492},[468,469],[224,8907,1716],{"className":8908},[549],[2674,8910,8911,8965,9106],{},[2677,8912,8913,8914,8964],{},"算术部分看到 ",[224,8915,8917,8934],{"className":8916,"translate":388},[392],[224,8918,8920],{"className":8919},[396],[398,8921,8922],{"xmlns":400},[403,8923,8924,8932],{},[406,8925,8926,8928,8930],{},[409,8927,371],{},[412,8929,278],{},[409,8931,375],{},[447,8933,5987],{"encoding":449},[224,8935,8937,8955],{"className":8936,"ariaHidden":455},[454],[224,8938,8940,8943,8946,8949,8952],{"className":8939},[459],[224,8941],{"className":8942,"style":839},[463],[224,8944,371],{"className":8945},[468,469],[224,8947],{"className":8948,"style":496},[473],[224,8950,278],{"className":8951},[500],[224,8953],{"className":8954,"style":496},[473],[224,8956,8958,8961],{"className":8957},[459],[224,8959],{"className":8960,"style":806},[463],[224,8962,375],{"className":8963,"style":492},[468,469],"，本身没有矛盾；",[2677,8966,8967,8968,9018,9019,2887,9062,9105],{},"EUF 部分如果不知道 ",[224,8969,8971,8988],{"className":8970,"translate":388},[392],[224,8972,8974],{"className":8973},[396],[398,8975,8976],{"xmlns":400},[403,8977,8978,8986],{},[406,8979,8980,8982,8984],{},[409,8981,371],{},[412,8983,278],{},[409,8985,375],{},[447,8987,5987],{"encoding":449},[224,8989,8991,9009],{"className":8990,"ariaHidden":455},[454],[224,8992,8994,8997,9000,9003,9006],{"className":8993},[459],[224,8995],{"className":8996,"style":839},[463],[224,8998,371],{"className":8999},[468,469],[224,9001],{"className":9002,"style":496},[473],[224,9004,278],{"className":9005},[500],[224,9007],{"className":9008,"style":496},[473],[224,9010,9012,9015],{"className":9011},[459],[224,9013],{"className":9014,"style":806},[463],[224,9016,375],{"className":9017,"style":492},[468,469],"，就会认为 ",[224,9020,9022,9041],{"className":9021,"translate":388},[392],[224,9023,9025],{"className":9024},[396],[398,9026,9027],{"xmlns":400},[403,9028,9029,9039],{},[406,9030,9031,9033,9035,9037],{},[409,9032,1705],{},[412,9034,239],{"stretchy":435},[409,9036,371],{},[412,9038,1716],{"stretchy":435},[447,9040,6466],{"encoding":449},[224,9042,9044],{"className":9043,"ariaHidden":455},[454],[224,9045,9047,9050,9053,9056,9059],{"className":9046},[459],[224,9048],{"className":9049,"style":535},[463],[224,9051,1705],{"className":9052,"style":1740},[468,469],[224,9054,239],{"className":9055},[542],[224,9057,371],{"className":9058},[468,469],[224,9060,1716],{"className":9061},[549],[224,9063,9065,9084],{"className":9064,"translate":388},[392],[224,9066,9068],{"className":9067},[396],[398,9069,9070],{"xmlns":400},[403,9071,9072,9082],{},[406,9073,9074,9076,9078,9080],{},[409,9075,1705],{},[412,9077,239],{"stretchy":435},[409,9079,375],{},[412,9081,1716],{"stretchy":435},[447,9083,6511],{"encoding":449},[224,9085,9087],{"className":9086,"ariaHidden":455},[454],[224,9088,9090,9093,9096,9099,9102],{"className":9089},[459],[224,9091],{"className":9092,"style":535},[463],[224,9094,1705],{"className":9095,"style":1740},[468,469],[224,9097,239],{"className":9098},[542],[224,9100,375],{"className":9101,"style":492},[468,469],[224,9103,1716],{"className":9104},[549]," 可以不同，也不会报错；",[2677,9107,9108,9109,9159,9160,6533,9240,9353],{},"一旦算术把等式 ",[224,9110,9112,9129],{"className":9111,"translate":388},[392],[224,9113,9115],{"className":9114},[396],[398,9116,9117],{"xmlns":400},[403,9118,9119,9127],{},[406,9120,9121,9123,9125],{},[409,9122,371],{},[412,9124,278],{},[409,9126,375],{},[447,9128,5987],{"encoding":449},[224,9130,9132,9150],{"className":9131,"ariaHidden":455},[454],[224,9133,9135,9138,9141,9144,9147],{"className":9134},[459],[224,9136],{"className":9137,"style":839},[463],[224,9139,371],{"className":9140},[468,469],[224,9142],{"className":9143,"style":496},[473],[224,9145,278],{"className":9146},[500],[224,9148],{"className":9149,"style":496},[473],[224,9151,9153,9156],{"className":9152},[459],[224,9154],{"className":9155,"style":806},[463],[224,9157,375],{"className":9158,"style":492},[468,469]," 传给 EUF，EUF 立刻推出 ",[224,9161,9163,9192],{"className":9162,"translate":388},[392],[224,9164,9166],{"className":9165},[396],[398,9167,9168],{"xmlns":400},[403,9169,9170,9190],{},[406,9171,9172,9174,9176,9178,9180,9182,9184,9186,9188],{},[409,9173,1705],{},[412,9175,239],{"stretchy":435},[409,9177,371],{},[412,9179,1716],{"stretchy":435},[412,9181,278],{},[409,9183,1705],{},[412,9185,239],{"stretchy":435},[409,9187,375],{},[412,9189,1716],{"stretchy":435},[447,9191,6737],{"encoding":449},[224,9193,9195,9222],{"className":9194,"ariaHidden":455},[454],[224,9196,9198,9201,9204,9207,9210,9213,9216,9219],{"className":9197},[459],[224,9199],{"className":9200,"style":535},[463],[224,9202,1705],{"className":9203,"style":1740},[468,469],[224,9205,239],{"className":9206},[542],[224,9208,371],{"className":9209},[468,469],[224,9211,1716],{"className":9212},[549],[224,9214],{"className":9215,"style":496},[473],[224,9217,278],{"className":9218},[500],[224,9220],{"className":9221,"style":496},[473],[224,9223,9225,9228,9231,9234,9237],{"className":9224},[459],[224,9226],{"className":9227,"style":535},[463],[224,9229,1705],{"className":9230,"style":1740},[468,469],[224,9232,239],{"className":9233},[542],[224,9235,375],{"className":9236,"style":492},[468,469],[224,9238,1716],{"className":9239},[549],[224,9241,9243,9272],{"className":9242,"translate":388},[392],[224,9244,9246],{"className":9245},[396],[398,9247,9248],{"xmlns":400},[403,9249,9250,9270],{},[406,9251,9252,9254,9256,9258,9260,9262,9264,9266,9268],{},[409,9253,1705],{},[412,9255,239],{"stretchy":435},[409,9257,371],{},[412,9259,1716],{"stretchy":435},[412,9261,2048],{"mathvariant":678},[409,9263,1705],{},[412,9265,239],{"stretchy":435},[409,9267,375],{},[412,9269,1716],{"stretchy":435},[447,9271,2926],{"encoding":449},[224,9273,9275,9335],{"className":9274,"ariaHidden":455},[454],[224,9276,9278,9281,9284,9287,9290,9293,9296,9332],{"className":9277},[459],[224,9279],{"className":9280,"style":535},[463],[224,9282,1705],{"className":9283,"style":1740},[468,469],[224,9285,239],{"className":9286},[542],[224,9288,371],{"className":9289},[468,469],[224,9291,1716],{"className":9292},[549],[224,9294],{"className":9295,"style":496},[473],[224,9297,9299,9326,9329],{"className":9298},[500],[224,9300,9302],{"className":9301},[500],[224,9303,9305],{"className":9304},[468,2553],[224,9306,9308],{"className":9307},[2557],[224,9309,9311,9314,9323],{"className":9310},[2561],[224,9312],{"className":9313,"style":2565},[463],[224,9315,9317],{"className":9316},[2569],[224,9318,9320],{"className":9319},[468],[224,9321,2576],{"className":9322},[500],[224,9324],{"className":9325},[2580],[224,9327],{"className":9328},[473,2584],[224,9330,278],{"className":9331},[500],[224,9333],{"className":9334,"style":496},[473],[224,9336,9338,9341,9344,9347,9350],{"className":9337},[459],[224,9339],{"className":9340,"style":535},[463],[224,9342,1705],{"className":9343,"style":1740},[468,469],[224,9345,239],{"className":9346},[542],[224,9348,375],{"className":9349,"style":492},[468,469],[224,9351,1716],{"className":9352},[549]," 冲突。",[11,9355,9356],{},"于是整个公式被判为不可满足。这个结论不是某一个理论单独得出的，而是两个理论「对了一下账」才对出来的。",[11,9358,9359],{},"再看一个反方向的例子：",[224,9361,9363],{"className":9362,"translate":388},[387],[224,9364,9366,9413],{"className":9365,"translate":388},[392],[224,9367,9369],{"className":9368},[396],[398,9370,9371],{"xmlns":400,"display":401},[403,9372,9373,9410],{},[406,9374,9375,9377,9379,9381,9383,9385,9387,9389,9391,9393,9395,9397,9399,9401,9403,9405,9407],{},[409,9376,371],{},[412,9378,319],{},[409,9380,375],{},[412,9382,325],{},[419,9384,421],{},[423,9386,425],{},[412,9388,428],{},[423,9390,425],{},[409,9392,371],{},[412,9394,278],{},[419,9396,3594],{},[423,9398,425],{},[412,9400,428],{},[423,9402,425],{},[409,9404,375],{},[412,9406,278],{},[419,9408,9409],{},"4",[447,9411,9412],{"encoding":449},"x + y > 10 \\;\\wedge\\; x = 3 \\;\\wedge\\; y = 4",[224,9414,9416,9434,9452,9476,9494,9518,9536],{"className":9415,"ariaHidden":455},[454],[224,9417,9419,9422,9425,9428,9431],{"className":9418},[459],[224,9420],{"className":9421,"style":464},[463],[224,9423,371],{"className":9424},[468,469],[224,9426],{"className":9427,"style":474},[473],[224,9429,319],{"className":9430},[478],[224,9432],{"className":9433,"style":474},[473],[224,9435,9437,9440,9443,9446,9449],{"className":9436},[459],[224,9438],{"className":9439,"style":488},[463],[224,9441,375],{"className":9442,"style":492},[468,469],[224,9444],{"className":9445,"style":496},[473],[224,9447,325],{"className":9448},[500],[224,9450],{"className":9451,"style":496},[473],[224,9453,9455,9458,9461,9464,9467,9470,9473],{"className":9454},[459],[224,9456],{"className":9457,"style":510},[463],[224,9459,421],{"className":9460},[468],[224,9462],{"className":9463,"style":496},[473],[224,9465],{"className":9466,"style":474},[473],[224,9468,428],{"className":9469},[478],[224,9471],{"className":9472,"style":496},[473],[224,9474],{"className":9475,"style":474},[473],[224,9477,9479,9482,9485,9488,9491],{"className":9478},[459],[224,9480],{"className":9481,"style":839},[463],[224,9483,371],{"className":9484},[468,469],[224,9486],{"className":9487,"style":496},[473],[224,9489,278],{"className":9490},[500],[224,9492],{"className":9493,"style":496},[473],[224,9495,9497,9500,9503,9506,9509,9512,9515],{"className":9496},[459],[224,9498],{"className":9499,"style":510},[463],[224,9501,3594],{"className":9502},[468],[224,9504],{"className":9505,"style":496},[473],[224,9507],{"className":9508,"style":474},[473],[224,9510,428],{"className":9511},[478],[224,9513],{"className":9514,"style":496},[473],[224,9516],{"className":9517,"style":474},[473],[224,9519,9521,9524,9527,9530,9533],{"className":9520},[459],[224,9522],{"className":9523,"style":806},[463],[224,9525,375],{"className":9526,"style":492},[468,469],[224,9528],{"className":9529,"style":496},[473],[224,9531,278],{"className":9532},[500],[224,9534],{"className":9535,"style":496},[473],[224,9537,9539,9542],{"className":9538},[459],[224,9540],{"className":9541,"style":510},[463],[224,9543,9409],{"className":9544},[468],[11,9546,9547,9548,9645],{},"这里算术部分自己就能算出 ",[224,9549,9551,9578],{"className":9550,"translate":388},[392],[224,9552,9554],{"className":9553},[396],[398,9555,9556],{"xmlns":400},[403,9557,9558,9575],{},[406,9559,9560,9562,9564,9566,9568,9571,9573],{},[419,9561,3594],{},[412,9563,319],{},[419,9565,9409],{},[412,9567,278],{},[419,9569,9570],{},"7",[412,9572,1577],{},[419,9574,421],{},[447,9576,9577],{"encoding":449},"3 + 4 = 7 \u003C 10",[224,9579,9581,9599,9617,9636],{"className":9580,"ariaHidden":455},[454],[224,9582,9584,9587,9590,9593,9596],{"className":9583},[459],[224,9585],{"className":9586,"style":7047},[463],[224,9588,3594],{"className":9589},[468],[224,9591],{"className":9592,"style":474},[473],[224,9594,319],{"className":9595},[478],[224,9597],{"className":9598,"style":474},[473],[224,9600,9602,9605,9608,9611,9614],{"className":9601},[459],[224,9603],{"className":9604,"style":510},[463],[224,9606,9409],{"className":9607},[468],[224,9609],{"className":9610,"style":496},[473],[224,9612,278],{"className":9613},[500],[224,9615],{"className":9616,"style":496},[473],[224,9618,9620,9624,9627,9630,9633],{"className":9619},[459],[224,9621],{"className":9622,"style":9623},[463],"height:0.6835em;vertical-align:-0.0391em;",[224,9625,9570],{"className":9626},[468],[224,9628],{"className":9629,"style":496},[473],[224,9631,1577],{"className":9632},[500],[224,9634],{"className":9635,"style":496},[473],[224,9637,9639,9642],{"className":9638},[459],[224,9640],{"className":9641,"style":510},[463],[224,9643,421],{"className":9644},[468],"，直接判 unsat，根本不需要 EUF 参与。",[11,9647,9648,9649],{},"现代求解器（Z3、cvc5 等）通常不严格照搬经典 Nelson–Oppen，而是在 DPLL(T) 框架下做「理论传播 + 冲突解释」。但核心思想没变：",[378,9650,9651],{},"每个理论只懂自己那一摊，通过交换少量关键事实来达成全局一致。",[11,9653,9654,9655,9739,9740,9841],{},"有一个细节值得注意：如果理论是「非凸」的，光交换等式还不够。比如在整数算术里，",[224,9656,9658,9676],{"className":9657,"translate":388},[392],[224,9659,9661],{"className":9660},[396],[398,9662,9663],{"xmlns":400},[403,9664,9665,9673],{},[406,9666,9667,9669,9671],{},[409,9668,371],{},[412,9670,2048],{"mathvariant":678},[419,9672,284],{},[447,9674,9675],{"encoding":449},"x \\ne 0",[224,9677,9679,9730],{"className":9678,"ariaHidden":455},[454],[224,9680,9682,9685,9688,9691,9727],{"className":9681},[459],[224,9683],{"className":9684,"style":2565},[463],[224,9686,371],{"className":9687},[468,469],[224,9689],{"className":9690,"style":496},[473],[224,9692,9694,9721,9724],{"className":9693},[500],[224,9695,9697],{"className":9696},[500],[224,9698,9700],{"className":9699},[468,2553],[224,9701,9703],{"className":9702},[2557],[224,9704,9706,9709,9718],{"className":9705},[2561],[224,9707],{"className":9708,"style":2565},[463],[224,9710,9712],{"className":9711},[2569],[224,9713,9715],{"className":9714},[468],[224,9716,2576],{"className":9717},[500],[224,9719],{"className":9720},[2580],[224,9722],{"className":9723},[473,2584],[224,9725,278],{"className":9726},[500],[224,9728],{"className":9729,"style":496},[473],[224,9731,9733,9736],{"className":9732},[459],[224,9734],{"className":9735,"style":510},[463],[224,9737,284],{"className":9738},[468]," 等价于 ",[224,9741,9743,9772],{"className":9742,"translate":388},[392],[224,9744,9746],{"className":9745},[396],[398,9747,9748],{"xmlns":400},[403,9749,9750,9769],{},[406,9751,9752,9754,9756,9758,9760,9762,9764,9767],{},[409,9753,371],{},[412,9755,2070],{},[412,9757,5268],{},[419,9759,1438],{},[412,9761,1131],{},[409,9763,371],{},[412,9765,9766],{},"≥",[419,9768,1438],{},[447,9770,9771],{"encoding":449},"x \\le -1 \\vee x \\ge 1",[224,9773,9775,9793,9814,9832],{"className":9774,"ariaHidden":455},[454],[224,9776,9778,9781,9784,9787,9790],{"className":9777},[459],[224,9779],{"className":9780,"style":3204},[463],[224,9782,371],{"className":9783},[468,469],[224,9785],{"className":9786,"style":496},[473],[224,9788,2070],{"className":9789},[500],[224,9791],{"className":9792,"style":496},[473],[224,9794,9796,9799,9802,9805,9808,9811],{"className":9795},[459],[224,9797],{"className":9798,"style":7047},[463],[224,9800,5268],{"className":9801},[468],[224,9803,1438],{"className":9804},[468],[224,9806],{"className":9807,"style":474},[473],[224,9809,1131],{"className":9810},[478],[224,9812],{"className":9813,"style":474},[473],[224,9815,9817,9820,9823,9826,9829],{"className":9816},[459],[224,9818],{"className":9819,"style":3204},[463],[224,9821,371],{"className":9822},[468,469],[224,9824],{"className":9825,"style":496},[473],[224,9827,9766],{"className":9828},[500],[224,9830],{"className":9831,"style":496},[473],[224,9833,9835,9838],{"className":9834},[459],[224,9836],{"className":9837,"style":510},[463],[224,9839,1438],{"className":9840},[468],"，这是一个析取，没法用一组等式表达。上层只能做情况分裂（case split），分别假设每一支再检查。这会让组合复杂度上升，也是混合理论求解有时会突然变慢的原因之一。",[613,9843,9845],{"id":9844},"_34-给出解释","3.4 给出解释",[11,9847,9848,9849,3117],{},"第四个策略是",[378,9850,9851],{},"不只回答是或否，还给出可操作的结果",[2674,9853,9854,9859,9864,9925],{},[2677,9855,9856,9858],{},[378,9857,599],{},"：求解器返回一个模型，给出每个变量的具体取值。对符号执行来说，这就是一条能触发目标路径的测试输入。",[2677,9860,9861,9863],{},[378,9862,6650],{},"：求解器可以返回一个 unsat core，也就是一组不可满足的子约束，告诉你「是这几条打架了」。",[2677,9865,9866,9869,9870,9924],{},[378,9867,9868],{},"插值","：从 ",[224,9871,9873,9893],{"className":9872,"translate":388},[392],[224,9874,9876],{"className":9875},[396],[398,9877,9878],{"xmlns":400},[403,9879,9880,9890],{},[406,9881,9882,9884,9887],{},[409,9883,1097],{},[412,9885,9886],{},"⇒",[409,9888,9889],{},"B",[447,9891,9892],{"encoding":449},"A \\Rightarrow B",[224,9894,9896,9914],{"className":9895,"ariaHidden":455},[454],[224,9897,9899,9902,9905,9908,9911],{"className":9898},[459],[224,9900],{"className":9901,"style":649},[463],[224,9903,1097],{"className":9904},[468,469],[224,9906],{"className":9907,"style":496},[473],[224,9909,9886],{"className":9910},[500],[224,9912],{"className":9913,"style":496},[473],[224,9915,9917,9920],{"className":9916},[459],[224,9918],{"className":9919,"style":649},[463],[224,9921,9889],{"className":9922,"style":9923},[468,469],"margin-right:0.0502em;"," 的不可满足性中提取一个只含共同变量的中间公式（Craig interpolant），用于程序验证中的不变式合成和抽象精化。",[2677,9926,9927,9930],{},[378,9928,9929],{},"证明","：部分求解器能输出 resolution 证明，供独立检查，避免「求解器说有漏洞，但你不知道为什么」的黑盒困境。",[11,9932,9933,9934,9936],{},"这一点经常被忽略，但它决定了 SMT 能不能真正嵌进工具链。一个只会输出 ",[221,9935,6650],{}," 的求解器，在调试时几乎没用；一个能告诉你「第 3、7、9 条约束冲突」的求解器，才能帮工程师定位问题。",[613,9938,9940],{"id":9939},"_35-增量求解","3.5 增量求解",[11,9942,9943,9944,9947],{},"第五个策略是",[378,9945,9946],{},"增量求解","（incremental solving）。",[11,9949,9950],{},"符号执行的路径是一棵树：父路径的约束是子路径约束的前缀。如果每换一条路径都从头求解，大量工作会被重复浪费。",[11,9952,9953,9954,3145,9957,2887,9960,9963],{},"SMT-LIB 提供了 ",[221,9955,9956],{},"push",[221,9958,9959],{},"pop",[221,9961,9962],{},"check-sat-assuming","，允许在已有约束的基础上临时加入假设、求解、再回退。求解器会保留之前学到的子句和理论状态。一条路径判 unsat 后回退到分叉点，下一条路径可以接着用。",[11,9965,9966],{},"KLEE、angr 这类工具都重度依赖这个特性。没有增量求解，符号执行的性能会差一个数量级。",[608,9968,9970],{"id":9969},"四dpllt","四、DPLL(T)",[613,9972,9974],{"id":9973},"_41-总体架构","4.1 总体架构",[11,9976,9977,9978,9981,9982,9985],{},"现代 SMT 求解器的标准架构叫 ",[378,9979,9980],{},"DPLL(T)","，更准确的名字是 ",[378,9983,9984],{},"CDCL(T)","：外层是带冲突学习的 SAT 求解器，内层挂一个或多个理论求解器。",[3101,9987,9988],{},[3104,9989],{"height":9990,"width":9991,"style":3108,"data-d2-theme":3109,"alt":186,"src":9992},320,2578,"data:image\u002Fsvg+xml;base64,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\u002Fmu6HotrM\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",[11,9994,9995],{},"用一句话概括这个循环：",[9997,9998,9999,10002,10005,10008,10011,10014],"ol",{},[2677,10000,10001],{},"把公式里的每个理论原子替换成一个命题变量，得到一个纯布尔公式；",[2677,10003,10004],{},"让 SAT 求解器找一个布尔模型；",[2677,10006,10007],{},"把布尔模型翻译回理论原子的真假，交给理论求解器检查；",[2677,10009,10010],{},"如果理论层说「一致」，就得到 sat 和模型；",[2677,10012,10013],{},"如果理论层说「冲突」，就把冲突总结成一条新的布尔子句，塞回 SAT 层，让它换个方向再搜；",[2677,10015,10016],{},"如果 SAT 层把所有可能性都穷尽了，就得到 unsat。",[11,10018,10019],{},"可以把它想象成一个总承包商和几个专业工种的协作：总承包商（SAT）负责排期和组合，电工、水管工（理论求解器）负责检查自己那部分是否可行。电工说「这个排期下电线会短路」，总承包商就把这条经验记下来，换一个排期重新试。",[613,10021,10023],{"id":10022},"_42-布尔抽象","4.2 布尔抽象",[11,10025,10026],{},"布尔抽象（也叫 propositional abstraction）是两层之间的接口。",[11,10028,10029],{},"以这个公式为例：",[224,10031,10033],{"className":10032,"translate":388},[387],[224,10034,10036,10090],{"className":10035,"translate":388},[392],[224,10037,10039],{"className":10038},[396],[398,10040,10041],{"xmlns":400,"display":401},[403,10042,10043,10087],{},[406,10044,10045,10047,10049,10051,10053,10055,10057,10059,10061,10063,10065,10067,10069,10071,10073,10075,10077,10079,10081,10083,10085],{},[409,10046,704],{},[412,10048,278],{},[412,10050,239],{"stretchy":435},[409,10052,371],{},[412,10054,325],{},[419,10056,284],{},[412,10058,1131],{},[409,10060,375],{},[412,10062,325],{},[419,10064,284],{},[412,10066,1716],{"stretchy":435},[423,10068,425],{},[412,10070,428],{},[423,10072,425],{},[412,10074,239],{"stretchy":435},[409,10076,371],{},[412,10078,319],{},[409,10080,375],{},[412,10082,1577],{},[419,10084,284],{},[412,10086,1716],{"stretchy":435},[447,10088,10089],{"encoding":449},"\\varphi = (x > 0 \\vee y > 0) \\;\\wedge\\; (x + y \u003C 0)",[224,10091,10093,10111,10132,10150,10168,10195,10216,10234],{"className":10092,"ariaHidden":455},[454],[224,10094,10096,10099,10102,10105,10108],{"className":10095},[459],[224,10097],{"className":10098,"style":806},[463],[224,10100,704],{"className":10101},[468,469],[224,10103],{"className":10104,"style":496},[473],[224,10106,278],{"className":10107},[500],[224,10109],{"className":10110,"style":496},[473],[224,10112,10114,10117,10120,10123,10126,10129],{"className":10113},[459],[224,10115],{"className":10116,"style":535},[463],[224,10118,239],{"className":10119},[542],[224,10121,371],{"className":10122},[468,469],[224,10124],{"className":10125,"style":496},[473],[224,10127,325],{"className":10128},[500],[224,10130],{"className":10131,"style":496},[473],[224,10133,10135,10138,10141,10144,10147],{"className":10134},[459],[224,10136],{"className":10137,"style":510},[463],[224,10139,284],{"className":10140},[468],[224,10142],{"className":10143,"style":474},[473],[224,10145,1131],{"className":10146},[478],[224,10148],{"className":10149,"style":474},[473],[224,10151,10153,10156,10159,10162,10165],{"className":10152},[459],[224,10154],{"className":10155,"style":488},[463],[224,10157,375],{"className":10158,"style":492},[468,469],[224,10160],{"className":10161,"style":496},[473],[224,10163,325],{"className":10164},[500],[224,10166],{"className":10167,"style":496},[473],[224,10169,10171,10174,10177,10180,10183,10186,10189,10192],{"className":10170},[459],[224,10172],{"className":10173,"style":535},[463],[224,10175,284],{"className":10176},[468],[224,10178,1716],{"className":10179},[549],[224,10181],{"className":10182,"style":496},[473],[224,10184],{"className":10185,"style":474},[473],[224,10187,428],{"className":10188},[478],[224,10190],{"className":10191,"style":496},[473],[224,10193],{"className":10194,"style":474},[473],[224,10196,10198,10201,10204,10207,10210,10213],{"className":10197},[459],[224,10199],{"className":10200,"style":535},[463],[224,10202,239],{"className":10203},[542],[224,10205,371],{"className":10206},[468,469],[224,10208],{"className":10209,"style":474},[473],[224,10211,319],{"className":10212},[478],[224,10214],{"className":10215,"style":474},[473],[224,10217,10219,10222,10225,10228,10231],{"className":10218},[459],[224,10220],{"className":10221,"style":488},[463],[224,10223,375],{"className":10224,"style":492},[468,469],[224,10226],{"className":10227,"style":496},[473],[224,10229,1577],{"className":10230},[500],[224,10232],{"className":10233,"style":496},[473],[224,10235,10237,10240,10243],{"className":10236},[459],[224,10238],{"className":10239,"style":535},[463],[224,10241,284],{"className":10242},[468],[224,10244,1716],{"className":10245},[549],[11,10247,10248],{},"引入三个命题变量：",[224,10250,10252],{"className":10251,"translate":388},[387],[224,10253,10255,10322],{"className":10254,"translate":388},[392],[224,10256,10258],{"className":10257},[396],[398,10259,10260],{"xmlns":400,"display":401},[403,10261,10262,10319],{},[406,10263,10264,10266,10269,10271,10273,10275,10277,10279,10281,10283,10285,10287,10289,10291,10293,10295,10297,10299,10301,10303,10305,10307,10309,10311,10313,10315,10317],{},[409,10265,11],{},[412,10267,10268],{},"≡",[412,10270,239],{"stretchy":435},[409,10272,371],{},[412,10274,325],{},[419,10276,284],{},[412,10278,1716],{"stretchy":435},[412,10280,1877],{"separator":455},[473,10282],{"width":4247},[409,10284,2023],{},[412,10286,10268],{},[412,10288,239],{"stretchy":435},[409,10290,375],{},[412,10292,325],{},[419,10294,284],{},[412,10296,1716],{"stretchy":435},[412,10298,1877],{"separator":455},[473,10300],{"width":4247},[409,10302,2061],{},[412,10304,10268],{},[412,10306,239],{"stretchy":435},[409,10308,371],{},[412,10310,319],{},[409,10312,375],{},[412,10314,1577],{},[419,10316,284],{},[412,10318,1716],{"stretchy":435},[447,10320,10321],{"encoding":449},"p \\equiv (x > 0), \\qquad q \\equiv (y > 0), \\qquad r \\equiv (x + y \u003C 0)",[224,10323,10325,10344,10365,10398,10419,10452,10473,10491],{"className":10324,"ariaHidden":455},[454],[224,10326,10328,10332,10335,10338,10341],{"className":10327},[459],[224,10329],{"className":10330,"style":10331},[463],"height:0.6582em;vertical-align:-0.1944em;",[224,10333,11],{"className":10334},[468,469],[224,10336],{"className":10337,"style":496},[473],[224,10339,10268],{"className":10340},[500],[224,10342],{"className":10343,"style":496},[473],[224,10345,10347,10350,10353,10356,10359,10362],{"className":10346},[459],[224,10348],{"className":10349,"style":535},[463],[224,10351,239],{"className":10352},[542],[224,10354,371],{"className":10355},[468,469],[224,10357],{"className":10358,"style":496},[473],[224,10360,325],{"className":10361},[500],[224,10363],{"className":10364,"style":496},[473],[224,10366,10368,10371,10374,10377,10380,10383,10386,10389,10392,10395],{"className":10367},[459],[224,10369],{"className":10370,"style":535},[463],[224,10372,284],{"className":10373},[468],[224,10375,1716],{"className":10376},[549],[224,10378,1877],{"className":10379},[1917],[224,10381],{"className":10382,"style":4416},[473],[224,10384],{"className":10385,"style":724},[473],[224,10387,2023],{"className":10388,"style":492},[468,469],[224,10390],{"className":10391,"style":496},[473],[224,10393,10268],{"className":10394},[500],[224,10396],{"className":10397,"style":496},[473],[224,10399,10401,10404,10407,10410,10413,10416],{"className":10400},[459],[224,10402],{"className":10403,"style":535},[463],[224,10405,239],{"className":10406},[542],[224,10408,375],{"className":10409,"style":492},[468,469],[224,10411],{"className":10412,"style":496},[473],[224,10414,325],{"className":10415},[500],[224,10417],{"className":10418,"style":496},[473],[224,10420,10422,10425,10428,10431,10434,10437,10440,10443,10446,10449],{"className":10421},[459],[224,10423],{"className":10424,"style":535},[463],[224,10426,284],{"className":10427},[468],[224,10429,1716],{"className":10430},[549],[224,10432,1877],{"className":10433},[1917],[224,10435],{"className":10436,"style":4416},[473],[224,10438],{"className":10439,"style":724},[473],[224,10441,2061],{"className":10442,"style":2473},[468,469],[224,10444],{"className":10445,"style":496},[473],[224,10447,10268],{"className":10448},[500],[224,10450],{"className":10451,"style":496},[473],[224,10453,10455,10458,10461,10464,10467,10470],{"className":10454},[459],[224,10456],{"className":10457,"style":535},[463],[224,10459,239],{"className":10460},[542],[224,10462,371],{"className":10463},[468,469],[224,10465],{"className":10466,"style":474},[473],[224,10468,319],{"className":10469},[478],[224,10471],{"className":10472,"style":474},[473],[224,10474,10476,10479,10482,10485,10488],{"className":10475},[459],[224,10477],{"className":10478,"style":488},[463],[224,10480,375],{"className":10481,"style":492},[468,469],[224,10483],{"className":10484,"style":496},[473],[224,10486,1577],{"className":10487},[500],[224,10489],{"className":10490,"style":496},[473],[224,10492,10494,10497,10500],{"className":10493},[459],[224,10495],{"className":10496,"style":535},[463],[224,10498,284],{"className":10499},[468],[224,10501,1716],{"className":10502},[549],[11,10504,10505],{},"布尔骨架变成：",[224,10507,10509],{"className":10508,"translate":388},[387],[224,10510,10512,10542],{"className":10511,"translate":388},[392],[224,10513,10515],{"className":10514},[396],[398,10516,10517],{"xmlns":400,"display":401},[403,10518,10519,10539],{},[406,10520,10521,10523,10525,10527,10529,10531,10533,10535,10537],{},[412,10522,239],{"stretchy":435},[409,10524,11],{},[412,10526,1131],{},[409,10528,2023],{},[412,10530,1716],{"stretchy":435},[423,10532,425],{},[412,10534,428],{},[423,10536,425],{},[409,10538,2061],{},[447,10540,10541],{"encoding":449},"(p \\vee q) \\;\\wedge\\; r",[224,10543,10545,10566,10593],{"className":10544,"ariaHidden":455},[454],[224,10546,10548,10551,10554,10557,10560,10563],{"className":10547},[459],[224,10549],{"className":10550,"style":535},[463],[224,10552,239],{"className":10553},[542],[224,10555,11],{"className":10556},[468,469],[224,10558],{"className":10559,"style":474},[473],[224,10561,1131],{"className":10562},[478],[224,10564],{"className":10565,"style":474},[473],[224,10567,10569,10572,10575,10578,10581,10584,10587,10590],{"className":10568},[459],[224,10570],{"className":10571,"style":535},[463],[224,10573,2023],{"className":10574,"style":492},[468,469],[224,10576,1716],{"className":10577},[549],[224,10579],{"className":10580,"style":496},[473],[224,10582],{"className":10583,"style":474},[473],[224,10585,428],{"className":10586},[478],[224,10588],{"className":10589,"style":496},[473],[224,10591],{"className":10592,"style":474},[473],[224,10594,10596,10599],{"className":10595},[459],[224,10597],{"className":10598,"style":839},[463],[224,10600,2061],{"className":10601,"style":2473},[468,469],[11,10603,10604,10605,10686,10687,10808],{},"SAT 求解器在这个骨架上的搜索空间只有 ",[224,10606,10608,10630],{"className":10607,"translate":388},[392],[224,10609,10611],{"className":10610},[396],[398,10612,10613],{"xmlns":400},[403,10614,10615,10627],{},[406,10616,10617,10623,10625],{},[3248,10618,10619,10621],{},[419,10620,1447],{},[419,10622,3594],{},[412,10624,278],{},[419,10626,5430],{},[447,10628,10629],{"encoding":449},"2^3 = 8",[224,10631,10633,10677],{"className":10632,"ariaHidden":455},[454],[224,10634,10636,10639,10668,10671,10674],{"className":10635},[459],[224,10637],{"className":10638,"style":5671},[463],[224,10640,10642,10645],{"className":10641},[468],[224,10643,1447],{"className":10644},[468],[224,10646,10648],{"className":10647},[750],[224,10649,10651],{"className":10650},[754],[224,10652,10654],{"className":10653},[759],[224,10655,10657],{"className":10656,"style":5671},[763],[224,10658,10659,10662],{"style":3288},[224,10660],{"className":10661,"style":772},[771],[224,10663,10665],{"className":10664},[776,777,778,779],[224,10666,3594],{"className":10667},[468,779],[224,10669],{"className":10670,"style":496},[473],[224,10672,278],{"className":10673},[500],[224,10675],{"className":10676,"style":496},[473],[224,10678,10680,10683],{"className":10679},[459],[224,10681],{"className":10682,"style":510},[463],[224,10684,5430],{"className":10685},[468]," 种赋值。它先试 ",[224,10688,10690,10724],{"className":10689,"translate":388},[392],[224,10691,10693],{"className":10692},[396],[398,10694,10695],{"xmlns":400},[403,10696,10697,10721],{},[406,10698,10699,10701,10703,10705,10707,10709,10711,10713,10715,10717,10719],{},[409,10700,11],{},[412,10702,278],{},[409,10704,1078],{"mathvariant":678},[412,10706,1877],{"separator":455},[409,10708,2023],{},[412,10710,278],{},[409,10712,1078],{"mathvariant":678},[412,10714,1877],{"separator":455},[409,10716,2061],{},[412,10718,278],{},[409,10720,1078],{"mathvariant":678},[447,10722,10723],{"encoding":449},"p = \\top, q = \\top, r = \\top",[224,10725,10727,10745,10772,10799],{"className":10726,"ariaHidden":455},[454],[224,10728,10730,10733,10736,10739,10742],{"className":10729},[459],[224,10731],{"className":10732,"style":806},[463],[224,10734,11],{"className":10735},[468,469],[224,10737],{"className":10738,"style":496},[473],[224,10740,278],{"className":10741},[500],[224,10743],{"className":10744,"style":496},[473],[224,10746,10748,10751,10754,10757,10760,10763,10766,10769],{"className":10747},[459],[224,10749],{"className":10750,"style":2565},[463],[224,10752,1078],{"className":10753},[468],[224,10755,1877],{"className":10756},[1917],[224,10758],{"className":10759,"style":724},[473],[224,10761,2023],{"className":10762,"style":492},[468,469],[224,10764],{"className":10765,"style":496},[473],[224,10767,278],{"className":10768},[500],[224,10770],{"className":10771,"style":496},[473],[224,10773,10775,10778,10781,10784,10787,10790,10793,10796],{"className":10774},[459],[224,10776],{"className":10777,"style":2565},[463],[224,10779,1078],{"className":10780},[468],[224,10782,1877],{"className":10783},[1917],[224,10785],{"className":10786,"style":724},[473],[224,10788,2061],{"className":10789,"style":2473},[468,469],[224,10791],{"className":10792,"style":496},[473],[224,10794,278],{"className":10795},[500],[224,10797],{"className":10798,"style":496},[473],[224,10800,10802,10805],{"className":10801},[459],[224,10803],{"className":10804,"style":7642},[463],[224,10806,1078],{"className":10807},[468],"，把这三个真值交回理论层。",[11,10810,10811,10812,10815],{},"理论层看到的是三个",[378,10813,10814],{},"文字","（literal）：",[224,10817,10819],{"className":10818,"translate":388},[387],[224,10820,10822,10864],{"className":10821,"translate":388},[392],[224,10823,10825],{"className":10824},[396],[398,10826,10827],{"xmlns":400,"display":401},[403,10828,10829,10861],{},[406,10830,10831,10833,10835,10837,10839,10841,10843,10845,10847,10849,10851,10853,10855,10857,10859],{},[409,10832,371],{},[412,10834,325],{},[419,10836,284],{},[412,10838,1877],{"separator":455},[473,10840],{"width":4247},[409,10842,375],{},[412,10844,325],{},[419,10846,284],{},[412,10848,1877],{"separator":455},[473,10850],{"width":4247},[409,10852,371],{},[412,10854,319],{},[409,10856,375],{},[412,10858,1577],{},[419,10860,284],{},[447,10862,10863],{"encoding":449},"x > 0, \\qquad y > 0, \\qquad x + y \u003C 0",[224,10865,10867,10885,10915,10945,10963],{"className":10866,"ariaHidden":455},[454],[224,10868,10870,10873,10876,10879,10882],{"className":10869},[459],[224,10871],{"className":10872,"style":2810},[463],[224,10874,371],{"className":10875},[468,469],[224,10877],{"className":10878,"style":496},[473],[224,10880,325],{"className":10881},[500],[224,10883],{"className":10884,"style":496},[473],[224,10886,10888,10891,10894,10897,10900,10903,10906,10909,10912],{"className":10887},[459],[224,10889],{"className":10890,"style":8688},[463],[224,10892,284],{"className":10893},[468],[224,10895,1877],{"className":10896},[1917],[224,10898],{"className":10899,"style":4416},[473],[224,10901],{"className":10902,"style":724},[473],[224,10904,375],{"className":10905,"style":492},[468,469],[224,10907],{"className":10908,"style":496},[473],[224,10910,325],{"className":10911},[500],[224,10913],{"className":10914,"style":496},[473],[224,10916,10918,10921,10924,10927,10930,10933,10936,10939,10942],{"className":10917},[459],[224,10919],{"className":10920,"style":8688},[463],[224,10922,284],{"className":10923},[468],[224,10925,1877],{"className":10926},[1917],[224,10928],{"className":10929,"style":4416},[473],[224,10931],{"className":10932,"style":724},[473],[224,10934,371],{"className":10935},[468,469],[224,10937],{"className":10938,"style":474},[473],[224,10940,319],{"className":10941},[478],[224,10943],{"className":10944,"style":474},[473],[224,10946,10948,10951,10954,10957,10960],{"className":10947},[459],[224,10949],{"className":10950,"style":488},[463],[224,10952,375],{"className":10953,"style":492},[468,469],[224,10955],{"className":10956,"style":496},[473],[224,10958,1577],{"className":10959},[500],[224,10961],{"className":10962,"style":496},[473],[224,10964,10966,10969],{"className":10965},[459],[224,10967],{"className":10968,"style":510},[463],[224,10970,284],{"className":10971},[468],[11,10973,10974,10975,11048,11049,3117],{},"线性算术求解器立刻发现：前两个推出 ",[224,10976,10978,11000],{"className":10977,"translate":388},[392],[224,10979,10981],{"className":10980},[396],[398,10982,10983],{"xmlns":400},[403,10984,10985,10997],{},[406,10986,10987,10989,10991,10993,10995],{},[409,10988,371],{},[412,10990,319],{},[409,10992,375],{},[412,10994,325],{},[419,10996,284],{},[447,10998,10999],{"encoding":449},"x + y > 0",[224,11001,11003,11021,11039],{"className":11002,"ariaHidden":455},[454],[224,11004,11006,11009,11012,11015,11018],{"className":11005},[459],[224,11007],{"className":11008,"style":464},[463],[224,11010,371],{"className":11011},[468,469],[224,11013],{"className":11014,"style":474},[473],[224,11016,319],{"className":11017},[478],[224,11019],{"className":11020,"style":474},[473],[224,11022,11024,11027,11030,11033,11036],{"className":11023},[459],[224,11025],{"className":11026,"style":488},[463],[224,11028,375],{"className":11029,"style":492},[468,469],[224,11031],{"className":11032,"style":496},[473],[224,11034,325],{"className":11035},[500],[224,11037],{"className":11038,"style":496},[473],[224,11040,11042,11045],{"className":11041},[459],[224,11043],{"className":11044,"style":510},[463],[224,11046,284],{"className":11047},[468],"，与第三个矛盾。这是一次",[378,11050,11051],{},"理论冲突",[11,11053,11054,11055,11106,11107,11173,11174,11225,11226,11360],{},"注意，理论层收到的不是「原子为真」，而是「带正负号的文字」。",[224,11056,11058,11076],{"className":11057,"translate":388},[392],[224,11059,11061],{"className":11060},[396],[398,11062,11063],{"xmlns":400},[403,11064,11065,11073],{},[406,11066,11067,11069,11071],{},[409,11068,2023],{},[412,11070,278],{},[409,11072,1088],{"mathvariant":678},[447,11074,11075],{"encoding":449},"q = \\bot",[224,11077,11079,11097],{"className":11078,"ariaHidden":455},[454],[224,11080,11082,11085,11088,11091,11094],{"className":11081},[459],[224,11083],{"className":11084,"style":806},[463],[224,11086,2023],{"className":11087,"style":492},[468,469],[224,11089],{"className":11090,"style":496},[473],[224,11092,278],{"className":11093},[500],[224,11095],{"className":11096,"style":496},[473],[224,11098,11100,11103],{"className":11099},[459],[224,11101],{"className":11102,"style":7642},[463],[224,11104,1088],{"className":11105},[468]," 对应的是 ",[224,11108,11110,11134],{"className":11109,"translate":388},[392],[224,11111,11113],{"className":11112},[396],[398,11114,11115],{"xmlns":400},[403,11116,11117,11131],{},[406,11118,11119,11121,11123,11125,11127,11129],{},[409,11120,1106],{"mathvariant":678},[412,11122,239],{"stretchy":435},[409,11124,375],{},[412,11126,325],{},[419,11128,284],{},[412,11130,1716],{"stretchy":435},[447,11132,11133],{"encoding":449},"\\neg(y > 0)",[224,11135,11137,11161],{"className":11136,"ariaHidden":455},[454],[224,11138,11140,11143,11146,11149,11152,11155,11158],{"className":11139},[459],[224,11141],{"className":11142,"style":535},[463],[224,11144,1106],{"className":11145},[468],[224,11147,239],{"className":11148},[542],[224,11150,375],{"className":11151,"style":492},[468,469],[224,11153],{"className":11154,"style":496},[473],[224,11156,325],{"className":11157},[500],[224,11159],{"className":11160,"style":496},[473],[224,11162,11164,11167,11170],{"className":11163},[459],[224,11165],{"className":11166,"style":535},[463],[224,11168,284],{"className":11169},[468],[224,11171,1716],{"className":11172},[549],"，也就是 ",[224,11175,11177,11195],{"className":11176,"translate":388},[392],[224,11178,11180],{"className":11179},[396],[398,11181,11182],{"xmlns":400},[403,11183,11184,11192],{},[406,11185,11186,11188,11190],{},[409,11187,375],{},[412,11189,2070],{},[419,11191,284],{},[447,11193,11194],{"encoding":449},"y \\le 0",[224,11196,11198,11216],{"className":11197,"ariaHidden":455},[454],[224,11199,11201,11204,11207,11210,11213],{"className":11200},[459],[224,11202],{"className":11203,"style":2644},[463],[224,11205,375],{"className":11206,"style":492},[468,469],[224,11208],{"className":11209,"style":496},[473],[224,11211,2070],{"className":11212},[500],[224,11214],{"className":11215,"style":496},[473],[224,11217,11219,11222],{"className":11218},[459],[224,11220],{"className":11221,"style":510},[463],[224,11223,284],{"className":11224},[468],"。理论求解器必须能处理这种否定形式，这也是为什么它通常直接操作形如 ",[224,11227,11229,11255],{"className":11228,"translate":388},[392],[224,11230,11232],{"className":11231},[396],[398,11233,11234],{"xmlns":400},[403,11235,11236,11252],{},[406,11237,11238,11244,11246],{},[695,11239,11240,11242],{},[409,11241,1435],{},[419,11243,1438],{},[412,11245,2070],{},[695,11247,11248,11250],{},[409,11249,1435],{},[419,11251,1447],{},[447,11253,11254],{"encoding":449},"t_1 \\le t_2",[224,11256,11258,11314],{"className":11257,"ariaHidden":455},[454],[224,11259,11261,11265,11305,11308,11311],{"className":11260},[459],[224,11262],{"className":11263,"style":11264},[463],"height:0.786em;vertical-align:-0.15em;",[224,11266,11268,11271],{"className":11267},[468],[224,11269,1435],{"className":11270},[468,469],[224,11272,11274],{"className":11273},[750],[224,11275,11277,11297],{"className":11276},[754,755],[224,11278,11280,11294],{"className":11279},[759],[224,11281,11283],{"className":11282,"style":1479},[763],[224,11284,11285,11288],{"style":767},[224,11286],{"className":11287,"style":772},[771],[224,11289,11291],{"className":11290},[776,777,778,779],[224,11292,1438],{"className":11293},[468,779],[224,11295,787],{"className":11296},[786],[224,11298,11300],{"className":11299},[759],[224,11301,11303],{"className":11302,"style":794},[763],[224,11304],{},[224,11306],{"className":11307,"style":496},[473],[224,11309,2070],{"className":11310},[500],[224,11312],{"className":11313,"style":496},[473],[224,11315,11317,11320],{"className":11316},[459],[224,11318],{"className":11319,"style":1460},[463],[224,11321,11323,11326],{"className":11322},[468],[224,11324,1435],{"className":11325},[468,469],[224,11327,11329],{"className":11328},[750],[224,11330,11332,11352],{"className":11331},[754,755],[224,11333,11335,11349],{"className":11334},[759],[224,11336,11338],{"className":11337,"style":1479},[763],[224,11339,11340,11343],{"style":767},[224,11341],{"className":11342,"style":772},[771],[224,11344,11346],{"className":11345},[776,777,778,779],[224,11347,1447],{"className":11348},[468,779],[224,11350,787],{"className":11351},[786],[224,11353,11355],{"className":11354},[759],[224,11356,11358],{"className":11357,"style":794},[763],[224,11359],{}," 的不等式，而不是只处理等式。",[613,11362,11364],{"id":11363},"_43-cdcl","4.3 CDCL",[11,11366,11367],{},"SAT 层用的是 CDCL（冲突驱动子句学习）。它的核心循环大致如下：",[215,11369,11374],{"className":11370,"code":11372,"language":11373},[11371],"language-text","loop:\n    conflict = unit_propagate()      # 单元传播\n    if conflict:\n        if decision_level == 0: return UNSAT\n        lemma = analyze(conflict)    # 冲突分析\n        backtrack()                  # 非时序回跳\n        add_clause(lemma)            # 学习子句\n    else:\n        if all_variables_assigned(): return SAT\n        decide()                     # 选一个未赋值的变量\n","text",[221,11375,11372],{"__ignoreMap":186},[11,11377,11378],{},"三个关键机制：",[2674,11380,11381,11387,11393],{},[2677,11382,11383,11386],{},[378,11384,11385],{},"单元传播","：一条子句里只剩一个未赋值的文字时，强制它取能让子句为真的值。",[2677,11388,11389,11392],{},[378,11390,11391],{},"冲突分析","：从冲突出发，反向找出是哪几个决策导致的，生成一条新的子句。这条子句是原公式的逻辑推论，加进去不会改变解集。",[2677,11394,11395,11398],{},[378,11396,11397],{},"非时序回跳","：不退回上一层，而是直接跳回冲突的真正源头，跳过大量无用的搜索。",[11,11400,11401],{},"CDCL 是过去二十年 SAT 求解器性能提升的主要来源。SMT 做的事情，基本上是把这套成熟的工程能力「复用」到理论层。",[613,11403,11405],{"id":11404},"_44-理论求解器","4.4 理论求解器",[11,11407,11408,11409,11412,11413,11441],{},"理论层要做的事情叫 ",[378,11410,11411],{},"T-一致性检查","：给定一组带正负号的原子，判断它们在理论 ",[224,11414,11416,11429],{"className":11415,"translate":388},[392],[224,11417,11419],{"className":11418},[396],[398,11420,11421],{"xmlns":400},[403,11422,11423,11427],{},[406,11424,11425],{},[409,11426,637],{},[447,11428,637],{"encoding":449},[224,11430,11432],{"className":11431,"ariaHidden":455},[454],[224,11433,11435,11438],{"className":11434},[459],[224,11436],{"className":11437,"style":649},[463],[224,11439,637],{"className":11440,"style":653},[468,469]," 下是否可能同时为真。",[11,11443,11444],{},"不同理论有不同的算法。",[11446,11447,11449],"h4",{"id":11448},"euf","EUF",[11,11451,11452],{},"EUF（Equality with Uninterpreted Functions）只关心相等关系和函数的一致性。核心是两件事：",[9997,11454,11455,11458],{},[2677,11456,11457],{},"用并查集维护变量的等价类；",[2677,11459,11460,11461,11512,11513,3117],{},"用同余闭包（congruence closure）保证：如果 ",[224,11462,11464,11482],{"className":11463,"translate":388},[392],[224,11465,11467],{"className":11466},[396],[398,11468,11469],{"xmlns":400},[403,11470,11471,11479],{},[406,11472,11473,11475,11477],{},[409,11474,15],{},[412,11476,278],{},[409,11478,4014],{},[447,11480,11481],{"encoding":449},"a = b",[224,11483,11485,11503],{"className":11484,"ariaHidden":455},[454],[224,11486,11488,11491,11494,11497,11500],{"className":11487},[459],[224,11489],{"className":11490,"style":839},[463],[224,11492,15],{"className":11493},[468,469],[224,11495],{"className":11496,"style":496},[473],[224,11498,278],{"className":11499},[500],[224,11501],{"className":11502,"style":496},[473],[224,11504,11506,11509],{"className":11505},[459],[224,11507],{"className":11508,"style":7642},[463],[224,11510,4014],{"className":11511},[468,469],"，那么 ",[224,11514,11516,11546],{"className":11515,"translate":388},[392],[224,11517,11519],{"className":11518},[396],[398,11520,11521],{"xmlns":400},[403,11522,11523,11543],{},[406,11524,11525,11527,11529,11531,11533,11535,11537,11539,11541],{},[409,11526,1705],{},[412,11528,239],{"stretchy":435},[409,11530,15],{},[412,11532,1716],{"stretchy":435},[412,11534,278],{},[409,11536,1705],{},[412,11538,239],{"stretchy":435},[409,11540,4014],{},[412,11542,1716],{"stretchy":435},[447,11544,11545],{"encoding":449},"f(a) = f(b)",[224,11547,11549,11576],{"className":11548,"ariaHidden":455},[454],[224,11550,11552,11555,11558,11561,11564,11567,11570,11573],{"className":11551},[459],[224,11553],{"className":11554,"style":535},[463],[224,11556,1705],{"className":11557,"style":1740},[468,469],[224,11559,239],{"className":11560},[542],[224,11562,15],{"className":11563},[468,469],[224,11565,1716],{"className":11566},[549],[224,11568],{"className":11569,"style":496},[473],[224,11571,278],{"className":11572},[500],[224,11574],{"className":11575,"style":496},[473],[224,11577,11579,11582,11585,11588,11591],{"className":11578},[459],[224,11580],{"className":11581,"style":535},[463],[224,11583,1705],{"className":11584,"style":1740},[468,469],[224,11586,239],{"className":11587},[542],[224,11589,4014],{"className":11590},[468,469],[224,11592,1716],{"className":11593},[549],[11,11595,11596],{},"例子：",[224,11598,11600],{"className":11599,"translate":388},[387],[224,11601,11603,11657],{"className":11602,"translate":388},[392],[224,11604,11606],{"className":11605},[396],[398,11607,11608],{"xmlns":400,"display":401},[403,11609,11610,11654],{},[406,11611,11612,11614,11616,11618,11620,11622,11624,11626,11628,11630,11632,11634,11636,11638,11640,11642,11644,11646,11648,11650,11652],{},[409,11613,15],{},[412,11615,278],{},[409,11617,4014],{},[423,11619,425],{},[412,11621,428],{},[423,11623,425],{},[409,11625,4014],{},[412,11627,278],{},[409,11629,219],{},[423,11631,425],{},[412,11633,428],{},[423,11635,425],{},[409,11637,1705],{},[412,11639,239],{"stretchy":435},[409,11641,15],{},[412,11643,1716],{"stretchy":435},[412,11645,2048],{"mathvariant":678},[409,11647,1705],{},[412,11649,239],{"stretchy":435},[409,11651,219],{},[412,11653,1716],{"stretchy":435},[447,11655,11656],{"encoding":449},"a = b \\;\\wedge\\; b = c \\;\\wedge\\; f(a) \\ne f(c)",[224,11658,11660,11678,11702,11720,11745,11805],{"className":11659,"ariaHidden":455},[454],[224,11661,11663,11666,11669,11672,11675],{"className":11662},[459],[224,11664],{"className":11665,"style":839},[463],[224,11667,15],{"className":11668},[468,469],[224,11670],{"className":11671,"style":496},[473],[224,11673,278],{"className":11674},[500],[224,11676],{"className":11677,"style":496},[473],[224,11679,11681,11684,11687,11690,11693,11696,11699],{"className":11680},[459],[224,11682],{"className":11683,"style":7642},[463],[224,11685,4014],{"className":11686},[468,469],[224,11688],{"className":11689,"style":496},[473],[224,11691],{"className":11692,"style":474},[473],[224,11694,428],{"className":11695},[478],[224,11697],{"className":11698,"style":496},[473],[224,11700],{"className":11701,"style":474},[473],[224,11703,11705,11708,11711,11714,11717],{"className":11704},[459],[224,11706],{"className":11707,"style":7642},[463],[224,11709,4014],{"className":11710},[468,469],[224,11712],{"className":11713,"style":496},[473],[224,11715,278],{"className":11716},[500],[224,11718],{"className":11719,"style":496},[473],[224,11721,11723,11727,11730,11733,11736,11739,11742],{"className":11722},[459],[224,11724],{"className":11725,"style":11726},[463],"height:0.5556em;",[224,11728,219],{"className":11729},[468,469],[224,11731],{"className":11732,"style":496},[473],[224,11734],{"className":11735,"style":474},[473],[224,11737,428],{"className":11738},[478],[224,11740],{"className":11741,"style":496},[473],[224,11743],{"className":11744,"style":474},[473],[224,11746,11748,11751,11754,11757,11760,11763,11766,11802],{"className":11747},[459],[224,11749],{"className":11750,"style":535},[463],[224,11752,1705],{"className":11753,"style":1740},[468,469],[224,11755,239],{"className":11756},[542],[224,11758,15],{"className":11759},[468,469],[224,11761,1716],{"className":11762},[549],[224,11764],{"className":11765,"style":496},[473],[224,11767,11769,11796,11799],{"className":11768},[500],[224,11770,11772],{"className":11771},[500],[224,11773,11775],{"className":11774},[468,2553],[224,11776,11778],{"className":11777},[2557],[224,11779,11781,11784,11793],{"className":11780},[2561],[224,11782],{"className":11783,"style":2565},[463],[224,11785,11787],{"className":11786},[2569],[224,11788,11790],{"className":11789},[468],[224,11791,2576],{"className":11792},[500],[224,11794],{"className":11795},[2580],[224,11797],{"className":11798},[473,2584],[224,11800,278],{"className":11801},[500],[224,11803],{"className":11804,"style":496},[473],[224,11806,11808,11811,11814,11817,11820],{"className":11807},[459],[224,11809],{"className":11810,"style":535},[463],[224,11812,1705],{"className":11813,"style":1740},[468,469],[224,11815,239],{"className":11816},[542],[224,11818,219],{"className":11819},[468,469],[224,11821,1716],{"className":11822},[549],[11,11824,11825,11826,11881,11882,2887,11926,11970,11971,12043],{},"并查集把 ",[224,11827,11829,11851],{"className":11828,"translate":388},[392],[224,11830,11832],{"className":11831},[396],[398,11833,11834],{"xmlns":400},[403,11835,11836,11848],{},[406,11837,11838,11840,11842,11844,11846],{},[409,11839,15],{},[412,11841,1877],{"separator":455},[409,11843,4014],{},[412,11845,1877],{"separator":455},[409,11847,219],{},[447,11849,11850],{"encoding":449},"a, b, c",[224,11852,11854],{"className":11853,"ariaHidden":455},[454],[224,11855,11857,11860,11863,11866,11869,11872,11875,11878],{"className":11856},[459],[224,11858],{"className":11859,"style":2565},[463],[224,11861,15],{"className":11862},[468,469],[224,11864,1877],{"className":11865},[1917],[224,11867],{"className":11868,"style":724},[473],[224,11870,4014],{"className":11871},[468,469],[224,11873,1877],{"className":11874},[1917],[224,11876],{"className":11877,"style":724},[473],[224,11879,219],{"className":11880},[468,469]," 合并成一个类；同余闭包发现 ",[224,11883,11885,11905],{"className":11884,"translate":388},[392],[224,11886,11888],{"className":11887},[396],[398,11889,11890],{"xmlns":400},[403,11891,11892,11902],{},[406,11893,11894,11896,11898,11900],{},[409,11895,1705],{},[412,11897,239],{"stretchy":435},[409,11899,15],{},[412,11901,1716],{"stretchy":435},[447,11903,11904],{"encoding":449},"f(a)",[224,11906,11908],{"className":11907,"ariaHidden":455},[454],[224,11909,11911,11914,11917,11920,11923],{"className":11910},[459],[224,11912],{"className":11913,"style":535},[463],[224,11915,1705],{"className":11916,"style":1740},[468,469],[224,11918,239],{"className":11919},[542],[224,11921,15],{"className":11922},[468,469],[224,11924,1716],{"className":11925},[549],[224,11927,11929,11949],{"className":11928,"translate":388},[392],[224,11930,11932],{"className":11931},[396],[398,11933,11934],{"xmlns":400},[403,11935,11936,11946],{},[406,11937,11938,11940,11942,11944],{},[409,11939,1705],{},[412,11941,239],{"stretchy":435},[409,11943,219],{},[412,11945,1716],{"stretchy":435},[447,11947,11948],{"encoding":449},"f(c)",[224,11950,11952],{"className":11951,"ariaHidden":455},[454],[224,11953,11955,11958,11961,11964,11967],{"className":11954},[459],[224,11956],{"className":11957,"style":535},[463],[224,11959,1705],{"className":11960,"style":1740},[468,469],[224,11962,239],{"className":11963},[542],[224,11965,219],{"className":11966},[468,469],[224,11968,1716],{"className":11969},[549]," 的参数相等，于是把 ",[224,11972,11974,12004],{"className":11973,"translate":388},[392],[224,11975,11977],{"className":11976},[396],[398,11978,11979],{"xmlns":400},[403,11980,11981,12001],{},[406,11982,11983,11985,11987,11989,11991,11993,11995,11997,11999],{},[409,11984,1705],{},[412,11986,239],{"stretchy":435},[409,11988,15],{},[412,11990,1716],{"stretchy":435},[412,11992,1877],{"separator":455},[409,11994,1705],{},[412,11996,239],{"stretchy":435},[409,11998,219],{},[412,12000,1716],{"stretchy":435},[447,12002,12003],{"encoding":449},"f(a), f(c)",[224,12005,12007],{"className":12006,"ariaHidden":455},[454],[224,12008,12010,12013,12016,12019,12022,12025,12028,12031,12034,12037,12040],{"className":12009},[459],[224,12011],{"className":12012,"style":535},[463],[224,12014,1705],{"className":12015,"style":1740},[468,469],[224,12017,239],{"className":12018},[542],[224,12020,15],{"className":12021},[468,469],[224,12023,1716],{"className":12024},[549],[224,12026,1877],{"className":12027},[1917],[224,12029],{"className":12030,"style":724},[473],[224,12032,1705],{"className":12033,"style":1740},[468,469],[224,12035,239],{"className":12036},[542],[224,12038,219],{"className":12039},[468,469],[224,12041,1716],{"className":12042},[549]," 也合并；但公式又要求它们不等，冲突，判 unsat。",[11,12045,12046],{},"EUF 在程序分析里非常有用，因为指针别名、对象身份、类型相等，本质上都是「未解释函数的相等性」。",[11446,12048,12049],{"id":12049},"线性算术",[11,12051,12052,12054],{},[221,12053,3144],{}," 用单纯形法（更准确地说，是 SMT 里常用的 general simplex）。它先维护一组满足所有等式的赋值，再通过枢轴变换（pivot）调整变量的取值，直到所有上下界约束也被满足；如果怎么调都满足不了，就报告冲突。",[11,12056,12057,12059,12060,12063,12064,2887,12127,12190],{},[221,12058,3148],{}," 在此基础上加",[378,12061,12062],{},"分支定界","：先按实数求解，如果解里某个整数变量取了小数，就分裂成两支，比如 ",[224,12065,12067,12091],{"className":12066,"translate":388},[392],[224,12068,12070],{"className":12069},[396],[398,12071,12072],{"xmlns":400},[403,12073,12074,12088],{},[406,12075,12076,12078,12080,12083,12085],{},[409,12077,371],{},[412,12079,2070],{},[412,12081,12082],{"stretchy":435},"⌊",[409,12084,1887],{},[412,12086,12087],{"stretchy":435},"⌋",[447,12089,12090],{"encoding":449},"x \\le \\lfloor v \\rfloor",[224,12092,12094,12112],{"className":12093,"ariaHidden":455},[454],[224,12095,12097,12100,12103,12106,12109],{"className":12096},[459],[224,12098],{"className":12099,"style":3204},[463],[224,12101,371],{"className":12102},[468,469],[224,12104],{"className":12105,"style":496},[473],[224,12107,2070],{"className":12108},[500],[224,12110],{"className":12111,"style":496},[473],[224,12113,12115,12118,12121,12124],{"className":12114},[459],[224,12116],{"className":12117,"style":535},[463],[224,12119,12082],{"className":12120},[542],[224,12122,1887],{"className":12123,"style":492},[468,469],[224,12125,12087],{"className":12126},[549],[224,12128,12130,12154],{"className":12129,"translate":388},[392],[224,12131,12133],{"className":12132},[396],[398,12134,12135],{"xmlns":400},[403,12136,12137,12151],{},[406,12138,12139,12141,12143,12146,12148],{},[409,12140,371],{},[412,12142,9766],{},[412,12144,12145],{"stretchy":435},"⌈",[409,12147,1887],{},[412,12149,12150],{"stretchy":435},"⌉",[447,12152,12153],{"encoding":449},"x \\ge \\lceil v \\rceil",[224,12155,12157,12175],{"className":12156,"ariaHidden":455},[454],[224,12158,12160,12163,12166,12169,12172],{"className":12159},[459],[224,12161],{"className":12162,"style":3204},[463],[224,12164,371],{"className":12165},[468,469],[224,12167],{"className":12168,"style":496},[473],[224,12170,9766],{"className":12171},[500],[224,12173],{"className":12174,"style":496},[473],[224,12176,12178,12181,12184,12187],{"className":12177},[459],[224,12179],{"className":12180,"style":535},[463],[224,12182,12145],{"className":12183},[542],[224,12185,1887],{"className":12186,"style":492},[468,469],[224,12188,12150],{"className":12189},[549],"，分别再解。",[11,12192,11596],{},[224,12194,12196],{"className":12195,"translate":388},[387],[224,12197,12199,12256],{"className":12198,"translate":388},[392],[224,12200,12202],{"className":12201},[396],[398,12203,12204],{"xmlns":400,"display":401},[403,12205,12206,12253],{},[406,12207,12208,12210,12212,12214,12216,12218,12220,12222,12224,12226,12228,12230,12232,12234,12236,12238,12240,12242,12244,12246,12249],{},[419,12209,3594],{},[409,12211,371],{},[412,12213,319],{},[419,12215,1447],{},[409,12217,375],{},[412,12219,278],{},[419,12221,445],{},[412,12223,1877],{"separator":455},[473,12225],{"width":4247},[409,12227,371],{},[412,12229,1877],{"separator":455},[409,12231,375],{},[412,12233,9766],{},[419,12235,284],{},[412,12237,1877],{"separator":455},[473,12239],{"width":4247},[409,12241,371],{},[412,12243,1877],{"separator":455},[409,12245,375],{},[412,12247,12248],{},"∈",[409,12250,12252],{"mathvariant":12251},"double-struck","Z",[447,12254,12255],{"encoding":449},"3x + 2y = 5, \\qquad x, y \\ge 0, \\qquad x, y \\in \\mathbb{Z}",[224,12257,12259,12280,12301,12340,12379],{"className":12258,"ariaHidden":455},[454],[224,12260,12262,12265,12268,12271,12274,12277],{"className":12261},[459],[224,12263],{"className":12264,"style":7047},[463],[224,12266,3594],{"className":12267},[468],[224,12269,371],{"className":12270},[468,469],[224,12272],{"className":12273,"style":474},[473],[224,12275,319],{"className":12276},[478],[224,12278],{"className":12279,"style":474},[473],[224,12281,12283,12286,12289,12292,12295,12298],{"className":12282},[459],[224,12284],{"className":12285,"style":8688},[463],[224,12287,1447],{"className":12288},[468],[224,12290,375],{"className":12291,"style":492},[468,469],[224,12293],{"className":12294,"style":496},[473],[224,12296,278],{"className":12297},[500],[224,12299],{"className":12300,"style":496},[473],[224,12302,12304,12307,12310,12313,12316,12319,12322,12325,12328,12331,12334,12337],{"className":12303},[459],[224,12305],{"className":12306,"style":8688},[463],[224,12308,445],{"className":12309},[468],[224,12311,1877],{"className":12312},[1917],[224,12314],{"className":12315,"style":4416},[473],[224,12317],{"className":12318,"style":724},[473],[224,12320,371],{"className":12321},[468,469],[224,12323,1877],{"className":12324},[1917],[224,12326],{"className":12327,"style":724},[473],[224,12329,375],{"className":12330,"style":492},[468,469],[224,12332],{"className":12333,"style":496},[473],[224,12335,9766],{"className":12336},[500],[224,12338],{"className":12339,"style":496},[473],[224,12341,12343,12346,12349,12352,12355,12358,12361,12364,12367,12370,12373,12376],{"className":12342},[459],[224,12344],{"className":12345,"style":8688},[463],[224,12347,284],{"className":12348},[468],[224,12350,1877],{"className":12351},[1917],[224,12353],{"className":12354,"style":4416},[473],[224,12356],{"className":12357,"style":724},[473],[224,12359,371],{"className":12360},[468,469],[224,12362,1877],{"className":12363},[1917],[224,12365],{"className":12366,"style":724},[473],[224,12368,375],{"className":12369,"style":492},[468,469],[224,12371],{"className":12372,"style":496},[473],[224,12374,12248],{"className":12375},[500],[224,12377],{"className":12378,"style":496},[473],[224,12380,12382,12386],{"className":12381},[459],[224,12383],{"className":12384,"style":12385},[463],"height:0.6889em;",[224,12387,12252],{"className":12388},[468,12389],"mathbb",[11,12391,12392,12393,12485,12486,12514],{},"实数松弛下，",[224,12394,12396,12427],{"className":12395,"translate":388},[392],[224,12397,12399],{"className":12398},[396],[398,12400,12401],{"xmlns":400},[403,12402,12403,12424],{},[406,12404,12405,12407,12409,12411,12414,12416,12418,12420,12422],{},[409,12406,371],{},[412,12408,278],{},[419,12410,445],{},[409,12412,12413],{"mathvariant":678},"\u002F",[419,12415,3594],{},[412,12417,1877],{"separator":455},[409,12419,375],{},[412,12421,278],{},[419,12423,284],{},[447,12425,12426],{"encoding":449},"x = 5\u002F3, y = 0",[224,12428,12430,12448,12476],{"className":12429,"ariaHidden":455},[454],[224,12431,12433,12436,12439,12442,12445],{"className":12432},[459],[224,12434],{"className":12435,"style":839},[463],[224,12437,371],{"className":12438},[468,469],[224,12440],{"className":12441,"style":496},[473],[224,12443,278],{"className":12444},[500],[224,12446],{"className":12447,"style":496},[473],[224,12449,12451,12454,12458,12461,12464,12467,12470,12473],{"className":12450},[459],[224,12452],{"className":12453,"style":535},[463],[224,12455,12457],{"className":12456},[468],"5\u002F3",[224,12459,1877],{"className":12460},[1917],[224,12462],{"className":12463,"style":724},[473],[224,12465,375],{"className":12466,"style":492},[468,469],[224,12468],{"className":12469,"style":496},[473],[224,12471,278],{"className":12472},[500],[224,12474],{"className":12475,"style":496},[473],[224,12477,12479,12482],{"className":12478},[459],[224,12480],{"className":12481,"style":510},[463],[224,12483,284],{"className":12484},[468]," 是一个解，但 ",[224,12487,12489,12502],{"className":12488,"translate":388},[392],[224,12490,12492],{"className":12491},[396],[398,12493,12494],{"xmlns":400},[403,12495,12496,12500],{},[406,12497,12498],{},[409,12499,371],{},[447,12501,371],{"encoding":449},[224,12503,12505],{"className":12504,"ariaHidden":455},[454],[224,12506,12508,12511],{"className":12507},[459],[224,12509],{"className":12510,"style":839},[463],[224,12512,371],{"className":12513},[468,469]," 不是整数。分支：",[2674,12516,12517,12865],{},[2677,12518,368,12519,12570,12571,12599,12600,12628,12629,6095,12657,12708,12709,12761,12762,12708,12813,12864],{},[224,12520,12522,12540],{"className":12521,"translate":388},[392],[224,12523,12525],{"className":12524},[396],[398,12526,12527],{"xmlns":400},[403,12528,12529,12537],{},[406,12530,12531,12533,12535],{},[409,12532,371],{},[412,12534,2070],{},[419,12536,1438],{},[447,12538,12539],{"encoding":449},"x \\le 1",[224,12541,12543,12561],{"className":12542,"ariaHidden":455},[454],[224,12544,12546,12549,12552,12555,12558],{"className":12545},[459],[224,12547],{"className":12548,"style":3204},[463],[224,12550,371],{"className":12551},[468,469],[224,12553],{"className":12554,"style":496},[473],[224,12556,2070],{"className":12557},[500],[224,12559],{"className":12560,"style":496},[473],[224,12562,12564,12567],{"className":12563},[459],[224,12565],{"className":12566,"style":510},[463],[224,12568,1438],{"className":12569},[468],"：",[224,12572,12574,12587],{"className":12573,"translate":388},[392],[224,12575,12577],{"className":12576},[396],[398,12578,12579],{"xmlns":400},[403,12580,12581,12585],{},[406,12582,12583],{},[409,12584,371],{},[447,12586,371],{"encoding":449},[224,12588,12590],{"className":12589,"ariaHidden":455},[454],[224,12591,12593,12596],{"className":12592},[459],[224,12594],{"className":12595,"style":839},[463],[224,12597,371],{"className":12598},[468,469]," 只能取 ",[224,12601,12603,12616],{"className":12602,"translate":388},[392],[224,12604,12606],{"className":12605},[396],[398,12607,12608],{"xmlns":400},[403,12609,12610,12614],{},[406,12611,12612],{},[419,12613,284],{},[447,12615,284],{"encoding":449},[224,12617,12619],{"className":12618,"ariaHidden":455},[454],[224,12620,12622,12625],{"className":12621},[459],[224,12623],{"className":12624,"style":510},[463],[224,12626,284],{"className":12627},[468]," 或 ",[224,12630,12632,12645],{"className":12631,"translate":388},[392],[224,12633,12635],{"className":12634},[396],[398,12636,12637],{"xmlns":400},[403,12638,12639,12643],{},[406,12640,12641],{},[419,12642,1438],{},[447,12644,1438],{"encoding":449},[224,12646,12648],{"className":12647,"ariaHidden":455},[454],[224,12649,12651,12654],{"className":12650},[459],[224,12652],{"className":12653,"style":510},[463],[224,12655,1438],{"className":12656},[468],[224,12658,12660,12678],{"className":12659,"translate":388},[392],[224,12661,12663],{"className":12662},[396],[398,12664,12665],{"xmlns":400},[403,12666,12667,12675],{},[406,12668,12669,12671,12673],{},[409,12670,371],{},[412,12672,278],{},[419,12674,284],{},[447,12676,12677],{"encoding":449},"x = 0",[224,12679,12681,12699],{"className":12680,"ariaHidden":455},[454],[224,12682,12684,12687,12690,12693,12696],{"className":12683},[459],[224,12685],{"className":12686,"style":839},[463],[224,12688,371],{"className":12689},[468,469],[224,12691],{"className":12692,"style":496},[473],[224,12694,278],{"className":12695},[500],[224,12697],{"className":12698,"style":496},[473],[224,12700,12702,12705],{"className":12701},[459],[224,12703],{"className":12704,"style":510},[463],[224,12706,284],{"className":12707},[468]," 时 ",[224,12710,12712,12731],{"className":12711,"translate":388},[392],[224,12713,12715],{"className":12714},[396],[398,12716,12717],{"xmlns":400},[403,12718,12719,12728],{},[406,12720,12721,12723,12725],{},[409,12722,375],{},[412,12724,278],{},[419,12726,12727],{},"2.5",[447,12729,12730],{"encoding":449},"y = 2.5",[224,12732,12734,12752],{"className":12733,"ariaHidden":455},[454],[224,12735,12737,12740,12743,12746,12749],{"className":12736},[459],[224,12738],{"className":12739,"style":806},[463],[224,12741,375],{"className":12742,"style":492},[468,469],[224,12744],{"className":12745,"style":496},[473],[224,12747,278],{"className":12748},[500],[224,12750],{"className":12751,"style":496},[473],[224,12753,12755,12758],{"className":12754},[459],[224,12756],{"className":12757,"style":510},[463],[224,12759,12727],{"className":12760},[468]," 仍不是整数，",[224,12763,12765,12783],{"className":12764,"translate":388},[392],[224,12766,12768],{"className":12767},[396],[398,12769,12770],{"xmlns":400},[403,12771,12772,12780],{},[406,12773,12774,12776,12778],{},[409,12775,371],{},[412,12777,278],{},[419,12779,1438],{},[447,12781,12782],{"encoding":449},"x = 1",[224,12784,12786,12804],{"className":12785,"ariaHidden":455},[454],[224,12787,12789,12792,12795,12798,12801],{"className":12788},[459],[224,12790],{"className":12791,"style":839},[463],[224,12793,371],{"className":12794},[468,469],[224,12796],{"className":12797,"style":496},[473],[224,12799,278],{"className":12800},[500],[224,12802],{"className":12803,"style":496},[473],[224,12805,12807,12810],{"className":12806},[459],[224,12808],{"className":12809,"style":510},[463],[224,12811,1438],{"className":12812},[468],[224,12814,12816,12834],{"className":12815,"translate":388},[392],[224,12817,12819],{"className":12818},[396],[398,12820,12821],{"xmlns":400},[403,12822,12823,12831],{},[406,12824,12825,12827,12829],{},[409,12826,375],{},[412,12828,278],{},[419,12830,1438],{},[447,12832,12833],{"encoding":449},"y = 1",[224,12835,12837,12855],{"className":12836,"ariaHidden":455},[454],[224,12838,12840,12843,12846,12849,12852],{"className":12839},[459],[224,12841],{"className":12842,"style":806},[463],[224,12844,375],{"className":12845,"style":492},[468,469],[224,12847],{"className":12848,"style":496},[473],[224,12850,278],{"className":12851},[500],[224,12853],{"className":12854,"style":496},[473],[224,12856,12858,12861],{"className":12857},[459],[224,12859],{"className":12860,"style":510},[463],[224,12862,1438],{"className":12863},[468],"，得到整数解，sat；",[2677,12866,12867,12868,12919],{},"另一支 ",[224,12869,12871,12889],{"className":12870,"translate":388},[392],[224,12872,12874],{"className":12873},[396],[398,12875,12876],{"xmlns":400},[403,12877,12878,12886],{},[406,12879,12880,12882,12884],{},[409,12881,371],{},[412,12883,9766],{},[419,12885,1447],{},[447,12887,12888],{"encoding":449},"x \\ge 2",[224,12890,12892,12910],{"className":12891,"ariaHidden":455},[454],[224,12893,12895,12898,12901,12904,12907],{"className":12894},[459],[224,12896],{"className":12897,"style":3204},[463],[224,12899,371],{"className":12900},[468,469],[224,12902],{"className":12903,"style":496},[473],[224,12905,9766],{"className":12906},[500],[224,12908],{"className":12909,"style":496},[473],[224,12911,12913,12916],{"className":12912},[459],[224,12914],{"className":12915,"style":510},[463],[224,12917,1447],{"className":12918},[468]," 就无须再看了。",[11,12921,12922],{},"如果两支都不可行，就判 unsat。实际求解器还会加上割平面（cut）和更强的预处理，避免分支树爆炸。",[11446,12924,7885],{"id":7885},[11,12926,12927,12929],{},[221,12928,3233],{}," 最常用的方法还是位爆破：把位向量运算展开成门电路，再交给 SAT 层。但因为位向量有固定宽度，这种展开是有限且可预测的。",[11,12931,12932],{},"现代求解器会先做字级重写，把明显恒真的部分消掉，比如：",[224,12934,12936],{"className":12935,"translate":388},[387],[224,12937,12939,12994],{"className":12938,"translate":388},[392],[224,12940,12942],{"className":12941},[396],[398,12943,12944],{"xmlns":400,"display":401},[403,12945,12946,12991],{},[406,12947,12948,12950,12952,12954,12956,12958,12960,12962,12964,12966,12968,12970,12972,12974,12976,12978,12980,12982,12985,12987,12989],{},[409,12949,371],{},[423,12951,682],{},[409,12953,7909],{"mathvariant":678},[423,12955,682],{},[419,12957,284],{},[412,12959,278],{},[419,12961,284],{},[412,12963,1877],{"separator":455},[473,12965],{"width":4247},[409,12967,371],{},[412,12969,5268],{},[409,12971,371],{},[412,12973,278],{},[419,12975,284],{},[412,12977,1877],{"separator":455},[473,12979],{"width":4247},[409,12981,371],{},[412,12983,12984],{},"≪",[419,12986,284],{},[412,12988,278],{},[409,12990,371],{},[447,12992,12993],{"encoding":449},"x \\,\\&\\, 0 = 0, \\qquad x - x = 0, \\qquad x \\ll 0 = x",[224,12995,12997,13027,13057,13075,13105,13123],{"className":12996,"ariaHidden":455},[454],[224,12998,13000,13003,13006,13009,13012,13015,13018,13021,13024],{"className":12999},[459],[224,13001],{"className":13002,"style":7642},[463],[224,13004,371],{"className":13005},[468,469],[224,13007],{"className":13008,"style":724},[473],[224,13010,7909],{"className":13011},[468],[224,13013],{"className":13014,"style":724},[473],[224,13016,284],{"className":13017},[468],[224,13019],{"className":13020,"style":496},[473],[224,13022,278],{"className":13023},[500],[224,13025],{"className":13026,"style":496},[473],[224,13028,13030,13033,13036,13039,13042,13045,13048,13051,13054],{"className":13029},[459],[224,13031],{"className":13032,"style":8688},[463],[224,13034,284],{"className":13035},[468],[224,13037,1877],{"className":13038},[1917],[224,13040],{"className":13041,"style":4416},[473],[224,13043],{"className":13044,"style":724},[473],[224,13046,371],{"className":13047},[468,469],[224,13049],{"className":13050,"style":474},[473],[224,13052,5268],{"className":13053},[478],[224,13055],{"className":13056,"style":474},[473],[224,13058,13060,13063,13066,13069,13072],{"className":13059},[459],[224,13061],{"className":13062,"style":839},[463],[224,13064,371],{"className":13065},[468,469],[224,13067],{"className":13068,"style":496},[473],[224,13070,278],{"className":13071},[500],[224,13073],{"className":13074,"style":496},[473],[224,13076,13078,13081,13084,13087,13090,13093,13096,13099,13102],{"className":13077},[459],[224,13079],{"className":13080,"style":8688},[463],[224,13082,284],{"className":13083},[468],[224,13085,1877],{"className":13086},[1917],[224,13088],{"className":13089,"style":4416},[473],[224,13091],{"className":13092,"style":724},[473],[224,13094,371],{"className":13095},[468,469],[224,13097],{"className":13098,"style":496},[473],[224,13100,12984],{"className":13101},[500],[224,13103],{"className":13104,"style":496},[473],[224,13106,13108,13111,13114,13117,13120],{"className":13107},[459],[224,13109],{"className":13110,"style":510},[463],[224,13112,284],{"className":13113},[468],[224,13115],{"className":13116,"style":496},[473],[224,13118,278],{"className":13119},[500],[224,13121],{"className":13122,"style":496},[473],[224,13124,13126,13129],{"className":13125},[459],[224,13127],{"className":13128,"style":839},[463],[224,13130,371],{"className":13131},[468,469],[11,13133,13134],{},"然后再决定哪些子表达式值得展开。乘法是位向量的难点，因为乘法器电路规模大，求解器通常会配合专门的乘法推理和等价类传播。",[11,13136,13137,13138,13141],{},"位向量的语义是",[378,13139,13140],{},"模运算","，这一点非常重要：",[224,13143,13145],{"className":13144,"translate":388},[387],[224,13146,13148,13212],{"className":13147,"translate":388},[392],[224,13149,13151],{"className":13150},[396],[398,13152,13153],{"xmlns":400,"display":401},[403,13154,13155,13209],{},[406,13156,13157,13159,13161,13163,13165,13167,13169,13171,13173,13183,13186,13193,13195,13197,13199,13201,13203,13205],{},[409,13158,371],{},[412,13160,319],{},[419,13162,1438],{},[412,13164,10268],{},[419,13166,284],{},[473,13168],{},[473,13170],{"width":691},[412,13172,239],{"stretchy":435},[406,13174,13175,13178,13180],{},[409,13176,13177],{"mathvariant":678},"m",[409,13179,4127],{"mathvariant":678},[409,13181,13182],{"mathvariant":678},"d",[473,13184],{"width":13185},"0.3333em",[3248,13187,13188,13190],{},[419,13189,1447],{},[419,13191,13192],{},"32",[412,13194,1716],{"stretchy":435},[423,13196,425],{},[412,13198,428],{},[423,13200,425],{},[409,13202,371],{},[412,13204,2048],{"mathvariant":678},[423,13206,13208],{"mathvariant":13207},"monospace","0xFFFFFFFF",[447,13210,13211],{"encoding":449},"x + 1 \\equiv 0 \\pmod{2^{32}} \\;\\wedge\\; x \\ne \\texttt{0xFFFFFFFF}",[224,13213,13215,13233,13251,13267,13341,13392],{"className":13214,"ariaHidden":455},[454],[224,13216,13218,13221,13224,13227,13230],{"className":13217},[459],[224,13219],{"className":13220,"style":464},[463],[224,13222,371],{"className":13223},[468,469],[224,13225],{"className":13226,"style":474},[473],[224,13228,319],{"className":13229},[478],[224,13231],{"className":13232,"style":474},[473],[224,13234,13236,13239,13242,13245,13248],{"className":13235},[459],[224,13237],{"className":13238,"style":510},[463],[224,13240,1438],{"className":13241},[468],[224,13243],{"className":13244,"style":496},[473],[224,13246,10268],{"className":13247},[500],[224,13249],{"className":13250,"style":496},[473],[224,13252,13254,13257,13260,13264],{"className":13253},[459],[224,13255],{"className":13256,"style":510},[463],[224,13258,284],{"className":13259},[468],[224,13261],{"className":13262},[473,13263],"allowbreak",[224,13265],{"className":13266,"style":734},[473],[224,13268,13270,13274,13277,13287,13291,13323,13326,13329,13332,13335,13338],{"className":13269},[459],[224,13271],{"className":13272,"style":13273},[463],"height:1.1141em;vertical-align:-0.25em;",[224,13275,239],{"className":13276},[542],[224,13278,13280],{"className":13279},[468],[224,13281,13283],{"className":13282},[468],[224,13284,13286],{"className":13285},[468,1906],"mod",[224,13288],{"className":13289,"style":13290},[473],"margin-right:0.3333em;",[224,13292,13294,13297],{"className":13293},[468],[224,13295,1447],{"className":13296},[468],[224,13298,13300],{"className":13299},[750],[224,13301,13303],{"className":13302},[754],[224,13304,13306],{"className":13305},[759],[224,13307,13309],{"className":13308,"style":5071},[763],[224,13310,13311,13314],{"style":5074},[224,13312],{"className":13313,"style":772},[771],[224,13315,13317],{"className":13316},[776,777,778,779],[224,13318,13320],{"className":13319},[468,779],[224,13321,13192],{"className":13322},[468,779],[224,13324,1716],{"className":13325},[549],[224,13327],{"className":13328,"style":496},[473],[224,13330],{"className":13331,"style":474},[473],[224,13333,428],{"className":13334},[478],[224,13336],{"className":13337,"style":496},[473],[224,13339],{"className":13340,"style":474},[473],[224,13342,13344,13347,13350,13353,13389],{"className":13343},[459],[224,13345],{"className":13346,"style":2565},[463],[224,13348,371],{"className":13349},[468,469],[224,13351],{"className":13352,"style":496},[473],[224,13354,13356,13383,13386],{"className":13355},[500],[224,13357,13359],{"className":13358},[500],[224,13360,13362],{"className":13361},[468,2553],[224,13363,13365],{"className":13364},[2557],[224,13366,13368,13371,13380],{"className":13367},[2561],[224,13369],{"className":13370,"style":2565},[463],[224,13372,13374],{"className":13373},[2569],[224,13375,13377],{"className":13376},[468],[224,13378,2576],{"className":13379},[500],[224,13381],{"className":13382},[2580],[224,13384],{"className":13385},[473,2584],[224,13387,278],{"className":13388},[500],[224,13390],{"className":13391,"style":496},[473],[224,13393,13395,13399],{"className":13394},[459],[224,13396],{"className":13397,"style":13398},[463],"height:0.6111em;",[224,13400,13402],{"className":13401},[468,11373],[224,13403,13208],{"className":13404},[468,13405],"texttt",[11,13407,13408,13409,13482,13483,13485,13486,13542,13543,13545,13546],{},"在 32 位无符号整数里，唯一满足 ",[224,13410,13412,13434],{"className":13411,"translate":388},[392],[224,13413,13415],{"className":13414},[396],[398,13416,13417],{"xmlns":400},[403,13418,13419,13431],{},[406,13420,13421,13423,13425,13427,13429],{},[409,13422,371],{},[412,13424,319],{},[419,13426,1438],{},[412,13428,10268],{},[419,13430,284],{},[447,13432,13433],{"encoding":449},"x + 1 \\equiv 0",[224,13435,13437,13455,13473],{"className":13436,"ariaHidden":455},[454],[224,13438,13440,13443,13446,13449,13452],{"className":13439},[459],[224,13441],{"className":13442,"style":464},[463],[224,13444,371],{"className":13445},[468,469],[224,13447],{"className":13448,"style":474},[473],[224,13450,319],{"className":13451},[478],[224,13453],{"className":13454,"style":474},[473],[224,13456,13458,13461,13464,13467,13470],{"className":13457},[459],[224,13459],{"className":13460,"style":510},[463],[224,13462,1438],{"className":13463},[468],[224,13465],{"className":13466,"style":496},[473],[224,13468,10268],{"className":13469},[500],[224,13471],{"className":13472,"style":496},[473],[224,13474,13476,13479],{"className":13475},[459],[224,13477],{"className":13478,"style":510},[463],[224,13480,284],{"className":13481},[468]," 的值就是 ",[221,13484,13208],{},"，所以这个公式是 unsat。如果把它当成数学整数，",[224,13487,13489,13509],{"className":13488,"translate":388},[392],[224,13490,13492],{"className":13491},[396],[398,13493,13494],{"xmlns":400},[403,13495,13496,13506],{},[406,13497,13498,13500,13502,13504],{},[409,13499,371],{},[412,13501,278],{},[412,13503,5268],{},[419,13505,1438],{},[447,13507,13508],{"encoding":449},"x = -1",[224,13510,13512,13530],{"className":13511,"ariaHidden":455},[454],[224,13513,13515,13518,13521,13524,13527],{"className":13514},[459],[224,13516],{"className":13517,"style":839},[463],[224,13519,371],{"className":13520},[468,469],[224,13522],{"className":13523,"style":496},[473],[224,13525,278],{"className":13526},[500],[224,13528],{"className":13529,"style":496},[473],[224,13531,13533,13536,13539],{"className":13532},[459],[224,13534],{"className":13535,"style":7047},[463],[224,13537,5268],{"className":13538},[468],[224,13540,1438],{"className":13541},[468]," 又不等于 ",[221,13544,13208],{},"，结论就完全不同。",[378,13547,13548],{},"选错理论，漏洞检测就会漏报或误报。",[11446,13550,7879],{"id":7879},[11,13552,13553,13554,2887,13556,13558],{},"数组理论的核心是 ",[221,13555,1907],{},[221,13557,2890],{}," 两条操作，以及前面提到的 read-over-write 公理。",[11,13560,13561,13562,13565,13566,13569],{},"求解器不会一次性展开所有下标，而是采用",[378,13563,13564],{},"惰性公理实例化","（lazy axiom instantiation）：先假设数组是一致的，当检查中发现某个 ",[221,13567,13568],{},"select(store(...))"," 需要用到公理时，才生成对应的引理加入公式。这叫「按需生成」，也叫 lemma-on-demand。",[11,13571,11596],{},[224,13573,13575],{"className":13574,"translate":388},[387],[224,13576,13578,13667],{"className":13577,"translate":388},[392],[224,13579,13581],{"className":13580},[396],[398,13582,13583],{"xmlns":400,"display":401},[403,13584,13585,13664],{},[406,13586,13587,13601,13603,13615,13617,13619,13621,13623,13625,13627,13629,13631,13633,13635,13637,13640,13642,13644,13646,13648,13650,13652,13654,13656,13658,13660,13662],{},[406,13588,13589,13591,13593,13595,13597,13599],{},[409,13590,1858],{"mathvariant":678},[409,13592,1861],{"mathvariant":678},[409,13594,1864],{"mathvariant":678},[409,13596,1861],{"mathvariant":678},[409,13598,219],{"mathvariant":678},[409,13600,1435],{"mathvariant":678},[412,13602,239],{"fence":455,"stretchy":455,"minsize":2004,"maxsize":2004},[406,13604,13605,13607,13609,13611,13613],{},[409,13606,1858],{"mathvariant":678},[409,13608,1435],{"mathvariant":678},[409,13610,4127],{"mathvariant":678},[409,13612,2061],{"mathvariant":678},[409,13614,1861],{"mathvariant":678},[412,13616,239],{"stretchy":435},[409,13618,15],{},[412,13620,1877],{"separator":455},[409,13622,1880],{},[412,13624,1877],{"separator":455},[409,13626,1887],{},[412,13628,1716],{"stretchy":435},[412,13630,1877],{"separator":455},[409,13632,8101],{},[412,13634,1716],{"fence":455,"stretchy":455,"minsize":2004,"maxsize":2004},[412,13636,278],{},[409,13638,13639],{},"w",[423,13641,425],{},[412,13643,428],{},[423,13645,425],{},[409,13647,1880],{},[412,13649,278],{},[409,13651,8101],{},[423,13653,425],{},[412,13655,428],{},[423,13657,425],{},[409,13659,13639],{},[412,13661,2048],{"mathvariant":678},[409,13663,1887],{},[447,13665,13666],{"encoding":449},"\\mathrm{select}\\bigl(\\mathrm{store}(a, i, v), j\\bigr) = w \\;\\wedge\\; i = j \\;\\wedge\\; w \\ne v",[224,13668,13670,13745,13770,13788,13813,13864],{"className":13669,"ariaHidden":455},[454],[224,13671,13673,13676,13682,13688,13694,13697,13700,13703,13706,13709,13712,13715,13718,13721,13724,13727,13730,13736,13739,13742],{"className":13672},[459],[224,13674],{"className":13675,"style":2663},[463],[224,13677,13679],{"className":13678},[468],[224,13680,1907],{"className":13681},[468,1906],[224,13683,13685],{"className":13684},[542],[224,13686,239],{"className":13687},[2363,2364],[224,13689,13691],{"className":13690},[468],[224,13692,2890],{"className":13693},[468,1906],[224,13695,239],{"className":13696},[542],[224,13698,15],{"className":13699},[468,469],[224,13701,1877],{"className":13702},[1917],[224,13704],{"className":13705,"style":724},[473],[224,13707,1880],{"className":13708},[468,469],[224,13710,1877],{"className":13711},[1917],[224,13713],{"className":13714,"style":724},[473],[224,13716,1887],{"className":13717,"style":492},[468,469],[224,13719,1716],{"className":13720},[549],[224,13722,1877],{"className":13723},[1917],[224,13725],{"className":13726,"style":724},[473],[224,13728,8101],{"className":13729,"style":8251},[468,469],[224,13731,13733],{"className":13732},[549],[224,13734,1716],{"className":13735},[2363,2364],[224,13737],{"className":13738,"style":496},[473],[224,13740,278],{"className":13741},[500],[224,13743],{"className":13744,"style":496},[473],[224,13746,13748,13751,13755,13758,13761,13764,13767],{"className":13747},[459],[224,13749],{"className":13750,"style":11726},[463],[224,13752,13639],{"className":13753,"style":13754},[468,469],"margin-right:0.0269em;",[224,13756],{"className":13757,"style":496},[473],[224,13759],{"className":13760,"style":474},[473],[224,13762,428],{"className":13763},[478],[224,13765],{"className":13766,"style":496},[473],[224,13768],{"className":13769,"style":474},[473],[224,13771,13773,13776,13779,13782,13785],{"className":13772},[459],[224,13774],{"className":13775,"style":7542},[463],[224,13777,1880],{"className":13778},[468,469],[224,13780],{"className":13781,"style":496},[473],[224,13783,278],{"className":13784},[500],[224,13786],{"className":13787,"style":496},[473],[224,13789,13791,13795,13798,13801,13804,13807,13810],{"className":13790},[459],[224,13792],{"className":13793,"style":13794},[463],"height:0.854em;vertical-align:-0.1944em;",[224,13796,8101],{"className":13797,"style":8251},[468,469],[224,13799],{"className":13800,"style":496},[473],[224,13802],{"className":13803,"style":474},[473],[224,13805,428],{"className":13806},[478],[224,13808],{"className":13809,"style":496},[473],[224,13811],{"className":13812,"style":474},[473],[224,13814,13816,13819,13822,13825,13861],{"className":13815},[459],[224,13817],{"className":13818,"style":2565},[463],[224,13820,13639],{"className":13821,"style":13754},[468,469],[224,13823],{"className":13824,"style":496},[473],[224,13826,13828,13855,13858],{"className":13827},[500],[224,13829,13831],{"className":13830},[500],[224,13832,13834],{"className":13833},[468,2553],[224,13835,13837],{"className":13836},[2557],[224,13838,13840,13843,13852],{"className":13839},[2561],[224,13841],{"className":13842,"style":2565},[463],[224,13844,13846],{"className":13845},[2569],[224,13847,13849],{"className":13848},[468],[224,13850,2576],{"className":13851},[500],[224,13853],{"className":13854},[2580],[224,13856],{"className":13857},[473,2584],[224,13859,278],{"className":13860},[500],[224,13862],{"className":13863,"style":496},[473],[224,13865,13867,13870],{"className":13866},[459],[224,13868],{"className":13869,"style":839},[463],[224,13871,1887],{"className":13872,"style":492},[468,469],[11,13874,13875,13876,13927,13928,13956,13957,6533,14008,14092],{},"由 ",[224,13877,13879,13897],{"className":13878,"translate":388},[392],[224,13880,13882],{"className":13881},[396],[398,13883,13884],{"xmlns":400},[403,13885,13886,13894],{},[406,13887,13888,13890,13892],{},[409,13889,1880],{},[412,13891,278],{},[409,13893,8101],{},[447,13895,13896],{"encoding":449},"i = j",[224,13898,13900,13918],{"className":13899,"ariaHidden":455},[454],[224,13901,13903,13906,13909,13912,13915],{"className":13902},[459],[224,13904],{"className":13905,"style":7542},[463],[224,13907,1880],{"className":13908},[468,469],[224,13910],{"className":13911,"style":496},[473],[224,13913,278],{"className":13914},[500],[224,13916],{"className":13917,"style":496},[473],[224,13919,13921,13924],{"className":13920},[459],[224,13922],{"className":13923,"style":13794},[463],[224,13925,8101],{"className":13926,"style":8251},[468,469]," 和 read-over-write 公理，左边等于 ",[224,13929,13931,13944],{"className":13930,"translate":388},[392],[224,13932,13934],{"className":13933},[396],[398,13935,13936],{"xmlns":400},[403,13937,13938,13942],{},[406,13939,13940],{},[409,13941,1887],{},[447,13943,1887],{"encoding":449},[224,13945,13947],{"className":13946,"ariaHidden":455},[454],[224,13948,13950,13953],{"className":13949},[459],[224,13951],{"className":13952,"style":839},[463],[224,13954,1887],{"className":13955,"style":492},[468,469],"；于是 ",[224,13958,13960,13978],{"className":13959,"translate":388},[392],[224,13961,13963],{"className":13962},[396],[398,13964,13965],{"xmlns":400},[403,13966,13967,13975],{},[406,13968,13969,13971,13973],{},[409,13970,1887],{},[412,13972,278],{},[409,13974,13639],{},[447,13976,13977],{"encoding":449},"v = w",[224,13979,13981,13999],{"className":13980,"ariaHidden":455},[454],[224,13982,13984,13987,13990,13993,13996],{"className":13983},[459],[224,13985],{"className":13986,"style":839},[463],[224,13988,1887],{"className":13989,"style":492},[468,469],[224,13991],{"className":13992,"style":496},[473],[224,13994,278],{"className":13995},[500],[224,13997],{"className":13998,"style":496},[473],[224,14000,14002,14005],{"className":14001},[459],[224,14003],{"className":14004,"style":839},[463],[224,14006,13639],{"className":14007,"style":13754},[468,469],[224,14009,14011,14029],{"className":14010,"translate":388},[392],[224,14012,14014],{"className":14013},[396],[398,14015,14016],{"xmlns":400},[403,14017,14018,14026],{},[406,14019,14020,14022,14024],{},[409,14021,13639],{},[412,14023,2048],{"mathvariant":678},[409,14025,1887],{},[447,14027,14028],{"encoding":449},"w \\ne v",[224,14030,14032,14083],{"className":14031,"ariaHidden":455},[454],[224,14033,14035,14038,14041,14044,14080],{"className":14034},[459],[224,14036],{"className":14037,"style":2565},[463],[224,14039,13639],{"className":14040,"style":13754},[468,469],[224,14042],{"className":14043,"style":496},[473],[224,14045,14047,14074,14077],{"className":14046},[500],[224,14048,14050],{"className":14049},[500],[224,14051,14053],{"className":14052},[468,2553],[224,14054,14056],{"className":14055},[2557],[224,14057,14059,14062,14071],{"className":14058},[2561],[224,14060],{"className":14061,"style":2565},[463],[224,14063,14065],{"className":14064},[2569],[224,14066,14068],{"className":14067},[468],[224,14069,2576],{"className":14070},[500],[224,14072],{"className":14073},[2580],[224,14075],{"className":14076},[473,2584],[224,14078,278],{"className":14079},[500],[224,14081],{"className":14082,"style":496},[473],[224,14084,14086,14089],{"className":14085},[459],[224,14087],{"className":14088,"style":839},[463],[224,14090,1887],{"className":14091,"style":492},[468,469]," 冲突，unsat。整个过程中，数组的长度、其他下标的值，全都不需要知道。",[11446,14094,14095],{"id":14095},"其他理论",[11,14097,14098,14099,14101],{},"字符串理论通常用自动机或序列理论处理，word equation 本身可判定，但加上长度约束、正则约束后很多片段会变得不可判定，工业求解器往往采用不完整但实用的策略。浮点理论通常把 IEEE 754 的位级语义编码成位向量，再交给 ",[221,14100,3233],{}," 后端。",[613,14103,14105],{"id":14104},"_45-冲突学习","4.5 冲突学习",[11,14107,14108,14109,14112],{},"理论冲突不会只让当前搜索失败一次，它会变成一条",[378,14110,14111],{},"理论引理","（theory lemma），永久加入布尔公式。",[11,14114,14115],{},"前面 4.2 节的例子：",[224,14117,14119],{"className":14118,"translate":388},[387],[224,14120,14122,14163],{"className":14121,"translate":388},[392],[224,14123,14125],{"className":14124},[396],[398,14126,14127],{"xmlns":400,"display":401},[403,14128,14129,14161],{},[406,14130,14131,14133,14135,14137,14139,14141,14143,14145,14147,14149,14151,14153,14155,14157,14159],{},[409,14132,371],{},[412,14134,325],{},[419,14136,284],{},[412,14138,1877],{"separator":455},[473,14140],{"width":4247},[409,14142,375],{},[412,14144,325],{},[419,14146,284],{},[412,14148,1877],{"separator":455},[473,14150],{"width":4247},[409,14152,371],{},[412,14154,319],{},[409,14156,375],{},[412,14158,1577],{},[419,14160,284],{},[447,14162,10863],{"encoding":449},[224,14164,14166,14184,14214,14244,14262],{"className":14165,"ariaHidden":455},[454],[224,14167,14169,14172,14175,14178,14181],{"className":14168},[459],[224,14170],{"className":14171,"style":2810},[463],[224,14173,371],{"className":14174},[468,469],[224,14176],{"className":14177,"style":496},[473],[224,14179,325],{"className":14180},[500],[224,14182],{"className":14183,"style":496},[473],[224,14185,14187,14190,14193,14196,14199,14202,14205,14208,14211],{"className":14186},[459],[224,14188],{"className":14189,"style":8688},[463],[224,14191,284],{"className":14192},[468],[224,14194,1877],{"className":14195},[1917],[224,14197],{"className":14198,"style":4416},[473],[224,14200],{"className":14201,"style":724},[473],[224,14203,375],{"className":14204,"style":492},[468,469],[224,14206],{"className":14207,"style":496},[473],[224,14209,325],{"className":14210},[500],[224,14212],{"className":14213,"style":496},[473],[224,14215,14217,14220,14223,14226,14229,14232,14235,14238,14241],{"className":14216},[459],[224,14218],{"className":14219,"style":8688},[463],[224,14221,284],{"className":14222},[468],[224,14224,1877],{"className":14225},[1917],[224,14227],{"className":14228,"style":4416},[473],[224,14230],{"className":14231,"style":724},[473],[224,14233,371],{"className":14234},[468,469],[224,14236],{"className":14237,"style":474},[473],[224,14239,319],{"className":14240},[478],[224,14242],{"className":14243,"style":474},[473],[224,14245,14247,14250,14253,14256,14259],{"className":14246},[459],[224,14248],{"className":14249,"style":488},[463],[224,14251,375],{"className":14252,"style":492},[468,469],[224,14254],{"className":14255,"style":496},[473],[224,14257,1577],{"className":14258},[500],[224,14260],{"className":14261,"style":496},[473],[224,14263,14265,14268],{"className":14264},[459],[224,14266],{"className":14267,"style":510},[463],[224,14269,284],{"className":14270},[468],[11,14272,14273],{},"这三条文字不可同时为真，所以下面这条子句是理论有效的：",[224,14275,14277],{"className":14276,"translate":388},[387],[224,14278,14280,14344],{"className":14279,"translate":388},[392],[224,14281,14283],{"className":14282},[396],[398,14284,14285],{"xmlns":400,"display":401},[403,14286,14287,14341],{},[406,14288,14289,14291,14293,14295,14297,14299,14301,14303,14305,14307,14309,14311,14313,14315,14317,14319,14321,14323,14325,14327,14329,14331,14333,14335,14337,14339],{},[409,14290,1106],{"mathvariant":678},[412,14292,239],{"stretchy":435},[409,14294,371],{},[412,14296,325],{},[419,14298,284],{},[412,14300,1716],{"stretchy":435},[423,14302,425],{},[412,14304,1131],{},[423,14306,425],{},[409,14308,1106],{"mathvariant":678},[412,14310,239],{"stretchy":435},[409,14312,375],{},[412,14314,325],{},[419,14316,284],{},[412,14318,1716],{"stretchy":435},[423,14320,425],{},[412,14322,1131],{},[423,14324,425],{},[409,14326,1106],{"mathvariant":678},[412,14328,239],{"stretchy":435},[409,14330,371],{},[412,14332,319],{},[409,14334,375],{},[412,14336,1577],{},[419,14338,284],{},[412,14340,1716],{"stretchy":435},[447,14342,14343],{"encoding":449},"\\neg(x > 0) \\;\\vee\\; \\neg(y > 0) \\;\\vee\\; \\neg(x + y \u003C 0)",[224,14345,14347,14371,14398,14422,14449,14473,14491],{"className":14346,"ariaHidden":455},[454],[224,14348,14350,14353,14356,14359,14362,14365,14368],{"className":14349},[459],[224,14351],{"className":14352,"style":535},[463],[224,14354,1106],{"className":14355},[468],[224,14357,239],{"className":14358},[542],[224,14360,371],{"className":14361},[468,469],[224,14363],{"className":14364,"style":496},[473],[224,14366,325],{"className":14367},[500],[224,14369],{"className":14370,"style":496},[473],[224,14372,14374,14377,14380,14383,14386,14389,14392,14395],{"className":14373},[459],[224,14375],{"className":14376,"style":535},[463],[224,14378,284],{"className":14379},[468],[224,14381,1716],{"className":14382},[549],[224,14384],{"className":14385,"style":496},[473],[224,14387],{"className":14388,"style":474},[473],[224,14390,1131],{"className":14391},[478],[224,14393],{"className":14394,"style":496},[473],[224,14396],{"className":14397,"style":474},[473],[224,14399,14401,14404,14407,14410,14413,14416,14419],{"className":14400},[459],[224,14402],{"className":14403,"style":535},[463],[224,14405,1106],{"className":14406},[468],[224,14408,239],{"className":14409},[542],[224,14411,375],{"className":14412,"style":492},[468,469],[224,14414],{"className":14415,"style":496},[473],[224,14417,325],{"className":14418},[500],[224,14420],{"className":14421,"style":496},[473],[224,14423,14425,14428,14431,14434,14437,14440,14443,14446],{"className":14424},[459],[224,14426],{"className":14427,"style":535},[463],[224,14429,284],{"className":14430},[468],[224,14432,1716],{"className":14433},[549],[224,14435],{"className":14436,"style":496},[473],[224,14438],{"className":14439,"style":474},[473],[224,14441,1131],{"className":14442},[478],[224,14444],{"className":14445,"style":496},[473],[224,14447],{"className":14448,"style":474},[473],[224,14450,14452,14455,14458,14461,14464,14467,14470],{"className":14451},[459],[224,14453],{"className":14454,"style":535},[463],[224,14456,1106],{"className":14457},[468],[224,14459,239],{"className":14460},[542],[224,14462,371],{"className":14463},[468,469],[224,14465],{"className":14466,"style":474},[473],[224,14468,319],{"className":14469},[478],[224,14471],{"className":14472,"style":474},[473],[224,14474,14476,14479,14482,14485,14488],{"className":14475},[459],[224,14477],{"className":14478,"style":488},[463],[224,14480,375],{"className":14481,"style":492},[468,469],[224,14483],{"className":14484,"style":496},[473],[224,14486,1577],{"className":14487},[500],[224,14489],{"className":14490,"style":496},[473],[224,14492,14494,14497,14500],{"className":14493},[459],[224,14495],{"className":14496,"style":535},[463],[224,14498,284],{"className":14499},[468],[224,14501,1716],{"className":14502},[549],[11,14504,14505],{},"把它翻译回命题变量，就是：",[224,14507,14509],{"className":14508,"translate":388},[387],[224,14510,14512,14548],{"className":14511,"translate":388},[392],[224,14513,14515],{"className":14514},[396],[398,14516,14517],{"xmlns":400,"display":401},[403,14518,14519,14545],{},[406,14520,14521,14523,14525,14527,14529,14531,14533,14535,14537,14539,14541,14543],{},[409,14522,1106],{"mathvariant":678},[409,14524,11],{},[423,14526,425],{},[412,14528,1131],{},[423,14530,425],{},[409,14532,1106],{"mathvariant":678},[409,14534,2023],{},[423,14536,425],{},[412,14538,1131],{},[423,14540,425],{},[409,14542,1106],{"mathvariant":678},[409,14544,2061],{},[447,14546,14547],{"encoding":449},"\\neg p \\;\\vee\\; \\neg q \\;\\vee\\; \\neg r",[224,14549,14551,14578,14605],{"className":14550,"ariaHidden":455},[454],[224,14552,14554,14557,14560,14563,14566,14569,14572,14575],{"className":14553},[459],[224,14555],{"className":14556,"style":1277},[463],[224,14558,1106],{"className":14559},[468],[224,14561,11],{"className":14562},[468,469],[224,14564],{"className":14565,"style":496},[473],[224,14567],{"className":14568,"style":474},[473],[224,14570,1131],{"className":14571},[478],[224,14573],{"className":14574,"style":496},[473],[224,14576],{"className":14577,"style":474},[473],[224,14579,14581,14584,14587,14590,14593,14596,14599,14602],{"className":14580},[459],[224,14582],{"className":14583,"style":1277},[463],[224,14585,1106],{"className":14586},[468],[224,14588,2023],{"className":14589,"style":492},[468,469],[224,14591],{"className":14592,"style":496},[473],[224,14594],{"className":14595,"style":474},[473],[224,14597,1131],{"className":14598},[478],[224,14600],{"className":14601,"style":496},[473],[224,14603],{"className":14604,"style":474},[473],[224,14606,14608,14611,14614],{"className":14607},[459],[224,14609],{"className":14610,"style":839},[463],[224,14612,1106],{"className":14613},[468],[224,14615,2061],{"className":14616,"style":2473},[468,469],[11,14618,14619,14620,14674,14675,3117],{},"SAT 层学到这条子句之后，就再也不会同时选 ",[224,14621,14623,14644],{"className":14622,"translate":388},[392],[224,14624,14626],{"className":14625},[396],[398,14627,14628],{"xmlns":400},[403,14629,14630,14642],{},[406,14631,14632,14634,14636,14638,14640],{},[409,14633,11],{},[412,14635,1877],{"separator":455},[409,14637,2023],{},[412,14639,1877],{"separator":455},[409,14641,2061],{},[447,14643,3068],{"encoding":449},[224,14645,14647],{"className":14646,"ariaHidden":455},[454],[224,14648,14650,14653,14656,14659,14662,14665,14668,14671],{"className":14649},[459],[224,14651],{"className":14652,"style":806},[463],[224,14654,11],{"className":14655},[468,469],[224,14657,1877],{"className":14658},[1917],[224,14660],{"className":14661,"style":724},[473],[224,14663,2023],{"className":14664,"style":492},[468,469],[224,14666,1877],{"className":14667},[1917],[224,14669],{"className":14670,"style":724},[473],[224,14672,2061],{"className":14673,"style":2473},[468,469]," 都为真了。这一步是 DPLL(T) 效率的关键：",[378,14676,14677],{},"理论层的「一次性检查」变成了布尔层的「永久知识」",[11,14679,14680],{},"如果没有冲突学习，SAT 层会在同一个理论冲突上反复撞墙；有了它，搜索空间被真正地剪掉了。",[613,14682,14684],{"id":14683},"_46-完整例子","4.6 完整例子",[11,14686,14687],{},"还是用这个公式：",[224,14689,14691],{"className":14690,"translate":388},[387],[224,14692,14694,14747],{"className":14693,"translate":388},[392],[224,14695,14697],{"className":14696},[396],[398,14698,14699],{"xmlns":400,"display":401},[403,14700,14701,14745],{},[406,14702,14703,14705,14707,14709,14711,14713,14715,14717,14719,14721,14723,14725,14727,14729,14731,14733,14735,14737,14739,14741,14743],{},[409,14704,704],{},[412,14706,278],{},[412,14708,239],{"stretchy":435},[409,14710,371],{},[412,14712,325],{},[419,14714,284],{},[412,14716,1131],{},[409,14718,375],{},[412,14720,325],{},[419,14722,284],{},[412,14724,1716],{"stretchy":435},[423,14726,425],{},[412,14728,428],{},[423,14730,425],{},[412,14732,239],{"stretchy":435},[409,14734,371],{},[412,14736,319],{},[409,14738,375],{},[412,14740,1577],{},[419,14742,284],{},[412,14744,1716],{"stretchy":435},[447,14746,10089],{"encoding":449},[224,14748,14750,14768,14789,14807,14825,14852,14873,14891],{"className":14749,"ariaHidden":455},[454],[224,14751,14753,14756,14759,14762,14765],{"className":14752},[459],[224,14754],{"className":14755,"style":806},[463],[224,14757,704],{"className":14758},[468,469],[224,14760],{"className":14761,"style":496},[473],[224,14763,278],{"className":14764},[500],[224,14766],{"className":14767,"style":496},[473],[224,14769,14771,14774,14777,14780,14783,14786],{"className":14770},[459],[224,14772],{"className":14773,"style":535},[463],[224,14775,239],{"className":14776},[542],[224,14778,371],{"className":14779},[468,469],[224,14781],{"className":14782,"style":496},[473],[224,14784,325],{"className":14785},[500],[224,14787],{"className":14788,"style":496},[473],[224,14790,14792,14795,14798,14801,14804],{"className":14791},[459],[224,14793],{"className":14794,"style":510},[463],[224,14796,284],{"className":14797},[468],[224,14799],{"className":14800,"style":474},[473],[224,14802,1131],{"className":14803},[478],[224,14805],{"className":14806,"style":474},[473],[224,14808,14810,14813,14816,14819,14822],{"className":14809},[459],[224,14811],{"className":14812,"style":488},[463],[224,14814,375],{"className":14815,"style":492},[468,469],[224,14817],{"className":14818,"style":496},[473],[224,14820,325],{"className":14821},[500],[224,14823],{"className":14824,"style":496},[473],[224,14826,14828,14831,14834,14837,14840,14843,14846,14849],{"className":14827},[459],[224,14829],{"className":14830,"style":535},[463],[224,14832,284],{"className":14833},[468],[224,14835,1716],{"className":14836},[549],[224,14838],{"className":14839,"style":496},[473],[224,14841],{"className":14842,"style":474},[473],[224,14844,428],{"className":14845},[478],[224,14847],{"className":14848,"style":496},[473],[224,14850],{"className":14851,"style":474},[473],[224,14853,14855,14858,14861,14864,14867,14870],{"className":14854},[459],[224,14856],{"className":14857,"style":535},[463],[224,14859,239],{"className":14860},[542],[224,14862,371],{"className":14863},[468,469],[224,14865],{"className":14866,"style":474},[473],[224,14868,319],{"className":14869},[478],[224,14871],{"className":14872,"style":474},[473],[224,14874,14876,14879,14882,14885,14888],{"className":14875},[459],[224,14877],{"className":14878,"style":488},[463],[224,14880,375],{"className":14881,"style":492},[468,469],[224,14883],{"className":14884,"style":496},[473],[224,14886,1577],{"className":14887},[500],[224,14889],{"className":14890,"style":496},[473],[224,14892,14894,14897,14900],{"className":14893},[459],[224,14895],{"className":14896,"style":535},[463],[224,14898,284],{"className":14899},[468],[224,14901,1716],{"className":14902},[549],[11,14904,14905,14908,14909,15029,15030,15195,15196,15268,15269,15357],{},[378,14906,14907],{},"第 1 轮","：SAT 层选择 ",[224,14910,14912,14945],{"className":14911,"translate":388},[392],[224,14913,14915],{"className":14914},[396],[398,14916,14917],{"xmlns":400},[403,14918,14919,14943],{},[406,14920,14921,14923,14925,14927,14929,14931,14933,14935,14937,14939,14941],{},[409,14922,11],{},[412,14924,278],{},[409,14926,1078],{"mathvariant":678},[412,14928,1877],{"separator":455},[409,14930,2023],{},[412,14932,278],{},[409,14934,1078],{"mathvariant":678},[412,14936,1877],{"separator":455},[409,14938,2061],{},[412,14940,278],{},[409,14942,1078],{"mathvariant":678},[447,14944,10723],{"encoding":449},[224,14946,14948,14966,14993,15020],{"className":14947,"ariaHidden":455},[454],[224,14949,14951,14954,14957,14960,14963],{"className":14950},[459],[224,14952],{"className":14953,"style":806},[463],[224,14955,11],{"className":14956},[468,469],[224,14958],{"className":14959,"style":496},[473],[224,14961,278],{"className":14962},[500],[224,14964],{"className":14965,"style":496},[473],[224,14967,14969,14972,14975,14978,14981,14984,14987,14990],{"className":14968},[459],[224,14970],{"className":14971,"style":2565},[463],[224,14973,1078],{"className":14974},[468],[224,14976,1877],{"className":14977},[1917],[224,14979],{"className":14980,"style":724},[473],[224,14982,2023],{"className":14983,"style":492},[468,469],[224,14985],{"className":14986,"style":496},[473],[224,14988,278],{"className":14989},[500],[224,14991],{"className":14992,"style":496},[473],[224,14994,14996,14999,15002,15005,15008,15011,15014,15017],{"className":14995},[459],[224,14997],{"className":14998,"style":2565},[463],[224,15000,1078],{"className":15001},[468],[224,15003,1877],{"className":15004},[1917],[224,15006],{"className":15007,"style":724},[473],[224,15009,2061],{"className":15010,"style":2473},[468,469],[224,15012],{"className":15013,"style":496},[473],[224,15015,278],{"className":15016},[500],[224,15018],{"className":15019,"style":496},[473],[224,15021,15023,15026],{"className":15022},[459],[224,15024],{"className":15025,"style":7642},[463],[224,15027,1078],{"className":15028},[468],"。理论层检查 ",[224,15031,15033,15081],{"className":15032,"translate":388},[392],[224,15034,15036],{"className":15035},[396],[398,15037,15038],{"xmlns":400},[403,15039,15040,15078],{},[406,15041,15042,15044,15046,15048,15050,15052,15055,15057,15059,15061,15063,15065,15067,15069,15071,15073,15075],{},[412,15043,8110],{"stretchy":435},[409,15045,371],{},[412,15047,325],{},[419,15049,284],{},[412,15051,1877],{"separator":455},[423,15053,15054],{}," ",[409,15056,375],{},[412,15058,325],{},[419,15060,284],{},[412,15062,1877],{"separator":455},[423,15064,15054],{},[409,15066,371],{},[412,15068,319],{},[409,15070,375],{},[412,15072,1577],{},[419,15074,284],{},[412,15076,15077],{"stretchy":435},"}",[447,15079,15080],{"encoding":449},"\\{x > 0,\\ y > 0,\\ x + y \u003C 0\\}",[224,15082,15084,15105,15135,15165,15183],{"className":15083,"ariaHidden":455},[454],[224,15085,15087,15090,15093,15096,15099,15102],{"className":15086},[459],[224,15088],{"className":15089,"style":535},[463],[224,15091,8110],{"className":15092},[542],[224,15094,371],{"className":15095},[468,469],[224,15097],{"className":15098,"style":496},[473],[224,15100,325],{"className":15101},[500],[224,15103],{"className":15104,"style":496},[473],[224,15106,15108,15111,15114,15117,15120,15123,15126,15129,15132],{"className":15107},[459],[224,15109],{"className":15110,"style":8688},[463],[224,15112,284],{"className":15113},[468],[224,15115,1877],{"className":15116},[1917],[224,15118,15054],{"className":15119},[473],[224,15121],{"className":15122,"style":724},[473],[224,15124,375],{"className":15125,"style":492},[468,469],[224,15127],{"className":15128,"style":496},[473],[224,15130,325],{"className":15131},[500],[224,15133],{"className":15134,"style":496},[473],[224,15136,15138,15141,15144,15147,15150,15153,15156,15159,15162],{"className":15137},[459],[224,15139],{"className":15140,"style":8688},[463],[224,15142,284],{"className":15143},[468],[224,15145,1877],{"className":15146},[1917],[224,15148,15054],{"className":15149},[473],[224,15151],{"className":15152,"style":724},[473],[224,15154,371],{"className":15155},[468,469],[224,15157],{"className":15158,"style":474},[473],[224,15160,319],{"className":15161},[478],[224,15163],{"className":15164,"style":474},[473],[224,15166,15168,15171,15174,15177,15180],{"className":15167},[459],[224,15169],{"className":15170,"style":488},[463],[224,15172,375],{"className":15173,"style":492},[468,469],[224,15175],{"className":15176,"style":496},[473],[224,15178,1577],{"className":15179},[500],[224,15181],{"className":15182,"style":496},[473],[224,15184,15186,15189,15192],{"className":15185},[459],[224,15187],{"className":15188,"style":535},[463],[224,15190,284],{"className":15191},[468],[224,15193,15077],{"className":15194},[549],"，发现 ",[224,15197,15199,15220],{"className":15198,"translate":388},[392],[224,15200,15202],{"className":15201},[396],[398,15203,15204],{"xmlns":400},[403,15205,15206,15218],{},[406,15207,15208,15210,15212,15214,15216],{},[409,15209,371],{},[412,15211,319],{},[409,15213,375],{},[412,15215,325],{},[419,15217,284],{},[447,15219,10999],{"encoding":449},[224,15221,15223,15241,15259],{"className":15222,"ariaHidden":455},[454],[224,15224,15226,15229,15232,15235,15238],{"className":15225},[459],[224,15227],{"className":15228,"style":464},[463],[224,15230,371],{"className":15231},[468,469],[224,15233],{"className":15234,"style":474},[473],[224,15236,319],{"className":15237},[478],[224,15239],{"className":15240,"style":474},[473],[224,15242,15244,15247,15250,15253,15256],{"className":15243},[459],[224,15245],{"className":15246,"style":488},[463],[224,15248,375],{"className":15249,"style":492},[468,469],[224,15251],{"className":15252,"style":496},[473],[224,15254,325],{"className":15255},[500],[224,15257],{"className":15258,"style":496},[473],[224,15260,15262,15265],{"className":15261},[459],[224,15263],{"className":15264,"style":510},[463],[224,15266,284],{"className":15267},[468],"，冲突。生成引理 ",[224,15270,15272,15300],{"className":15271,"translate":388},[392],[224,15273,15275],{"className":15274},[396],[398,15276,15277],{"xmlns":400},[403,15278,15279,15297],{},[406,15280,15281,15283,15285,15287,15289,15291,15293,15295],{},[409,15282,1106],{"mathvariant":678},[409,15284,11],{},[412,15286,1131],{},[409,15288,1106],{"mathvariant":678},[409,15290,2023],{},[412,15292,1131],{},[409,15294,1106],{"mathvariant":678},[409,15296,2061],{},[447,15298,15299],{"encoding":449},"\\neg p \\vee \\neg q \\vee \\neg r",[224,15301,15303,15324,15345],{"className":15302,"ariaHidden":455},[454],[224,15304,15306,15309,15312,15315,15318,15321],{"className":15305},[459],[224,15307],{"className":15308,"style":1277},[463],[224,15310,1106],{"className":15311},[468],[224,15313,11],{"className":15314},[468,469],[224,15316],{"className":15317,"style":474},[473],[224,15319,1131],{"className":15320},[478],[224,15322],{"className":15323,"style":474},[473],[224,15325,15327,15330,15333,15336,15339,15342],{"className":15326},[459],[224,15328],{"className":15329,"style":1277},[463],[224,15331,1106],{"className":15332},[468],[224,15334,2023],{"className":15335,"style":492},[468,469],[224,15337],{"className":15338,"style":474},[473],[224,15340,1131],{"className":15341},[478],[224,15343],{"className":15344,"style":474},[473],[224,15346,15348,15351,15354],{"className":15347},[459],[224,15349],{"className":15350,"style":839},[463],[224,15352,1106],{"className":15353},[468],[224,15355,2061],{"className":15356,"style":2473},[468,469]," 并回跳。",[11,15359,15360,15363,15364,15485],{},[378,15361,15362],{},"第 2 轮","：SAT 层在引理约束下重新搜索，可能选择 ",[224,15365,15367,15401],{"className":15366,"translate":388},[392],[224,15368,15370],{"className":15369},[396],[398,15371,15372],{"xmlns":400},[403,15373,15374,15398],{},[406,15375,15376,15378,15380,15382,15384,15386,15388,15390,15392,15394,15396],{},[409,15377,11],{},[412,15379,278],{},[409,15381,1078],{"mathvariant":678},[412,15383,1877],{"separator":455},[409,15385,2023],{},[412,15387,278],{},[409,15389,1088],{"mathvariant":678},[412,15391,1877],{"separator":455},[409,15393,2061],{},[412,15395,278],{},[409,15397,1078],{"mathvariant":678},[447,15399,15400],{"encoding":449},"p = \\top, q = \\bot, r = \\top",[224,15402,15404,15422,15449,15476],{"className":15403,"ariaHidden":455},[454],[224,15405,15407,15410,15413,15416,15419],{"className":15406},[459],[224,15408],{"className":15409,"style":806},[463],[224,15411,11],{"className":15412},[468,469],[224,15414],{"className":15415,"style":496},[473],[224,15417,278],{"className":15418},[500],[224,15420],{"className":15421,"style":496},[473],[224,15423,15425,15428,15431,15434,15437,15440,15443,15446],{"className":15424},[459],[224,15426],{"className":15427,"style":2565},[463],[224,15429,1078],{"className":15430},[468],[224,15432,1877],{"className":15433},[1917],[224,15435],{"className":15436,"style":724},[473],[224,15438,2023],{"className":15439,"style":492},[468,469],[224,15441],{"className":15442,"style":496},[473],[224,15444,278],{"className":15445},[500],[224,15447],{"className":15448,"style":496},[473],[224,15450,15452,15455,15458,15461,15464,15467,15470,15473],{"className":15451},[459],[224,15453],{"className":15454,"style":2565},[463],[224,15456,1088],{"className":15457},[468],[224,15459,1877],{"className":15460},[1917],[224,15462],{"className":15463,"style":724},[473],[224,15465,2061],{"className":15466,"style":2473},[468,469],[224,15468],{"className":15469,"style":496},[473],[224,15471,278],{"className":15472},[500],[224,15474],{"className":15475,"style":496},[473],[224,15477,15479,15482],{"className":15478},[459],[224,15480],{"className":15481,"style":7642},[463],[224,15483,1078],{"className":15484},[468],"。理论层检查：",[224,15487,15489],{"className":15488,"translate":388},[387],[224,15490,15492,15534],{"className":15491,"translate":388},[392],[224,15493,15495],{"className":15494},[396],[398,15496,15497],{"xmlns":400,"display":401},[403,15498,15499,15531],{},[406,15500,15501,15503,15505,15507,15509,15511,15513,15515,15517,15519,15521,15523,15525,15527,15529],{},[409,15502,371],{},[412,15504,325],{},[419,15506,284],{},[412,15508,1877],{"separator":455},[473,15510],{"width":4247},[409,15512,375],{},[412,15514,2070],{},[419,15516,284],{},[412,15518,1877],{"separator":455},[473,15520],{"width":4247},[409,15522,371],{},[412,15524,319],{},[409,15526,375],{},[412,15528,1577],{},[419,15530,284],{},[447,15532,15533],{"encoding":449},"x > 0, \\qquad y \\le 0, \\qquad x + y \u003C 0",[224,15535,15537,15555,15585,15615,15633],{"className":15536,"ariaHidden":455},[454],[224,15538,15540,15543,15546,15549,15552],{"className":15539},[459],[224,15541],{"className":15542,"style":2810},[463],[224,15544,371],{"className":15545},[468,469],[224,15547],{"className":15548,"style":496},[473],[224,15550,325],{"className":15551},[500],[224,15553],{"className":15554,"style":496},[473],[224,15556,15558,15561,15564,15567,15570,15573,15576,15579,15582],{"className":15557},[459],[224,15559],{"className":15560,"style":8688},[463],[224,15562,284],{"className":15563},[468],[224,15565,1877],{"className":15566},[1917],[224,15568],{"className":15569,"style":4416},[473],[224,15571],{"className":15572,"style":724},[473],[224,15574,375],{"className":15575,"style":492},[468,469],[224,15577],{"className":15578,"style":496},[473],[224,15580,2070],{"className":15581},[500],[224,15583],{"className":15584,"style":496},[473],[224,15586,15588,15591,15594,15597,15600,15603,15606,15609,15612],{"className":15587},[459],[224,15589],{"className":15590,"style":8688},[463],[224,15592,284],{"className":15593},[468],[224,15595,1877],{"className":15596},[1917],[224,15598],{"className":15599,"style":4416},[473],[224,15601],{"className":15602,"style":724},[473],[224,15604,371],{"className":15605},[468,469],[224,15607],{"className":15608,"style":474},[473],[224,15610,319],{"className":15611},[478],[224,15613],{"className":15614,"style":474},[473],[224,15616,15618,15621,15624,15627,15630],{"className":15617},[459],[224,15619],{"className":15620,"style":488},[463],[224,15622,375],{"className":15623,"style":492},[468,469],[224,15625],{"className":15626,"style":496},[473],[224,15628,1577],{"className":15629},[500],[224,15631],{"className":15632,"style":496},[473],[224,15634,15636,15639],{"className":15635},[459],[224,15637],{"className":15638,"style":510},[463],[224,15640,284],{"className":15641},[468],[11,15643,15644,15645,15736],{},"取 ",[224,15646,15648,15676],{"className":15647,"translate":388},[392],[224,15649,15651],{"className":15650},[396],[398,15652,15653],{"xmlns":400},[403,15654,15655,15673],{},[406,15656,15657,15659,15661,15663,15665,15667,15669,15671],{},[409,15658,371],{},[412,15660,278],{},[419,15662,1438],{},[412,15664,1877],{"separator":455},[409,15666,375],{},[412,15668,278],{},[412,15670,5268],{},[419,15672,1447],{},[447,15674,15675],{"encoding":449},"x = 1, y = -2",[224,15677,15679,15697,15724],{"className":15678,"ariaHidden":455},[454],[224,15680,15682,15685,15688,15691,15694],{"className":15681},[459],[224,15683],{"className":15684,"style":839},[463],[224,15686,371],{"className":15687},[468,469],[224,15689],{"className":15690,"style":496},[473],[224,15692,278],{"className":15693},[500],[224,15695],{"className":15696,"style":496},[473],[224,15698,15700,15703,15706,15709,15712,15715,15718,15721],{"className":15699},[459],[224,15701],{"className":15702,"style":8688},[463],[224,15704,1438],{"className":15705},[468],[224,15707,1877],{"className":15708},[1917],[224,15710],{"className":15711,"style":724},[473],[224,15713,375],{"className":15714,"style":492},[468,469],[224,15716],{"className":15717,"style":496},[473],[224,15719,278],{"className":15720},[500],[224,15722],{"className":15723,"style":496},[473],[224,15725,15727,15730,15733],{"className":15726},[459],[224,15728],{"className":15729,"style":7047},[463],[224,15731,5268],{"className":15732},[468],[224,15734,1447],{"className":15735},[468],"，三条同时成立。理论层返回一致，求解器输出：",[224,15738,15740],{"className":15739,"translate":388},[387],[224,15741,15743,15780],{"className":15742,"translate":388},[392],[224,15744,15746],{"className":15745},[396],[398,15747,15748],{"xmlns":400,"display":401},[403,15749,15750,15777],{},[406,15751,15752,15755,15757,15759,15761,15763,15765,15767,15769,15771,15773,15775],{},[423,15753,599],{"mathvariant":15754},"bold",[412,15756,1877],{"separator":455},[473,15758],{"width":4247},[409,15760,371],{},[412,15762,278],{},[419,15764,1438],{},[412,15766,1877],{"separator":455},[423,15768,15054],{},[409,15770,375],{},[412,15772,278],{},[412,15774,5268],{},[419,15776,1447],{},[447,15778,15779],{"encoding":449},"\\textbf{sat}, \\qquad x = 1,\\ y = -2",[224,15781,15783,15818,15848],{"className":15782,"ariaHidden":455},[454],[224,15784,15786,15790,15797,15800,15803,15806,15809,15812,15815],{"className":15785},[459],[224,15787],{"className":15788,"style":15789},[463],"height:0.8294em;vertical-align:-0.1944em;",[224,15791,15793],{"className":15792},[468,11373],[224,15794,599],{"className":15795},[468,15796],"textbf",[224,15798,1877],{"className":15799},[1917],[224,15801],{"className":15802,"style":4416},[473],[224,15804],{"className":15805,"style":724},[473],[224,15807,371],{"className":15808},[468,469],[224,15810],{"className":15811,"style":496},[473],[224,15813,278],{"className":15814},[500],[224,15816],{"className":15817,"style":496},[473],[224,15819,15821,15824,15827,15830,15833,15836,15839,15842,15845],{"className":15820},[459],[224,15822],{"className":15823,"style":8688},[463],[224,15825,1438],{"className":15826},[468],[224,15828,1877],{"className":15829},[1917],[224,15831,15054],{"className":15832},[473],[224,15834],{"className":15835,"style":724},[473],[224,15837,375],{"className":15838,"style":492},[468,469],[224,15840],{"className":15841,"style":496},[473],[224,15843,278],{"className":15844},[500],[224,15846],{"className":15847,"style":496},[473],[224,15849,15851,15854,15857],{"className":15850},[459],[224,15852],{"className":15853,"style":7047},[463],[224,15855,5268],{"className":15856},[468],[224,15858,1447],{"className":15859},[468],[11,15861,15862],{},"这就是一个完整的 DPLL(T) 循环：一次冲突、一次学习、一次成功。",[11,15864,15865,15866,15869],{},"在真实求解器里，理论层还会做",[378,15867,15868],{},"理论传播","：不等 SAT 层给出完整赋值，就提前推出某些文字必须为真或为假，从而更早剪枝。这让循环通常只需要很少几轮。",[11,15871,15872],{},"理论正确不代表跑得快。工业级求解器还依赖大量工程优化：",[2674,15874,15875,15881,15886,15892,15898,15904],{},[2677,15876,15877,15880],{},[378,15878,15879],{},"预处理","：化简公式、消除冗余变量、把公式规范化成求解器偏好的形式。",[2677,15882,15883,15885],{},[378,15884,15868],{},"：理论层主动推导文字，而不是被动等待完整赋值。",[2677,15887,15888,15891],{},[378,15889,15890],{},"重启策略","：按 Luby 序列或指数策略重启，避免陷入坏的搜索区域。",[2677,15893,15894,15897],{},[378,15895,15896],{},"子句删除","：学习子句太多会拖慢传播，需要定期清理。",[2677,15899,15900,15903],{},[378,15901,15902],{},"多引擎组合","：同一个问题用不同策略并行求解（portfolio），谁先出结果用谁。",[2677,15905,15906,15909],{},[378,15907,15908],{},"专用推理","：针对乘法、数组、字符串等难点，加入专门的推理规则。",[613,15911,15913],{"id":15912},"_47-复杂度边界","4.7 复杂度边界",[11,15915,15916,15917,3117],{},"必须强调：",[378,15918,15919],{},"可判定不等于可解",[2674,15921,15922,15925,15934,15939,15943],{},[2677,15923,15924],{},"SAT 是 NP-complete；",[2677,15926,15927,372,15929,372,15931,15933],{},[221,15928,3233],{},[221,15930,3148],{},[221,15932,3134],{}," 也都是 NP-complete；",[2677,15935,15936,15938],{},[221,15937,3144],{}," 虽然是多项式时间，但实际常数很大；",[2677,15940,15941,7663],{},[221,15942,3221],{},[2677,15944,15945,15947],{},[221,15946,3224],{}," 直接不可判定。",[11,15949,15950],{},"这意味着，工业问题的最坏情况仍然是指数级的。求解器的成功，很大程度上依赖问题结构、编码方式和启发式。遇到超时，工具通常只能返回「未知」，而不是「安全」。",[11,15952,15953],{},"这也是为什么实际系统里，SMT 很少单独使用，而是和抽象解释、fuzzing、测试生成等方法组合。",[608,15955,15957],{"id":15956},"五漏洞检测","五、漏洞检测",[613,15959,15961],{"id":15960},"_51-问题转化","5.1 问题转化",[11,15963,15964],{},"软件漏洞检测的核心问题可以写成一句话：",[24,15966,15967],{},[11,15968,15969],{},"是否存在一个输入，让程序走到某个危险位置，并且违反某条安全性质？",[11,15971,15972],{},"形式化之后，它就是一个存在性查询：",[224,15974,15976],{"className":15975,"translate":388},[387],[224,15977,15979,16096],{"className":15978,"translate":388},[392],[224,15980,15982],{"className":15981},[396],[398,15983,15984],{"xmlns":400,"display":401},[403,15985,15986,16093],{},[406,15987,15988,15990,15992,16004,16006,16008,16043,16045,16047,16049],{},[409,15989,679],{"mathvariant":678},[423,15991,682],{},[406,15993,15994,15996,15998,16000,16002],{},[409,15995,1880],{},[409,15997,3254],{},[409,15999,11],{},[409,16001,4130],{},[409,16003,1435],{},[409,16005,688],{"mathvariant":678},[473,16007],{"width":691},[1965,16009,16010,16040],{},[1965,16011,16012,16038],{},[406,16013,16014,16022,16024,16036],{},[406,16015,16016,16019],{},[409,16017,16018],{"mathvariant":678},"P",[409,16020,16021],{"mathvariant":678},"C",[412,16023,239],{"stretchy":435},[406,16025,16026,16028,16030,16032,16034],{},[409,16027,1880],{},[409,16029,3254],{},[409,16031,11],{},[409,16033,4130],{},[409,16035,1435],{},[412,16037,1716],{"stretchy":435},[412,16039,1987],{"stretchy":455},[423,16041,16042],{},"路径条件",[423,16044,425],{},[412,16046,428],{},[423,16048,425],{},[1965,16050,16051,16090],{},[1965,16052,16053,16088],{},[406,16054,16055,16072,16074,16086],{},[406,16056,16057,16060,16062,16064,16066,16068,16070],{},[409,16058,16059],{"mathvariant":678},"V",[409,16061,1880],{"mathvariant":678},[409,16063,4127],{"mathvariant":678},[409,16065,1864],{"mathvariant":678},[409,16067,15],{"mathvariant":678},[409,16069,1435],{"mathvariant":678},[409,16071,1861],{"mathvariant":678},[412,16073,239],{"stretchy":435},[406,16075,16076,16078,16080,16082,16084],{},[409,16077,1880],{},[409,16079,3254],{},[409,16081,11],{},[409,16083,4130],{},[409,16085,1435],{},[412,16087,1716],{"stretchy":435},[412,16089,1987],{"stretchy":455},[423,16091,16092],{},"安全性质取反",[447,16094,16095],{"encoding":449},"\\exists\\, \\mathit{input}.\\quad \\underbrace{\\mathrm{PC}(\\mathit{input})}_{\\text{路径条件}} \\;\\wedge\\; \\underbrace{\\mathrm{Violate}(\\mathit{input})}_{\\text{安全性质取反}}",[224,16097,16099,16272],{"className":16098,"ariaHidden":455},[454],[224,16100,16102,16106,16109,16112,16120,16123,16126,16129,16257,16260,16263,16266,16269],{"className":16101},[459],[224,16103],{"className":16104,"style":16105},[463],"height:2.3263em;vertical-align:-1.5763em;",[224,16107,679],{"className":16108},[468],[224,16110],{"className":16111,"style":724},[473],[224,16113,16115],{"className":16114},[468],[224,16116,16119],{"className":16117},[468,16118],"mathit","input",[224,16121,688],{"className":16122},[468],[224,16124],{"className":16125,"style":734},[473],[224,16127],{"className":16128,"style":724},[473],[224,16130,16132],{"className":16131},[2091,1965],[224,16133,16135,16248],{"className":16134},[754,755],[224,16136,16138,16245],{"className":16137},[759],[224,16139,16141,16160],{"className":16140,"style":2101},[763],[224,16142,16144,16147],{"style":16143},"top:-1.4237em;",[224,16145],{"className":16146,"style":2108},[771],[224,16148,16150],{"className":16149},[776,777,778,779],[224,16151,16153],{"className":16152},[468,779],[224,16154,16156],{"className":16155},[468,11373,779],[224,16157,16042],{"className":16158},[468,16159,779],"cjk_fallback",[224,16161,16162,16165],{"style":2120},[224,16163],{"className":16164,"style":2108},[771],[224,16166,16168],{"className":16167},[2091,1965],[224,16169,16171,16237],{"className":16170},[754,755],[224,16172,16174,16234],{"className":16173},[759],[224,16175,16177,16207],{"className":16176,"style":2101},[763],[224,16178,16180,16183],{"className":16179,"style":2140},[2139],[224,16181],{"className":16182,"style":2108},[771],[224,16184,16186,16193,16200],{"className":16185,"style":2148},[2147],[224,16187,16189],{"className":16188,"style":2153},[2152],[2155,16190,16191],{"xmlns":2157,"width":2158,"height":2159,"viewBox":2160,"preserveAspectRatio":2161},[2163,16192],{"d":2165},[224,16194,16196],{"className":16195,"style":2153},[2169],[2155,16197,16198],{"xmlns":2157,"width":2158,"height":2159,"viewBox":2160,"preserveAspectRatio":2172},[2163,16199],{"d":2175},[224,16201,16203],{"className":16202,"style":2153},[2179],[2155,16204,16205],{"xmlns":2157,"width":2158,"height":2159,"viewBox":2160,"preserveAspectRatio":2182},[2163,16206],{"d":2185},[224,16208,16209,16212],{"style":2120},[224,16210],{"className":16211,"style":2108},[771],[224,16213,16215,16222,16225,16231],{"className":16214},[468],[224,16216,16218],{"className":16217},[468],[224,16219,16221],{"className":16220},[468,1906],"PC",[224,16223,239],{"className":16224},[542],[224,16226,16228],{"className":16227},[468],[224,16229,16119],{"className":16230},[468,16118],[224,16232,1716],{"className":16233},[549],[224,16235,787],{"className":16236},[786],[224,16238,16240],{"className":16239},[759],[224,16241,16243],{"className":16242,"style":2236},[763],[224,16244],{},[224,16246,787],{"className":16247},[786],[224,16249,16251],{"className":16250},[759],[224,16252,16255],{"className":16253,"style":16254},[763],"height:1.5763em;",[224,16256],{},[224,16258],{"className":16259,"style":496},[473],[224,16261],{"className":16262,"style":474},[473],[224,16264,428],{"className":16265},[478],[224,16267],{"className":16268,"style":496},[473],[224,16270],{"className":16271,"style":474},[473],[224,16273,16275,16278],{"className":16274},[459],[224,16276],{"className":16277,"style":16105},[463],[224,16279,16281],{"className":16280},[2091,1965],[224,16282,16284,16395],{"className":16283},[754,755],[224,16285,16287,16392],{"className":16286},[759],[224,16288,16290,16307],{"className":16289,"style":2101},[763],[224,16291,16292,16295],{"style":16143},[224,16293],{"className":16294,"style":2108},[771],[224,16296,16298],{"className":16297},[776,777,778,779],[224,16299,16301],{"className":16300},[468,779],[224,16302,16304],{"className":16303},[468,11373,779],[224,16305,16092],{"className":16306},[468,16159,779],[224,16308,16309,16312],{"style":2120},[224,16310],{"className":16311,"style":2108},[771],[224,16313,16315],{"className":16314},[2091,1965],[224,16316,16318,16384],{"className":16317},[754,755],[224,16319,16321,16381],{"className":16320},[759],[224,16322,16324,16354],{"className":16323,"style":2101},[763],[224,16325,16327,16330],{"className":16326,"style":2140},[2139],[224,16328],{"className":16329,"style":2108},[771],[224,16331,16333,16340,16347],{"className":16332,"style":2148},[2147],[224,16334,16336],{"className":16335,"style":2153},[2152],[2155,16337,16338],{"xmlns":2157,"width":2158,"height":2159,"viewBox":2160,"preserveAspectRatio":2161},[2163,16339],{"d":2165},[224,16341,16343],{"className":16342,"style":2153},[2169],[2155,16344,16345],{"xmlns":2157,"width":2158,"height":2159,"viewBox":2160,"preserveAspectRatio":2172},[2163,16346],{"d":2175},[224,16348,16350],{"className":16349,"style":2153},[2179],[2155,16351,16352],{"xmlns":2157,"width":2158,"height":2159,"viewBox":2160,"preserveAspectRatio":2182},[2163,16353],{"d":2185},[224,16355,16356,16359],{"style":2120},[224,16357],{"className":16358,"style":2108},[771],[224,16360,16362,16369,16372,16378],{"className":16361},[468],[224,16363,16365],{"className":16364},[468],[224,16366,16368],{"className":16367},[468,1906],"Violate",[224,16370,239],{"className":16371},[542],[224,16373,16375],{"className":16374},[468],[224,16376,16119],{"className":16377},[468,16118],[224,16379,1716],{"className":16380},[549],[224,16382,787],{"className":16383},[786],[224,16385,16387],{"className":16386},[759],[224,16388,16390],{"className":16389,"style":2236},[763],[224,16391],{},[224,16393,787],{"className":16394},[786],[224,16396,16398],{"className":16397},[759],[224,16399,16401],{"className":16400,"style":16254},[763],[224,16402],{},[2674,16404,16405,16442],{},[2677,16406,16407,16441],{},[224,16408,16410,16426],{"className":16409,"translate":388},[392],[224,16411,16413],{"className":16412},[396],[398,16414,16415],{"xmlns":400},[403,16416,16417,16423],{},[406,16418,16419,16421],{},[409,16420,16018],{"mathvariant":678},[409,16422,16021],{"mathvariant":678},[447,16424,16425],{"encoding":449},"\\mathrm{PC}",[224,16427,16429],{"className":16428,"ariaHidden":455},[454],[224,16430,16432,16435],{"className":16431},[459],[224,16433],{"className":16434,"style":649},[463],[224,16436,16438],{"className":16437},[468],[224,16439,16221],{"className":16440},[468,1906]," 是路径条件，描述「走到这里需要满足哪些约束」；",[2677,16443,16444,16488],{},[224,16445,16447,16473],{"className":16446,"translate":388},[392],[224,16448,16450],{"className":16449},[396],[398,16451,16452],{"xmlns":400},[403,16453,16454,16470],{},[406,16455,16456,16458,16460,16462,16464,16466,16468],{},[409,16457,16059],{"mathvariant":678},[409,16459,1880],{"mathvariant":678},[409,16461,4127],{"mathvariant":678},[409,16463,1864],{"mathvariant":678},[409,16465,15],{"mathvariant":678},[409,16467,1435],{"mathvariant":678},[409,16469,1861],{"mathvariant":678},[447,16471,16472],{"encoding":449},"\\mathrm{Violate}",[224,16474,16476],{"className":16475,"ariaHidden":455},[454],[224,16477,16479,16482],{"className":16478},[459],[224,16480],{"className":16481,"style":7642},[463],[224,16483,16485],{"className":16484},[468],[224,16486,16368],{"className":16487},[468,1906]," 是安全性质的否定，描述「怎样才算越界\u002F溢出\u002F空指针解引用」。",[11,16490,16491,16492,16494,16495,16497],{},"如果这个公式 ",[378,16493,599],{},"，求解器给出的模型就是一条能触发漏洞的输入，漏洞可达；如果 ",[378,16496,6650],{},"，说明在这条路径上不可能违反性质，可以安全剪枝。",[11,16499,16500,16501,3117],{},"这个转化是整个应用的基石：",[378,16502,16503],{},"「找漏洞」变成了「解约束」",[613,16505,16507],{"id":16506},"_52-缓冲区溢出","5.2 缓冲区溢出",[11,16509,16510],{},"先看一个最直观的例子：",[215,16512,16514],{"className":217,"code":16513,"language":219,"meta":186,"style":186},"void process(char *input, int len) {\n    char buf[8];\n    if (len > 4 && len \u003C 12) {\n        memcpy(buf, input, len);   \u002F\u002F 可疑点\n    }\n}\n",[221,16515,16516,16542,16557,16581,16592,16596],{"__ignoreMap":186},[224,16517,16518,16520,16523,16525,16528,16531,16533,16535,16537,16540],{"class":226,"line":227},[224,16519,231],{"class":230},[224,16521,16522],{"class":234}," process",[224,16524,239],{"class":238},[224,16526,16527],{"class":230},"char",[224,16529,16530],{"class":230}," *",[224,16532,16119],{"class":245},[224,16534,249],{"class":238},[224,16536,242],{"class":230},[224,16538,16539],{"class":245}," len",[224,16541,257],{"class":238},[224,16543,16544,16547,16550,16552,16554],{"class":226,"line":187},[224,16545,16546],{"class":230},"    char",[224,16548,16549],{"class":245}," buf",[224,16551,268],{"class":238},[224,16553,5430],{"class":271},[224,16555,16556],{"class":238},"];\n",[224,16558,16559,16561,16564,16566,16569,16571,16574,16576,16579],{"class":226,"line":290},[224,16560,313],{"class":230},[224,16562,16563],{"class":238}," (len ",[224,16565,325],{"class":230},[224,16567,16568],{"class":271}," 4",[224,16570,331],{"class":230},[224,16572,16573],{"class":238}," len ",[224,16575,1577],{"class":230},[224,16577,16578],{"class":271}," 12",[224,16580,257],{"class":238},[224,16582,16583,16586,16589],{"class":226,"line":310},[224,16584,16585],{"class":234},"        memcpy",[224,16587,16588],{"class":238},"(buf, input, len);",[224,16590,16591],{"class":352},"   \u002F\u002F 可疑点\n",[224,16593,16594],{"class":226,"line":349},[224,16595,359],{"class":238},[224,16597,16598],{"class":226,"line":356},[224,16599,365],{"class":238},[11,16601,16602,16605,16606,16609,16610,16613,16614,16616],{},[221,16603,16604],{},"buf"," 只有 8 字节，而 ",[221,16607,16608],{},"memcpy"," 拷贝 ",[221,16611,16612],{},"len"," 字节。如果 ",[221,16615,16612],{}," 能大于 8，就会发生栈溢出。",[11,16618,16619,16620,2887,16622,16624,16625,16627],{},"符号执行引擎把 ",[221,16621,16119],{},[221,16623,16612],{}," 当作符号变量，沿路径走到 ",[221,16626,16608],{}," 时，收集到的路径条件是：",[224,16629,16631],{"className":16630,"translate":388},[387],[224,16632,16634,16673],{"className":16633,"translate":388},[392],[224,16635,16637],{"className":16636},[396],[398,16638,16639],{"xmlns":400,"display":401},[403,16640,16641,16670],{},[406,16642,16643,16645,16647,16649,16651,16653,16655,16657,16659,16661,16663,16665,16667],{},[409,16644,1864],{},[409,16646,1861],{},[409,16648,3254],{},[412,16650,325],{},[419,16652,9409],{},[423,16654,425],{},[412,16656,428],{},[423,16658,425],{},[409,16660,1864],{},[409,16662,1861],{},[409,16664,3254],{},[412,16666,1577],{},[419,16668,16669],{},"12",[447,16671,16672],{"encoding":449},"len > 4 \\;\\wedge\\; len \u003C 12",[224,16674,16676,16702,16726,16750],{"className":16675,"ariaHidden":455},[454],[224,16677,16679,16683,16687,16690,16693,16696,16699],{"className":16678},[459],[224,16680],{"className":16681,"style":16682},[463],"height:0.7335em;vertical-align:-0.0391em;",[224,16684,1864],{"className":16685,"style":16686},[468,469],"margin-right:0.0197em;",[224,16688,1861],{"className":16689},[468,469],[224,16691,3254],{"className":16692},[468,469],[224,16694],{"className":16695,"style":496},[473],[224,16697,325],{"className":16698},[500],[224,16700],{"className":16701,"style":496},[473],[224,16703,16705,16708,16711,16714,16717,16720,16723],{"className":16704},[459],[224,16706],{"className":16707,"style":510},[463],[224,16709,9409],{"className":16710},[468],[224,16712],{"className":16713,"style":496},[473],[224,16715],{"className":16716,"style":474},[473],[224,16718,428],{"className":16719},[478],[224,16721],{"className":16722,"style":496},[473],[224,16724],{"className":16725,"style":474},[473],[224,16727,16729,16732,16735,16738,16741,16744,16747],{"className":16728},[459],[224,16730],{"className":16731,"style":16682},[463],[224,16733,1864],{"className":16734,"style":16686},[468,469],[224,16736,1861],{"className":16737},[468,469],[224,16739,3254],{"className":16740},[468,469],[224,16742],{"className":16743,"style":496},[473],[224,16745,1577],{"className":16746},[500],[224,16748],{"className":16749,"style":496},[473],[224,16751,16753,16756],{"className":16752},[459],[224,16754],{"className":16755,"style":510},[463],[224,16757,16669],{"className":16758},[468],[11,16760,16761],{},"要判断溢出是否可达，就在路径条件上再加一条「写入长度超过缓冲区大小」：",[224,16763,16765],{"className":16764,"translate":388},[387],[224,16766,16768,16822],{"className":16767,"translate":388},[392],[224,16769,16771],{"className":16770},[396],[398,16772,16773],{"xmlns":400,"display":401},[403,16774,16775,16819],{},[406,16776,16777,16779,16781,16783,16785,16787,16789,16791,16793,16795,16797,16799,16801,16803,16805,16807,16809,16811,16813,16815,16817],{},[409,16778,1864],{},[409,16780,1861],{},[409,16782,3254],{},[412,16784,325],{},[419,16786,9409],{},[423,16788,425],{},[412,16790,428],{},[423,16792,425],{},[409,16794,1864],{},[409,16796,1861],{},[409,16798,3254],{},[412,16800,1577],{},[419,16802,16669],{},[423,16804,425],{},[412,16806,428],{},[423,16808,425],{},[409,16810,1864],{},[409,16812,1861],{},[409,16814,3254],{},[412,16816,325],{},[419,16818,5430],{},[447,16820,16821],{"encoding":449},"len > 4 \\;\\wedge\\; len \u003C 12 \\;\\wedge\\; len > 8",[224,16823,16825,16849,16873,16897,16921,16945],{"className":16824,"ariaHidden":455},[454],[224,16826,16828,16831,16834,16837,16840,16843,16846],{"className":16827},[459],[224,16829],{"className":16830,"style":16682},[463],[224,16832,1864],{"className":16833,"style":16686},[468,469],[224,16835,1861],{"className":16836},[468,469],[224,16838,3254],{"className":16839},[468,469],[224,16841],{"className":16842,"style":496},[473],[224,16844,325],{"className":16845},[500],[224,16847],{"className":16848,"style":496},[473],[224,16850,16852,16855,16858,16861,16864,16867,16870],{"className":16851},[459],[224,16853],{"className":16854,"style":510},[463],[224,16856,9409],{"className":16857},[468],[224,16859],{"className":16860,"style":496},[473],[224,16862],{"className":16863,"style":474},[473],[224,16865,428],{"className":16866},[478],[224,16868],{"className":16869,"style":496},[473],[224,16871],{"className":16872,"style":474},[473],[224,16874,16876,16879,16882,16885,16888,16891,16894],{"className":16875},[459],[224,16877],{"className":16878,"style":16682},[463],[224,16880,1864],{"className":16881,"style":16686},[468,469],[224,16883,1861],{"className":16884},[468,469],[224,16886,3254],{"className":16887},[468,469],[224,16889],{"className":16890,"style":496},[473],[224,16892,1577],{"className":16893},[500],[224,16895],{"className":16896,"style":496},[473],[224,16898,16900,16903,16906,16909,16912,16915,16918],{"className":16899},[459],[224,16901],{"className":16902,"style":510},[463],[224,16904,16669],{"className":16905},[468],[224,16907],{"className":16908,"style":496},[473],[224,16910],{"className":16911,"style":474},[473],[224,16913,428],{"className":16914},[478],[224,16916],{"className":16917,"style":496},[473],[224,16919],{"className":16920,"style":474},[473],[224,16922,16924,16927,16930,16933,16936,16939,16942],{"className":16923},[459],[224,16925],{"className":16926,"style":16682},[463],[224,16928,1864],{"className":16929,"style":16686},[468,469],[224,16931,1861],{"className":16932},[468,469],[224,16934,3254],{"className":16935},[468,469],[224,16937],{"className":16938,"style":496},[473],[224,16940,325],{"className":16941},[500],[224,16943],{"className":16944,"style":496},[473],[224,16946,16948,16951],{"className":16947},[459],[224,16949],{"className":16950,"style":510},[463],[224,16952,5430],{"className":16953},[468],[11,16955,16956],{},"写成 SMT-LIB：",[215,16958,16962],{"className":16959,"code":16960,"language":16961,"meta":186,"style":186},"language-smt shiki shiki-themes github-light github-light github-dark","(set-logic QF_LIA)\n(declare-const len Int)\n(assert (and (> len 4) (\u003C len 12) (> len 8)))\n(check-sat)\n(get-model)\n","smt",[221,16963,16964,16969,16974,16979,16984],{"__ignoreMap":186},[224,16965,16966],{"class":226,"line":227},[224,16967,16968],{},"(set-logic QF_LIA)\n",[224,16970,16971],{"class":226,"line":187},[224,16972,16973],{},"(declare-const len Int)\n",[224,16975,16976],{"class":226,"line":290},[224,16977,16978],{},"(assert (and (> len 4) (\u003C len 12) (> len 8)))\n",[224,16980,16981],{"class":226,"line":310},[224,16982,16983],{},"(check-sat)\n",[224,16985,16986],{"class":226,"line":349},[224,16987,16988],{},"(get-model)\n",[11,16990,16991],{},"求解器返回：",[215,16993,16996],{"className":16994,"code":16995,"language":11373},[11371],"sat\n(model\n  (define-fun len () Int\n    9)\n)\n",[221,16997,16995],{"__ignoreMap":186},[11,16999,17000,17003,17004,17078,17079,17130,17131,17133],{},[221,17001,17002],{},"len = 9"," 是一个合法解：它满足 ",[224,17005,17007,17030],{"className":17006,"translate":388},[392],[224,17008,17010],{"className":17009},[396],[398,17011,17012],{"xmlns":400},[403,17013,17014,17027],{},[406,17015,17016,17018,17020,17023,17025],{},[419,17017,9409],{},[412,17019,1577],{},[419,17021,17022],{},"9",[412,17024,1577],{},[419,17026,16669],{},[447,17028,17029],{"encoding":449},"4 \u003C 9 \u003C 12",[224,17031,17033,17051,17069],{"className":17032,"ariaHidden":455},[454],[224,17034,17036,17039,17042,17045,17048],{"className":17035},[459],[224,17037],{"className":17038,"style":9623},[463],[224,17040,9409],{"className":17041},[468],[224,17043],{"className":17044,"style":496},[473],[224,17046,1577],{"className":17047},[500],[224,17049],{"className":17050,"style":496},[473],[224,17052,17054,17057,17060,17063,17066],{"className":17053},[459],[224,17055],{"className":17056,"style":9623},[463],[224,17058,17022],{"className":17059},[468],[224,17061],{"className":17062,"style":496},[473],[224,17064,1577],{"className":17065},[500],[224,17067],{"className":17068,"style":496},[473],[224,17070,17072,17075],{"className":17071},[459],[224,17073],{"className":17074,"style":510},[463],[224,17076,16669],{"className":17077},[468],"，同时 ",[224,17080,17082,17100],{"className":17081,"translate":388},[392],[224,17083,17085],{"className":17084},[396],[398,17086,17087],{"xmlns":400},[403,17088,17089,17097],{},[406,17090,17091,17093,17095],{},[419,17092,17022],{},[412,17094,325],{},[419,17096,5430],{},[447,17098,17099],{"encoding":449},"9 > 8",[224,17101,17103,17121],{"className":17102,"ariaHidden":455},[454],[224,17104,17106,17109,17112,17115,17118],{"className":17105},[459],[224,17107],{"className":17108,"style":9623},[463],[224,17110,17022],{"className":17111},[468],[224,17113],{"className":17114,"style":496},[473],[224,17116,325],{"className":17117},[500],[224,17119],{"className":17120,"style":496},[473],[224,17122,17124,17127],{"className":17123},[459],[224,17125],{"className":17126,"style":510},[463],[224,17128,5430],{"className":17129},[468],"。这条路径可达，",[221,17132,16608],{}," 会往 8 字节的缓冲区里写 9 字节，溢出 1 字节。是否真的覆盖到返回地址取决于栈布局，但「越界写」这个事实本身已经成立。",[11,17135,17136],{},"如果把代码改成：",[215,17138,17140],{"className":217,"code":17139,"language":219,"meta":186,"style":186},"if (len > 4 && len \u003C= 8) {\n    memcpy(buf, input, len);\n}\n",[221,17141,17142,17165,17173],{"__ignoreMap":186},[224,17143,17144,17147,17149,17151,17153,17155,17157,17160,17163],{"class":226,"line":227},[224,17145,17146],{"class":230},"if",[224,17148,16563],{"class":238},[224,17150,325],{"class":230},[224,17152,16568],{"class":271},[224,17154,331],{"class":230},[224,17156,16573],{"class":238},[224,17158,17159],{"class":230},"\u003C=",[224,17161,17162],{"class":271}," 8",[224,17164,257],{"class":238},[224,17166,17167,17170],{"class":226,"line":187},[224,17168,17169],{"class":234},"    memcpy",[224,17171,17172],{"class":238},"(buf, input, len);\n",[224,17174,17175],{"class":226,"line":290},[224,17176,365],{"class":238},[11,17178,17179],{},"约束就变成：",[224,17181,17183],{"className":17182,"translate":388},[387],[224,17184,17186,17240],{"className":17185,"translate":388},[392],[224,17187,17189],{"className":17188},[396],[398,17190,17191],{"xmlns":400,"display":401},[403,17192,17193,17237],{},[406,17194,17195,17197,17199,17201,17203,17205,17207,17209,17211,17213,17215,17217,17219,17221,17223,17225,17227,17229,17231,17233,17235],{},[409,17196,1864],{},[409,17198,1861],{},[409,17200,3254],{},[412,17202,325],{},[419,17204,9409],{},[423,17206,425],{},[412,17208,428],{},[423,17210,425],{},[409,17212,1864],{},[409,17214,1861],{},[409,17216,3254],{},[412,17218,2070],{},[419,17220,5430],{},[423,17222,425],{},[412,17224,428],{},[423,17226,425],{},[409,17228,1864],{},[409,17230,1861],{},[409,17232,3254],{},[412,17234,325],{},[419,17236,5430],{},[447,17238,17239],{"encoding":449},"len > 4 \\;\\wedge\\; len \\le 8 \\;\\wedge\\; len > 8",[224,17241,17243,17267,17291,17316,17340,17364],{"className":17242,"ariaHidden":455},[454],[224,17244,17246,17249,17252,17255,17258,17261,17264],{"className":17245},[459],[224,17247],{"className":17248,"style":16682},[463],[224,17250,1864],{"className":17251,"style":16686},[468,469],[224,17253,1861],{"className":17254},[468,469],[224,17256,3254],{"className":17257},[468,469],[224,17259],{"className":17260,"style":496},[473],[224,17262,325],{"className":17263},[500],[224,17265],{"className":17266,"style":496},[473],[224,17268,17270,17273,17276,17279,17282,17285,17288],{"className":17269},[459],[224,17271],{"className":17272,"style":510},[463],[224,17274,9409],{"className":17275},[468],[224,17277],{"className":17278,"style":496},[473],[224,17280],{"className":17281,"style":474},[473],[224,17283,428],{"className":17284},[478],[224,17286],{"className":17287,"style":496},[473],[224,17289],{"className":17290,"style":474},[473],[224,17292,17294,17298,17301,17304,17307,17310,17313],{"className":17293},[459],[224,17295],{"className":17296,"style":17297},[463],"height:0.8304em;vertical-align:-0.136em;",[224,17299,1864],{"className":17300,"style":16686},[468,469],[224,17302,1861],{"className":17303},[468,469],[224,17305,3254],{"className":17306},[468,469],[224,17308],{"className":17309,"style":496},[473],[224,17311,2070],{"className":17312},[500],[224,17314],{"className":17315,"style":496},[473],[224,17317,17319,17322,17325,17328,17331,17334,17337],{"className":17318},[459],[224,17320],{"className":17321,"style":510},[463],[224,17323,5430],{"className":17324},[468],[224,17326],{"className":17327,"style":496},[473],[224,17329],{"className":17330,"style":474},[473],[224,17332,428],{"className":17333},[478],[224,17335],{"className":17336,"style":496},[473],[224,17338],{"className":17339,"style":474},[473],[224,17341,17343,17346,17349,17352,17355,17358,17361],{"className":17342},[459],[224,17344],{"className":17345,"style":16682},[463],[224,17347,1864],{"className":17348,"style":16686},[468,469],[224,17350,1861],{"className":17351},[468,469],[224,17353,3254],{"className":17354},[468,469],[224,17356],{"className":17357,"style":496},[473],[224,17359,325],{"className":17360},[500],[224,17362],{"className":17363,"style":496},[473],[224,17365,17367,17370],{"className":17366},[459],[224,17368],{"className":17369,"style":510},[463],[224,17371,5430],{"className":17372},[468],[11,17374,17375,17376,17378],{},"求解器返回 ",[378,17377,6650],{},"。这条溢出路径不可达，工具不会报告漏洞。",[11,17380,17381,17382,17384,17385,17387,17388,17390,17391,12570],{},"注意这里用的是 ",[221,17383,3148],{},"（整数线性算术）。真实的 C 代码里 ",[221,17386,16612],{}," 是 32 位 ",[221,17389,242],{},"，更精确的建模应该用 ",[221,17392,3233],{},[215,17394,17396],{"className":16959,"code":17395,"language":16961,"meta":186,"style":186},"(set-logic QF_BV)\n(declare-const len (_ BitVec 32))\n(assert (and (bvsgt len (_ bv4 32))\n             (bvslt len (_ bv12 32))\n             (bvsgt len (_ bv8 32))))\n(check-sat)\n(get-model)\n",[221,17397,17398,17403,17408,17413,17418,17423,17427],{"__ignoreMap":186},[224,17399,17400],{"class":226,"line":227},[224,17401,17402],{},"(set-logic QF_BV)\n",[224,17404,17405],{"class":226,"line":187},[224,17406,17407],{},"(declare-const len (_ BitVec 32))\n",[224,17409,17410],{"class":226,"line":290},[224,17411,17412],{},"(assert (and (bvsgt len (_ bv4 32))\n",[224,17414,17415],{"class":226,"line":310},[224,17416,17417],{},"             (bvslt len (_ bv12 32))\n",[224,17419,17420],{"class":226,"line":349},[224,17421,17422],{},"             (bvsgt len (_ bv8 32))))\n",[224,17424,17425],{"class":226,"line":356},[224,17426,16983],{},[224,17428,17429],{"class":226,"line":362},[224,17430,16988],{},[11,17432,17433,17434,3145,17437,17440,17441,17443,17444,17447],{},"注意这里用的是有符号比较（",[221,17435,17436],{},"bvsgt",[221,17438,17439],{},"bvslt","），因为 C 里的 ",[221,17442,242],{}," 是有符号的。一个可能的模型是 ",[221,17445,17446],{},"len = #x00000009","。对这个例子来说，两种建模的结论一样；但下一个例子就会看到，理论选错，结论可能完全相反。",[613,17449,17451],{"id":17450},"_53-整数溢出","5.3 整数溢出",[215,17453,17455],{"className":217,"code":17454,"language":219,"meta":186,"style":186},"void copy(char *src, unsigned int off, unsigned int len) {\n    char buf[64];\n    if (off + len \u003C= 64) {\n        memcpy(buf + off, src, len);\n    }\n}\n",[221,17456,17457,17494,17507,17525,17537,17541],{"__ignoreMap":186},[224,17458,17459,17461,17464,17466,17468,17470,17473,17475,17478,17481,17484,17486,17488,17490,17492],{"class":226,"line":227},[224,17460,231],{"class":230},[224,17462,17463],{"class":234}," copy",[224,17465,239],{"class":238},[224,17467,16527],{"class":230},[224,17469,16530],{"class":230},[224,17471,17472],{"class":245},"src",[224,17474,249],{"class":238},[224,17476,17477],{"class":230},"unsigned",[224,17479,17480],{"class":230}," int",[224,17482,17483],{"class":245}," off",[224,17485,249],{"class":238},[224,17487,17477],{"class":230},[224,17489,17480],{"class":230},[224,17491,16539],{"class":245},[224,17493,257],{"class":238},[224,17495,17496,17498,17500,17502,17505],{"class":226,"line":187},[224,17497,16546],{"class":230},[224,17499,16549],{"class":245},[224,17501,268],{"class":238},[224,17503,17504],{"class":271},"64",[224,17506,16556],{"class":238},[224,17508,17509,17511,17514,17516,17518,17520,17523],{"class":226,"line":290},[224,17510,313],{"class":230},[224,17512,17513],{"class":238}," (off ",[224,17515,319],{"class":230},[224,17517,16573],{"class":238},[224,17519,17159],{"class":230},[224,17521,17522],{"class":271}," 64",[224,17524,257],{"class":238},[224,17526,17527,17529,17532,17534],{"class":226,"line":310},[224,17528,16585],{"class":234},[224,17530,17531],{"class":238},"(buf ",[224,17533,319],{"class":230},[224,17535,17536],{"class":238}," off, src, len);\n",[224,17538,17539],{"class":226,"line":349},[224,17540,359],{"class":238},[224,17542,17543],{"class":226,"line":356},[224,17544,365],{"class":238},[11,17546,17547,17548,17551,17552,2887,17555,17557,17558,3298],{},"看起来检查很严：只有 ",[221,17549,17550],{},"off + len"," 不超过 64 才拷贝。但 ",[221,17553,17554],{},"off",[221,17556,16612],{}," 都是 32 位无符号整数，加法是",[378,17559,17560,17561],{},"模 ",[224,17562,17564,17582],{"className":17563,"translate":388},[392],[224,17565,17567],{"className":17566},[396],[398,17568,17569],{"xmlns":400},[403,17570,17571,17579],{},[406,17572,17573],{},[3248,17574,17575,17577],{},[419,17576,1447],{},[419,17578,13192],{},[447,17580,17581],{"encoding":449},"2^{32}",[224,17583,17585],{"className":17584,"ariaHidden":455},[454],[224,17586,17588,17591],{"className":17587},[459],[224,17589],{"className":17590,"style":5671},[463],[224,17592,17594,17597],{"className":17593},[468],[224,17595,1447],{"className":17596},[468],[224,17598,17600],{"className":17599},[750],[224,17601,17603],{"className":17602},[754],[224,17604,17606],{"className":17605},[759],[224,17607,17609],{"className":17608,"style":5671},[763],[224,17610,17611,17614],{"style":3288},[224,17612],{"className":17613,"style":772},[771],[224,17615,17617],{"className":17616},[776,777,778,779],[224,17618,17620],{"className":17619},[468,779],[224,17621,13192],{"className":17622},[468,779],[11,17624,17625],{},"攻击者可以取：",[224,17627,17629],{"className":17628,"translate":388},[387],[224,17630,17632,17662],{"className":17631,"translate":388},[392],[224,17633,17635],{"className":17634},[396],[398,17636,17637],{"xmlns":400,"display":401},[403,17638,17639,17659],{},[406,17640,17641,17643,17645,17648,17650,17652,17654,17656],{},[423,17642,17554],{},[412,17644,278],{},[423,17646,17647],{"mathvariant":13207},"0xFFFFFFF0",[412,17649,1877],{"separator":455},[473,17651],{"width":4247},[423,17653,16612],{},[412,17655,278],{},[423,17657,17658],{"mathvariant":13207},"0x20",[447,17660,17661],{"encoding":449},"\\text{off} = \\texttt{0xFFFFFFF0}, \\qquad \\text{len} = \\texttt{0x20}",[224,17663,17665,17686,17722],{"className":17664,"ariaHidden":455},[454],[224,17666,17668,17671,17677,17680,17683],{"className":17667},[459],[224,17669],{"className":17670,"style":7642},[463],[224,17672,17674],{"className":17673},[468,11373],[224,17675,17554],{"className":17676},[468],[224,17678],{"className":17679,"style":496},[473],[224,17681,278],{"className":17682},[500],[224,17684],{"className":17685,"style":496},[473],[224,17687,17689,17692,17698,17701,17704,17707,17713,17716,17719],{"className":17688},[459],[224,17690],{"className":17691,"style":2565},[463],[224,17693,17695],{"className":17694},[468,11373],[224,17696,17647],{"className":17697},[468,13405],[224,17699,1877],{"className":17700},[1917],[224,17702],{"className":17703,"style":4416},[473],[224,17705],{"className":17706,"style":724},[473],[224,17708,17710],{"className":17709},[468,11373],[224,17711,16612],{"className":17712},[468],[224,17714],{"className":17715,"style":496},[473],[224,17717,278],{"className":17718},[500],[224,17720],{"className":17721,"style":496},[473],[224,17723,17725,17728],{"className":17724},[459],[224,17726],{"className":17727,"style":13398},[463],[224,17729,17731],{"className":17730},[468,11373],[224,17732,17658],{"className":17733},[468,13405],[11,17735,17736],{},"计算一下：",[224,17738,17740],{"className":17739,"translate":388},[387],[224,17741,17743,17856],{"className":17742,"translate":388},[392],[224,17744,17746],{"className":17745},[396],[398,17747,17748],{"xmlns":400,"display":401},[403,17749,17750,17853],{},[5209,17751,17752,17782,17803],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,17753,17754,17766],{},[5218,17755,17756],{},[5221,17757,17758],{"scriptlevel":284,"displaystyle":455},[406,17759,17760,17762,17764],{},[423,17761,17554],{},[412,17763,319],{},[423,17765,16612],{},[5218,17767,17768],{},[5221,17769,17770],{"scriptlevel":284,"displaystyle":455},[406,17771,17772,17774,17776,17778,17780],{},[406,17773],{},[412,17775,278],{},[423,17777,17647],{"mathvariant":13207},[412,17779,319],{},[423,17781,17658],{"mathvariant":13207},[5215,17783,17784,17790],{},[5218,17785,17786],{},[5221,17787,17788],{"scriptlevel":284,"displaystyle":455},[406,17789],{},[5218,17791,17792],{},[5221,17793,17794],{"scriptlevel":284,"displaystyle":455},[406,17795,17796,17798,17800],{},[406,17797],{},[412,17799,278],{},[423,17801,17802],{"mathvariant":13207},"0x100000010",[5215,17804,17805,17811],{},[5218,17806,17807],{},[5221,17808,17809],{"scriptlevel":284,"displaystyle":455},[406,17810],{},[5218,17812,17813],{},[5221,17814,17815],{"scriptlevel":284,"displaystyle":455},[406,17816,17817,17819,17821,17824,17826,17829,17831,17833,17835,17843,17845,17851],{},[406,17818],{},[412,17820,10268],{},[423,17822,17823],{"mathvariant":13207},"0x10",[412,17825,278],{},[419,17827,17828],{},"16",[473,17830],{},[473,17832],{"width":691},[412,17834,239],{"stretchy":435},[406,17836,17837,17839,17841],{},[409,17838,13177],{"mathvariant":678},[409,17840,4127],{"mathvariant":678},[409,17842,13182],{"mathvariant":678},[473,17844],{"width":13185},[3248,17846,17847,17849],{},[419,17848,1447],{},[419,17850,13192],{},[412,17852,1716],{"stretchy":435},[447,17854,17855],{"encoding":449},"\\begin{aligned}\n\\text{off} + \\text{len}\n&= \\texttt{0xFFFFFFF0} + \\texttt{0x20} \\\\\n&= \\texttt{0x100000010} \\\\\n&\\equiv \\texttt{0x10} = 16 \\pmod{2^{32}}\n\\end{aligned}",[224,17857,17859],{"className":17858,"ariaHidden":455},[454],[224,17860,17862,17866],{"className":17861},[459],[224,17863],{"className":17864,"style":17865},[463],"height:4.2241em;vertical-align:-1.8621em;",[224,17867,17869],{"className":17868},[468],[224,17870,17872,17944],{"className":17871},[5209],[224,17873,17875],{"className":17874},[5331],[224,17876,17878,17935],{"className":17877},[754,755],[224,17879,17881,17932],{"className":17880},[759],[224,17882,17885,17914,17923],{"className":17883,"style":17884},[763],"height:2.3621em;",[224,17886,17887,17890],{"style":5344},[224,17888],{"className":17889,"style":2108},[771],[224,17891,17893,17899,17902,17905,17908],{"className":17892},[468],[224,17894,17896],{"className":17895},[468,11373],[224,17897,17554],{"className":17898},[468],[224,17900],{"className":17901,"style":474},[473],[224,17903,319],{"className":17904},[478],[224,17906],{"className":17907,"style":474},[473],[224,17909,17911],{"className":17910},[468,11373],[224,17912,16612],{"className":17913},[468],[224,17915,17917,17920],{"style":17916},"top:-3.0221em;",[224,17918],{"className":17919,"style":2108},[771],[224,17921],{"className":17922},[468],[224,17924,17926,17929],{"style":17925},"top:-1.4979em;",[224,17927],{"className":17928,"style":2108},[771],[224,17930],{"className":17931},[468],[224,17933,787],{"className":17934},[786],[224,17936,17938],{"className":17937},[759],[224,17939,17942],{"className":17940,"style":17941},[763],"height:1.8621em;",[224,17943],{},[224,17945,17947],{"className":17946},[5399],[224,17948,17950,18120],{"className":17949},[754,755],[224,17951,17953,18117],{"className":17952},[759],[224,17954,17956,17997,18023],{"className":17955,"style":17884},[763],[224,17957,17958,17961],{"style":5344},[224,17959],{"className":17960,"style":2108},[771],[224,17962,17964,17967,17970,17973,17976,17982,17985,17988,17991],{"className":17963},[468],[224,17965],{"className":17966},[468],[224,17968],{"className":17969,"style":496},[473],[224,17971,278],{"className":17972},[500],[224,17974],{"className":17975,"style":496},[473],[224,17977,17979],{"className":17978},[468,11373],[224,17980,17647],{"className":17981},[468,13405],[224,17983],{"className":17984,"style":474},[473],[224,17986,319],{"className":17987},[478],[224,17989],{"className":17990,"style":474},[473],[224,17992,17994],{"className":17993},[468,11373],[224,17995,17658],{"className":17996},[468,13405],[224,17998,17999,18002],{"style":17916},[224,18000],{"className":18001,"style":2108},[771],[224,18003,18005,18008,18011,18014,18017],{"className":18004},[468],[224,18006],{"className":18007},[468],[224,18009],{"className":18010,"style":496},[473],[224,18012,278],{"className":18013},[500],[224,18015],{"className":18016,"style":496},[473],[224,18018,18020],{"className":18019},[468,11373],[224,18021,17802],{"className":18022},[468,13405],[224,18024,18025,18028],{"style":17925},[224,18026],{"className":18027,"style":2108},[771],[224,18029,18031,18034,18037,18040,18043,18049,18052,18055,18058,18061,18064,18067,18070,18079,18082,18114],{"className":18030},[468],[224,18032],{"className":18033},[468],[224,18035],{"className":18036,"style":496},[473],[224,18038,10268],{"className":18039},[500],[224,18041],{"className":18042,"style":496},[473],[224,18044,18046],{"className":18045},[468,11373],[224,18047,17823],{"className":18048},[468,13405],[224,18050],{"className":18051,"style":496},[473],[224,18053,278],{"className":18054},[500],[224,18056],{"className":18057,"style":496},[473],[224,18059,17828],{"className":18060},[468],[224,18062],{"className":18063},[473,13263],[224,18065],{"className":18066,"style":734},[473],[224,18068,239],{"className":18069},[542],[224,18071,18073],{"className":18072},[468],[224,18074,18076],{"className":18075},[468],[224,18077,13286],{"className":18078},[468,1906],[224,18080],{"className":18081,"style":13290},[473],[224,18083,18085,18088],{"className":18084},[468],[224,18086,1447],{"className":18087},[468],[224,18089,18091],{"className":18090},[750],[224,18092,18094],{"className":18093},[754],[224,18095,18097],{"className":18096},[759],[224,18098,18100],{"className":18099,"style":5071},[763],[224,18101,18102,18105],{"style":5074},[224,18103],{"className":18104,"style":772},[771],[224,18106,18108],{"className":18107},[776,777,778,779],[224,18109,18111],{"className":18110},[468,779],[224,18112,13192],{"className":18113},[468,779],[224,18115,1716],{"className":18116},[549],[224,18118,787],{"className":18119},[786],[224,18121,18123],{"className":18122},[759],[224,18124,18126],{"className":18125,"style":17941},[763],[224,18127],{},[11,18129,18130,18182,18183,18186,18187,18189,18190,3117],{},[224,18131,18133,18151],{"className":18132,"translate":388},[392],[224,18134,18136],{"className":18135},[396],[398,18137,18138],{"xmlns":400},[403,18139,18140,18148],{},[406,18141,18142,18144,18146],{},[419,18143,17828],{},[412,18145,2070],{},[419,18147,17504],{},[447,18149,18150],{"encoding":449},"16 \\le 64",[224,18152,18154,18173],{"className":18153,"ariaHidden":455},[454],[224,18155,18157,18161,18164,18167,18170],{"className":18156},[459],[224,18158],{"className":18159,"style":18160},[463],"height:0.7804em;vertical-align:-0.136em;",[224,18162,17828],{"className":18163},[468],[224,18165],{"className":18166,"style":496},[473],[224,18168,2070],{"className":18169},[500],[224,18171],{"className":18172,"style":496},[473],[224,18174,18176,18179],{"className":18175},[459],[224,18177],{"className":18178,"style":510},[463],[224,18180,17504],{"className":18181},[468],"，检查通过。但 ",[221,18184,18185],{},"buf + off"," 指向的是缓冲区外大约 4GB 的位置，",[221,18188,16608],{}," 会往一个完全无关的地址写 32 字节。这是一个典型的",[378,18191,18192],{},"整数溢出导致的任意地址写",[11,18194,18195],{},"用位向量理论来问：",[215,18197,18199],{"className":16959,"code":18198,"language":16961,"meta":186,"style":186},"(set-logic QF_BV)\n(declare-const off (_ BitVec 32))\n(declare-const len (_ BitVec 32))\n(assert (bvule (bvadd off len) (_ bv64 32)))\n(assert (bvugt off (_ bv64 32)))\n(assert (bvule len (_ bv64 32)))\n(check-sat)\n(get-model)\n",[221,18200,18201,18205,18210,18214,18219,18224,18229,18233],{"__ignoreMap":186},[224,18202,18203],{"class":226,"line":227},[224,18204,17402],{},[224,18206,18207],{"class":226,"line":187},[224,18208,18209],{},"(declare-const off (_ BitVec 32))\n",[224,18211,18212],{"class":226,"line":290},[224,18213,17407],{},[224,18215,18216],{"class":226,"line":310},[224,18217,18218],{},"(assert (bvule (bvadd off len) (_ bv64 32)))\n",[224,18220,18221],{"class":226,"line":349},[224,18222,18223],{},"(assert (bvugt off (_ bv64 32)))\n",[224,18225,18226],{"class":226,"line":356},[224,18227,18228],{},"(assert (bvule len (_ bv64 32)))\n",[224,18230,18231],{"class":226,"line":362},[224,18232,16983],{},[224,18234,18236],{"class":226,"line":18235},8,[224,18237,16988],{},[11,18239,18240],{},"求解器返回 sat，模型可能是：",[215,18242,18245],{"className":18243,"code":18244,"language":11373},[11371],"off = #xfffffff0\nlen = #x00000020\n",[221,18246,18244],{"__ignoreMap":186},[11,18248,18249],{},"漏洞可达。",[11,18251,18252,18253,2887,18255,18257],{},"如果换一种建模，把 ",[221,18254,17554],{},[221,18256,16612],{}," 当成数学整数：",[224,18259,18261],{"className":18260,"translate":388},[387],[224,18262,18264,18310],{"className":18263,"translate":388},[392],[224,18265,18267],{"className":18266},[396],[398,18268,18269],{"xmlns":400,"display":401},[403,18270,18271,18307],{},[406,18272,18273,18275,18277,18279,18281,18283,18285,18287,18289,18291,18293,18295,18297,18299,18301,18303,18305],{},[423,18274,17554],{},[412,18276,319],{},[423,18278,16612],{},[412,18280,2070],{},[419,18282,17504],{},[423,18284,425],{},[412,18286,428],{},[423,18288,425],{},[423,18290,17554],{},[412,18292,325],{},[419,18294,17504],{},[423,18296,425],{},[412,18298,428],{},[423,18300,425],{},[423,18302,16612],{},[412,18304,9766],{},[419,18306,284],{},[447,18308,18309],{"encoding":449},"\\text{off} + \\text{len} \\le 64 \\;\\wedge\\; \\text{off} > 64 \\;\\wedge\\; \\text{len} \\ge 0",[224,18311,18313,18334,18355,18379,18400,18424,18445],{"className":18312,"ariaHidden":455},[454],[224,18314,18316,18319,18325,18328,18331],{"className":18315},[459],[224,18317],{"className":18318,"style":4348},[463],[224,18320,18322],{"className":18321},[468,11373],[224,18323,17554],{"className":18324},[468],[224,18326],{"className":18327,"style":474},[473],[224,18329,319],{"className":18330},[478],[224,18332],{"className":18333,"style":474},[473],[224,18335,18337,18340,18346,18349,18352],{"className":18336},[459],[224,18338],{"className":18339,"style":17297},[463],[224,18341,18343],{"className":18342},[468,11373],[224,18344,16612],{"className":18345},[468],[224,18347],{"className":18348,"style":496},[473],[224,18350,2070],{"className":18351},[500],[224,18353],{"className":18354,"style":496},[473],[224,18356,18358,18361,18364,18367,18370,18373,18376],{"className":18357},[459],[224,18359],{"className":18360,"style":510},[463],[224,18362,17504],{"className":18363},[468],[224,18365],{"className":18366,"style":496},[473],[224,18368],{"className":18369,"style":474},[473],[224,18371,428],{"className":18372},[478],[224,18374],{"className":18375,"style":496},[473],[224,18377],{"className":18378,"style":474},[473],[224,18380,18382,18385,18391,18394,18397],{"className":18381},[459],[224,18383],{"className":18384,"style":16682},[463],[224,18386,18388],{"className":18387},[468,11373],[224,18389,17554],{"className":18390},[468],[224,18392],{"className":18393,"style":496},[473],[224,18395,325],{"className":18396},[500],[224,18398],{"className":18399,"style":496},[473],[224,18401,18403,18406,18409,18412,18415,18418,18421],{"className":18402},[459],[224,18404],{"className":18405,"style":510},[463],[224,18407,17504],{"className":18408},[468],[224,18410],{"className":18411,"style":496},[473],[224,18413],{"className":18414,"style":474},[473],[224,18416,428],{"className":18417},[478],[224,18419],{"className":18420,"style":496},[473],[224,18422],{"className":18423,"style":474},[473],[224,18425,18427,18430,18436,18439,18442],{"className":18426},[459],[224,18428],{"className":18429,"style":17297},[463],[224,18431,18433],{"className":18432},[468,11373],[224,18434,16612],{"className":18435},[468],[224,18437],{"className":18438,"style":496},[473],[224,18440,9766],{"className":18441},[500],[224,18443],{"className":18444,"style":496},[473],[224,18446,18448,18451],{"className":18447},[459],[224,18449],{"className":18450,"style":510},[463],[224,18452,284],{"className":18453},[468],[11,18455,18456,18457,18459,18460,18539],{},"这个公式是 ",[378,18458,6650],{},"，因为 ",[224,18461,18463,18485],{"className":18462,"translate":388},[392],[224,18464,18466],{"className":18465},[396],[398,18467,18468],{"xmlns":400},[403,18469,18470,18482],{},[406,18471,18472,18474,18476,18478,18480],{},[423,18473,17554],{},[412,18475,319],{},[423,18477,16612],{},[412,18479,325],{},[419,18481,17504],{},[447,18483,18484],{"encoding":449},"\\text{off} + \\text{len} > 64",[224,18486,18488,18509,18530],{"className":18487,"ariaHidden":455},[454],[224,18489,18491,18494,18500,18503,18506],{"className":18490},[459],[224,18492],{"className":18493,"style":4348},[463],[224,18495,18497],{"className":18496},[468,11373],[224,18498,17554],{"className":18499},[468],[224,18501],{"className":18502,"style":474},[473],[224,18504,319],{"className":18505},[478],[224,18507],{"className":18508,"style":474},[473],[224,18510,18512,18515,18521,18524,18527],{"className":18511},[459],[224,18513],{"className":18514,"style":16682},[463],[224,18516,18518],{"className":18517},[468,11373],[224,18519,16612],{"className":18520},[468],[224,18522],{"className":18523,"style":496},[473],[224,18525,325],{"className":18526},[500],[224,18528],{"className":18529,"style":496},[473],[224,18531,18533,18536],{"className":18532},[459],[224,18534],{"className":18535,"style":510},[463],[224,18537,17504],{"className":18538},[468]," 直接和第一条矛盾。工具会得出结论「没有漏洞」，而实际上漏洞就在那里。",[11,18541,18542,18545,18546,18549],{},[378,18543,18544],{},"同一个程序，两种理论，两个相反的答案。"," 这就是为什么漏洞检测工具必须精确建模目标语言的整数语义：C 的 ",[221,18547,18548],{},"unsigned int"," 是模运算，不是数学整数。",[613,18551,18553],{"id":18552},"_54-空指针","5.4 空指针",[11,18555,18556],{},"静态分析经常报「可能的空指针解引用」。这类告警里有很多是误报，SMT 可以用来过滤。",[215,18558,18560],{"className":217,"code":18559,"language":219,"meta":186,"style":186},"int read_flag(struct node *p) {\n    if (p == NULL) return 0;\n    return p->flag;     \u002F\u002F 静态分析：p 可能为空？\n}\n",[221,18561,18562,18584,18607,18618],{"__ignoreMap":186},[224,18563,18564,18566,18569,18571,18574,18577,18580,18582],{"class":226,"line":227},[224,18565,242],{"class":230},[224,18567,18568],{"class":234}," read_flag",[224,18570,239],{"class":238},[224,18572,18573],{"class":230},"struct",[224,18575,18576],{"class":238}," node ",[224,18578,18579],{"class":230},"*",[224,18581,11],{"class":245},[224,18583,257],{"class":238},[224,18585,18586,18588,18591,18593,18596,18599,18602,18605],{"class":226,"line":187},[224,18587,313],{"class":230},[224,18589,18590],{"class":238}," (p ",[224,18592,342],{"class":230},[224,18594,18595],{"class":271}," NULL",[224,18597,18598],{"class":238},") ",[224,18600,18601],{"class":230},"return",[224,18603,18604],{"class":271}," 0",[224,18606,307],{"class":238},[224,18608,18609,18612,18615],{"class":226,"line":290},[224,18610,18611],{"class":230},"    return",[224,18613,18614],{"class":238}," p->flag;",[224,18616,18617],{"class":352},"     \u002F\u002F 静态分析：p 可能为空？\n",[224,18619,18620],{"class":226,"line":310},[224,18621,365],{"class":238},[11,18623,18624,18625,18628,18629,18632],{},"工具在 ",[221,18626,18627],{},"p->flag"," 处收集路径条件。走到这一行，必然经过了 ",[221,18630,18631],{},"p == NULL"," 为假的分支，所以路径条件里包含：",[224,18634,18636],{"className":18635,"translate":388},[387],[224,18637,18639,18658],{"className":18638,"translate":388},[392],[224,18640,18642],{"className":18641},[396],[398,18643,18644],{"xmlns":400,"display":401},[403,18645,18646,18655],{},[406,18647,18648,18650,18652],{},[409,18649,11],{},[412,18651,2048],{"mathvariant":678},[423,18653,18654],{"mathvariant":13207},"NULL",[447,18656,18657],{"encoding":449},"p \\ne \\texttt{NULL}",[224,18659,18661,18712],{"className":18660,"ariaHidden":455},[454],[224,18662,18664,18667,18670,18673,18709],{"className":18663},[459],[224,18665],{"className":18666,"style":2565},[463],[224,18668,11],{"className":18669},[468,469],[224,18671],{"className":18672,"style":496},[473],[224,18674,18676,18703,18706],{"className":18675},[500],[224,18677,18679],{"className":18678},[500],[224,18680,18682],{"className":18681},[468,2553],[224,18683,18685],{"className":18684},[2557],[224,18686,18688,18691,18700],{"className":18687},[2561],[224,18689],{"className":18690,"style":2565},[463],[224,18692,18694],{"className":18693},[2569],[224,18695,18697],{"className":18696},[468],[224,18698,2576],{"className":18699},[500],[224,18701],{"className":18702},[2580],[224,18704],{"className":18705},[473,2584],[224,18707,278],{"className":18708},[500],[224,18710],{"className":18711,"style":496},[473],[224,18713,18715,18718],{"className":18714},[459],[224,18716],{"className":18717,"style":13398},[463],[224,18719,18721],{"className":18720},[468,11373],[224,18722,18654],{"className":18723},[468,13405],[11,18725,18726,18727,18729],{},"静态分析如果只做局部启发式，可能会说「",[221,18728,11],{}," 可能是空指针」。这时问 SMT：",[224,18731,18733],{"className":18732,"translate":388},[387],[224,18734,18736,18766],{"className":18735,"translate":388},[392],[224,18737,18739],{"className":18738},[396],[398,18740,18741],{"xmlns":400,"display":401},[403,18742,18743,18763],{},[406,18744,18745,18747,18749,18751,18753,18755,18757,18759,18761],{},[409,18746,11],{},[412,18748,278],{},[423,18750,18654],{"mathvariant":13207},[423,18752,425],{},[412,18754,428],{},[423,18756,425],{},[409,18758,11],{},[412,18760,2048],{"mathvariant":678},[423,18762,18654],{"mathvariant":13207},[447,18764,18765],{"encoding":449},"p = \\texttt{NULL} \\;\\wedge\\; p \\ne \\texttt{NULL}",[224,18767,18769,18787,18814,18865],{"className":18768,"ariaHidden":455},[454],[224,18770,18772,18775,18778,18781,18784],{"className":18771},[459],[224,18773],{"className":18774,"style":806},[463],[224,18776,11],{"className":18777},[468,469],[224,18779],{"className":18780,"style":496},[473],[224,18782,278],{"className":18783},[500],[224,18785],{"className":18786,"style":496},[473],[224,18788,18790,18793,18799,18802,18805,18808,18811],{"className":18789},[459],[224,18791],{"className":18792,"style":13398},[463],[224,18794,18796],{"className":18795},[468,11373],[224,18797,18654],{"className":18798},[468,13405],[224,18800],{"className":18801,"style":496},[473],[224,18803],{"className":18804,"style":474},[473],[224,18806,428],{"className":18807},[478],[224,18809],{"className":18810,"style":496},[473],[224,18812],{"className":18813,"style":474},[473],[224,18815,18817,18820,18823,18826,18862],{"className":18816},[459],[224,18818],{"className":18819,"style":2565},[463],[224,18821,11],{"className":18822},[468,469],[224,18824],{"className":18825,"style":496},[473],[224,18827,18829,18856,18859],{"className":18828},[500],[224,18830,18832],{"className":18831},[500],[224,18833,18835],{"className":18834},[468,2553],[224,18836,18838],{"className":18837},[2557],[224,18839,18841,18844,18853],{"className":18840},[2561],[224,18842],{"className":18843,"style":2565},[463],[224,18845,18847],{"className":18846},[2569],[224,18848,18850],{"className":18849},[468],[224,18851,2576],{"className":18852},[500],[224,18854],{"className":18855},[2580],[224,18857],{"className":18858},[473,2584],[224,18860,278],{"className":18861},[500],[224,18863],{"className":18864,"style":496},[473],[224,18866,18868,18871],{"className":18867},[459],[224,18869],{"className":18870,"style":13398},[463],[224,18872,18874],{"className":18873},[468,11373],[224,18875,18654],{"className":18876},[468,13405],[11,18878,18879,18880,18882],{},"这是明显的矛盾，返回 ",[378,18881,6650],{},"，告警被排除。",[11,18884,18885],{},"反过来，如果代码是：",[215,18887,18889],{"className":217,"code":18888,"language":219,"meta":186,"style":186},"int read_flag(struct node *p) {\n    if (p->flag == 1) {   \u002F\u002F 这里没有检查 p\n        return 1;\n    }\n    return 0;\n}\n",[221,18890,18891,18909,18927,18936,18940,18948],{"__ignoreMap":186},[224,18892,18893,18895,18897,18899,18901,18903,18905,18907],{"class":226,"line":227},[224,18894,242],{"class":230},[224,18896,18568],{"class":234},[224,18898,239],{"class":238},[224,18900,18573],{"class":230},[224,18902,18576],{"class":238},[224,18904,18579],{"class":230},[224,18906,11],{"class":245},[224,18908,257],{"class":238},[224,18910,18911,18913,18916,18918,18921,18924],{"class":226,"line":187},[224,18912,313],{"class":230},[224,18914,18915],{"class":238}," (p->flag ",[224,18917,342],{"class":230},[224,18919,18920],{"class":271}," 1",[224,18922,18923],{"class":238},") {",[224,18925,18926],{"class":352},"   \u002F\u002F 这里没有检查 p\n",[224,18928,18929,18932,18934],{"class":226,"line":290},[224,18930,18931],{"class":230},"        return",[224,18933,18920],{"class":271},[224,18935,307],{"class":238},[224,18937,18938],{"class":226,"line":310},[224,18939,359],{"class":238},[224,18941,18942,18944,18946],{"class":226,"line":349},[224,18943,18611],{"class":230},[224,18945,18604],{"class":271},[224,18947,307],{"class":238},[224,18949,18950],{"class":226,"line":356},[224,18951,365],{"class":238},[11,18953,18954,18955,19041],{},"路径条件里没有 ",[224,18956,18958,18975],{"className":18957,"translate":388},[392],[224,18959,18961],{"className":18960},[396],[398,18962,18963],{"xmlns":400},[403,18964,18965,18973],{},[406,18966,18967,18969,18971],{},[409,18968,11],{},[412,18970,2048],{"mathvariant":678},[423,18972,18654],{"mathvariant":13207},[447,18974,18657],{"encoding":449},[224,18976,18978,19029],{"className":18977,"ariaHidden":455},[454],[224,18979,18981,18984,18987,18990,19026],{"className":18980},[459],[224,18982],{"className":18983,"style":2565},[463],[224,18985,11],{"className":18986},[468,469],[224,18988],{"className":18989,"style":496},[473],[224,18991,18993,19020,19023],{"className":18992},[500],[224,18994,18996],{"className":18995},[500],[224,18997,18999],{"className":18998},[468,2553],[224,19000,19002],{"className":19001},[2557],[224,19003,19005,19008,19017],{"className":19004},[2561],[224,19006],{"className":19007,"style":2565},[463],[224,19009,19011],{"className":19010},[2569],[224,19012,19014],{"className":19013},[468],[224,19015,2576],{"className":19016},[500],[224,19018],{"className":19019},[2580],[224,19021],{"className":19022},[473,2584],[224,19024,278],{"className":19025},[500],[224,19027],{"className":19028,"style":496},[473],[224,19030,19032,19035],{"className":19031},[459],[224,19033],{"className":19034,"style":13398},[463],[224,19036,19038],{"className":19037},[468,11373],[224,19039,18654],{"className":19040},[468,13405],"，查询：",[224,19043,19045],{"className":19044,"translate":388},[387],[224,19046,19048,19066],{"className":19047,"translate":388},[392],[224,19049,19051],{"className":19050},[396],[398,19052,19053],{"xmlns":400,"display":401},[403,19054,19055,19063],{},[406,19056,19057,19059,19061],{},[409,19058,11],{},[412,19060,278],{},[423,19062,18654],{"mathvariant":13207},[447,19064,19065],{"encoding":449},"p = \\texttt{NULL}",[224,19067,19069,19087],{"className":19068,"ariaHidden":455},[454],[224,19070,19072,19075,19078,19081,19084],{"className":19071},[459],[224,19073],{"className":19074,"style":806},[463],[224,19076,11],{"className":19077},[468,469],[224,19079],{"className":19080,"style":496},[473],[224,19082,278],{"className":19083},[500],[224,19085],{"className":19086,"style":496},[473],[224,19088,19090,19093],{"className":19089},[459],[224,19091],{"className":19092,"style":13398},[463],[224,19094,19096],{"className":19095},[468,11373],[224,19097,18654],{"className":19098},[468,13405],[11,19100,19101,19102,19104,19105,19108],{},"返回 ",[378,19103,599],{},"，模型给出 ",[221,19106,19107],{},"p = NULL","。这是一条真实可达的空指针路径，告警保留。",[11,19110,19111,19112,19115],{},"这个用法体现了一个重要原则：",[378,19113,19114],{},"SMT 不负责发现漏洞，它负责判断「这条路径到底可不可能」","。发现可疑点靠前端分析，排除误报靠 SMT。",[613,19117,19119],{"id":19118},"_55-数组越界","5.5 数组越界",[215,19121,19123],{"className":217,"code":19122,"language":219,"meta":186,"style":186},"int get(int *a, int i) {\n    if (i \u003C 100) {\n        return a[i];     \u002F\u002F i 可能为负\n    }\n    return -1;\n}\n",[221,19124,19125,19149,19163,19175,19179,19190],{"__ignoreMap":186},[224,19126,19127,19129,19132,19134,19136,19138,19140,19142,19144,19147],{"class":226,"line":227},[224,19128,242],{"class":230},[224,19130,19131],{"class":234}," get",[224,19133,239],{"class":238},[224,19135,242],{"class":230},[224,19137,16530],{"class":230},[224,19139,15],{"class":245},[224,19141,249],{"class":238},[224,19143,242],{"class":230},[224,19145,19146],{"class":245}," i",[224,19148,257],{"class":238},[224,19150,19151,19153,19156,19158,19161],{"class":226,"line":187},[224,19152,313],{"class":230},[224,19154,19155],{"class":238}," (i ",[224,19157,1577],{"class":230},[224,19159,19160],{"class":271}," 100",[224,19162,257],{"class":238},[224,19164,19165,19167,19169,19172],{"class":226,"line":290},[224,19166,18931],{"class":230},[224,19168,265],{"class":245},[224,19170,19171],{"class":238},"[i];",[224,19173,19174],{"class":352},"     \u002F\u002F i 可能为负\n",[224,19176,19177],{"class":226,"line":310},[224,19178,359],{"class":238},[224,19180,19181,19183,19186,19188],{"class":226,"line":349},[224,19182,18611],{"class":230},[224,19184,19185],{"class":230}," -",[224,19187,1438],{"class":271},[224,19189,307],{"class":238},[224,19191,19192],{"class":226,"line":356},[224,19193,365],{"class":238},[11,19195,19196],{},"只检查了上界，没检查下界。问 SMT：",[224,19198,19200],{"className":19199,"translate":388},[387],[224,19201,19203,19233],{"className":19202,"translate":388},[392],[224,19204,19206],{"className":19205},[396],[398,19207,19208],{"xmlns":400,"display":401},[403,19209,19210,19230],{},[406,19211,19212,19214,19216,19218,19220,19222,19224,19226,19228],{},[409,19213,1880],{},[412,19215,1577],{},[419,19217,272],{},[423,19219,425],{},[412,19221,428],{},[423,19223,425],{},[409,19225,1880],{},[412,19227,1577],{},[419,19229,284],{},[447,19231,19232],{"encoding":449},"i \u003C 100 \\;\\wedge\\; i \u003C 0",[224,19234,19236,19255,19279,19297],{"className":19235,"ariaHidden":455},[454],[224,19237,19239,19243,19246,19249,19252],{"className":19238},[459],[224,19240],{"className":19241,"style":19242},[463],"height:0.6986em;vertical-align:-0.0391em;",[224,19244,1880],{"className":19245},[468,469],[224,19247],{"className":19248,"style":496},[473],[224,19250,1577],{"className":19251},[500],[224,19253],{"className":19254,"style":496},[473],[224,19256,19258,19261,19264,19267,19270,19273,19276],{"className":19257},[459],[224,19259],{"className":19260,"style":510},[463],[224,19262,272],{"className":19263},[468],[224,19265],{"className":19266,"style":496},[473],[224,19268],{"className":19269,"style":474},[473],[224,19271,428],{"className":19272},[478],[224,19274],{"className":19275,"style":496},[473],[224,19277],{"className":19278,"style":474},[473],[224,19280,19282,19285,19288,19291,19294],{"className":19281},[459],[224,19283],{"className":19284,"style":19242},[463],[224,19286,1880],{"className":19287},[468,469],[224,19289],{"className":19290,"style":496},[473],[224,19292,1577],{"className":19293},[500],[224,19295],{"className":19296,"style":496},[473],[224,19298,19300,19303],{"className":19299},[459],[224,19301],{"className":19302,"style":510},[463],[224,19304,284],{"className":19305},[468],[11,19307,19308,19309,3117,19312,19315],{},"返回 sat，例如 ",[221,19310,19311],{},"i = -1",[221,19313,19314],{},"a[-1]"," 是一次越界读。",[11,19317,19318],{},"如果加上下界检查：",[215,19320,19322],{"className":217,"code":19321,"language":219,"meta":186,"style":186},"if (i >= 0 && i \u003C 100) {\n    return a[i];\n}\n",[221,19323,19324,19346,19355],{"__ignoreMap":186},[224,19325,19326,19328,19330,19333,19335,19337,19340,19342,19344],{"class":226,"line":227},[224,19327,17146],{"class":230},[224,19329,19155],{"class":238},[224,19331,19332],{"class":230},">=",[224,19334,18604],{"class":271},[224,19336,331],{"class":230},[224,19338,19339],{"class":238}," i ",[224,19341,1577],{"class":230},[224,19343,19160],{"class":271},[224,19345,257],{"class":238},[224,19347,19348,19350,19352],{"class":226,"line":187},[224,19349,18611],{"class":230},[224,19351,265],{"class":245},[224,19353,19354],{"class":238},"[i];\n",[224,19356,19357],{"class":226,"line":290},[224,19358,365],{"class":238},[11,19360,19361],{},"约束变成：",[224,19363,19365],{"className":19364,"translate":388},[387],[224,19366,19368,19410],{"className":19367,"translate":388},[392],[224,19369,19371],{"className":19370},[396],[398,19372,19373],{"xmlns":400,"display":401},[403,19374,19375,19407],{},[406,19376,19377,19379,19381,19383,19385,19387,19389,19391,19393,19395,19397,19399,19401,19403,19405],{},[409,19378,1880],{},[412,19380,9766],{},[419,19382,284],{},[423,19384,425],{},[412,19386,428],{},[423,19388,425],{},[409,19390,1880],{},[412,19392,1577],{},[419,19394,272],{},[423,19396,425],{},[412,19398,428],{},[423,19400,425],{},[409,19402,1880],{},[412,19404,1577],{},[419,19406,284],{},[447,19408,19409],{"encoding":449},"i \\ge 0 \\;\\wedge\\; i \u003C 100 \\;\\wedge\\; i \u003C 0",[224,19411,19413,19432,19456,19474,19498,19516],{"className":19412,"ariaHidden":455},[454],[224,19414,19416,19420,19423,19426,19429],{"className":19415},[459],[224,19417],{"className":19418,"style":19419},[463],"height:0.7955em;vertical-align:-0.136em;",[224,19421,1880],{"className":19422},[468,469],[224,19424],{"className":19425,"style":496},[473],[224,19427,9766],{"className":19428},[500],[224,19430],{"className":19431,"style":496},[473],[224,19433,19435,19438,19441,19444,19447,19450,19453],{"className":19434},[459],[224,19436],{"className":19437,"style":510},[463],[224,19439,284],{"className":19440},[468],[224,19442],{"className":19443,"style":496},[473],[224,19445],{"className":19446,"style":474},[473],[224,19448,428],{"className":19449},[478],[224,19451],{"className":19452,"style":496},[473],[224,19454],{"className":19455,"style":474},[473],[224,19457,19459,19462,19465,19468,19471],{"className":19458},[459],[224,19460],{"className":19461,"style":19242},[463],[224,19463,1880],{"className":19464},[468,469],[224,19466],{"className":19467,"style":496},[473],[224,19469,1577],{"className":19470},[500],[224,19472],{"className":19473,"style":496},[473],[224,19475,19477,19480,19483,19486,19489,19492,19495],{"className":19476},[459],[224,19478],{"className":19479,"style":510},[463],[224,19481,272],{"className":19482},[468],[224,19484],{"className":19485,"style":496},[473],[224,19487],{"className":19488,"style":474},[473],[224,19490,428],{"className":19491},[478],[224,19493],{"className":19494,"style":496},[473],[224,19496],{"className":19497,"style":474},[473],[224,19499,19501,19504,19507,19510,19513],{"className":19500},[459],[224,19502],{"className":19503,"style":19242},[463],[224,19505,1880],{"className":19506},[468,469],[224,19508],{"className":19509,"style":496},[473],[224,19511,1577],{"className":19512},[500],[224,19514],{"className":19515,"style":496},[473],[224,19517,19519,19522],{"className":19518},[459],[224,19520],{"className":19521,"style":510},[463],[224,19523,284],{"className":19524},[468],[11,19526,19101,19527,19529],{},[378,19528,6650],{},"，越界路径被排除。",[11,19531,19532,19533,19536,19537,19540],{},"这个例子还可以扩展成带数组理论的版本。比如要验证「下标不同时，写入 ",[221,19534,19535],{},"a[j]"," 不会影响对 ",[221,19538,19539],{},"a[i]"," 的读取」，可以把「影响了」写成公式，再问它是否可满足：",[224,19542,19544],{"className":19543,"translate":388},[387],[224,19545,19547,19645],{"className":19546,"translate":388},[392],[224,19548,19550],{"className":19549},[396],[398,19551,19552],{"xmlns":400,"display":401},[403,19553,19554,19642],{},[406,19555,19556,19558,19560,19562,19564,19566,19568,19582,19584,19596,19598,19600,19602,19604,19606,19608,19610,19612,19614,19616,19618,19632,19634,19636,19638,19640],{},[409,19557,1880],{},[412,19559,2048],{"mathvariant":678},[409,19561,8101],{},[423,19563,425],{},[412,19565,428],{},[423,19567,425],{},[406,19569,19570,19572,19574,19576,19578,19580],{},[409,19571,1858],{"mathvariant":678},[409,19573,1861],{"mathvariant":678},[409,19575,1864],{"mathvariant":678},[409,19577,1861],{"mathvariant":678},[409,19579,219],{"mathvariant":678},[409,19581,1435],{"mathvariant":678},[412,19583,239],{"fence":455,"stretchy":455,"minsize":2004,"maxsize":2004},[406,19585,19586,19588,19590,19592,19594],{},[409,19587,1858],{"mathvariant":678},[409,19589,1435],{"mathvariant":678},[409,19591,4127],{"mathvariant":678},[409,19593,2061],{"mathvariant":678},[409,19595,1861],{"mathvariant":678},[412,19597,239],{"stretchy":435},[409,19599,15],{},[412,19601,1877],{"separator":455},[409,19603,8101],{},[412,19605,1877],{"separator":455},[409,19607,1887],{},[412,19609,1716],{"stretchy":435},[412,19611,1877],{"separator":455},[409,19613,1880],{},[412,19615,1716],{"fence":455,"stretchy":455,"minsize":2004,"maxsize":2004},[412,19617,2048],{"mathvariant":678},[406,19619,19620,19622,19624,19626,19628,19630],{},[409,19621,1858],{"mathvariant":678},[409,19623,1861],{"mathvariant":678},[409,19625,1864],{"mathvariant":678},[409,19627,1861],{"mathvariant":678},[409,19629,219],{"mathvariant":678},[409,19631,1435],{"mathvariant":678},[412,19633,239],{"stretchy":435},[409,19635,15],{},[412,19637,1877],{"separator":455},[409,19639,1880],{},[412,19641,1716],{"stretchy":435},[447,19643,19644],{"encoding":449},"i \\ne j \\;\\wedge\\; \\mathrm{select}\\bigl(\\mathrm{store}(a, j, v), i\\bigr) \\ne \\mathrm{select}(a, i)",[224,19646,19648,19699,19723,19831],{"className":19647,"ariaHidden":455},[454],[224,19649,19651,19654,19657,19660,19696],{"className":19650},[459],[224,19652],{"className":19653,"style":2565},[463],[224,19655,1880],{"className":19656},[468,469],[224,19658],{"className":19659,"style":496},[473],[224,19661,19663,19690,19693],{"className":19662},[500],[224,19664,19666],{"className":19665},[500],[224,19667,19669],{"className":19668},[468,2553],[224,19670,19672],{"className":19671},[2557],[224,19673,19675,19678,19687],{"className":19674},[2561],[224,19676],{"className":19677,"style":2565},[463],[224,19679,19681],{"className":19680},[2569],[224,19682,19684],{"className":19683},[468],[224,19685,2576],{"className":19686},[500],[224,19688],{"className":19689},[2580],[224,19691],{"className":19692},[473,2584],[224,19694,278],{"className":19695},[500],[224,19697],{"className":19698,"style":496},[473],[224,19700,19702,19705,19708,19711,19714,19717,19720],{"className":19701},[459],[224,19703],{"className":19704,"style":13794},[463],[224,19706,8101],{"className":19707,"style":8251},[468,469],[224,19709],{"className":19710,"style":496},[473],[224,19712],{"className":19713,"style":474},[473],[224,19715,428],{"className":19716},[478],[224,19718],{"className":19719,"style":496},[473],[224,19721],{"className":19722,"style":474},[473],[224,19724,19726,19729,19735,19741,19747,19750,19753,19756,19759,19762,19765,19768,19771,19774,19777,19780,19783,19789,19792,19828],{"className":19725},[459],[224,19727],{"className":19728,"style":2663},[463],[224,19730,19732],{"className":19731},[468],[224,19733,1907],{"className":19734},[468,1906],[224,19736,19738],{"className":19737},[542],[224,19739,239],{"className":19740},[2363,2364],[224,19742,19744],{"className":19743},[468],[224,19745,2890],{"className":19746},[468,1906],[224,19748,239],{"className":19749},[542],[224,19751,15],{"className":19752},[468,469],[224,19754,1877],{"className":19755},[1917],[224,19757],{"className":19758,"style":724},[473],[224,19760,8101],{"className":19761,"style":8251},[468,469],[224,19763,1877],{"className":19764},[1917],[224,19766],{"className":19767,"style":724},[473],[224,19769,1887],{"className":19770,"style":492},[468,469],[224,19772,1716],{"className":19773},[549],[224,19775,1877],{"className":19776},[1917],[224,19778],{"className":19779,"style":724},[473],[224,19781,1880],{"className":19782},[468,469],[224,19784,19786],{"className":19785},[549],[224,19787,1716],{"className":19788},[2363,2364],[224,19790],{"className":19791,"style":496},[473],[224,19793,19795,19822,19825],{"className":19794},[500],[224,19796,19798],{"className":19797},[500],[224,19799,19801],{"className":19800},[468,2553],[224,19802,19804],{"className":19803},[2557],[224,19805,19807,19810,19819],{"className":19806},[2561],[224,19808],{"className":19809,"style":2565},[463],[224,19811,19813],{"className":19812},[2569],[224,19814,19816],{"className":19815},[468],[224,19817,2576],{"className":19818},[500],[224,19820],{"className":19821},[2580],[224,19823],{"className":19824},[473,2584],[224,19826,278],{"className":19827},[500],[224,19829],{"className":19830,"style":496},[473],[224,19832,19834,19837,19843,19846,19849,19852,19855,19858],{"className":19833},[459],[224,19835],{"className":19836,"style":535},[463],[224,19838,19840],{"className":19839},[468],[224,19841,1907],{"className":19842},[468,1906],[224,19844,239],{"className":19845},[542],[224,19847,15],{"className":19848},[468,469],[224,19850,1877],{"className":19851},[1917],[224,19853],{"className":19854,"style":724},[473],[224,19856,1880],{"className":19857},[468,469],[224,19859,1716],{"className":19860},[549],[11,19862,19863,19864,19948,19949,19951],{},"数组理论会展开 read-over-write 公理，算术理论会处理 ",[224,19865,19867,19885],{"className":19866,"translate":388},[392],[224,19868,19870],{"className":19869},[396],[398,19871,19872],{"xmlns":400},[403,19873,19874,19882],{},[406,19875,19876,19878,19880],{},[409,19877,1880],{},[412,19879,2048],{"mathvariant":678},[409,19881,8101],{},[447,19883,19884],{"encoding":449},"i \\ne j",[224,19886,19888,19939],{"className":19887,"ariaHidden":455},[454],[224,19889,19891,19894,19897,19900,19936],{"className":19890},[459],[224,19892],{"className":19893,"style":2565},[463],[224,19895,1880],{"className":19896},[468,469],[224,19898],{"className":19899,"style":496},[473],[224,19901,19903,19930,19933],{"className":19902},[500],[224,19904,19906],{"className":19905},[500],[224,19907,19909],{"className":19908},[468,2553],[224,19910,19912],{"className":19911},[2557],[224,19913,19915,19918,19927],{"className":19914},[2561],[224,19916],{"className":19917,"style":2565},[463],[224,19919,19921],{"className":19920},[2569],[224,19922,19924],{"className":19923},[468],[224,19925,2576],{"className":19926},[500],[224,19928],{"className":19929},[2580],[224,19931],{"className":19932},[473,2584],[224,19934,278],{"className":19935},[500],[224,19937],{"className":19938,"style":496},[473],[224,19940,19942,19945],{"className":19941},[459],[224,19943],{"className":19944,"style":13794},[463],[224,19946,8101],{"className":19947,"style":8251},[468,469],"，两个理论协作给出 ",[378,19950,6650],{},"——也就是说，「写入影响了另一个下标」的情况不可能发生。这正是 3.3 节里理论组合的实际用武之地。",[613,19953,19955],{"id":19954},"_56-完整流程","5.6 完整流程",[11,19957,19958],{},"把上面的步骤串起来，一个典型的漏洞检测流程是这样的：",[3101,19960,19961],{},[3104,19962],{"height":19963,"width":19964,"style":3108,"data-d2-theme":3109,"alt":186,"src":19965},240,2053,"data:image\u002Fsvg+xml;base64,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\u002Fm+WItjwvdGV4dD48L2c+PGcgY2xhc3M9ImMzbHQgc3RlcCI+PHBhdGggZmlsbD0iI2VlZjRmZiIgc3Ryb2tlPSIjNGE2ZmE1IiBkPSJNMjk0IDgxaDE3NnY4MkgyOTR6IiBjbGFzcz0ic2hhcGUiIHN0eWxlPSJzdHJva2Utd2lkdGg6MiIvPjx0ZXh0IHg9IjM4MiIgeT0iMTE5LjUiIGZpbGw9IiMwYTBmMjUiIGNsYXNzPSJ0ZXh0LWJvbGQgZmlsbC1OMSIgc3R5bGU9InRleHQtYW5jaG9yOm1pZGRsZTtmb250L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只负责中间那个方框。前面「怎么把程序变成公式」是符号执行引擎的事，后面「怎么确认漏洞」是测试和人工的事。",[378,19970,19971],{},"公式建得对不对，直接决定 SMT 的结论有没有意义。",[11,19973,19974,19975,19978],{},"所以，SMT 在漏洞检测里的正确定位是：",[378,19976,19977],{},"一个强大但需要被正确使用的判定引擎","。它能把「这条路径可不可能」回答得很准，但「该问哪些问题」「怎么把程序翻译成问题」，仍然依赖前端工具和工程师的判断。",[608,19980,19982],{"id":19981},"六小结","六、小结",[11,19984,19985,19986,20081,20082,20085],{},"回到开头那道小学题。SMT 的价值，不是把 ",[224,19987,19989,20015],{"className":19988,"translate":388},[392],[224,19990,19992],{"className":19991},[396],[398,19993,19994],{"xmlns":400},[403,19995,19996,20012],{},[406,19997,19998,20000,20002,20004,20006,20008,20010],{},[409,19999,371],{},[412,20001,325],{},[419,20003,3594],{},[412,20005,428],{},[409,20007,371],{},[412,20009,1577],{},[419,20011,445],{},[447,20013,20014],{"encoding":449},"x > 3 \\wedge x \u003C 5",[224,20016,20018,20036,20054,20072],{"className":20017,"ariaHidden":455},[454],[224,20019,20021,20024,20027,20030,20033],{"className":20020},[459],[224,20022],{"className":20023,"style":2810},[463],[224,20025,371],{"className":20026},[468,469],[224,20028],{"className":20029,"style":496},[473],[224,20031,325],{"className":20032},[500],[224,20034],{"className":20035,"style":496},[473],[224,20037,20039,20042,20045,20048,20051],{"className":20038},[459],[224,20040],{"className":20041,"style":510},[463],[224,20043,3594],{"className":20044},[468],[224,20046],{"className":20047,"style":474},[473],[224,20049,428],{"className":20050},[478],[224,20052],{"className":20053,"style":474},[473],[224,20055,20057,20060,20063,20066,20069],{"className":20056},[459],[224,20058],{"className":20059,"style":2810},[463],[224,20061,371],{"className":20062},[468,469],[224,20064],{"className":20065,"style":496},[473],[224,20067,1577],{"className":20068},[500],[224,20070],{"className":20071,"style":496},[473],[224,20073,20075,20078],{"className":20074},[459],[224,20076],{"className":20077,"style":510},[463],[224,20079,445],{"className":20080},[468]," 算得更快，而是让机器",[378,20083,20084],{},"在保留「整数」「数组」「位向量」这些概念的前提下","，仍然能自动地回答「有没有解」。",[11,20087,20088,20091,20092,20095],{},[378,20089,20090],{},"SMT = SAT + 背景理论。"," 布尔层负责组合，理论层负责语义，两者通过「原子真值」这一层接口通信。 ",[378,20093,20094],{},"它的核心取舍是用表达力换可判定性。"," 把主战场放在无量词的可判定片段，换来一个在片段内保证终止、且能处理工业规模混合约束的自动判定过程。",[20097,20098,20099],"style",{},"html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .snPdu, html code.shiki .snPdu{--shiki-light:#6F42C1;--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sP4rz, html code.shiki .sP4rz{--shiki-light:#E36209;--shiki-default:#E36209;--shiki-dark:#FFAB70}html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sCsY4, html code.shiki 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把 SAT 的布尔搜索与多个理论求解器组合起来，使得混合理论约束可以被统一、自动地判定。",[20145],"logic","2026-10-29T09:03:03+08:00","1C-2B31uaFHKLW1GJ3_3j1Cbaf3yP10u6PKSP7bBHZ8",{"id":20149,"title":20150,"body":20151,"description":152574,"draft":190,"enableCodeHighlight":190,"enableCodeHightlight":190,"enableComment":191,"enableD2":190,"enableKatex":191,"extension":192,"image":186,"important":190,"location":193,"meta":152575,"navigation":191,"ogImage":193,"path":152576,"seo":152577,"stem":152578,"summary":152579,"tags":152580,"time":152583,"__hash__":152584},"blog\u002Fblog\u002F2026\u002F0011-about-the-f-divergence.md","f-散度族及其推论",{"type":8,"value":20152,"toc":152514},[20153,20185,20284,20597,20600,20604,20819,20981,21054,22355,22359,23188,23853,24289,24728,25212,25216,25537,25580,25584,25719,25897,25951,26473,26645,27121,27333,27337,28380,28714,28812,28816,29057,29329,29364,29832,29986,30513,30516,30520,30622,30924,30929,31890,32554,33365,34208,34340,35268,35816,36113,36453,36987,37293,37909,39020,39839,40380,40725,40978,41216,41247,41343,41347,41379,41382,41385,41402,41405,41409,41412,41500,41736,42276,42441,42985,42988,43278,43563,44124,44127,44425,45115,45479,45483,45645,46120,46391,46787,47498,47501,47938,47941,48378,48647,49611,49864,50505,50743,50911,51097,51507,52011,52727,53453,53508,53752,54013,54430,54434,54436,54603,54672,55311,55706,56098,56107,56453,57068,57072,57074,57205,57273,57801,57808,58181,58554,59502,60349,60353,60355,60536,60851,61390,61552,61960,62023,62500,63117,64029,64220,64646,64650,64734,65075,65520,65896,66105,66377,66563,67101,67640,68066,68069,68924,69013,69282,69286,69289,69857,70323,70533,70661,71202,71415,71424,71926,72394,72398,74816,74820,75115,75119,75373,76040,76044,77001,77692,78194,78198,78550,78874,78979,79712,80585,80589,81408,81443,81694,81860,82906,83559,83565,83831,84329,84522,86712,87726,88036,88311,88823,90511,91401,93171,93934,93937,94348,94351,94354,94358,94662,95000,95765,95768,96129,96160,96649,96712,97507,97886,97934,97938,98818,99124,100266,101960,101964,103780,103786,103930,104332,104876,106003,106944,106950,107417,108101,108508,108581,108620,109018,109347,109599,110345,110410,111691,112336,116837,116844,116848,116851,118763,119022,119191,119195,119199,119202,119605,119811,120599,120725,121237,121241,121655,121913,122373,123079,123083,123144,123276,123663,124011,125241,125245,125431,125786,125789,126267,126657,127169,127452,127873,128133,128137,128140,128474,128478,128482,131442,134501,134504,134508,134561,134808,135021,135147,135508,135783,136160,136281,136286,136290,137325,137328,137332,137335,137394,137397,137401,137591,137595,137598,138125,138480,138484,138487,139059,139179,139183,139231,139634,139637,139641,139706,139710,139713,140067,140070,140074,140155,140508,140784,140942,141179,141445,141768,142139,142549,143424,143956,144226,144229,144233,144237,144588,144593,145332,145785,146123,146125,146726,146888,146894,147240,147756,147762,148336,148578,148582,149175,149566,149986,150334,150808,150812,151069,151384,151490,152024,152027,152031,152034,152511],[11,20154,20155,20156,20184],{},"f-散度族是概率分布之间差异的一大类度量，它用一个凸函数 ",[224,20157,20159,20172],{"className":20158,"translate":388},[392],[224,20160,20162],{"className":20161},[396],[398,20163,20164],{"xmlns":400},[403,20165,20166,20170],{},[406,20167,20168],{},[409,20169,1705],{},[447,20171,1705],{"encoding":449},[224,20173,20175],{"className":20174,"ariaHidden":455},[454],[224,20176,20178,20181],{"className":20177},[459],[224,20179],{"className":20180,"style":2565},[463],[224,20182,1705],{"className":20183,"style":1740},[468,469]," 来统一描述 KL 散度、JS 散度、全变差、卡方散度等许多常见散度。可以把它理解为「散度」的通用模板。",[11,20186,20187,20188,20216,20217,20283],{},"用一个凸函数 ",[224,20189,20191,20204],{"className":20190,"translate":388},[392],[224,20192,20194],{"className":20193},[396],[398,20195,20196],{"xmlns":400},[403,20197,20198,20202],{},[406,20199,20200],{},[409,20201,1705],{},[447,20203,1705],{"encoding":449},[224,20205,20207],{"className":20206,"ariaHidden":455},[454],[224,20208,20210,20213],{"className":20209},[459],[224,20211],{"className":20212,"style":2565},[463],[224,20214,1705],{"className":20215,"style":1740},[468,469],"（满足 ",[224,20218,20220,20244],{"className":20219,"translate":388},[392],[224,20221,20223],{"className":20222},[396],[398,20224,20225],{"xmlns":400},[403,20226,20227,20241],{},[406,20228,20229,20231,20233,20235,20237,20239],{},[409,20230,1705],{},[412,20232,239],{"stretchy":435},[419,20234,1438],{},[412,20236,1716],{"stretchy":435},[412,20238,278],{},[419,20240,284],{},[447,20242,20243],{"encoding":449},"f(1)=0",[224,20245,20247,20274],{"className":20246,"ariaHidden":455},[454],[224,20248,20250,20253,20256,20259,20262,20265,20268,20271],{"className":20249},[459],[224,20251],{"className":20252,"style":535},[463],[224,20254,1705],{"className":20255,"style":1740},[468,469],[224,20257,239],{"className":20258},[542],[224,20260,1438],{"className":20261},[468],[224,20263,1716],{"className":20264},[549],[224,20266],{"className":20267,"style":496},[473],[224,20269,278],{"className":20270},[500],[224,20272],{"className":20273,"style":496},[473],[224,20275,20277,20280],{"className":20276},[459],[224,20278],{"className":20279,"style":510},[463],[224,20281,284],{"className":20282},[468],"）定义的散度模板：",[224,20285,20287],{"className":20286,"translate":388},[387],[224,20288,20290,20359],{"className":20289,"translate":388},[392],[224,20291,20293],{"className":20292},[396],[398,20294,20295],{"xmlns":400,"display":401},[403,20296,20297,20356],{},[406,20298,20299,20306,20308,20310,20313,20316,20318,20320,20327],{},[695,20300,20301,20304],{},[409,20302,20303],{},"D",[409,20305,1705],{},[412,20307,239],{"stretchy":435},[409,20309,16018],{},[409,20311,20312],{"mathvariant":678},"∥",[409,20314,20315],{},"Q",[412,20317,1716],{"stretchy":435},[412,20319,278],{},[695,20321,20322,20325],{},[409,20323,20324],{"mathvariant":12251},"E",[409,20326,20315],{},[406,20328,20329,20331,20333,20354],{},[412,20330,268],{"fence":455},[409,20332,1705],{},[406,20334,20335,20337,20352],{},[412,20336,239],{"fence":455},[20338,20339,20340,20346],"mfrac",{},[406,20341,20342,20344],{},[409,20343,13182],{},[409,20345,16018],{},[406,20347,20348,20350],{},[409,20349,13182],{},[409,20351,20315],{},[412,20353,1716],{"fence":455},[412,20355,440],{"fence":455},[447,20357,20358],{"encoding":449},"D_f(P\\|Q)\n=\\mathbb E_Q\\left[\nf\\left(\\frac{dP}{dQ}\\right)\\right]",[224,20360,20362,20436],{"className":20361,"ariaHidden":455},[454],[224,20363,20365,20369,20412,20415,20418,20421,20424,20427,20430,20433],{"className":20364},[459],[224,20366],{"className":20367,"style":20368},[463],"height:1.0361em;vertical-align:-0.2861em;",[224,20370,20372,20375],{"className":20371},[468],[224,20373,20303],{"className":20374,"style":2473},[468,469],[224,20376,20378],{"className":20377},[750],[224,20379,20381,20403],{"className":20380},[754,755],[224,20382,20384,20400],{"className":20383},[759],[224,20385,20388],{"className":20386,"style":20387},[763],"height:0.3361em;",[224,20389,20391,20394],{"style":20390},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[224,20392],{"className":20393,"style":772},[771],[224,20395,20397],{"className":20396},[776,777,778,779],[224,20398,1705],{"className":20399,"style":1740},[468,469,779],[224,20401,787],{"className":20402},[786],[224,20404,20406],{"className":20405},[759],[224,20407,20410],{"className":20408,"style":20409},[763],"height:0.2861em;",[224,20411],{},[224,20413,239],{"className":20414},[542],[224,20416,16018],{"className":20417,"style":653},[468,469],[224,20419,20312],{"className":20420},[468],[224,20422,20315],{"className":20423},[468,469],[224,20425,1716],{"className":20426},[549],[224,20428],{"className":20429,"style":496},[473],[224,20431,278],{"className":20432},[500],[224,20434],{"className":20435,"style":496},[473],[224,20437,20439,20443,20483,20486],{"className":20438},[459],[224,20440],{"className":20441,"style":20442},[463],"height:2.4em;vertical-align:-0.95em;",[224,20444,20446,20449],{"className":20445},[468],[224,20447,20324],{"className":20448},[468,12389],[224,20450,20452],{"className":20451},[750],[224,20453,20455,20475],{"className":20454},[754,755],[224,20456,20458,20472],{"className":20457},[759],[224,20459,20461],{"className":20460,"style":764},[763],[224,20462,20463,20466],{"style":767},[224,20464],{"className":20465,"style":772},[771],[224,20467,20469],{"className":20468},[776,777,778,779],[224,20470,20315],{"className":20471},[468,469,779],[224,20473,787],{"className":20474},[786],[224,20476,20478],{"className":20477},[759],[224,20479,20481],{"className":20480,"style":20409},[763],[224,20482],{},[224,20484],{"className":20485,"style":724},[473],[224,20487,20489,20495,20498,20501,20591],{"className":20488},[2091],[224,20490,20492],{"className":20491,"style":8281},[542,8280],[224,20493,268],{"className":20494},[2363,778],[224,20496,1705],{"className":20497,"style":1740},[468,469],[224,20499],{"className":20500,"style":724},[473],[224,20502,20504,20510,20585],{"className":20503},[2091],[224,20505,20507],{"className":20506,"style":8281},[542,8280],[224,20508,239],{"className":20509},[2363,778],[224,20511,20513,20516,20582],{"className":20512},[468],[224,20514],{"className":20515},[542,8479],[224,20517,20519],{"className":20518},[20338],[224,20520,20522,20573],{"className":20521},[754,755],[224,20523,20525,20570],{"className":20524},[759],[224,20526,20529,20544,20555],{"className":20527,"style":20528},[763],"height:1.3714em;",[224,20530,20532,20535],{"style":20531},"top:-2.314em;",[224,20533],{"className":20534,"style":2108},[771],[224,20536,20538,20541],{"className":20537},[468],[224,20539,13182],{"className":20540},[468,469],[224,20542,20315],{"className":20543},[468,469],[224,20545,20547,20550],{"style":20546},"top:-3.23em;",[224,20548],{"className":20549,"style":2108},[771],[224,20551],{"className":20552,"style":20554},[20553],"frac-line","border-bottom-width:0.04em;",[224,20556,20558,20561],{"style":20557},"top:-3.677em;",[224,20559],{"className":20560,"style":2108},[771],[224,20562,20564,20567],{"className":20563},[468],[224,20565,13182],{"className":20566},[468,469],[224,20568,16018],{"className":20569,"style":653},[468,469],[224,20571,787],{"className":20572},[786],[224,20574,20576],{"className":20575},[759],[224,20577,20580],{"className":20578,"style":20579},[763],"height:0.8804em;",[224,20581],{},[224,20583],{"className":20584},[549,8479],[224,20586,20588],{"className":20587,"style":8281},[549,8280],[224,20589,1716],{"className":20590},[2363,778],[224,20592,20594],{"className":20593,"style":8281},[549,8280],[224,20595,440],{"className":20596},[2363,778],[11,20598,20599],{},"它统一了 KL、JS、全变差、卡方、Hellinger 等众多散度，具有非负性、联合凸性、变分对偶表示和数据处理不等式等核心性质，是信息论和生成模型理论中的基础工具。",[608,20601,20603],{"id":20602},"_1-定义","1. 定义",[11,20605,20606,20607,20650,20651,20707,20708,20736,20737,20765,20766,20818],{},"设 ",[224,20608,20610,20628],{"className":20609,"translate":388},[392],[224,20611,20613],{"className":20612},[396],[398,20614,20615],{"xmlns":400},[403,20616,20617,20625],{},[406,20618,20619,20621,20623],{},[409,20620,16018],{},[412,20622,1877],{"separator":455},[409,20624,20315],{},[447,20626,20627],{"encoding":449},"P,Q",[224,20629,20631],{"className":20630,"ariaHidden":455},[454],[224,20632,20634,20638,20641,20644,20647],{"className":20633},[459],[224,20635],{"className":20636,"style":20637},[463],"height:0.8778em;vertical-align:-0.1944em;",[224,20639,16018],{"className":20640,"style":653},[468,469],[224,20642,1877],{"className":20643},[1917],[224,20645],{"className":20646,"style":724},[473],[224,20648,20315],{"className":20649},[468,469]," 是同一个可测空间 ",[224,20652,20654,20678],{"className":20653,"translate":388},[392],[224,20655,20657],{"className":20656},[396],[398,20658,20659],{"xmlns":400},[403,20660,20661,20675],{},[406,20662,20663,20665,20669,20671,20673],{},[412,20664,239],{"stretchy":435},[409,20666,20668],{"mathvariant":20667},"script","X",[412,20670,1877],{"separator":455},[409,20672,1097],{"mathvariant":20667},[412,20674,1716],{"stretchy":435},[447,20676,20677],{"encoding":449},"(\\mathcal X,\\mathcal A)",[224,20679,20681],{"className":20680,"ariaHidden":455},[454],[224,20682,20684,20687,20690,20695,20698,20701,20704],{"className":20683},[459],[224,20685],{"className":20686,"style":535},[463],[224,20688,239],{"className":20689},[542],[224,20691,20668],{"className":20692,"style":20694},[468,20693],"mathcal","margin-right:0.1464em;",[224,20696,1877],{"className":20697},[1917],[224,20699],{"className":20700,"style":724},[473],[224,20702,1097],{"className":20703},[468,20693],[224,20705,1716],{"className":20706},[549]," 上的两个概率分布，且 ",[224,20709,20711,20724],{"className":20710,"translate":388},[392],[224,20712,20714],{"className":20713},[396],[398,20715,20716],{"xmlns":400},[403,20717,20718,20722],{},[406,20719,20720],{},[409,20721,16018],{},[447,20723,16018],{"encoding":449},[224,20725,20727],{"className":20726,"ariaHidden":455},[454],[224,20728,20730,20733],{"className":20729},[459],[224,20731],{"className":20732,"style":649},[463],[224,20734,16018],{"className":20735,"style":653},[468,469]," 关于 ",[224,20738,20740,20753],{"className":20739,"translate":388},[392],[224,20741,20743],{"className":20742},[396],[398,20744,20745],{"xmlns":400},[403,20746,20747,20751],{},[406,20748,20749],{},[409,20750,20315],{},[447,20752,20315],{"encoding":449},[224,20754,20756],{"className":20755,"ariaHidden":455},[454],[224,20757,20759,20762],{"className":20758},[459],[224,20760],{"className":20761,"style":20637},[463],[224,20763,20315],{"className":20764},[468,469]," 绝对连续（即 ",[224,20767,20769,20787],{"className":20768,"translate":388},[392],[224,20770,20772],{"className":20771},[396],[398,20773,20774],{"xmlns":400},[403,20775,20776,20784],{},[406,20777,20778,20780,20782],{},[409,20779,16018],{},[412,20781,12984],{},[409,20783,20315],{},[447,20785,20786],{"encoding":449},"P\\ll Q",[224,20788,20790,20809],{"className":20789,"ariaHidden":455},[454],[224,20791,20793,20797,20800,20803,20806],{"className":20792},[459],[224,20794],{"className":20795,"style":20796},[463],"height:0.7224em;vertical-align:-0.0391em;",[224,20798,16018],{"className":20799,"style":653},[468,469],[224,20801],{"className":20802,"style":496},[473],[224,20804,12984],{"className":20805},[500],[224,20807],{"className":20808,"style":496},[473],[224,20810,20812,20815],{"className":20811},[459],[224,20813],{"className":20814,"style":20637},[463],[224,20816,20315],{"className":20817},[468,469],"），记",[224,20820,20822],{"className":20821,"translate":388},[387],[224,20823,20825,20867],{"className":20824,"translate":388},[392],[224,20826,20828],{"className":20827},[396],[398,20829,20830],{"xmlns":400,"display":401},[403,20831,20832,20864],{},[406,20833,20834,20836,20838,20840,20842,20844,20858,20860,20862],{},[409,20835,1435],{},[412,20837,239],{"stretchy":435},[409,20839,371],{},[412,20841,1716],{"stretchy":435},[412,20843,278],{},[20338,20845,20846,20852],{},[406,20847,20848,20850],{},[409,20849,13182],{},[409,20851,16018],{},[406,20853,20854,20856],{},[409,20855,13182],{},[409,20857,20315],{},[412,20859,239],{"stretchy":435},[409,20861,371],{},[412,20863,1716],{"stretchy":435},[447,20865,20866],{"encoding":449},"t(x)=\\frac{dP}{dQ}(x)",[224,20868,20870,20897],{"className":20869,"ariaHidden":455},[454],[224,20871,20873,20876,20879,20882,20885,20888,20891,20894],{"className":20872},[459],[224,20874],{"className":20875,"style":535},[463],[224,20877,1435],{"className":20878},[468,469],[224,20880,239],{"className":20881},[542],[224,20883,371],{"className":20884},[468,469],[224,20886,1716],{"className":20887},[549],[224,20889],{"className":20890,"style":496},[473],[224,20892,278],{"className":20893},[500],[224,20895],{"className":20896,"style":496},[473],[224,20898,20900,20904,20972,20975,20978],{"className":20899},[459],[224,20901],{"className":20902,"style":20903},[463],"height:2.2519em;vertical-align:-0.8804em;",[224,20905,20907,20910,20969],{"className":20906},[468],[224,20908],{"className":20909},[542,8479],[224,20911,20913],{"className":20912},[20338],[224,20914,20916,20961],{"className":20915},[754,755],[224,20917,20919,20958],{"className":20918},[759],[224,20920,20922,20936,20944],{"className":20921,"style":20528},[763],[224,20923,20924,20927],{"style":20531},[224,20925],{"className":20926,"style":2108},[771],[224,20928,20930,20933],{"className":20929},[468],[224,20931,13182],{"className":20932},[468,469],[224,20934,20315],{"className":20935},[468,469],[224,20937,20938,20941],{"style":20546},[224,20939],{"className":20940,"style":2108},[771],[224,20942],{"className":20943,"style":20554},[20553],[224,20945,20946,20949],{"style":20557},[224,20947],{"className":20948,"style":2108},[771],[224,20950,20952,20955],{"className":20951},[468],[224,20953,13182],{"className":20954},[468,469],[224,20956,16018],{"className":20957,"style":653},[468,469],[224,20959,787],{"className":20960},[786],[224,20962,20964],{"className":20963},[759],[224,20965,20967],{"className":20966,"style":20579},[763],[224,20968],{},[224,20970],{"className":20971},[549,8479],[224,20973,239],{"className":20974},[542],[224,20976,371],{"className":20977},[468,469],[224,20979,1716],{"className":20980},[549],[11,20982,20983,20984,21053],{},"为 Radon–Nikodym 导数（在离散情形就是概率质量之比，在密度存在时就是密度之比 ",[224,20985,20987,21017],{"className":20986,"translate":388},[392],[224,20988,20990],{"className":20989},[396],[398,20991,20992],{"xmlns":400},[403,20993,20994,21014],{},[406,20995,20996,20998,21000,21002,21004,21006,21008,21010,21012],{},[409,20997,11],{},[412,20999,239],{"stretchy":435},[409,21001,371],{},[412,21003,1716],{"stretchy":435},[409,21005,12413],{"mathvariant":678},[409,21007,2023],{},[412,21009,239],{"stretchy":435},[409,21011,371],{},[412,21013,1716],{"stretchy":435},[447,21015,21016],{"encoding":449},"p(x)\u002Fq(x)",[224,21018,21020],{"className":21019,"ariaHidden":455},[454],[224,21021,21023,21026,21029,21032,21035,21038,21041,21044,21047,21050],{"className":21022},[459],[224,21024],{"className":21025,"style":535},[463],[224,21027,11],{"className":21028},[468,469],[224,21030,239],{"className":21031},[542],[224,21033,371],{"className":21034},[468,469],[224,21036,1716],{"className":21037},[549],[224,21039,12413],{"className":21040},[468],[224,21042,2023],{"className":21043,"style":492},[468,469],[224,21045,239],{"className":21046},[542],[224,21048,371],{"className":21049},[468,469],[224,21051,1716],{"className":21052},[549],"）。",[21055,21056,21058,21287,21683,21715,22295],"theorem",{"title":21057},"定义 1.1（f-散度）",[11,21059,21060,21061,21159,21160,21225,21226,20736,21254,21282,21283,21286],{},"给定一个凸函数 ",[224,21062,21064,21096],{"className":21063,"translate":388},[392],[224,21065,21067],{"className":21066},[396],[398,21068,21069],{"xmlns":400},[403,21070,21071,21093],{},[406,21072,21073,21075,21077,21079,21081,21083,21086,21088,21090],{},[409,21074,1705],{},[412,21076,1071],{},[412,21078,268],{"stretchy":435},[419,21080,284],{},[412,21082,1877],{"separator":455},[409,21084,21085],{"mathvariant":678},"∞",[412,21087,1716],{"stretchy":435},[412,21089,1144],{},[409,21091,21092],{"mathvariant":12251},"R",[447,21094,21095],{"encoding":449},"f:[0,\\infty)\\to\\mathbb R",[224,21097,21099,21117,21150],{"className":21098,"ariaHidden":455},[454],[224,21100,21102,21105,21108,21111,21114],{"className":21101},[459],[224,21103],{"className":21104,"style":2565},[463],[224,21106,1705],{"className":21107,"style":1740},[468,469],[224,21109],{"className":21110,"style":496},[473],[224,21112,1071],{"className":21113},[500],[224,21115],{"className":21116,"style":496},[473],[224,21118,21120,21123,21126,21129,21132,21135,21138,21141,21144,21147],{"className":21119},[459],[224,21121],{"className":21122,"style":535},[463],[224,21124,268],{"className":21125},[542],[224,21127,284],{"className":21128},[468],[224,21130,1877],{"className":21131},[1917],[224,21133],{"className":21134,"style":724},[473],[224,21136,21085],{"className":21137},[468],[224,21139,1716],{"className":21140},[549],[224,21142],{"className":21143,"style":496},[473],[224,21145,1144],{"className":21146},[500],[224,21148],{"className":21149,"style":496},[473],[224,21151,21153,21156],{"className":21152},[459],[224,21154],{"className":21155,"style":12385},[463],[224,21157,21092],{"className":21158},[468,12389],"，且满足 ",[224,21161,21163,21186],{"className":21162,"translate":388},[392],[224,21164,21166],{"className":21165},[396],[398,21167,21168],{"xmlns":400},[403,21169,21170,21184],{},[406,21171,21172,21174,21176,21178,21180,21182],{},[409,21173,1705],{},[412,21175,239],{"stretchy":435},[419,21177,1438],{},[412,21179,1716],{"stretchy":435},[412,21181,278],{},[419,21183,284],{},[447,21185,20243],{"encoding":449},[224,21187,21189,21216],{"className":21188,"ariaHidden":455},[454],[224,21190,21192,21195,21198,21201,21204,21207,21210,21213],{"className":21191},[459],[224,21193],{"className":21194,"style":535},[463],[224,21196,1705],{"className":21197,"style":1740},[468,469],[224,21199,239],{"className":21200},[542],[224,21202,1438],{"className":21203},[468],[224,21205,1716],{"className":21206},[549],[224,21208],{"className":21209,"style":496},[473],[224,21211,278],{"className":21212},[500],[224,21214],{"className":21215,"style":496},[473],[224,21217,21219,21222],{"className":21218},[459],[224,21220],{"className":21221,"style":510},[463],[224,21223,284],{"className":21224},[468],"，则 ",[224,21227,21229,21242],{"className":21228,"translate":388},[392],[224,21230,21232],{"className":21231},[396],[398,21233,21234],{"xmlns":400},[403,21235,21236,21240],{},[406,21237,21238],{},[409,21239,16018],{},[447,21241,16018],{"encoding":449},[224,21243,21245],{"className":21244,"ariaHidden":455},[454],[224,21246,21248,21251],{"className":21247},[459],[224,21249],{"className":21250,"style":649},[463],[224,21252,16018],{"className":21253,"style":653},[468,469],[224,21255,21257,21270],{"className":21256,"translate":388},[392],[224,21258,21260],{"className":21259},[396],[398,21261,21262],{"xmlns":400},[403,21263,21264,21268],{},[406,21265,21266],{},[409,21267,20315],{},[447,21269,20315],{"encoding":449},[224,21271,21273],{"className":21272,"ariaHidden":455},[454],[224,21274,21276,21279],{"className":21275},[459],[224,21277],{"className":21278,"style":20637},[463],[224,21280,20315],{"className":21281},[468,469]," 的 ",[378,21284,21285],{},"f-散度"," 定义为",[224,21288,21290],{"className":21289,"translate":388},[387],[224,21291,21293,21391],{"className":21292,"translate":388},[392],[224,21294,21296],{"className":21295},[396],[398,21297,21298],{"xmlns":400,"display":401},[403,21299,21300,21388],{},[5209,21301,21302],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,21303,21304,21326],{},[5218,21305,21306],{},[5221,21307,21308],{"scriptlevel":284,"displaystyle":455},[406,21309,21310,21316,21318,21320,21322,21324],{},[695,21311,21312,21314],{},[409,21313,20303],{},[409,21315,1705],{},[412,21317,239],{"stretchy":435},[409,21319,16018],{},[409,21321,20312],{"mathvariant":678},[409,21323,20315],{},[412,21325,1716],{"stretchy":435},[5218,21327,21328],{},[5221,21329,21330],{"scriptlevel":284,"displaystyle":455},[406,21331,21332,21334,21336,21339,21341,21343,21345,21347,21349,21351,21354,21382,21384,21386],{},[406,21333],{},[412,21335,278],{},[412,21337,21338],{},"∫",[409,21340,2023],{},[412,21342,239],{"stretchy":435},[409,21344,371],{},[412,21346,1716],{"stretchy":435},[423,21348,682],{},[409,21350,1705],{},[423,21352,21353],{}," ⁣",[406,21355,21356,21358,21380],{},[412,21357,239],{"fence":455},[20338,21359,21360,21370],{},[406,21361,21362,21364,21366,21368],{},[409,21363,11],{},[412,21365,239],{"stretchy":435},[409,21367,371],{},[412,21369,1716],{"stretchy":435},[406,21371,21372,21374,21376,21378],{},[409,21373,2023],{},[412,21375,239],{"stretchy":435},[409,21377,371],{},[412,21379,1716],{"stretchy":435},[412,21381,1716],{"fence":455},[409,21383,13182],{},[409,21385,371],{},[412,21387,1877],{"separator":455},[447,21389,21390],{"encoding":449},"\\begin{aligned}\nD_f(P\\|Q)\n&=\\int q(x)\\,\nf\\!\\left(\\frac{p(x)}{q(x)}\\right)dx,\n\\end{aligned}",[224,21392,21394],{"className":21393,"ariaHidden":455},[454],[224,21395,21397,21400],{"className":21396},[459],[224,21398],{"className":21399,"style":20442},[463],[224,21401,21403],{"className":21402},[468],[224,21404,21406,21496],{"className":21405},[5209],[224,21407,21409],{"className":21408},[5331],[224,21410,21412,21487],{"className":21411},[754,755],[224,21413,21415,21484],{"className":21414},[759],[224,21416,21419],{"className":21417,"style":21418},[763],"height:1.45em;",[224,21420,21422,21426],{"style":21421},"top:-3.45em;",[224,21423],{"className":21424,"style":21425},[771],"height:3.45em;",[224,21427,21429,21469,21472,21475,21478,21481],{"className":21428},[468],[224,21430,21432,21435],{"className":21431},[468],[224,21433,20303],{"className":21434,"style":2473},[468,469],[224,21436,21438],{"className":21437},[750],[224,21439,21441,21461],{"className":21440},[754,755],[224,21442,21444,21458],{"className":21443},[759],[224,21445,21447],{"className":21446,"style":20387},[763],[224,21448,21449,21452],{"style":20390},[224,21450],{"className":21451,"style":772},[771],[224,21453,21455],{"className":21454},[776,777,778,779],[224,21456,1705],{"className":21457,"style":1740},[468,469,779],[224,21459,787],{"className":21460},[786],[224,21462,21464],{"className":21463},[759],[224,21465,21467],{"className":21466,"style":20409},[763],[224,21468],{},[224,21470,239],{"className":21471},[542],[224,21473,16018],{"className":21474,"style":653},[468,469],[224,21476,20312],{"className":21477},[468],[224,21479,20315],{"className":21480},[468,469],[224,21482,1716],{"className":21483},[549],[224,21485,787],{"className":21486},[786],[224,21488,21490],{"className":21489},[759],[224,21491,21494],{"className":21492,"style":21493},[763],"height:0.95em;",[224,21495],{},[224,21497,21499],{"className":21498},[5399],[224,21500,21502,21675],{"className":21501},[754,755],[224,21503,21505,21672],{"className":21504},[759],[224,21506,21508],{"className":21507,"style":21418},[763],[224,21509,21510,21513],{"style":21421},[224,21511],{"className":21512,"style":21425},[771],[224,21514,21516,21519,21522,21525,21528,21535,21538,21541,21544,21547,21550,21553,21556,21560,21563,21660,21663,21666,21669],{"className":21515},[468],[224,21517],{"className":21518},[468],[224,21520],{"className":21521,"style":496},[473],[224,21523,278],{"className":21524},[500],[224,21526],{"className":21527,"style":496},[473],[224,21529,21338],{"className":21530,"style":21534},[21531,21532,21533],"mop","op-symbol","large-op","margin-right:0.4445em;position:relative;top:-0.0011em;",[224,21536],{"className":21537,"style":724},[473],[224,21539,2023],{"className":21540,"style":492},[468,469],[224,21542,239],{"className":21543},[542],[224,21545,371],{"className":21546},[468,469],[224,21548,1716],{"className":21549},[549],[224,21551],{"className":21552,"style":724},[473],[224,21554,1705],{"className":21555,"style":1740},[468,469],[224,21557],{"className":21558,"style":21559},[473],"margin-right:-0.1667em;",[224,21561],{"className":21562,"style":724},[473],[224,21564,21566,21572,21654],{"className":21565},[2091],[224,21567,21569],{"className":21568,"style":8281},[542,8280],[224,21570,239],{"className":21571},[2363,778],[224,21573,21575,21578,21651],{"className":21574},[468],[224,21576],{"className":21577},[542,8479],[224,21579,21581],{"className":21580},[20338],[224,21582,21584,21642],{"className":21583},[754,755],[224,21585,21587,21639],{"className":21586},[759],[224,21588,21591,21611,21619],{"className":21589,"style":21590},[763],"height:1.427em;",[224,21592,21593,21596],{"style":20531},[224,21594],{"className":21595,"style":2108},[771],[224,21597,21599,21602,21605,21608],{"className":21598},[468],[224,21600,2023],{"className":21601,"style":492},[468,469],[224,21603,239],{"className":21604},[542],[224,21606,371],{"className":21607},[468,469],[224,21609,1716],{"className":21610},[549],[224,21612,21613,21616],{"style":20546},[224,21614],{"className":21615,"style":2108},[771],[224,21617],{"className":21618,"style":20554},[20553],[224,21620,21621,21624],{"style":20557},[224,21622],{"className":21623,"style":2108},[771],[224,21625,21627,21630,21633,21636],{"className":21626},[468],[224,21628,11],{"className":21629},[468,469],[224,21631,239],{"className":21632},[542],[224,21634,371],{"className":21635},[468,469],[224,21637,1716],{"className":21638},[549],[224,21640,787],{"className":21641},[786],[224,21643,21645],{"className":21644},[759],[224,21646,21649],{"className":21647,"style":21648},[763],"height:0.936em;",[224,21650],{},[224,21652],{"className":21653},[549,8479],[224,21655,21657],{"className":21656,"style":8281},[549,8280],[224,21658,1716],{"className":21659},[2363,778],[224,21661],{"className":21662,"style":724},[473],[224,21664,13182],{"className":21665},[468,469],[224,21667,371],{"className":21668},[468,469],[224,21670,1877],{"className":21671},[1917],[224,21673,787],{"className":21674},[786],[224,21676,21678],{"className":21677},[759],[224,21679,21681],{"className":21680,"style":21493},[763],[224,21682],{},[11,21684,21685,21686,21714],{},"更一般地写成关于 ",[224,21687,21689,21702],{"className":21688,"translate":388},[392],[224,21690,21692],{"className":21691},[396],[398,21693,21694],{"xmlns":400},[403,21695,21696,21700],{},[406,21697,21698],{},[409,21699,20315],{},[447,21701,20315],{"encoding":449},[224,21703,21705],{"className":21704,"ariaHidden":455},[454],[224,21706,21708,21711],{"className":21707},[459],[224,21709],{"className":21710,"style":20637},[463],[224,21712,20315],{"className":21713},[468,469]," 的期望形式",[224,21716,21718],{"className":21717,"translate":388},[387],[224,21719,21721,21855],{"className":21720,"translate":388},[392],[224,21722,21724],{"className":21723},[396],[398,21725,21726],{"xmlns":400,"display":401},[403,21727,21728,21852],{},[5209,21729,21730,21802],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,21731,21732,21754],{},[5218,21733,21734],{},[5221,21735,21736],{"scriptlevel":284,"displaystyle":455},[406,21737,21738,21744,21746,21748,21750,21752],{},[695,21739,21740,21742],{},[409,21741,20303],{},[409,21743,1705],{},[412,21745,239],{"stretchy":435},[409,21747,16018],{},[409,21749,20312],{"mathvariant":678},[409,21751,20315],{},[412,21753,1716],{"stretchy":435},[5218,21755,21756],{},[5221,21757,21758],{"scriptlevel":284,"displaystyle":455},[406,21759,21760,21762,21764,21770,21772],{},[406,21761],{},[412,21763,278],{},[695,21765,21766,21768],{},[409,21767,20324],{"mathvariant":12251},[409,21769,20315],{},[423,21771,21353],{},[406,21773,21774,21776,21778,21780,21800],{},[412,21775,268],{"fence":455},[409,21777,1705],{},[423,21779,21353],{},[406,21781,21782,21784,21798],{},[412,21783,239],{"fence":455},[20338,21785,21786,21792],{},[406,21787,21788,21790],{},[409,21789,13182],{},[409,21791,16018],{},[406,21793,21794,21796],{},[409,21795,13182],{},[409,21797,20315],{},[412,21799,1716],{"fence":455},[412,21801,440],{"fence":455},[5215,21803,21804,21810],{},[5218,21805,21806],{},[5221,21807,21808],{"scriptlevel":284,"displaystyle":455},[406,21809],{},[5218,21811,21812],{},[5221,21813,21814],{"scriptlevel":284,"displaystyle":455},[406,21815,21816,21818,21820,21822,21824,21826,21846,21848,21850],{},[406,21817],{},[412,21819,278],{},[412,21821,21338],{},[409,21823,1705],{},[423,21825,21353],{},[406,21827,21828,21830,21844],{},[412,21829,239],{"fence":455},[20338,21831,21832,21838],{},[406,21833,21834,21836],{},[409,21835,13182],{},[409,21837,16018],{},[406,21839,21840,21842],{},[409,21841,13182],{},[409,21843,20315],{},[412,21845,1716],{"fence":455},[409,21847,13182],{},[409,21849,20315],{},[412,21851,1877],{"separator":455},[447,21853,21854],{"encoding":449},"\\begin{aligned}\nD_f(P\\|Q)\n&=\\mathbb E_Q\\!\\left[\nf\\!\\left(\\frac{dP}{dQ}\\right)\\right]\\\\\n&=\\int f\\!\\left(\\frac{dP}{dQ}\\right)dQ,\n\\end{aligned}",[224,21856,21858],{"className":21857,"ariaHidden":455},[454],[224,21859,21861,21865],{"className":21860},[459],[224,21862],{"className":21863,"style":21864},[463],"height:5.1001em;vertical-align:-2.3em;",[224,21866,21868],{"className":21867},[468],[224,21869,21871,21969],{"className":21870},[5209],[224,21872,21874],{"className":21873},[5331],[224,21875,21877,21960],{"className":21876},[754,755],[224,21878,21880,21957],{"className":21879},[759],[224,21881,21884,21948],{"className":21882,"style":21883},[763],"height:2.8em;",[224,21885,21887,21890],{"style":21886},"top:-4.8em;",[224,21888],{"className":21889,"style":21425},[771],[224,21891,21893,21933,21936,21939,21942,21945],{"className":21892},[468],[224,21894,21896,21899],{"className":21895},[468],[224,21897,20303],{"className":21898,"style":2473},[468,469],[224,21900,21902],{"className":21901},[750],[224,21903,21905,21925],{"className":21904},[754,755],[224,21906,21908,21922],{"className":21907},[759],[224,21909,21911],{"className":21910,"style":20387},[763],[224,21912,21913,21916],{"style":20390},[224,21914],{"className":21915,"style":772},[771],[224,21917,21919],{"className":21918},[776,777,778,779],[224,21920,1705],{"className":21921,"style":1740},[468,469,779],[224,21923,787],{"className":21924},[786],[224,21926,21928],{"className":21927},[759],[224,21929,21931],{"className":21930,"style":20409},[763],[224,21932],{},[224,21934,239],{"className":21935},[542],[224,21937,16018],{"className":21938,"style":653},[468,469],[224,21940,20312],{"className":21941},[468],[224,21943,20315],{"className":21944},[468,469],[224,21946,1716],{"className":21947},[549],[224,21949,21951,21954],{"style":21950},"top:-2.1em;",[224,21952],{"className":21953,"style":21425},[771],[224,21955],{"className":21956},[468],[224,21958,787],{"className":21959},[786],[224,21961,21963],{"className":21962},[759],[224,21964,21967],{"className":21965,"style":21966},[763],"height:2.3em;",[224,21968],{},[224,21970,21972],{"className":21971},[5399],[224,21973,21975,22287],{"className":21974},[754,755],[224,21976,21978,22284],{"className":21977},[759],[224,21979,21981,22154],{"className":21980,"style":21883},[763],[224,21982,21983,21986],{"style":21886},[224,21984],{"className":21985,"style":21425},[771],[224,21987,21989,21992,21995,21998,22001,22041,22044,22047],{"className":21988},[468],[224,21990],{"className":21991},[468],[224,21993],{"className":21994,"style":496},[473],[224,21996,278],{"className":21997},[500],[224,21999],{"className":22000,"style":496},[473],[224,22002,22004,22007],{"className":22003},[468],[224,22005,20324],{"className":22006},[468,12389],[224,22008,22010],{"className":22009},[750],[224,22011,22013,22033],{"className":22012},[754,755],[224,22014,22016,22030],{"className":22015},[759],[224,22017,22019],{"className":22018,"style":764},[763],[224,22020,22021,22024],{"style":767},[224,22022],{"className":22023,"style":772},[771],[224,22025,22027],{"className":22026},[776,777,778,779],[224,22028,20315],{"className":22029},[468,469,779],[224,22031,787],{"className":22032},[786],[224,22034,22036],{"className":22035},[759],[224,22037,22039],{"className":22038,"style":20409},[763],[224,22040],{},[224,22042],{"className":22043,"style":21559},[473],[224,22045],{"className":22046,"style":724},[473],[224,22048,22050,22056,22059,22062,22065,22148],{"className":22049},[2091],[224,22051,22053],{"className":22052,"style":8281},[542,8280],[224,22054,268],{"className":22055},[2363,778],[224,22057,1705],{"className":22058,"style":1740},[468,469],[224,22060],{"className":22061,"style":21559},[473],[224,22063],{"className":22064,"style":724},[473],[224,22066,22068,22074,22142],{"className":22067},[2091],[224,22069,22071],{"className":22070,"style":8281},[542,8280],[224,22072,239],{"className":22073},[2363,778],[224,22075,22077,22080,22139],{"className":22076},[468],[224,22078],{"className":22079},[542,8479],[224,22081,22083],{"className":22082},[20338],[224,22084,22086,22131],{"className":22085},[754,755],[224,22087,22089,22128],{"className":22088},[759],[224,22090,22092,22106,22114],{"className":22091,"style":20528},[763],[224,22093,22094,22097],{"style":20531},[224,22095],{"className":22096,"style":2108},[771],[224,22098,22100,22103],{"className":22099},[468],[224,22101,13182],{"className":22102},[468,469],[224,22104,20315],{"className":22105},[468,469],[224,22107,22108,22111],{"style":20546},[224,22109],{"className":22110,"style":2108},[771],[224,22112],{"className":22113,"style":20554},[20553],[224,22115,22116,22119],{"style":20557},[224,22117],{"className":22118,"style":2108},[771],[224,22120,22122,22125],{"className":22121},[468],[224,22123,13182],{"className":22124},[468,469],[224,22126,16018],{"className":22127,"style":653},[468,469],[224,22129,787],{"className":22130},[786],[224,22132,22134],{"className":22133},[759],[224,22135,22137],{"className":22136,"style":20579},[763],[224,22138],{},[224,22140],{"className":22141},[549,8479],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是 ",[224,22327,22329,22342],{"className":22328,"translate":388},[392],[224,22330,22332],{"className":22331},[396],[398,22333,22334],{"xmlns":400},[403,22335,22336,22340],{},[406,22337,22338],{},[409,22339,20315],{},[447,22341,20315],{"encoding":449},[224,22343,22345],{"className":22344,"ariaHidden":455},[454],[224,22346,22348,22351],{"className":22347},[459],[224,22349],{"className":22350,"style":20637},[463],[224,22352,20315],{"className":22353},[468,469]," 的密度。",[613,22356,22358],{"id":22357},"_11-不依赖参考测度","1.1 不依赖参考测度",[11,22360,22361,22362,22390,22391,22421,22422,22450,22451,22479,22480,22644,22645,7098,22782,22919,22920,22981,22982,23187],{},"上面的定义里出现了「密度」，这会让读者误以为它依赖 Lebesgue 测度。其实 f-散度只依赖两个分布本身。取任一 ",[224,22363,22365,22378],{"className":22364,"translate":388},[392],[224,22366,22368],{"className":22367},[396],[398,22369,22370],{"xmlns":400},[403,22371,22372,22376],{},[406,22373,22374],{},[409,22375,685],{},[447,22377,829],{"encoding":449},[224,22379,22381],{"className":22380,"ariaHidden":455},[454],[224,22382,22384,22387],{"className":22383},[459],[224,22385],{"className":22386,"style":839},[463],[224,22388,685],{"className":22389,"style":492},[468,469],"-有限测度 ",[224,22392,22394,22409],{"className":22393,"translate":388},[392],[224,22395,22397],{"className":22396},[396],[398,22398,22399],{"xmlns":400},[403,22400,22401,22406],{},[406,22402,22403],{},[409,22404,22405],{},"μ",[447,22407,22408],{"encoding":449},"\\mu",[224,22410,22412],{"className":22411,"ariaHidden":455},[454],[224,22413,22415,22418],{"className":22414},[459],[224,22416],{"className":22417,"style":806},[463],[224,22419,22405],{"className":22420},[468,469]," 同时控制 ",[224,22423,22425,22438],{"className":22424,"translate":388},[392],[224,22426,22428],{"className":22427},[396],[398,22429,22430],{"xmlns":400},[403,22431,22432,22436],{},[406,22433,22434],{},[409,22435,16018],{},[447,22437,16018],{"encoding":449},[224,22439,22441],{"className":22440,"ariaHidden":455},[454],[224,22442,22444,22447],{"className":22443},[459],[224,22445],{"className":22446,"style":649},[463],[224,22448,16018],{"className":22449,"style":653},[468,469]," 与 ",[224,22452,22454,22467],{"className":22453,"translate":388},[392],[224,22455,22457],{"className":22456},[396],[398,22458,22459],{"xmlns":400},[403,22460,22461,22465],{},[406,22462,22463],{},[409,22464,20315],{},[447,22466,20315],{"encoding":449},[224,22468,22470],{"className":22469,"ariaHidden":455},[454],[224,22471,22473,22476],{"className":22472},[459],[224,22474],{"className":22475,"style":20637},[463],[224,22477,20315],{"className":22478},[468,469],"（例如 ",[224,22481,22483,22517],{"className":22482,"translate":388},[392],[224,22484,22486],{"className":22485},[396],[398,22487,22488],{"xmlns":400},[403,22489,22490,22514],{},[406,22491,22492,22494,22496,22504,22506,22508,22510,22512],{},[409,22493,22405],{},[412,22495,278],{},[5221,22497,22498],{"scriptlevel":284,"displaystyle":435},[20338,22499,22500,22502],{},[419,22501,1438],{},[419,22503,1447],{},[412,22505,239],{"stretchy":435},[409,22507,16018],{},[412,22509,319],{},[409,22511,20315],{},[412,22513,1716],{"stretchy":435},[447,22515,22516],{"encoding":449},"\\mu=\\tfrac12(P+Q)",[224,22518,22520,22538,22632],{"className":22519,"ariaHidden":455},[454],[224,22521,22523,22526,22529,22532,22535],{"className":22522},[459],[224,22524],{"className":22525,"style":806},[463],[224,22527,22405],{"className":22528},[468,469],[224,22530],{"className":22531,"style":496},[473],[224,22533,278],{"className":22534},[500],[224,22536],{"className":22537,"style":496},[473],[224,22539,22541,22545,22617,22620,22623,22626,22629],{"className":22540},[459],[224,22542],{"className":22543,"style":22544},[463],"height:1.1901em;vertical-align:-0.345em;",[224,22546,22548,22551,22614],{"className":22547},[468],[224,22549],{"className":22550},[542,8479],[224,22552,22554],{"className":22553},[20338],[224,22555,22557,22605],{"className":22556},[754,755],[224,22558,22560,22602],{"className":22559},[759],[224,22561,22564,22579,22587],{"className":22562,"style":22563},[763],"height:0.8451em;",[224,22565,22567,22570],{"style":22566},"top:-2.655em;",[224,22568],{"className":22569,"style":2108},[771],[224,22571,22573],{"className":22572},[776,777,778,779],[224,22574,22576],{"className":22575},[468,779],[224,22577,1447],{"className":22578},[468,779],[224,22580,22581,22584],{"style":20546},[224,22582],{"className":22583,"style":2108},[771],[224,22585],{"className":22586,"style":20554},[20553],[224,22588,22590,22593],{"style":22589},"top:-3.394em;",[224,22591],{"className":22592,"style":2108},[771],[224,22594,22596],{"className":22595},[776,777,778,779],[224,22597,22599],{"className":22598},[468,779],[224,22600,1438],{"className":22601},[468,779],[224,22603,787],{"className":22604},[786],[224,22606,22608],{"className":22607},[759],[224,22609,22612],{"className":22610,"style":22611},[763],"height:0.345em;",[224,22613],{},[224,22615],{"className":22616},[549,8479],[224,22618,239],{"className":22619},[542],[224,22621,16018],{"className":22622,"style":653},[468,469],[224,22624],{"className":22625,"style":474},[473],[224,22627,319],{"className":22628},[478],[224,22630],{"className":22631,"style":474},[473],[224,22633,22635,22638,22641],{"className":22634},[459],[224,22636],{"className":22637,"style":535},[463],[224,22639,20315],{"className":22640},[468,469],[224,22642,1716],{"className":22643},[549],"），令 ",[224,22646,22648,22678],{"className":22647,"translate":388},[392],[224,22649,22651],{"className":22650},[396],[398,22652,22653],{"xmlns":400},[403,22654,22655,22675],{},[406,22656,22657,22659,22661],{},[409,22658,11],{},[412,22660,278],{},[20338,22662,22663,22669],{},[406,22664,22665,22667],{},[409,22666,13182],{},[409,22668,16018],{},[406,22670,22671,22673],{},[409,22672,13182],{},[409,22674,22405],{},[447,22676,22677],{"encoding":449},"p=\\frac{dP}{d\\mu}",[224,22679,22681,22699],{"className":22680,"ariaHidden":455},[454],[224,22682,22684,22687,22690,22693,22696],{"className":22683},[459],[224,22685],{"className":22686,"style":806},[463],[224,22688,11],{"className":22689},[468,469],[224,22691],{"className":22692,"style":496},[473],[224,22694,278],{"className":22695},[500],[224,22697],{"className":22698,"style":496},[473],[224,22700,22702,22706],{"className":22701},[459],[224,22703],{"className":22704,"style":22705},[463],"height:1.3612em;vertical-align:-0.4811em;",[224,22707,22709,22712,22779],{"className":22708},[468],[224,22710],{"className":22711},[542,8479],[224,22713,22715],{"className":22714},[20338],[224,22716,22718,22770],{"className":22717},[754,755],[224,22719,22721,22767],{"className":22720},[759],[224,22722,22725,22742,22750],{"className":22723,"style":22724},[763],"height:0.8801em;",[224,22726,22727,22730],{"style":22566},[224,22728],{"className":22729,"style":2108},[771],[224,22731,22733],{"className":22732},[776,777,778,779],[224,22734,22736,22739],{"className":22735},[468,779],[224,22737,13182],{"className":22738},[468,469,779],[224,22740,22405],{"className":22741},[468,469,779],[224,22743,22744,22747],{"style":20546},[224,22745],{"className":22746,"style":2108},[771],[224,22748],{"className":22749,"style":20554},[20553],[224,22751,22752,22755],{"style":22589},[224,22753],{"className":22754,"style":2108},[771],[224,22756,22758],{"className":22757},[776,777,778,779],[224,22759,22761,22764],{"className":22760},[468,779],[224,22762,13182],{"className":22763},[468,469,779],[224,22765,16018],{"className":22766,"style":653},[468,469,779],[224,22768,787],{"className":22769},[786],[224,22771,22773],{"className":22772},[759],[224,22774,22777],{"className":22775,"style":22776},[763],"height:0.4811em;",[224,22778],{},[224,22780],{"className":22781},[549,8479],[224,22783,22785,22815],{"className":22784,"translate":388},[392],[224,22786,22788],{"className":22787},[396],[398,22789,22790],{"xmlns":400},[403,22791,22792,22812],{},[406,22793,22794,22796,22798],{},[409,22795,2023],{},[412,22797,278],{},[20338,22799,22800,22806],{},[406,22801,22802,22804],{},[409,22803,13182],{},[409,22805,20315],{},[406,22807,22808,22810],{},[409,22809,13182],{},[409,22811,22405],{},[447,22813,22814],{"encoding":449},"q=\\frac{dQ}{d\\mu}",[224,22816,22818,22836],{"className":22817,"ariaHidden":455},[454],[224,22819,22821,22824,22827,22830,22833],{"className":22820},[459],[224,22822],{"className":22823,"style":806},[463],[224,22825,2023],{"className":22826,"style":492},[468,469],[224,22828],{"className":22829,"style":496},[473],[224,22831,278],{"className":22832},[500],[224,22834],{"className":22835,"style":496},[473],[224,22837,22839,22843],{"className":22838},[459],[224,22840],{"className":22841,"style":22842},[463],"height:1.4133em;vertical-align:-0.4811em;",[224,22844,22846,22849,22916],{"className":22845},[468],[224,22847],{"className":22848},[542,8479],[224,22850,22852],{"className":22851},[20338],[224,22853,22855,22908],{"className":22854},[754,755],[224,22856,22858,22905],{"className":22857},[759],[224,22859,22862,22879,22887],{"className":22860,"style":22861},[763],"height:0.9322em;",[224,22863,22864,22867],{"style":22566},[224,22865],{"className":22866,"style":2108},[771],[224,22868,22870],{"className":22869},[776,777,778,779],[224,22871,22873,22876],{"className":22872},[468,779],[224,22874,13182],{"className":22875},[468,469,779],[224,22877,22405],{"className":22878},[468,469,779],[224,22880,22881,22884],{"style":20546},[224,22882],{"className":22883,"style":2108},[771],[224,22885],{"className":22886,"style":20554},[20553],[224,22888,22890,22893],{"style":22889},"top:-3.4461em;",[224,22891],{"className":22892,"style":2108},[771],[224,22894,22896],{"className":22895},[776,777,778,779],[224,22897,22899,22902],{"className":22898},[468,779],[224,22900,13182],{"className":22901},[468,469,779],[224,22903,20315],{"className":22904},[468,469,779],[224,22906,787],{"className":22907},[786],[224,22909,22911],{"className":22910},[759],[224,22912,22914],{"className":22913,"style":22776},[763],[224,22915],{},[224,22917],{"className":22918},[549,8479],"。在集合 ",[224,22921,22923,22945],{"className":22922,"translate":388},[392],[224,22924,22926],{"className":22925},[396],[398,22927,22928],{"xmlns":400},[403,22929,22930,22942],{},[406,22931,22932,22934,22936,22938,22940],{},[412,22933,8110],{"stretchy":435},[409,22935,2023],{},[412,22937,325],{},[419,22939,284],{},[412,22941,15077],{"stretchy":435},[447,22943,22944],{"encoding":449},"\\{q>0\\}",[224,22946,22948,22969],{"className":22947,"ariaHidden":455},[454],[224,22949,22951,22954,22957,22960,22963,22966],{"className":22950},[459],[224,22952],{"className":22953,"style":535},[463],[224,22955,8110],{"className":22956},[542],[224,22958,2023],{"className":22959,"style":492},[468,469],[224,22961],{"className":22962,"style":496},[473],[224,22964,325],{"className":22965},[500],[224,22967],{"className":22968,"style":496},[473],[224,22970,22972,22975,22978],{"className":22971},[459],[224,22973],{"className":22974,"style":535},[463],[224,22976,284],{"className":22977},[468],[224,22979,15077],{"className":22980},[549]," 上有 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0\\,d\\mu\n\\end{aligned}",[224,23344,23346],{"className":23345,"ariaHidden":455},[454],[224,23347,23349,23353],{"className":23348},[459],[224,23350],{"className":23351,"style":23352},[463],"height:5.2839em;vertical-align:-2.392em;",[224,23354,23356],{"className":23355},[468],[224,23357,23359,23457],{"className":23358},[5209],[224,23360,23362],{"className":23361},[5331],[224,23363,23365,23448],{"className":23364},[754,755],[224,23366,23368,23445],{"className":23367},[759],[224,23369,23372,23436],{"className":23370,"style":23371},[763],"height:2.892em;",[224,23373,23375,23378],{"style":23374},"top:-4.892em;",[224,23376],{"className":23377,"style":21425},[771],[224,23379,23381,23421,23424,23427,23430,23433],{"className":23380},[468],[224,23382,23384,23387],{"className":23383},[468],[224,23385,20303],{"className":23386,"style":2473},[468,469],[224,23388,23390],{"className":23389},[750],[224,23391,23393,23413],{"className":23392},[754,755],[224,23394,23396,23410],{"className":23395},[759],[224,23397,23399],{"className":23398,"style":20387},[763],[224,23400,23401,23404],{"style":20390},[224,23402],{"className":23403,"style":772},[771],[224,23405,23407],{"className":23406},[776,777,778,779],[224,23408,1705],{"className":23409,"style":1740},[468,469,779],[224,23411,787],{"className":23412},[786],[224,23414,23416],{"className":23415},[759],[224,23417,23419],{"className":23418,"style":20409},[763],[224,23420],{},[224,23422,239],{"className":23423},[542],[224,23425,16018],{"className":23426,"style":653},[468,469],[224,23428,20312],{"className":23429},[468],[224,23431,20315],{"className":23432},[468,469],[224,23434,1716],{"className":23435},[549],[224,23437,23439,23442],{"style":23438},"top:-2.145em;",[224,23440],{"className":23441,"style":21425},[771],[224,23443],{"className":23444},[468],[224,23446,787],{"className":23447},[786],[224,23449,23451],{"className":23450},[759],[224,23452,23455],{"className":23453,"style":23454},[763],"height:2.392em;",[224,23456],{},[224,23458,23460],{"className":23459},[5399],[224,23461,23463,23845],{"className":23462},[754,755],[224,23464,23466,23842],{"className":23465},[759],[224,23467,23469,23652],{"className":23468,"style":23371},[763],[224,23470,23471,23474],{"style":23374},[224,23472],{"className":23473,"style":21425},[771],[224,23475,23477,23480,23483,23486,23489,23547,23550,23553,23556,23559,23637,23640,23643,23646,23649],{"className":23476},[468],[224,23478],{"className":23479},[468],[224,23481],{"className":23482,"style":496},[473],[224,23484,278],{"className":23485},[500],[224,23487],{"className":23488,"style":496},[473],[224,23490,23492,23495],{"className":23491},[21531],[224,23493,21338],{"className":23494,"style":21534},[21531,21532,21533],[224,23496,23498],{"className":23497},[750],[224,23499,23501,23538],{"className":23500},[754,755],[224,23502,23504,23535],{"className":23503},[759],[224,23505,23508],{"className":23506,"style":23507},[763],"height:-0.3869em;",[224,23509,23511,23514],{"style":23510},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[224,23512],{"className":23513,"style":772},[771],[224,23515,23517],{"className":23516},[776,777,778,779],[224,23518,23520,23523,23526,23529,23532],{"className":23519},[468,779],[224,23521,8110],{"className":23522},[542,779],[224,23524,2023],{"className":23525,"style":492},[468,469,779],[224,23527,325],{"className":23528},[500,779],[224,23530,284],{"className":23531},[468,779],[224,23533,15077],{"className":23534},[549,779],[224,23536,787],{"className":23537},[786],[224,23539,23541],{"className":23540},[759],[224,23542,23545],{"className":23543,"style":23544},[763],"height:1.0869em;",[224,23546],{},[224,23548],{"className":23549,"style":724},[473],[224,23551,1705],{"className":23552,"style":1740},[468,469],[224,23554],{"className":23555,"style":21559},[473],[224,23557],{"className":23558,"style":724},[473],[224,23560,23562,23568,23631],{"className":23561},[2091],[224,23563,23565],{"className":23564,"style":8281},[542,8280],[224,23566,239],{"className":23567},[2363,778],[224,23569,23571,23574,23628],{"className":23570},[468],[224,23572],{"className":23573},[542,8479],[224,23575,23577],{"className":23576},[20338],[224,23578,23580,23620],{"className":23579},[754,755],[224,23581,23583,23617],{"className":23582},[759],[224,23584,23587,23598,23606],{"className":23585,"style":23586},[763],"height:1.1076em;",[224,23588,23589,23592],{"style":20531},[224,23590],{"className":23591,"style":2108},[771],[224,23593,23595],{"className":23594},[468],[224,23596,2023],{"className":23597,"style":492},[468,469],[224,23599,23600,23603],{"style":20546},[224,23601],{"className":23602,"style":2108},[771],[224,23604],{"className":23605,"style":20554},[20553],[224,23607,23608,23611],{"style":20557},[224,23609],{"className":23610,"style":2108},[771],[224,23612,23614],{"className":23613},[468],[224,23615,11],{"className":23616},[468,469],[224,23618,787],{"className":23619},[786],[224,23621,23623],{"className":23622},[759],[224,23624,23626],{"clas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",[224,24005,24007,24020],{"className":24006,"translate":388},[392],[224,24008,24010],{"className":24009},[396],[398,24011,24012],{"xmlns":400},[403,24013,24014,24018],{},[406,24015,24016],{},[409,24017,1705],{},[447,24019,1705],{"encoding":449},[224,24021,24023],{"className":24022,"ariaHidden":455},[454],[224,24024,24026,24029],{"className":24025},[459],[224,24027],{"className":24028,"style":2565},[463],[224,24030,1705],{"className":24031,"style":1740},[468,469]," 无界，则这一项是否发散决定了 ",[224,24034,24036,24054],{"className":24035,"translate":388},[392],[224,24037,24039],{"className":24038},[396],[398,24040,24041],{"xmlns":400},[403,24042,24043,24051],{},[406,24044,24045],{},[695,24046,24047,24049],{},[409,24048,20303],{},[409,24050,1705],{},[447,24052,24053],{"encoding":449},"D_f",[224,24055,24057],{"className":24056,"ariaHidden":455},[454],[224,24058,24060,24064],{"className":24059},[459],[224,24061],{"className":24062,"style":24063},[463],"height:0.9694em;vertical-align:-0.2861em;",[224,24065,24067,24070],{"className":24066},[468],[224,24068,20303],{"className":24069,"style":2473},[468,469],[224,24071,24073],{"className":24072},[750],[224,24074,24076,24096],{"className":24075},[754,755],[224,24077,24079,24093],{"className":24078},[759],[224,24080,24082],{"className":24081,"style":20387},[763],[224,24083,24084,24087],{"style":20390},[224,24085],{"className":24086,"style":772},[771],[224,24088,24090],{"className":24089},[776,777,778,779],[224,24091,1705],{"className":24092,"style":1740},[468,469,779],[224,24094,787],{"className":24095},[786],[224,24097,24099],{"className":24098},[759],[224,24100,24102],{"className":24101,"style":20409},[763],[224,24103],{}," 是否有限。例如对 KL 的 ",[224,24106,24108,24140],{"className":24107,"translate":388},[392],[224,24109,24111],{"className":24110},[396],[398,24112,24113],{"xmlns":400},[403,24114,24115,24137],{},[406,24116,24117,24119,24121,24123,24125,24127,24129,24132,24135],{},[409,24118,1705],{},[412,24120,239],{"stretchy":435},[409,24122,1435],{},[412,24124,1716],{"stretchy":435},[412,24126,278],{},[409,24128,1435],{},[409,24130,24131],{},"log",[412,24133,24134],{},"⁡",[409,24136,1435],{},[447,24138,24139],{"encoding":449},"f(t)=t\\log t",[224,24141,24143,24170],{"className":24142,"ariaHidden":455},[454],[224,24144,24146,24149,24152,24155,24158,24161,24164,24167],{"className":24145},[459],[224,24147],{"className":24148,"style":535},[463],[224,24150,1705],{"className":24151,"style":1740},[468,469],[224,24153,239],{"className":24154},[542],[224,24156,1435],{"className":24157},[468,469],[224,24159,1716],{"className":24160},[549],[224,24162],{"className":24163,"style":496},[473],[224,24165,278],{"className":24166},[500],[224,24168],{"className":24169,"style":496},[473],[224,24171,24173,24176,24179,24182,24190,24193],{"className":24172},[459],[224,24174],{"className":24175,"style":2565},[463],[224,24177,1435],{"className":24178},[468,469],[224,24180],{"className":24181,"style":724},[473],[224,24183,24185,24186],{"className":24184},[21531],"lo",[224,24187,24189],{"style":24188},"margin-right:0.0139em;","g",[224,24191],{"className":24192,"style":724},[473],[224,24194,1435],{"className":24195},[468,469],"，当 ",[224,24198,24200,24213],{"className":24199,"translate":388},[392],[224,24201,24203],{"className":24202},[396],[398,24204,24205],{"xmlns":400},[403,24206,24207,24211],{},[406,24208,24209],{},[409,24210,16018],{},[447,24212,16018],{"encoding":449},[224,24214,24216],{"className":24215,"ariaHidden":455},[454],[224,24217,24219,24222],{"className":24218},[459],[224,24220],{"className":24221,"style":649},[463],[224,24223,16018],{"className":24224,"style":653},[468,469],[224,24226,24228,24241],{"className":24227,"translate":388},[392],[224,24229,24231],{"className":24230},[396],[398,24232,24233],{"xmlns":400},[403,24234,24235,24239],{},[406,24236,24237],{},[409,24238,20315],{},[447,24240,20315],{"encoding":449},[224,24242,24244],{"className":24243,"ariaHidden":455},[454],[224,24245,24247,24250],{"className":24246},[459],[224,24248],{"className":24249,"style":20637},[463],[224,24251,20315],{"className":24252},[468,469]," 的零测集上放质量时这一项为 ",[224,24255,24257,24273],{"className":24256,"translate":388},[392],[224,24258,24260],{"className":24259},[396],[398,24261,24262],{"xmlns":400},[403,24263,24264,24270],{},[406,24265,24266,24268],{},[412,24267,319],{},[409,24269,21085],{"mathvariant":678},[447,24271,24272],{"encoding":449},"+\\infty",[224,24274,24276],{"className":24275,"ariaHidden":455},[454],[224,24277,24279,24282,24285],{"className":24278},[459],[224,24280],{"className":24281,"style":464},[463],[224,24283,319],{"className":24284},[468],[224,24286,21085],{"className":24287},[468],"。因此约定：",[224,24290,24292],{"className":24291,"translate":388},[387],[224,24293,24295,24404],{"className":24294,"translate":388},[392],[224,24296,24298],{"className":24297},[396],[398,24299,24300],{"xmlns":400,"display":401},[403,24301,24302,24401],{},[406,24303,24304,24306,24309,24311,24313,24316,24318,24324,24326,24328,24330,24332,24334,24336],{},[409,24305,16018],{},[412,24307,24308],{},"≪̸",[409,24310,20315],{},[423,24312,15054],{},[412,24314,24315],{},"⟹",[423,24317,15054],{},[695,24319,24320,24322],{},[409,24321,20303],{},[409,24323,1705],{},[412,24325,239],{"stretchy":435},[409,24327,16018],{},[409,24329,20312],{"mathvariant":678},[409,24331,20315],{},[412,24333,1716],{"stretchy":435},[412,24335,278],{},[406,24337,24338,24340],{},[412,24339,8110],{"fence":455},[5209,24341,24342,24377],{"rowspacing":8113,"columnalign":8114,"columnspacing":691},[5215,24343,24344,24356],{},[5218,24345,24346],{},[5221,24347,24348],{"scriptlevel":284,"displaystyle":435},[406,24349,24350,24352,24354],{},[412,24351,319],{},[409,24353,21085],{"mathvariant":678},[412,24355,1877],{"separator":455},[5218,24357,24358],{},[5221,24359,24360],{"scriptlevel":284,"displaystyle":435},[406,24361,24362,24365,24367,24370,24372,24375],{},[423,24363,24364],{},"若 ",[409,24366,1705],{},[423,24368,24369],{}," 在 ",[409,24371,21085],{"mathvariant":678},[423,24373,24374],{}," 处增长",[412,24376,1877],{"separator":455},[5215,24378,24379,24390],{},[5218,24380,24381],{},[5221,24382,24383],{"scriptlevel":284,"displaystyle":435},[406,24384,24385,24388],{},[423,24386,24387],{},"有限极限值",[412,24389,1877],{"separator":455},[5218,24391,24392],{},[5221,24393,24394],{"scriptlevel":284,"displaystyle":435},[406,24395,24396,24399],{},[423,24397,24398],{},"否则",[409,24400,688],{"mathvariant":678},[447,24402,24403],{"encoding":449},"P\\not\\ll Q\\ \\Longrightarrow\\\nD_f(P\\|Q)=\n\\begin{cases}\n+\\infty, & \\text{若 } f \\text{ 在 } \\infty\n\\text{ 处增长},\\\\[2pt]\n\\text{有限极限值}, & \\text{否则}.\n\\end{cases}",[224,24405,24407,24455,24479,24549],{"className":24406,"ariaHidden":455},[454],[224,24408,24410,24413,24416,24419,24446,24449,24452],{"className":24409},[459],[224,24411],{"className":24412,"style":2565},[463],[224,24414,16018],{"className":24415,"style":653},[468,469],[224,24417],{"className":24418,"style":496},[473],[224,24420,24422],{"className":24421},[500],[224,24423,24425],{"className":24424},[468,2553],[224,24426,24428],{"className":24427},[2557],[224,24429,24431,24434,24443],{"className":24430},[2561],[224,24432],{"className":24433,"style":2565},[463],[224,24435,24437],{"className":24436},[2569],[224,24438,24440],{"className":24439},[468],[224,24441,2576],{"className":24442},[500],[224,24444],{"className":24445},[2580],[224,24447],{"className":24448},[473,2584],[224,24450,12984],{"className":24451},[500],[224,24453],{"className":24454,"style":496},[473],[224,24456,24458,24461,24464,24467,24470,24473,24476],{"className":24457},[459],[224,24459],{"className":24460,"style":20637},[463],[224,24462,20315],{"className":24463},[468,469],[224,24465,15054],{"className":24466},[473],[224,24468],{"className":24469,"style":496},[473],[224,24471,24315],{"className":24472},[500],[224,24474,15054],{"className":24475},[473],[224,24477],{"className":24478,"style":496},[473],[224,24480,24482,24485,24525,24528,24531,24534,24537,24540,24543,24546],{"className":24481},[459],[224,24483],{"className":24484,"style":20368},[463],[224,24486,24488,24491],{"className":24487},[468],[224,24489,20303],{"className":24490,"style":2473},[468,469],[224,24492,24494],{"className":24493},[750],[224,24495,24497,24517],{"className":24496},[754,755],[224,24498,24500,24514],{"className":24499},[759],[224,24501,24503],{"className":24502,"style":20387},[763],[224,24504,24505,24508],{"style":20390},[224,24506],{"className":24507,"style":772},[771],[224,24509,24511],{"className":24510},[776,777,778,779],[224,24512,1705],{"className":24513,"style":1740},[468,469,779],[224,24515,787],{"className":24516},[786],[224,24518,24520],{"className":24519},[759],[224,24521,24523],{"className":24522,"style":20409},[763],[224,24524],{},[224,24526,239],{"className":24527},[542],[224,24529,16018],{"className":24530,"style":653},[468,469],[224,24532,20312],{"className":24533},[468],[224,24535,20315],{"className":24536},[468,469],[224,24538,1716],{"className":24539},[549],[224,24541],{"className":24542,"style":496},[473],[224,24544,278],{"className":24545},[500],[224,24547],{"className":24548,"style":496},[473],[224,24550,24552,24556],{"className":24551},[459],[224,24553],{"className":24554,"style":24555},[463],"height:3.08em;vertical-align:-1.29em;",[224,24557,24559,24565,24725],{"className":24558},[2091],[224,24560,24562],{"className":24561,"style":8281},[542,8280],[224,24563,8110],{"className":24564},[2363,8285],[224,24566,24568],{"className":24567},[468],[224,24569,24571,24632,24635],{"className":24570},[5209],[224,24572,24574],{"className":24573},[5399],[224,24575,24577,24623],{"className":24576},[754,755],[224,24578,24580,24620],{"className":24579},[759],[224,24581,24584,24602],{"className":24582,"style":24583},[763],"height:1.79em;",[224,24585,24587,24590],{"style":24586},"top:-3.79em;",[224,24588],{"className":24589,"style":8311},[771],[224,24591,24593,24596,24599],{"className":24592},[468],[224,24594,319],{"className":24595},[468],[224,24597,21085],{"className":24598},[468],[224,24600,1877],{"className":24601},[1917],[224,24603,24605,24608],{"style":24604},"top:-2.15em;",[224,24606],{"className":24607,"style":8311},[771],[224,24609,24611,24617],{"className":24610},[468],[224,24612,24614],{"className":24613},[468,11373],[224,24615,24387],{"className":24616},[468,16159],[224,24618,1877],{"className":24619},[1917],[224,24621,787],{"className":24622},[786],[224,24624,24626],{"className":24625},[759],[224,24627,24630],{"className":24628,"style":24629},[763],"height:1.29em;",[224,24631],{},[224,24633],{"className":24634,"style":8373},[8372],[224,24636,24638],{"className":24637},[5399],[224,24639,24641,24717],{"className":24640},[754,755],[224,24642,24644,24714],{"className":24643},[759],[224,24645,24647,24697],{"className":24646,"style":24583},[763],[224,24648,24649,24652],{"style":24586},[224,24650],{"className":24651,"style":8311},[771],[224,24653,24655,24665,24668,24681,24684,24694],{"className":24654},[468],[224,24656,24658,24662],{"className":24657},[468,11373],[224,24659,24661],{"className":24660},[468,16159],"若",[224,24663,15054],{"className":24664},[468],[224,24666,1705],{"className":24667,"style":1740},[468,469],[224,24669,24671,24674,24678],{"className":24670},[468,11373],[224,24672,15054],{"className":24673},[468],[224,24675,24677],{"className":24676},[468,16159],"在",[224,24679,15054],{"className":24680},[468],[224,24682,21085],{"className":24683},[468],[224,24685,24687,24690],{"className":24686},[468,11373],[224,24688,15054],{"className":24689},[468],[224,24691,24693],{"className":24692},[468,16159],"处增长",[224,24695,1877],{"className":24696},[1917],[224,24698,24699,24702],{"style":24604},[224,24700],{"className":24701,"style":8311},[771],[224,24703,24705,24711],{"className":24704},[468],[224,24706,24708],{"className":24707},[468,11373],[224,24709,24398],{"className":24710},[468,16159],[224,24712,688],{"className":24713},[468],[224,24715,787],{"className":24716},[786],[224,24718,24720],{"className":24719},[759],[224,24721,24723],{"className":24722,"style":24629},[763],[224,24724],{},[224,24726],{"className":24727},[549,8479],[11,24729,24730,24731,23884,24759,24787,24788,24962,24963,25032,25033,25083,25084,25153,25154,25211],{},"其中「否则」指 ",[224,24732,24734,24747],{"className":24733,"translate":388},[392],[224,24735,24737],{"className":24736},[396],[398,24738,24739],{"xmlns":400},[403,24740,24741,24745],{},[406,24742,24743],{},[409,24744,1705],{},[447,24746,1705],{"encoding":449},[224,24748,24750],{"className":24749,"ariaHidden":455},[454],[224,24751,24753,24756],{"className":24752},[459],[224,24754],{"className":24755,"style":2565},[463],[224,24757,1705],{"className":24758,"style":1740},[468,469],[224,24760,24762,24775],{"className":24761,"translate":388},[392],[224,24763,24765],{"className":24764},[396],[398,24766,24767],{"xmlns":400},[403,24768,24769,24773],{},[406,24770,24771],{},[409,24772,21085],{"mathvariant":678},[447,24774,23901],{"encoding":449},[224,24776,24778],{"className":24777,"ariaHidden":455},[454],[224,24779,24781,24784],{"className":24780},[459],[224,24782],{"className":24783,"style":839},[463],[224,24785,21085],{"className":24786},[468]," 处有界（如 ",[224,24789,24791,24831],{"className":24790,"translate":388},[392],[224,24792,24794],{"className":24793},[396],[398,24795,24796],{"xmlns":400},[403,24797,24798,24828],{},[406,24799,24800,24802,24804,24806,24808,24810,24818,24820,24822,24824,24826],{},[409,24801,1705],{},[412,24803,239],{"stretchy":435},[409,24805,1435],{},[412,24807,1716],{"stretchy":435},[412,24809,278],{},[5221,24811,24812],{"scriptlevel":284,"displaystyle":435},[20338,24813,24814,24816],{},[419,24815,1438],{},[419,24817,1447],{},[412,24819,1083],{"stretchy":435},[409,24821,1435],{},[412,24823,5268],{},[419,24825,1438],{},[412,24827,1083],{"stretchy":435},[447,24829,24830],{"encoding":449},"f(t)=\\tfrac12\\lvert t-1\\rvert",[224,24832,24834,24861,24950],{"className":24833,"ariaHidden":455},[454],[224,24835,24837,24840,24843,24846,24849,24852,24855,24858],{"className":24836},[459],[224,24838],{"className":24839,"style":535},[463],[224,24841,1705],{"className":24842,"style":1740},[468,469],[224,24844,239],{"className":24845},[542],[224,24847,1435],{"className":24848},[468,469],[224,24850,1716],{"className":24851},[549],[224,24853],{"className":24854,"style":496},[473],[224,24856,278],{"className":24857},[500],[224,24859],{"className":24860,"style":496},[473],[224,24862,24864,24867,24935,24938,24941,24944,24947],{"className":24863},[459],[224,24865],{"className":24866,"style":22544},[463],[224,24868,24870,24873,24932],{"className":24869},[468],[224,24871],{"className":24872},[542,8479],[224,24874,24876],{"className":24875},[20338],[224,24877,24879,24924],{"className":24878},[754,755],[224,24880,24882,24921],{"className":24881},[759],[224,24883,24885,24899,24907],{"className":24884,"style":22563},[763],[224,24886,24887,24890],{"style":22566},[224,24888],{"className":24889,"style":2108},[771],[224,24891,24893],{"className":24892},[776,777,778,779],[224,24894,24896],{"className":24895},[468,779],[224,24897,1447],{"className":24898},[468,779],[224,24900,24901,24904],{"style":20546},[224,24902],{"className":24903,"style":2108},[771],[224,24905],{"className":24906,"style":20554},[20553],[224,24908,24909,24912],{"style":22589},[224,24910],{"className":24911,"style":2108},[771],[224,24913,24915],{"className":24914},[776,777,778,779],[224,24916,24918],{"className":24917},[468,779],[224,24919,1438],{"className":24920},[468,779],[224,24922,787],{"className":24923},[786],[224,24925,24927],{"className":24926},[759],[224,24928,24930],{"className":24929,"style":22611},[763],[224,24931],{},[224,24933],{"className":24934},[549,8479],[224,24936,1083],{"className":24937},[542],[224,24939,1435],{"className":24940},[468,469],[224,24942],{"className":24943,"style":474},[473],[224,24945,5268],{"className":24946},[478],[224,24948],{"className":24949,"style":474},[473],[224,24951,24953,24956,24959],{"className":24952},[459],[224,24954],{"className":24955,"style":535},[463],[224,24957,1438],{"className":24958},[468],[224,24960,1083],{"className":24961},[549],"），此时 ",[224,24964,24966,24983],{"className":24965,"translate":388},[392],[224,24967,24969],{"className":24968},[396],[398,24970,24971],{"xmlns":400},[403,24972,24973,24981],{},[406,24974,24975],{},[695,24976,24977,24979],{},[409,24978,20303],{},[409,24980,1705],{},[447,24982,24053],{"encoding":449},[224,24984,24986],{"className":24985,"ariaHidden":455},[454],[224,24987,24989,24992],{"className":24988},[459],[224,24990],{"className":24991,"style":24063},[463],[224,24993,24995,24998],{"className":24994},[468],[224,24996,20303],{"className":24997,"style":2473},[468,469],[224,24999,25001],{"className":25000},[750],[224,25002,25004,25024],{"className":25003},[754,755],[224,25005,25007,25021],{"className":25006},[759],[224,25008,25010],{"className":25009,"style":20387},[763],[224,25011,25012,25015],{"style":20390},[224,25013],{"className":25014,"style":772},[771],[224,25016,25018],{"className":25017},[776,777,778,779],[224,25019,1705],{"className":25020,"style":1740},[468,469,779],[224,25022,787],{"className":25023},[786],[224,25025,25027],{"className":25026},[759],[224,25028,25030],{"className":25029,"style":20409},[763],[224,25031],{}," 取有限极限值。下文如无特别说明，默认 ",[224,25034,25036,25053],{"className":25035,"translate":388},[392],[224,25037,25039],{"className":25038},[396],[398,25040,25041],{"xmlns":400},[403,25042,25043,25051],{},[406,25044,25045,25047,25049],{},[409,25046,16018],{},[412,25048,12984],{},[409,25050,20315],{},[447,25052,20786],{"encoding":449},[224,25054,25056,25074],{"className":25055,"ariaHidden":455},[454],[224,25057,25059,25062,25065,25068,25071],{"className":25058},[459],[224,25060],{"className":25061,"style":20796},[463],[224,25063,16018],{"className":25064,"style":653},[468,469],[224,25066],{"className":25067,"style":496},[473],[224,25069,12984],{"className":25070},[500],[224,25072],{"className":25073,"style":496},[473],[224,25075,25077,25080],{"className":25076},[459],[224,25078],{"className":25079,"style":20637},[463],[224,25081,20315],{"className":25082},[468,469],"，并把 ",[224,25085,25087,25104],{"className":25086,"translate":388},[392],[224,25088,25090],{"className":25089},[396],[398,25091,25092],{"xmlns":400},[403,25093,25094,25102],{},[406,25095,25096],{},[695,25097,25098,25100],{},[409,25099,20303],{},[409,25101,1705],{},[447,25103,24053],{"encoding":449},[224,25105,25107],{"className":25106,"ariaHidden":455},[454],[224,25108,25110,25113],{"className":25109},[459],[224,25111],{"className":25112,"style":24063},[463],[224,25114,25116,25119],{"className":25115},[468],[224,25117,20303],{"className":25118,"style":2473},[468,469],[224,25120,25122],{"className":25121},[750],[224,25123,25125,25145],{"className":25124},[754,755],[224,25126,25128,25142],{"className":25127},[759],[224,25129,25131],{"className":25130,"style":20387},[763],[224,25132,25133,25136],{"style":20390},[224,25134],{"className":25135,"style":772},[771],[224,25137,25139],{"className":25138},[776,777,778,779],[224,25140,1705],{"className":25141,"style":1740},[468,469,779],[224,25143,787],{"className":25144},[786],[224,25146,25148],{"className":25147},[759],[224,25149,25151],{"className":25150,"style":20409},[763],[224,25152],{}," 看成 ",[224,25155,25157,25181],{"className":25156,"translate":388},[392],[224,25158,25160],{"className":25159},[396],[398,25161,25162],{"xmlns":400},[403,25163,25164,25178],{},[406,25165,25166,25168,25170,25172,25174,25176],{},[412,25167,268],{"stretchy":435},[419,25169,284],{},[412,25171,1877],{"separator":455},[412,25173,319],{},[409,25175,21085],{"mathvariant":678},[412,25177,440],{"stretchy":435},[447,25179,25180],{"encoding":449},"[0,+\\infty]",[224,25182,25184],{"className":25183,"ariaHidden":455},[454],[224,25185,25187,25190,25193,25196,25199,25202,25205,25208],{"className":25186},[459],[224,25188],{"className":25189,"style":535},[463],[224,25191,268],{"className":25192},[542],[224,25194,284],{"className":25195},[468],[224,25197,1877],{"className":25198},[1917],[224,25200],{"className":25201,"style":724},[473],[224,25203,319],{"className":25204},[468],[224,25206,21085],{"className":25207},[468],[224,25209,440],{"className":25210},[549]," 中的值。",[613,25213,25215],{"id":25214},"_12-只依赖似然比","1.2 只依赖似然比",[11,25217,25218,25219,25314,25315,25387,25388,25464,25465,25493,25494,25536],{},"由定义立刻看出：",[224,25220,25222,25250],{"className":25221,"translate":388},[392],[224,25223,25225],{"className":25224},[396],[398,25226,25227],{"xmlns":400},[403,25228,25229,25247],{},[406,25230,25231,25237,25239,25241,25243,25245],{},[695,25232,25233,25235],{},[409,25234,20303],{},[409,25236,1705],{},[412,25238,239],{"stretchy":435},[409,25240,16018],{},[409,25242,20312],{"mathvariant":678},[409,25244,20315],{},[412,25246,1716],{"stretchy":435},[447,25248,25249],{"encoding":449},"D_f(P\\|Q)",[224,25251,25253],{"className":25252,"ariaHidden":455},[454],[224,25254,25256,25259,25299,25302,25305,25308,25311],{"className":25255},[459],[224,25257],{"className":25258,"style":20368},[463],[224,25260,25262,25265],{"className":25261},[468],[224,25263,20303],{"className":25264,"style":2473},[468,469],[224,25266,25268],{"className":25267},[750],[224,25269,25271,25291],{"className":25270},[754,755],[224,25272,25274,25288],{"className":25273},[759],[224,25275,25277],{"className":25276,"style":20387},[763],[224,25278,25279,25282],{"style":20390},[224,25280],{"className":25281,"style":772},[771],[224,25283,25285],{"className":25284},[776,777,778,779],[224,25286,1705],{"className":25287,"style":1740},[468,469,779],[224,25289,787],{"className":25290},[786],[224,25292,25294],{"className":25293},[759],[224,25295,25297],{"className":25296,"style":20409},[763],[224,25298],{},[224,25300,239],{"className":25301},[542],[224,25303,16018],{"className":25304,"style":653},[468,469],[224,25306,20312],{"className":25307},[468],[224,25309,20315],{"className":25310},[468,469],[224,25312,1716],{"className":25313},[549]," 只通过似然比 ",[224,25316,25318,25344],{"className":25317,"translate":388},[392],[224,25319,25321],{"className":25320},[396],[398,25322,25323],{"xmlns":400},[403,25324,25325,25341],{},[406,25326,25327,25329,25331,25333,25335,25337,25339],{},[409,25328,1435],{},[412,25330,278],{},[409,25332,13182],{},[409,25334,16018],{},[409,25336,12413],{"mathvariant":678},[409,25338,13182],{},[409,25340,20315],{},[447,25342,25343],{"encoding":449},"t=dP\u002FdQ",[224,25345,25347,25366],{"className":25346,"ariaHidden":455},[454],[224,25348,25350,25354,25357,25360,25363],{"className":25349},[459],[224,25351],{"className":25352,"style":25353},[463],"height:0.6151em;",[224,25355,1435],{"className":25356},[468,469],[224,25358],{"className":25359,"style":496},[473],[224,25361,278],{"className":25362},[500],[224,25364],{"className":25365,"style":496},[473],[224,25367,25369,25372,25375,25378,25381,25384],{"className":25368},[459],[224,25370],{"className":25371,"style":535},[463],[224,25373,13182],{"className":25374},[468,469],[224,25376,16018],{"className":25377,"style":653},[468,469],[224,25379,12413],{"className":25380},[468],[224,25382,13182],{"className":25383},[468,469],[224,25385,20315],{"className":25386},[468,469]," 起作用。也就是说，对任意可测映射 ",[224,25389,25391,25414],{"className":25390,"translate":388},[392],[224,25392,25394],{"className":25393},[396],[398,25395,25396],{"xmlns":400},[403,25397,25398,25411],{},[406,25399,25400,25402,25404,25406,25408],{},[409,25401,637],{},[412,25403,1071],{},[409,25405,20668],{"mathvariant":20667},[412,25407,1144],{},[409,25409,25410],{"mathvariant":20667},"Y",[447,25412,25413],{"encoding":449},"T:\\mathcal X\\to\\mathcal Y",[224,25415,25417,25435,25453],{"className":25416,"ariaHidden":455},[454],[224,25418,25420,25423,25426,25429,25432],{"className":25419},[459],[224,25421],{"className":25422,"style":649},[463],[224,25424,637],{"className":25425,"style":653},[468,469],[224,25427],{"className":25428,"style":496},[473],[224,25430,1071],{"className":25431},[500],[224,25433],{"className":25434,"style":496},[473],[224,25436,25438,25441,25444,25447,25450],{"className":25437},[459],[224,25439],{"className":25440,"style":649},[463],[224,25442,20668],{"className":25443,"style":20694},[468,20693],[224,25445],{"className":25446,"style":496},[473],[224,25448,1144],{"className":25449},[500],[224,25451],{"className":25452,"style":496},[473],[224,25454,25456,25460],{"className":25455},[459],[224,25457],{"className":25458,"style":25459},[463],"height:0.7805em;vertical-align:-0.0972em;",[224,25461,25410],{"className":25462,"style":25463},[468,20693],"margin-right:0.0822em;","，它是 ",[224,25466,25468,25481],{"className":25467,"translate":388},[392],[224,25469,25471],{"className":25470},[396],[398,25472,25473],{"xmlns":400},[403,25474,25475,25479],{},[406,25476,25477],{},[409,25478,1435],{},[447,25480,1435],{"encoding":449},[224,25482,25484],{"className":25483,"ariaHidden":455},[454],[224,25485,25487,25490],{"className":25486},[459],[224,25488],{"className":25489,"style":25353},[463],[224,25491,1435],{"className":25492},[468,469]," 的函数，而不是 ",[224,25495,25497,25515],{"className":25496,"translate":388},[392],[224,25498,25500],{"className":25499},[396],[398,25501,25502],{"xmlns":400},[403,25503,25504,25512],{},[406,25505,25506,25508,25510],{},[409,25507,11],{},[412,25509,1877],{"separator":455},[409,25511,2023],{},[447,25513,25514],{"encoding":449},"p,q",[224,25516,25518],{"className":25517,"ariaHidden":455},[454],[224,25519,25521,25524,25527,25530,25533],{"className":25520},[459],[224,25522],{"className":25523,"style":806},[463],[224,25525,11],{"className":25526},[468,469],[224,25528,1877],{"className":25529},[1917],[224,25531],{"className":25532,"style":724},[473],[224,25534,2023],{"className":25535,"style":492},[468,469]," 各自的函数。",[11,25538,25539,25540,25579],{},"这一点是 f-散度「缺少几何」的根源，也是它与 Wasserstein 距离分道扬镳的地方（见第 6 节）：只要保持比值 ",[224,25541,25543,25561],{"className":25542,"translate":388},[392],[224,25544,25546],{"className":25545},[396],[398,25547,25548],{"xmlns":400},[403,25549,25550,25558],{},[406,25551,25552,25554,25556],{},[409,25553,11],{},[409,25555,12413],{"mathvariant":678},[409,25557,2023],{},[447,25559,25560],{"encoding":449},"p\u002Fq",[224,25562,25564],{"className":25563,"ariaHidden":455},[454],[224,25565,25567,25570,25573,25576],{"className":25566},[459],[224,25568],{"className":25569,"style":535},[463],[224,25571,11],{"className":25572},[468,469],[224,25574,12413],{"className":25575},[468],[224,25577,2023],{"className":25578,"style":492},[468,469]," 不变，把两个分布同时「搬到别处」，f-散度的值完全不动。",[613,25581,25583],{"id":25582},"_13-非单射与仿射自由度","1.3 非单射与仿射自由度",[11,25585,25586,25587,372,25654,25718],{},"注意 ",[224,25588,25590,25612],{"className":25589,"translate":388},[392],[224,25591,25593],{"className":25592},[396],[398,25594,25595],{"xmlns":400},[403,25596,25597,25609],{},[406,25598,25599,25601,25603,25605,25607],{},[412,25600,21338],{},[409,25602,13182],{},[409,25604,20315],{},[412,25606,278],{},[419,25608,1438],{},[447,25610,25611],{"encoding":449},"\\int dQ=1",[224,25613,25615,25645],{"className":25614,"ariaHidden":455},[454],[224,25616,25618,25622,25627,25630,25633,25636,25639,25642],{"className":25617},[459],[224,25619],{"className":25620,"style":25621},[463],"height:1.1111em;vertical-align:-0.3061em;",[224,25623,21338],{"className":25624,"style":25626},[21531,21532,25625],"small-op","margin-right:0.1945em;position:relative;top:-0.0006em;",[224,25628],{"className":25629,"style":724},[473],[224,25631,13182],{"className":25632},[468,469],[224,25634,20315],{"className":25635},[468,469],[224,25637],{"className":25638,"style":496},[473],[224,25640,278],{"className":25641},[500],[224,25643],{"className":25644,"style":496},[473],[224,25646,25648,25651],{"className":25647},[459],[224,25649],{"className":25650,"style":510},[463],[224,25652,1438],{"className":25653},[468],[224,25655,25657,25679],{"className":25656,"translate":388},[392],[224,25658,25660],{"className":25659},[396],[398,25661,25662],{"xmlns":400},[403,25663,25664,25676],{},[406,25665,25666,25668,25670,25672,25674],{},[412,25667,21338],{},[409,25669,13182],{},[409,25671,16018],{},[412,25673,278],{},[419,25675,1438],{},[447,25677,25678],{"encoding":449},"\\int dP=1",[224,25680,25682,25709],{"className":25681,"ariaHidden":455},[454],[224,25683,25685,25688,25691,25694,25697,25700,25703,25706],{"className":25684},[459],[224,25686],{"className":25687,"style":25621},[463],[224,25689,21338],{"className":25690,"style":25626},[21531,21532,25625],[224,25692],{"className":25693,"style":724},[473],[224,25695,13182],{"className":25696},[468,469],[224,25698,16018],{"className":25699,"style":653},[468,469],[224,25701],{"className":25702,"style":496},[473],[224,25704,278],{"className":25705},[500],[224,25707],{"className":25708,"style":496},[473],[224,25710,25712,25715],{"className":25711},[459],[224,25713],{"className":25714,"style":510},[463],[224,25716,1438],{"className":25717},[468],"，因此",[224,25720,25722],{"className":25721,"translate":388},[387],[224,25723,25725,25776],{"className":25724,"translate":388},[392],[224,25726,25728],{"className":25727},[396],[398,25729,25730],{"xmlns":400,"display":401},[403,25731,25732,25773],{},[406,25733,25734,25736,25738,25740,25742,25744,25746,25748,25750,25752,25754,25756,25758,25760,25762,25764,25766,25768,25770],{},[412,25735,21338],{},[412,25737,239],{"stretchy":435},[409,25739,1435],{},[412,25741,5268],{},[419,25743,1438],{},[412,25745,1716],{"stretchy":435},[423,25747,682],{},[409,25749,13182],{},[409,25751,20315],{},[412,25753,278],{},[412,25755,21338],{},[409,25757,13182],{},[409,25759,16018],{},[412,25761,5268],{},[412,25763,21338],{},[409,25765,13182],{},[409,25767,20315],{},[412,25769,278],{},[419,25771,25772],{},"0.",[447,25774,25775],{"encoding":449},"\\int (t-1)\\,dQ=\\int dP-\\int dQ=0 .",[224,25777,25779,25804,25834,25861,25888],{"className":25778,"ariaHidden":455},[454],[224,25780,25782,25786,25789,25792,25795,25798,25801],{"className":25781},[459],[224,25783],{"className":25784,"style":25785},[463],"height:2.2222em;vertical-align:-0.8622em;",[224,25787,21338],{"className":25788,"style":21534},[21531,21532,21533],[224,25790,239],{"className":25791},[542],[224,25793,1435],{"className":25794},[468,469],[224,25796],{"className":25797,"style":474},[473],[224,25799,5268],{"className":25800},[478],[224,25802],{"className":25803,"style":474},[473],[224,25805,25807,25810,25813,25816,25819,25822,25825,25828,25831],{"className":25806},[459],[224,25808],{"className":25809,"style":535},[463],[224,25811,1438],{"className":25812},[468],[224,25814,1716],{"className":25815},[549],[224,25817],{"className":25818,"style":724},[473],[224,25820,13182],{"className":25821},[468,469],[224,25823,20315],{"className":25824},[468,469],[224,25826],{"className":25827,"style":496},[473],[224,25829,278],{"className":25830},[500],[224,25832],{"className":25833,"style":496},[473],[224,25835,25837,25840,25843,25846,25849,25852,25855,25858],{"className":25836},[459],[224,25838],{"className":25839,"style":25785},[463],[224,25841,21338],{"className":25842,"style":21534},[21531,21532,21533],[224,25844],{"className":25845,"style":724},[473],[224,25847,13182],{"className":25848},[468,469],[224,25850,16018],{"className":25851,"style":653},[468,469],[224,25853],{"className":25854,"style":474},[473],[224,25856,5268],{"className":25857},[478],[224,25859],{"className":25860,"style":474},[473],[224,25862,25864,25867,25870,25873,25876,25879,25882,25885],{"className":25863},[459],[224,25865],{"className":25866,"style":25785},[463],[224,25868,21338],{"className":25869,"style":21534},[21531,21532,21533],[224,25871],{"className":25872,"style":724},[473],[224,25874,13182],{"className":25875},[468,469],[224,25877,20315],{"className":25878},[468,469],[224,25880],{"className":25881,"style":496},[473],[224,25883,278],{"className":25884},[500],[224,25886],{"className":25887,"style":496},[473],[224,25889,25891,25894],{"className":25890},[459],[224,25892],{"className":25893,"style":510},[463],[224,25895,25772],{"className":25896},[468],[11,25898,25899,25900,7098],{},"于是对任意常数 ",[224,25901,25903,25921],{"className":25902,"translate":388},[392],[224,25904,25906],{"className":25905},[396],[398,25907,25908],{"xmlns":400},[403,25909,25910,25918],{},[406,25911,25912,25914,25916],{},[409,25913,219],{},[412,25915,12248],{},[409,25917,21092],{"mathvariant":12251},[447,25919,25920],{"encoding":449},"c\\in\\mathbb R",[224,25922,25924,25942],{"className":25923,"ariaHidden":455},[454],[224,25925,25927,25930,25933,25936,25939],{"className":25926},[459],[224,25928],{"className":25929,"style":2810},[463],[224,25931,219],{"className":25932},[468,469],[224,25934],{"className":25935,"style":496},[473],[224,25937,12248],{"className":25938},[500],[224,25940],{"className":25941,"style":496},[473],[224,25943,25945,25948],{"className":25944},[459],[224,25946],{"className":25947,"style":12385},[463],[224,25949,21092],{"className":25950},[468,12389],[224,25952,25954],{"className":25953,"translate":388},[387],[224,25955,25957,26073],{"className":25956,"translate":388},[392],[224,25958,25960],{"className":25959},[396],[398,25961,25962],{"xmlns":400,"display":401},[403,25963,25964,26070],{},[406,25965,25966,25988,25990,25992,25994,25996,25998,26000,26006,26008,26010,26012,26014,26016,26018,26020,26050,26052,26058,26060,26062,26064,26066,26068],{},[695,25967,25968,25970],{},[409,25969,20303],{},[406,25971,25972,25974,25976,25978,25980,25982,25984,25986],{},[409,25973,1705],{},[412,25975,319],{},[409,25977,219],{},[412,25979,239],{"stretchy":435},[409,25981,1435],{},[412,25983,5268],{},[419,25985,1438],{},[412,25987,1716],{"stretchy":435},[412,25989,239],{"stretchy":435},[409,25991,16018],{},[409,25993,20312],{"mathvariant":678},[409,25995,20315],{},[412,25997,1716],{"stretchy":435},[412,25999,278],{},[695,26001,26002,26004],{},[409,26003,20303],{},[409,26005,1705],{},[412,26007,239],{"stretchy":435},[409,26009,16018],{},[409,26011,20312],{"mathvariant":678},[409,26013,20315],{},[412,26015,1716],{"stretchy":435},[412,26017,319],{},[409,26019,219],{},[1965,26021,26022,26044],{},[1965,26023,26024,26042],{},[406,26025,26026,26028,26030,26032,26034,26036,26038,26040],{},[412,26027,21338],{},[412,26029,239],{"stretchy":435},[409,26031,1435],{},[412,26033,5268],{},[419,26035,1438],{},[412,26037,1716],{"stretchy":435},[409,26039,13182],{},[409,26041,20315],{},[412,26043,1987],{"stretchy":455},[406,26045,26046,26048],{},[412,26047,278],{},[419,26049,284],{},[412,26051,278],{},[695,26053,26054,26056],{},[409,26055,20303],{},[409,26057,1705],{},[412,26059,239],{"stretchy":435},[409,26061,16018],{},[409,26063,20312],{"mathvariant":678},[409,26065,20315],{},[412,26067,1716],{"stretchy":435},[409,26069,688],{"mathvariant":678},[447,26071,26072],{"encoding":449},"D_{f+c(t-1)}(P\\|Q)\n=D_f(P\\|Q)+c\\underbrace{\\int(t-1)dQ}_{=0}\n=D_f(P\\|Q).",[224,26074,26076,26174,26244,26409],{"className":26075,"ariaHidden":455},[454],[224,26077,26079,26083,26150,26153,26156,26159,26162,26165,26168,26171],{"className":26078},[459],[224,26080],{"className":26081,"style":26082},[463],"height:1.1052em;vertical-align:-0.3552em;",[224,26084,26086,26089],{"className":26085},[468],[224,26087,20303],{"className":26088,"style":2473},[468,469],[224,26090,26092],{"className":26091},[750],[224,26093,26095,26141],{"className":26094},[754,755],[224,26096,26098,26138],{"className":26097},[759],[224,26099,26102],{"className":26100,"style":26101},[763],"height:0.3448em;",[224,26103,26105,26108],{"style":26104},"top:-2.5198em;margin-left:-0.0278em;margin-right:0.05em;",[224,26106],{"className":26107,"style":772},[771],[224,26109,26111],{"className":26110},[776,777,778,779],[224,26112,26114,26117,26120,26123,26126,26129,26132,26135],{"className":26113},[468,779],[224,26115,1705],{"className":26116,"style":1740},[468,469,779],[224,26118,319],{"className":26119},[478,779],[224,26121,219],{"className":26122},[468,469,779],[224,26124,239],{"className":26125},[542,779],[224,26127,1435],{"className":26128},[468,469,779],[224,26130,5268],{"className":26131},[478,779],[224,26133,1438],{"className":26134},[468,779],[224,26136,1716],{"className":26137},[549,779],[224,26139,787],{"className":26140},[786],[224,26142,26144],{"className":26143},[759],[224,26145,26148],{"className":26146,"style":26147},[763],"height:0.3552em;",[224,26149],{},[224,26151,239],{"className":26152},[542],[224,26154,16018],{"className":26155,"style":653},[468,469],[224,26157,20312],{"className":26158},[468],[224,26160,20315],{"className":26161},[468,469],[224,26163,1716],{"className":26164},[549],[224,26166],{"className":26167,"style":496},[473],[224,26169,278],{"className":26170},[500],[224,26172],{"className":26173,"style":496},[473],[224,26175,26177,26180,26220,26223,26226,26229,26232,26235,26238,26241],{"className":26176},[459],[224,26178],{"className":26179,"style":20368},[463],[224,26181,26183,26186],{"className":26182},[468],[224,26184,20303],{"className":26185,"style":2473},[468,469],[224,26187,26189],{"className":26188},[750],[224,26190,26192,26212],{"className":26191},[754,755],[224,26193,26195,26209],{"className":26194},[759],[224,26196,26198],{"className":26197,"style":20387},[763],[224,26199,26200,26203],{"style":20390},[224,26201],{"className":26202,"style":772},[771],[224,26204,26206],{"className":26205},[776,777,778,779],[224,26207,1705],{"className":26208,"style":1740},[468,469,779],[224,26210,787],{"className":26211},[786],[224,26213,26215],{"className":26214},[759],[224,26216,26218],{"className":26217,"style":20409},[763],[224,26219],{},[224,26221,239],{"className":26222},[542],[224,26224,16018],{"className":26225,"style":653},[468,469],[224,26227,20312],{"className":26228},[468],[224,26230,20315],{"className":26231},[468,469],[224,26233,1716],{"className":26234},[549],[224,26236],{"className":26237,"style":474},[473],[224,26239,319],{"className":26240},[478],[224,26242],{"className":26243,"style":474},[473],[224,26245,26247,26251,26254,26257,26400,26403,26406],{"className":26246},[459],[224,26248],{"className":26249,"style":26250},[463],"height:3.5214em;vertical-align:-2.1614em;",[224,26252,219],{"className":26253},[468,469],[224,26255],{"className":26256,"style":724},[473],[224,26258,26260],{"className":26259},[2091,1965],[224,26261,26263,26391],{"className":26262},[754,755],[224,26264,26266,26388],{"className":26265},[759],[224,26267,26270,26289],{"className":26268,"style":26269},[763],"height:1.36em;",[224,26271,26273,26277],{"style":26272},"top:-1.1986em;",[224,26274],{"className":26275,"style":26276},[771],"height:3.36em;",[224,26278,26280],{"className":26279},[776,777,778,779],[224,26281,26283,26286],{"className":26282},[468,779],[224,26284,278],{"className":26285},[500,779],[224,26287,284],{"className":26288},[468,779],[224,26290,26292,26295],{"style":26291},"top:-3.36em;",[224,26293],{"className":26294,"style":26276},[771],[224,26296,26298],{"className":26297},[2091,1965],[224,26299,26301,26379],{"className":26300},[754,755],[224,26302,26304,26376],{"className":26303},[759],[224,26305,26307,26338],{"className":26306,"style":26269},[763],[224,26308,26311,26314],{"className":26309,"style":26310},[2139],"top:-1.8498em;",[224,26312],{"className":26313,"style":26276},[771],[224,26315,26317,26324,26331],{"className":26316,"style":2148},[2147],[224,26318,26320],{"className":26319,"style":2153},[2152],[2155,26321,26322],{"xmlns":2157,"width":2158,"height":2159,"viewBox":2160,"preserveAspectRatio":2161},[2163,26323],{"d":2165},[224,26325,26327],{"className":26326,"style":2153},[2169],[2155,26328,26329],{"xmlns":2157,"width":2158,"height":2159,"viewBox":2160,"preserveAspectRatio":2172},[2163,26330],{"d":2175},[224,26332,26334],{"className":26333,"style":2153},[2179],[2155,26335,26336],{"xmlns":2157,"width":2158,"height":2159,"viewBox":2160,"preserveAspectRatio":2182},[2163,26337],{"d":2185},[224,26339,26340,26343],{"style":26291},[224,26341],{"className":26342,"style":26276},[771],[224,26344,26346,26349,26352,26355,26358,26361,26364,26367,26370,26373],{"className":26345},[468],[224,26347,21338],{"className":26348,"style":21534},[21531,21532,21533],[224,26350,239],{"className":26351},[542],[224,26353,1435],{"className":26354},[468,469],[224,26356],{"className":26357,"style":474},[473],[224,26359,5268],{"className":26360},[478],[224,26362],{"className":26363,"style":474},[473],[224,26365,1438],{"className":26366},[468],[224,26368,1716],{"className":26369},[549],[224,26371,13182],{"className":26372},[468,469],[224,26374,20315],{"className":26375},[468,469],[224,26377,787],{"className":26378},[786],[224,26380,26382],{"className":26381},[759],[224,26383,26386],{"className":26384,"style":26385},[763],"height:1.5103em;",[224,26387],{},[224,26389,787],{"className":26390},[786],[224,26392,26394],{"className":26393},[759],[224,26395,26398],{"className":26396,"style":26397},[763],"height:2.1614em;",[224,26399],{},[224,26401],{"className":26402,"style":496},[473],[224,26404,278],{"className":26405},[500],[224,26407],{"className":26408,"style":496},[473],[224,26410,26412,26415,26455,26458,26461,26464,26467,26470],{"className":26411},[459],[224,26413],{"className":26414,"style":20368},[463],[224,26416,26418,26421],{"className":26417},[468],[224,26419,20303],{"className":26420,"style":2473},[468,469],[224,26422,26424],{"className":26423},[750],[224,26425,26427,26447],{"className":26426},[754,755],[224,26428,26430,26444],{"className":26429},[759],[224,26431,26433],{"className":26432,"style":20387},[763],[224,26434,26435,26438],{"style":20390},[224,26436],{"className":26437,"style":772},[771],[224,26439,26441],{"className":26440},[776,777,778,779],[224,26442,1705],{"className":26443,"style":1740},[468,469,779],[224,26445,787],{"className":26446},[786],[224,26448,26450],{"className":26449},[759],[224,26451,26453],{"className":26452,"style":20409},[763],[224,26454],{},[224,26456,239],{"className":26457},[542],[224,26459,16018],{"className":26460,"style":653},[468,469],[224,26462,20312],{"className":26463},[468],[224,26465,20315],{"className":26466},[468,469],[224,26468,1716],{"className":26469},[549],[224,26471,688],{"className":26472},[468],[11,26474,26475,26478,26479,26548,26549,26577,26578,26644],{},[378,26476,26477],{},"结论："," ",[224,26480,26482,26499],{"className":26481,"translate":388},[392],[224,26483,26485],{"className":26484},[396],[398,26486,26487],{"xmlns":400},[403,26488,26489,26497],{},[406,26490,26491],{},[695,26492,26493,26495],{},[409,26494,20303],{},[409,26496,1705],{},[447,26498,24053],{"encoding":449},[224,26500,26502],{"className":26501,"ariaHidden":455},[454],[224,26503,26505,26508],{"className":26504},[459],[224,26506],{"className":26507,"style":24063},[463],[224,26509,26511,26514],{"className":26510},[468],[224,26512,20303],{"className":26513,"style":2473},[468,469],[224,26515,26517],{"className":26516},[750],[224,26518,26520,26540],{"className":26519},[754,755],[224,26521,26523,26537],{"className":26522},[759],[224,26524,26526],{"className":26525,"style":20387},[763],[224,26527,26528,26531],{"style":20390},[224,26529],{"className":26530,"style":772},[771],[224,26532,26534],{"className":26533},[776,777,778,779],[224,26535,1705],{"className":26536,"style":1740},[468,469,779],[224,26538,787],{"className":26539},[786],[224,26541,26543],{"className":26542},[759],[224,26544,26546],{"className":26545,"style":20409},[763],[224,26547],{}," 只依赖 ",[224,26550,26552,26565],{"className":26551,"translate":388},[392],[224,26553,26555],{"className":26554},[396],[398,26556,26557],{"xmlns":400},[403,26558,26559,26563],{},[406,26560,26561],{},[409,26562,1705],{},[447,26564,1705],{"encoding":449},[224,26566,26568],{"className":26567,"ariaHidden":455},[454],[224,26569,26571,26574],{"className":26570},[459],[224,26572],{"className":26573,"style":2565},[463],[224,26575,1705],{"className":26576,"style":1740},[468,469]," 在「模去 ",[224,26579,26581,26605],{"className":26580,"translate":388},[392],[224,26582,26584],{"className":26583},[396],[398,26585,26586],{"xmlns":400},[403,26587,26588,26602],{},[406,26589,26590,26592,26594,26596,26598,26600],{},[409,26591,219],{},[412,26593,239],{"stretchy":435},[409,26595,1435],{},[412,26597,5268],{},[419,26599,1438],{},[412,26601,1716],{"stretchy":435},[447,26603,26604],{"encoding":449},"c(t-1)",[224,26606,26608,26632],{"className":26607,"ariaHidden":455},[454],[224,26609,26611,26614,26617,26620,26623,26626,26629],{"className":26610},[459],[224,26612],{"className":26613,"style":535},[463],[224,26615,219],{"className":26616},[468,469],[224,26618,239],{"className":26619},[542],[224,26621,1435],{"className":26622},[468,469],[224,26624],{"className":26625,"style":474},[473],[224,26627,5268],{"className":26628},[478],[224,26630],{"className":26631,"style":474},[473],[224,26633,26635,26638,26641],{"className":26634},[459],[224,26636],{"className":26637,"style":535},[463],[224,26639,1438],{"className":26640},[468],[224,26642,1716],{"className":26643},[549],"」之后的等价类。两个常用规范化是：",[9997,26646,26647,26715],{},[2677,26648,26649,26714],{},[224,26650,26652,26675],{"className":26651,"translate":388},[392],[224,26653,26655],{"className":26654},[396],[398,26656,26657],{"xmlns":400},[403,26658,26659,26673],{},[406,26660,26661,26663,26665,26667,26669,26671],{},[409,26662,1705],{},[412,26664,239],{"stretchy":435},[419,26666,1438],{},[412,26668,1716],{"stretchy":435},[412,26670,278],{},[419,26672,284],{},[447,26674,20243],{"encoding":449},[224,26676,26678,26705],{"className":26677,"ariaHidden":455},[454],[224,26679,26681,26684,26687,26690,26693,26696,26699,26702],{"className":26680},[459],[224,26682],{"className":26683,"style":535},[463],[224,26685,1705],{"className":26686,"style":1740},[468,469],[224,26688,239],{"className":26689},[542],[224,26691,1438],{"className":26692},[468],[224,26694,1716],{"className":26695},[549],[224,26697],{"className":26698,"style":496},[473],[224,26700,278],{"className":26701},[500],[224,26703],{"className":26704,"style":496},[473],[224,26706,26708,26711],{"className":26707},[459],[224,26709],{"className":26710,"style":510},[463],[224,26712,284],{"className":26713},[468],"（本文采用）；",[2677,26716,26717,7755,26782,26881,26882,7098,27092,27120],{},[224,26718,26720,26743],{"className":26719,"translate":388},[392],[224,26721,26723],{"className":26722},[396],[398,26724,26725],{"xmlns":400},[403,26726,26727,26741],{},[406,26728,26729,26731,26733,26735,26737,26739],{},[409,26730,1705],{},[412,26732,239],{"stretchy":435},[419,26734,1438],{},[412,26736,1716],{"stretchy":435},[412,26738,278],{},[419,26740,284],{},[447,26742,20243],{"encoding":449},[224,26744,26746,26773],{"className":26745,"ariaHidden":455},[454],[224,26747,26749,26752,26755,26758,26761,26764,26767,26770],{"className":26748},[459],[224,26750],{"className":26751,"style":535},[463],[224,26753,1705],{"className":26754,"style":1740},[468,469],[224,26756,239],{"className":26757},[542],[224,26759,1438],{"className":26760},[468],[224,26762,1716],{"className":26763},[549],[224,26765],{"className":26766,"style":496},[473],[224,26768,278],{"className":26769},[500],[224,26771],{"className":26772,"style":496},[473],[224,26774,26776,26779],{"className":26775},[459],[224,26777],{"className":26778,"style":510},[463],[224,26780,284],{"className":26781},[468],[224,26783,26785,26813],{"className":26784,"translate":388},[392],[224,26786,26788],{"className":26787},[396],[398,26789,26790],{"xmlns":400},[403,26791,26792,26810],{},[406,26793,26794,26800,26802,26804,26806,26808],{},[3248,26795,26796,26798],{},[409,26797,1705],{},[412,26799,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,26801,239],{"stretchy":435},[419,26803,1438],{},[412,26805,1716],{"stretchy":435},[412,26807,278],{},[419,26809,284],{},[447,26811,26812],{"encoding":449},"f'(1)=0",[224,26814,26816,26872],{"className":26815,"ariaHidden":455},[454],[224,26817,26819,26822,26854,26857,26860,26863,26866,26869],{"className":26818},[459],[224,26820],{"className":26821,"style":6966},[463],[224,26823,26825,26828],{"className":26824},[468],[224,26826,1705],{"className":26827,"style":1740},[468,469],[224,26829,26831],{"className":26830},[750],[224,26832,26834],{"className":26833},[754],[224,26835,26837],{"className":26836},[759],[224,26838,26840],{"className":26839,"style":6992},[763],[224,26841,26842,26845],{"style":3288},[224,26843],{"className":26844,"style":772},[771],[224,26846,26848],{"className":26847},[776,777,778,779],[224,26849,26851],{"className":26850},[468,779],[224,26852,6933],{"className":26853},[468,779],[224,26855,239],{"className":26856},[542],[224,26858,1438],{"className":26859},[468],[224,26861,1716],{"className":26862},[549],[224,26864],{"className":26865,"style":496},[473],[224,26867,278],{"className":26868},[500],[224,26870],{"className":26871,"style":496},[473],[224,26873,26875,26878],{"className":26874},[459],[224,26876],{"className":26877,"style":510},[463],[224,26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",[224,26883,26885,26943],{"className":26884,"translate":388},[392],[224,26886,26888],{"className":26887},[396],[398,26889,26890],{"xmlns":400},[403,26891,26892,26940],{},[406,26893,26894,26896,26898,26900,26902,26904,26906,26908,26910,26912,26914,26920,26922,26924,26926,26928,26930,26932,26934,26936,26938],{},[409,26895,1705],{},[412,26897,239],{"stretchy":435},[409,26899,1435],{},[412,26901,1716],{"stretchy":435},[412,26903,9766],{},[409,26905,1705],{},[412,26907,239],{"stretchy":435},[419,26909,1438],{},[412,26911,1716],{"stretchy":435},[412,26913,319],{},[3248,26915,26916,26918],{},[409,26917,1705],{},[412,26919,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,26921,239],{"stretchy":435},[419,26923,1438],{},[412,26925,1716],{"stretchy":435},[412,26927,239],{"stretchy":435},[409,26929,1435],{},[412,26931,5268],{},[419,26933,1438],{},[412,26935,1716],{"stretchy":435},[412,26937,278],{},[419,26939,284],{},[447,26941,26942],{"encoding":449},"f(t)\\ge f(1)+f'(1)(t-1)=0",[224,26944,26946,26973,27000,27062,27083],{"className":26945,"ariaHidden":455},[454],[224,26947,26949,26952,26955,26958,26961,26964,26967,26970],{"className":26948},[459],[224,26950],{"className":26951,"style":535},[463],[224,26953,1705],{"className":26954,"style":1740},[468,469],[224,26956,239],{"className":26957},[542],[224,26959,1435],{"className":26960},[468,469],[224,26962,1716],{"className":26963},[549],[224,26965],{"className":26966,"style":496},[473],[224,26968,9766],{"className":26969},[500],[224,26971],{"className":26972,"style":496},[473],[224,26974,26976,26979,26982,26985,26988,26991,26994,26997],{"className":26975},[459],[224,26977],{"className":26978,"style":535},[463],[224,26980,1705],{"className":26981,"style":1740},[468,469],[224,26983,239],{"className":26984},[542],[224,26986,1438],{"className":26987},[468],[224,26989,1716],{"className":26990},[549],[224,26992],{"className":26993,"style":474},[473],[224,26995,319],{"className":26996},[478],[224,26998],{"className":26999,"style":474},[473],[224,27001,27003,27006,27038,27041,27044,27047,27050,27053,27056,27059],{"className":27002},[459],[224,27004],{"className":27005,"style":6966},[463],[224,27007,27009,27012],{"className":27008},[468],[224,27010,1705],{"className":27011,"style":1740},[468,469],[224,27013,27015],{"className":27014},[750],[224,27016,27018],{"className":27017},[754],[224,27019,27021],{"className":27020},[759],[224,27022,27024],{"className":27023,"style":6992},[763],[224,27025,27026,27029],{"style":3288},[224,27027],{"className":27028,"style":772},[771],[224,27030,27032],{"className":27031},[776,777,778,779],[224,27033,27035],{"className":27034},[468,779],[224,27036,6933],{"className":27037},[468,779],[224,27039,239],{"className":27040},[542],[224,27042,1438],{"className":27043},[468],[224,27045,1716],{"className":27046},[549],[224,27048,239],{"className":27049},[542],[224,27051,1435],{"className":27052},[468,469],[224,27054],{"className":27055,"style":474},[473],[224,27057,5268],{"className":27058},[478],[224,27060],{"className":27061,"style":474},[473],[224,27063,27065,27068,27071,27074,27077,27080],{"className":27064},[459],[224,27066],{"className":27067,"style":535},[463],[224,27069,1438],{"className":27070},[468],[224,27072,1716],{"className":27073},[549],[224,27075],{"className":27076,"style":496},[473],[224,27078,278],{"className":27079},[500],[224,27081],{"className":27082,"style":496},[473],[224,27084,27086,27089],{"className":27085},[459],[224,27087],{"className":27088,"style":510},[463],[224,27090,284],{"className":27091},[468],[224,27093,27095,27108],{"className":27094,"translate":388},[392],[224,27096,27098],{"className":27097},[396],[398,27099,27100],{"xmlns":400},[403,27101,27102,27106],{},[406,27103,27104],{},[409,27105,1705],{},[447,27107,1705],{"encoding":449},[224,27109,27111],{"className":27110,"ariaHidden":455},[454],[224,27112,27114,27117],{"className":27113},[459],[224,27115],{"className":27116,"style":2565},[463],[224,27118,1705],{"className":27119,"style":1740},[468,469]," 在整个定义域上非负。",[11,27122,27123,27124,27209,27210,27262,27263,27332],{},"第 3 节里的 KL 的 ",[224,27125,27127,27156],{"className":27126,"translate":388},[392],[224,27128,27130],{"className":27129},[396],[398,27131,27132],{"xmlns":400},[403,27133,27134,27154],{},[406,27135,27136,27138,27140,27142,27144,27146,27148,27150,27152],{},[409,27137,1705],{},[412,27139,239],{"stretchy":435},[409,27141,1435],{},[412,27143,1716],{"stretchy":435},[412,27145,278],{},[409,27147,1435],{},[409,27149,24131],{},[412,27151,24134],{},[409,27153,1435],{},[447,27155,24139],{"encoding":449},[224,27157,27159,27186],{"className":27158,"ariaHidden":455},[454],[224,27160,27162,27165,27168,27171,27174,27177,27180,27183],{"className":27161},[459],[224,27163],{"className":27164,"style":535},[463],[224,27166,1705],{"className":27167,"style":1740},[468,469],[224,27169,239],{"className":27170},[542],[224,27172,1435],{"className":27173},[468,469],[224,27175,1716],{"className":27176},[549],[224,27178],{"className":27179,"style":496},[473],[224,27181,278],{"className":27182},[500],[224,27184],{"className":27185,"style":496},[473],[224,27187,27189,27192,27195,27198,27203,27206],{"className":27188},[459],[224,27190],{"className":27191,"style":2565},[463],[224,27193,1435],{"className":27194},[468,469],[224,27196],{"className":27197,"style":724},[473],[224,27199,24185,27201],{"className":27200},[21531],[224,27202,24189],{"style":24188},[224,27204],{"className":27205,"style":724},[473],[224,27207,1435],{"className":27208},[468,469]," 不满足第 2 条（它在 ",[224,27211,27213,27231],{"className":27212,"translate":388},[392],[224,27214,27216],{"className":27215},[396],[398,27217,27218],{"xmlns":400},[403,27219,27220,27228],{},[406,27221,27222,27224,27226],{},[409,27223,1435],{},[412,27225,1577],{},[419,27227,1438],{},[447,27229,27230],{"encoding":449},"t\u003C1",[224,27232,27234,27253],{"className":27233,"ariaHidden":455},[454],[224,27235,27237,27241,27244,27247,27250],{"className":27236},[459],[224,27238],{"className":27239,"style":27240},[463],"height:0.6542em;vertical-align:-0.0391em;",[224,27242,1435],{"className":27243},[468,469],[224,27245],{"className":27246,"style":496},[473],[224,27248,1577],{"className":27249},[500],[224,27251],{"className":27252,"style":496},[473],[224,27254,27256,27259],{"className":27255},[459],[224,27257],{"className":27258,"style":510},[463],[224,27260,1438],{"className":27261},[468]," 时取负值），但这无妨：",[224,27264,27266,27283],{"className":27265,"translate":388},[392],[224,27267,27269],{"className":27268},[396],[398,27270,27271],{"xmlns":400},[403,27272,27273,27281],{},[406,27274,27275],{},[695,27276,27277,27279],{},[409,27278,20303],{},[409,27280,1705],{},[447,27282,24053],{"encoding":449},[224,27284,27286],{"className":27285,"ariaHidden":455},[454],[224,27287,27289,27292],{"className":27288},[459],[224,27290],{"className":27291,"style":24063},[463],[224,27293,27295,27298],{"className":27294},[468],[224,27296,20303],{"className":27297,"style":2473},[468,469],[224,27299,27301],{"className":27300},[750],[224,27302,27304,27324],{"className":27303},[754,755],[224,27305,27307,27321],{"className":27306},[759],[224,27308,27310],{"className":27309,"style":20387},[763],[224,27311,27312,27315],{"style":20390},[224,27313],{"className":27314,"style":772},[771],[224,27316,27318],{"className":27317},[776,777,778,779],[224,27319,1705],{"className":27320,"style":1740},[468,469,779],[224,27322,787],{"className":27323},[786],[224,27325,27327],{"className":27326},[759],[224,27328,27330],{"className":27329,"style":20409},[763],[224,27331],{}," 仍然非负（见 4.1），因为「逐点非负」从来不是 f-散度非负性的必要条件。",[613,27334,27336],{"id":27335},"_14-等价书写形式","1.4 等价书写形式",[224,27338,27340],{"className":27339,"translate":388},[387],[224,27341,27343,27573],{"className":27342,"translate":388},[392],[224,27344,27346],{"className":27345},[396],[398,27347,27348],{"xmlns":400,"display":401},[403,27349,27350,27570],{},[5209,27351,27352,27424,27496],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,27353,27354,27376],{},[5218,27355,27356],{},[5221,27357,27358],{"scriptlevel":284,"displaystyle":455},[406,27359,27360,27366,27368,27370,27372,27374],{},[695,27361,27362,27364],{},[409,27363,20303],{},[409,27365,1705],{},[412,27367,239],{"stretchy":435},[409,27369,16018],{},[409,27371,20312],{"mathvariant":678},[409,27373,20315],{},[412,27375,1716],{"stretchy":435},[5218,27377,27378],{},[5221,27379,27380],{"scriptlevel":284,"displaystyle":455},[406,27381,27382,27384,27386,27392,27394],{},[406,27383],{},[412,27385,278],{},[695,27387,27388,27390],{},[409,27389,20324],{"mathvariant":12251},[409,27391,20315],{},[423,27393,21353],{},[406,27395,27396,27398,27400,27402,27422],{},[412,27397,268],{"fence":455},[409,27399,1705],{},[423,27401,21353],{},[406,27403,27404,27406,27420],{},[412,27405,239],{"fence":455},[20338,27407,27408,27414],{},[406,27409,27410,27412],{},[409,27411,13182],{},[409,27413,16018],{},[406,27415,27416,27418],{},[409,27417,13182],{},[409,27419,20315],{},[412,27421,1716],{"fence":455},[412,27423,440],{"fence":455},[5215,27425,27426,27432],{},[5218,27427,27428],{},[5221,27429,27430],{"scriptlevel":284,"displaystyle":455},[406,27431],{},[5218,27433,27434],{},[5221,27435,27436],{"scriptlevel":284,"displaystyle":455},[406,27437,27438,27440,27442,27448,27450],{},[406,27439],{},[412,27441,278],{},[695,27443,27444,27446],{},[409,27445,20324],{"mathvariant":12251},[409,27447,16018],{},[423,27449,21353],{},[406,27451,27452,27454,27468,27470,27472,27474,27494],{},[412,27453,268],{"fence":455},[20338,27455,27456,27462],{},[406,27457,27458,27460],{},[409,27459,13182],{},[409,27461,20315],{},[406,27463,27464,27466],{},[409,27465,13182],{},[409,27467,16018],{},[423,27469,682],{},[409,27471,1705],{},[423,27473,21353],{},[406,27475,27476,27478,27492],{},[412,27477,239],{"fence":455},[20338,27479,27480,27486],{},[406,27481,27482,27484],{},[409,27483,13182],{},[409,27485,16018],{},[406,27487,27488,27490],{},[409,27489,13182],{},[409,27491,20315],{},[412,27493,1716],{"fence":455},[412,27495,440],{"fence":455},[5215,27497,27498,27504],{},[5218,27499,27500],{},[5221,27501,27502],{"scriptlevel":284,"displaystyle":455},[406,27503],{},[5218,27505,27506],{},[5221,27507,27508],{"scriptlevel":284,"displaystyle":455},[406,27509,27510,27512,27514,27520,27522,27568],{},[406,27511],{},[412,27513,278],{},[695,27515,27516,27518],{},[409,27517,20324],{"mathvariant":12251},[409,27519,16018],{},[423,27521,21353],{},[406,27523,27524,27526,27544,27546,27566],{},[412,27525,268],{"fence":455},[27527,27528,27529,27531,27541],"msubsup",{},[409,27530,1705],{},[406,27532,27533,27535,27537,27539],{},[409,27534,1858],{},[409,27536,7630],{},[409,27538,1861],{},[409,27540,13639],{},[412,27542,27543],{"lspace":5213,"rspace":5213},"∗",[423,27545,21353],{},[406,27547,27548,27550,27564],{},[412,27549,239],{"fence":455},[20338,27551,27552,27558],{},[406,27553,27554,27556],{},[409,27555,13182],{},[409,27557,20315],{},[406,27559,27560,27562],{},[409,27561,13182],{},[409,27563,16018],{},[412,27565,1716],{"fence":455},[412,27567,440],{"fence":455},[412,27569,1877],{"separator":455},[447,27571,27572],{"encoding":449},"\\begin{aligned}\nD_f(P\\|Q)\n&=\\mathbb E_Q\\!\\left[\nf\\!\\left(\\frac{dP}{dQ}\\right)\\right]\\\\[2pt]\n&=\\mathbb E_P\\!\\left[\\frac{dQ}{dP}\\,\nf\\!\\left(\\frac{dP}{dQ}\\right)\\right]\\\\\n&=\\mathbb 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^{*}_{skew}(t)=t\\,f(1\u002Ft)",[224,28438,28440,28534],{"className":28439,"ariaHidden":455},[454],[224,28441,28443,28447,28516,28519,28522,28525,28528,28531],{"className":28442},[459],[224,28444],{"className":28445,"style":28446},[463],"height:1.0331em;vertical-align:-0.2831em;",[224,28448,28450,28453],{"className":28449},[468],[224,28451,1705],{"className":28452,"style":1740},[468,469],[224,28454,28456],{"className":28455},[750],[224,28457,28459,28507],{"className":28458},[754,755],[224,28460,28462,28504],{"className":28461},[759],[224,28463,28466,28490],{"className":28464,"style":28465},[763],"height:0.6887em;",[224,28467,28469,28472],{"style":28468},"top:-2.4169em;margin-left:-0.1076em;margin-right:0.05em;",[224,28470],{"className":28471,"style":772},[771],[224,28473,28475],{"className":28474},[776,777,778,779],[224,28476,28478,28481,28484,28487],{"className":28477},[468,779],[224,28479,1858],{"className":28480},[468,469,779],[224,28482,7630],{"className":28483,"style":7646},[468,469,779],[224,28485,1861],{"className":28486},[468,469,779],[224,28488,13639],{"className":28489,"style":13754},[468,469,779],[224,28491,28492,28495],{"style":3288},[224,28493],{"className":28494,"style":772},[771],[224,28496,28498],{"className":28497},[776,777,778,779],[224,28499,28501],{"className":28500},[468,779],[224,28502,27543],{"className":28503},[468,779],[224,28505,787],{"className":28506},[786],[224,28508,28510],{"className":28509},[759],[224,28511,28514],{"className":28512,"style":28513},[763],"height:0.2831em;",[224,28515],{},[224,28517,239],{"className":28518},[542],[224,28520,1435],{"className":28521},[468,469],[224,28523,1716],{"className":28524},[549],[224,28526],{"className":28527,"style":496},[473],[224,28529,278],{"className":28530},[500],[224,28532],{"className":28533,"style":496},[473],[224,28535,28537,28540,28543,28546,28549,28552,28556,28559],{"className":28536},[459],[224,28538],{"className":28539,"style":535},[463],[224,28541,1435],{"className":28542},[468,469],[224,28544],{"className":28545,"style":724},[473],[224,28547,1705],{"className":28548,"style":1740},[468,469],[224,28550,239],{"className":28551},[542],[224,28553,28555],{"className":28554},[468],"1\u002F",[224,28557,1435],{"className":28558},[468,469],[224,28560,1716],{"className":28561},[549]," 是后面 4.3 节的「斜共轭」。最后一个等式说明：交换 ",[224,28564,28566,28583],{"className":28565,"translate":388},[392],[224,28567,28569],{"className":28568},[396],[398,28570,28571],{"xmlns":400},[403,28572,28573,28581],{},[406,28574,28575,28577,28579],{},[409,28576,16018],{},[412,28578,1877],{"separator":455},[409,28580,20315],{},[447,28582,20627],{"encoding":449},[224,28584,28586],{"className":28585,"ariaHidden":455},[454],[224,28587,28589,28592,28595,28598,28601],{"className":28588},[459],[224,28590],{"className":28591,"style":20637},[463],[224,28593,16018],{"className":28594,"style":653},[468,469],[224,28596,1877],{"className":28597},[1917],[224,28599],{"className":28600,"style":724},[473],[224,28602,20315],{"className":28603},[468,469]," 只是把 ",[224,28606,28608,28621],{"className":28607,"translate":388},[392],[224,28609,28611],{"className":28610},[396],[398,28612,28613],{"xmlns":400},[403,28614,28615,28619],{},[406,28616,28617],{},[409,28618,1705],{},[447,28620,1705],{"encoding":449},[224,28622,28624],{"className":28623,"ariaHidden":455},[454],[224,28625,28627,28630],{"className":28626},[459],[224,28628],{"className":28629,"style":2565},[463],[224,28631,1705],{"className":28632,"style":1740},[468,469],[224,28634,28636,28667],{"className":28635,"translate":388},[392],[224,28637,28639],{"className":28638},[396],[398,28640,28641],{"xmlns":400},[403,28642,28643,28664],{},[406,28644,28645,28647,28650,28652,28654,28656,28658,28660,28662],{},[409,28646,1435],{},[412,28648,28649],{},"↦",[409,28651,1435],{},[409,28653,1705],{},[412,28655,239],{"stretchy":435},[419,28657,1438],{},[409,28659,12413],{"mathvariant":678},[409,28661,1435],{},[412,28663,1716],{"stretchy":435},[447,28665,28666],{"encoding":449},"t\\mapsto t f(1\u002Ft)",[224,28668,28670,28689],{"className":28669,"ariaHidden":455},[454],[224,28671,28673,28677,28680,28683,28686],{"className":28672},[459],[224,28674],{"className":28675,"style":28676},[463],"height:0.6261em;vertical-align:-0.011em;",[224,28678,1435],{"className":28679},[468,469],[224,28681],{"className":28682,"style":496},[473],[224,28684,28649],{"className":28685},[500],[224,28687],{"className":28688,"style":496},[473],[224,28690,28692,28695,28698,28701,28704,28707,28710],{"className":28691},[459],[224,28693],{"className":28694,"style":535},[463],[224,28696,1435],{"className":28697},[468,469],[224,28699,1705],{"className":28700,"style":1740},[468,469],[224,28702,239],{"className":28703},[542],[224,28705,28555],{"className":28706},[468],[224,28708,1435],{"className":28709},[468,469],[224,28711,1716],{"className":28712},[549],"，这对推导「反向 KL」之类的对称性非常方便。",[608,28715,28717,28718,28746,28747],{"id":28716},"_2-为何-fff-凸且-f10f10f10","2. 为何 ",[224,28719,28721,28734],{"className":28720,"translate":388},[392],[224,28722,28724],{"className":28723},[396],[398,28725,28726],{"xmlns":400},[403,28727,28728,28732],{},[406,28729,28730],{},[409,28731,1705],{},[447,28733,1705],{"encoding":449},[224,28735,28737],{"className":28736,"ariaHidden":455},[454],[224,28738,28740,28743],{"className":28739},[459],[224,28741],{"className":28742,"style":2565},[463],[224,28744,1705],{"className":28745,"style":1740},[468,469]," 凸且 ",[224,28748,28750,28773],{"className":28749,"translate":388},[392],[224,28751,28753],{"className":28752},[396],[398,28754,28755],{"xmlns":400},[403,28756,28757,28771],{},[406,28758,28759,28761,28763,28765,28767,28769],{},[409,28760,1705],{},[412,28762,239],{"stretchy":435},[419,28764,1438],{},[412,28766,1716],{"stretchy":435},[412,28768,278],{},[419,28770,284],{},[447,28772,20243],{"encoding":449},[224,28774,28776,28803],{"className":28775,"ariaHidden":455},[454],[224,28777,28779,28782,28785,28788,28791,28794,28797,28800],{"className":28778},[459],[224,28780],{"className":28781,"style":535},[463],[224,28783,1705],{"className":28784,"style":1740},[468,469],[224,28786,239],{"className":28787},[542],[224,28789,1438],{"className":28790},[468],[224,28792,1716],{"className":28793},[549],[224,28795],{"className":28796,"style":496},[473],[224,28798,278],{"className":28799},[500],[224,28801],{"className":28802,"style":496},[473],[224,28804,28806,28809],{"className":28805},[459],[224,28807],{"className":28808,"style":510},[463],[224,28810,284],{"className":28811},[468],[613,28813,28815],{"id":28814},"_21-非负性与-jensen","2.1 非负性与 Jensen",[11,28817,20606,28818,21225,28871,28915,28916,28985,28986,29056],{},[224,28819,28821,28840],{"className":28820,"translate":388},[392],[224,28822,28824],{"className":28823},[396],[398,28825,28826],{"xmlns":400},[403,28827,28828,28837],{},[406,28829,28830,28832,28835],{},[409,28831,20668],{},[412,28833,28834],{},"∼",[409,28836,20315],{},[447,28838,28839],{"encoding":449},"X\\sim Q",[224,28841,28843,28862],{"className":28842,"ariaHidden":455},[454],[224,28844,28846,28849,28853,28856,28859],{"className":28845},[459],[224,28847],{"className":28848,"style":649},[463],[224,28850,20668],{"className":28851,"style":28852},[468,469],"margin-right:0.0785em;",[224,28854],{"className":28855,"style":496},[473],[224,28857,28834],{"className":28858},[500],[224,28860],{"className":28861,"style":496},[473],[224,28863,28865,28868],{"className":28864},[459],[224,28866],{"className":28867,"style":20637},[463],[224,28869,20315],{"className":28870},[468,469],[224,28872,28874,28894],{"className":28873,"translate":388},[392],[224,28875,28877],{"className":28876},[396],[398,28878,28879],{"xmlns":400},[403,28880,28881,28891],{},[406,28882,28883,28885,28887,28889],{},[409,28884,1705],{},[412,28886,239],{"stretchy":435},[409,28888,20668],{},[412,28890,1716],{"stretchy":435},[447,28892,28893],{"encoding":449},"f(X)",[224,28895,28897],{"className":28896,"ariaHidden":455},[454],[224,28898,28900,28903,28906,28909,28912],{"className":28899},[459],[224,28901],{"className":28902,"style":535},[463],[224,28904,1705],{"className":28905,"style":1740},[468,469],[224,28907,239],{"className":28908},[542],[224,28910,20668],{"className":28911,"style":28852},[468,469],[224,28913,1716],{"className":28914},[549]," 的期望就是 ",[224,28917,28919,28936],{"className":28918,"translate":388},[392],[224,28920,28922],{"className":28921},[396],[398,28923,28924],{"xmlns":400},[403,28925,28926,28934],{},[406,28927,28928],{},[695,28929,28930,28932],{},[409,28931,20303],{},[409,28933,1705],{},[447,28935,24053],{"encoding":449},[224,28937,28939],{"className":28938,"ariaHidden":455},[454],[224,28940,28942,28945],{"className":28941},[459],[224,28943],{"className":28944,"style":24063},[463],[224,28946,28948,28951],{"className":28947},[468],[224,28949,20303],{"className":28950,"style":2473},[468,469],[224,28952,28954],{"className":28953},[750],[224,28955,28957,28977],{"className":28956},[754,755],[224,28958,28960,28974],{"className":28959},[759],[224,28961,28963],{"className":28962,"style":20387},[763],[224,28964,28965,28968],{"style":20390},[224,28966],{"className":28967,"style":772},[771],[224,28969,28971],{"className":28970},[776,777,778,779],[224,28972,1705],{"className":28973,"style":1740},[468,469,779],[224,28975,787],{"className":28976},[786],[224,28978,28980],{"className":28979},[759],[224,28981,28983],{"className":28982,"style":20409},[763],[224,28984],{},"。因为 ",[224,28987,28989,29014],{"className":28988,"translate":388},[392],[224,28990,28992],{"className":28991},[396],[398,28993,28994],{"xmlns":400},[403,28995,28996,29012],{},[406,28997,28998,29000,29002,29004,29006,29008,29010],{},[409,28999,1435],{},[412,29001,278],{},[409,29003,13182],{},[409,29005,16018],{},[409,29007,12413],{"mathvariant":678},[409,29009,13182],{},[409,29011,20315],{},[447,29013,25343],{"encoding":449},[224,29015,29017,29035],{"className":29016,"ariaHidden":455},[454],[224,29018,29020,29023,29026,29029,29032],{"className":29019},[459],[224,29021],{"className":29022,"style":25353},[463],[224,29024,1435],{"className":29025},[468,469],[224,29027],{"className":29028,"style":496},[473],[224,29030,278],{"className":29031},[500],[224,29033],{"className":29034,"style":496},[473],[224,29036,29038,29041,29044,29047,29050,29053],{"className":29037},[459],[224,29039],{"className":29040,"style":535},[463],[224,29042,13182],{"className":29043},[468,469],[224,29045,16018],{"className":29046,"style":653},[468,469],[224,29048,12413],{"className":29049},[468],[224,29051,13182],{"className":29052},[468,469],[224,29054,20315],{"className":29055},[468,469]," 满足",[224,29058,29060],{"className":29059,"translate":388},[387],[224,29061,29063,29125],{"className":29062,"translate":388},[392],[224,29064,29066],{"className":29065},[396],[398,29067,29068],{"xmlns":400,"display":401},[403,29069,29070,29122],{},[406,29071,29072,29078,29080,29082,29084,29086,29088,29102,29104,29106,29108,29110,29112,29114,29116,29118,29120],{},[695,29073,29074,29076],{},[409,29075,20324],{"mathvariant":12251},[409,29077,20315],{},[412,29079,268],{"stretchy":435},[409,29081,1435],{},[412,29083,440],{"stretchy":435},[412,29085,278],{},[412,29087,21338],{},[20338,29089,29090,29096],{},[406,29091,29092,29094],{},[409,29093,13182],{},[409,29095,16018],{},[406,29097,29098,29100],{},[409,29099,13182],{},[409,29101,20315],{},[423,29103,682],{},[409,29105,13182],{},[409,29107,20315],{},[412,29109,278],{},[412,29111,21338],{},[409,29113,13182],{},[409,29115,16018],{},[412,29117,278],{},[419,29119,1438],{},[412,29121,1877],{"separator":455},[447,29123,29124],{"encoding":449},"\\mathbb E_Q[t]=\\int \\frac{dP}{dQ}\\,dQ=\\int dP=1,",[224,29126,29128,29192,29290,29317],{"className":29127,"ariaHidden":455},[454],[224,29129,29131,29134,29174,29177,29180,29183,29186,29189],{"className":29130},[459],[224,29132],{"className":29133,"style":20368},[463],[224,29135,29137,29140],{"className":29136},[468],[224,29138,20324],{"className":29139},[468,12389],[224,29141,29143],{"className":29142},[750],[224,29144,29146,29166],{"className":29145},[754,755],[224,29147,29149,29163],{"className":29148},[759],[224,29150,29152],{"className":29151,"style":764},[763],[224,29153,29154,29157],{"style":767},[224,29155],{"className":29156,"style":772},[771],[224,29158,29160],{"className":29159},[776,777,778,779],[224,29161,20315],{"className":29162},[468,469,779],[224,29164,787],{"className":29165},[786],[224,29167,29169],{"className":29168},[759],[224,29170,29172],{"className":29171,"style":20409},[763],[224,29173],{},[224,29175,268],{"className":29176},[542],[224,29178,1435],{"className":29179},[468,469],[224,29181,440],{"className":29182},[549],[224,29184],{"className":29185,"style":496},[473],[224,29187,278],{"className":29188},[500],[224,29190],{"className":29191,"style":496},[473],[224,29193,29195,29198,29201,29204,29272,29275,29278,29281,29284,29287],{"className":29194},[459],[224,29196],{"className":29197,"style":20903},[463],[224,29199,21338],{"className":29200,"style":21534},[21531,21532,21533],[224,29202],{"className":29203,"style":724},[473],[224,29205,29207,29210,29269],{"className":29206},[468],[224,29208],{"className":29209},[542,8479],[224,29211,29213],{"className":29212},[20338],[224,29214,29216,29261],{"className":29215},[754,755],[224,29217,29219,29258],{"className":29218},[759],[224,29220,29222,29236,29244],{"className":29221,"style":20528},[763],[224,29223,29224,29227],{"style":20531},[224,29225],{"className":29226,"style":2108},[771],[224,29228,29230,29233],{"className":29229},[468],[224,29231,13182],{"className":29232},[468,469],[224,29234,20315],{"className":29235},[468,469],[224,29237,29238,29241],{"style":20546},[224,29239],{"className":29240,"style":2108},[771],[224,29242],{"className":29243,"style":20554},[20553],[224,29245,29246,29249],{"style":20557},[224,29247],{"className":29248,"style":2108},[771],[224,29250,29252,29255],{"className":29251},[468],[224,29253,13182],{"className":29254},[468,469],[224,29256,16018],{"className":29257,"style":653},[468,469],[224,29259,787],{"className":29260},[786],[224,29262,29264],{"className":29263},[759],[224,29265,29267],{"className":29266,"style":20579},[763],[224,29268],{},[224,29270],{"className":29271},[549,8479],[224,29273],{"className":29274,"style":724},[473],[224,29276,13182],{"className":29277},[468,469],[224,29279,20315],{"className":29280},[468,469],[224,29282],{"className":29283,"style":496},[473],[224,29285,278],{"className":29286},[500],[224,29288],{"className":29289,"style":496},[473],[224,29291,29293,29296,29299,29302,29305,29308,29311,29314],{"className":29292},[459],[224,29294],{"className":29295,"style":25785},[463],[224,29297,21338],{"className":29298,"style":21534},[21531,21532,21533],[224,29300],{"className":29301,"style":724},[473],[224,29303,13182],{"className":29304},[468,469],[224,29306,16018],{"className":29307,"style":653},[468,469],[224,29309],{"className":29310,"style":496},[473],[224,29312,278],{"className":29313},[500],[224,29315],{"className":29316,"style":496},[473],[224,29318,29320,29323,29326],{"className":29319},[459],[224,29321],{"className":29322,"style":8688},[463],[224,29324,1438],{"className":29325},[468],[224,29327,1877],{"className":29328},[1917],[11,29330,29331,29332,29360,29361,12570],{},"对凸函数 ",[224,29333,29335,29348],{"className":29334,"translate":388},[392],[224,29336,29338],{"className":29337},[396],[398,29339,29340],{"xmlns":400},[403,29341,29342,29346],{},[406,29343,29344],{},[409,29345,1705],{},[447,29347,1705],{"encoding":449},[224,29349,29351],{"className":29350,"ariaHidden":455},[454],[224,29352,29354,29357],{"className":29353},[459],[224,29355],{"className":29356,"style":2565},[463],[224,29358,1705],{"className":29359,"style":1740},[468,469]," 用 ",[378,29362,29363],{},"Jensen 不等式",[224,29365,29367],{"className":29366,"translate":388},[387],[224,29368,29370,29488],{"className":29369,"translate":388},[392],[224,29371,29373],{"className":29372},[396],[398,29374,29375],{"xmlns":400,"display":401},[403,29376,29377,29485],{},[5209,29378,29379,29455],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,29380,29381,29423],{},[5218,29382,29383],{},[5221,29384,29385],{"scriptlevel":284,"displaystyle":455},[406,29386,29387,29393,29395,29397,29399,29401,29403,29405,29411,29413,29415,29417,29419,29421],{},[695,29388,29389,29391],{},[409,29390,20303],{},[409,29392,1705],{},[412,29394,239],{"stretchy":435},[409,29396,16018],{},[409,29398,20312],{"mathvariant":678},[409,29400,20315],{},[412,29402,1716],{"stretchy":435},[412,29404,278],{},[695,29406,29407,29409],{},[409,29408,20324],{"mathvariant":12251},[409,29410,20315],{},[412,29412,268],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,29414,1705],{},[412,29416,239],{"stretchy":435},[409,29418,1435],{},[412,29420,1716],{"stretchy":435},[412,29422,440],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[5218,29424,29425],{},[5221,29426,29427],{"scriptlevel":284,"displaystyle":455},[406,29428,29429,29431,29433,29435,29437,29439,29441,29447,29449,29451,29453],{},[406,29430],{},[423,29432,15054],{},[412,29434,9766],{},[423,29436,15054],{},[409,29438,1705],{},[412,29440,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[695,29442,29443,29445],{},[409,29444,20324],{"mathvariant":12251},[409,29446,20315],{},[412,29448,268],{"stretchy":435},[409,29450,1435],{},[412,29452,440],{"stretchy":435},[412,29454,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[5215,29456,29457,29463],{},[5218,29458,29459],{},[5221,29460,29461],{"scriptlevel":284,"displaystyle":455},[406,29462],{},[5218,29464,29465],{},[5221,29466,29467],{"scriptlevel":284,"displaystyle":455},[406,29468,29469,29471,29473,29475,29477,29479,29481,29483],{},[406,29470],{},[412,29472,278],{},[409,29474,1705],{},[412,29476,239],{"stretchy":435},[419,29478,1438],{},[412,29480,1716],{"stretchy":435},[412,29482,278],{},[419,29484,25772],{},[447,29486,29487],{"encoding":449},"\\begin{aligned}\nD_f(P\\|Q)=\\mathbb E_Q\\big[f(t)\\big]\n&\\ \\ge\\ f\\big(\\mathbb E_Q[t]\\big)\\\\[2pt]\n&=f(1)=0 .\n\\end{aligned}",[224,29489,29491],{"className":29490,"ariaHidden":455},[454],[224,29492,29494,29498],{"className":29493},[459],[224,29495],{"className":29496,"style":29497},[463],"height:2.91em;vertical-align:-1.205em;",[224,29499,29501],{"className":29500},[468],[224,29502,29504,29675],{"className":29503},[5209],[224,29505,29507],{"className":29506},[5331],[224,29508,29510,29666],{"className":29509},[754,755],[224,29511,29513,29663],{"className":29512},[759],[224,29514,29517,29654],{"className":29515,"style":29516},[763],"height:1.705em;",[224,29518,29520,29523],{"style":29519},"top:-3.855em;",[224,29521],{"className":29522,"style":2108},[771],[224,29524,29526,29566,29569,29572,29575,29578,29581,29584,29587,29590,29630,29636,29639,29642,29645,29648],{"className":29525},[468],[224,29527,29529,29532],{"className":29528},[468],[224,29530,20303],{"className":29531,"style":2473},[468,469],[224,29533,29535],{"className":29534},[750],[224,29536,29538,29558],{"className":29537},[754,755],[224,29539,29541,29555],{"className":29540},[759],[224,29542,29544],{"className":29543,"style":20387},[763],[224,29545,29546,29549],{"style":20390},[224,29547],{"className":29548,"style":772},[771],[224,29550,29552],{"className":29551},[776,777,778,779],[224,29553,1705],{"className":29554,"style":1740},[468,469,779],[224,29556,787],{"className":29557},[786],[224,29559,29561],{"className":29560},[759],[224,29562,29564],{"className":29563,"style":20409},[763],[224,29565],{},[224,29567,239],{"className":29568},[542],[224,29570,16018],{"className":29571,"style":653},[468,469],[224,29573,20312],{"className":29574},[468],[224,29576,20315],{"className":29577},[468,469],[224,29579,1716],{"className":29580},[549],[224,29582],{"className":29583,"style":496},[473],[224,29585,278],{"className":29586},[500],[224,29588],{"className":29589,"style":496},[473],[224,29591,29593,29596],{"className":29592},[468],[224,29594,20324],{"className":29595},[468,12389],[224,29597,29599],{"className":29598},[750],[224,29600,29602,29622],{"className":29601},[754,755],[224,29603,29605,29619],{"className":29604},[759],[224,29606,29608],{"className":29607,"style":764},[763],[224,29609,29610,29613],{"style":767},[224,29611],{"className":29612,"style":772},[771],[224,29614,29616],{"className":29615},[776,777,778,779],[224,29617,20315],{"className":29618},[468,469,779],[224,29620,787],{"className":29621},[786],[224,29623,29625],{"className":29624},[759],[224,29626,29628],{"className":29627,"style":20409},[763],[224,29629],{},[224,29631,29633],{"className":29632},[468],[224,29634,268],{"className":29635},[2363,2364],[224,29637,1705],{"className":29638,"style":1740},[468,469],[224,29640,239],{"className":29641},[542],[224,29643,1435],{"className":29644},[468,469],[224,29646,1716],{"className":29647},[549],[224,29649,29651],{"className":29650},[468],[224,29652,440],{"className":29653},[2363,2364],[224,29655,29657,29660],{"style":29656},"top:-2.155em;",[224,29658],{"className":29659,"style":2108},[771],[224,29661],{"className":29662},[468],[224,29664,787],{"className":29665},[786],[224,29667,29669],{"className":29668},[759],[224,29670,29673],{"className":29671,"style":29672},[763],"height:1.205em;",[224,29674],{},[224,29676,29678],{"className":29677},[5399],[224,29679,29681,29824],{"className":29680},[754,755],[224,29682,29684,29821],{"className":29683},[759],[224,29685,29687,29777],{"className":29686,"style":29516},[763],[224,29688,29689,29692],{"style":29519},[224,29690],{"className":29691,"style":2108},[771],[224,29693,29695,29698,29701,29704,29707,29710,29713,29716,29722,29762,29765,29768,29771],{"className":29694},[468],[224,29696],{"className":29697},[468],[224,29699,15054],{"className":29700},[473],[224,29702],{"className":29703,"style":496},[473],[224,29705,9766],{"className":29706},[500],[224,29708,15054],{"className":29709},[473],[224,29711],{"className":29712,"style":496},[473],[224,29714,1705],{"className":29715,"style":1740},[468,469],[224,29717,29719],{"className":29718},[468],[224,29720,239],{"className":29721},[2363,2364],[224,29723,29725,29728],{"className":29724},[468],[224,29726,20324],{"className":29727},[468,12389],[224,29729,29731],{"className":29730},[750],[224,29732,29734,29754],{"className":29733},[754,755],[224,29735,29737,29751],{"className":29736},[759],[224,29738,29740],{"className":29739,"style":764},[763],[224,29741,29742,29745],{"style":767},[224,29743],{"className":29744,"style":772},[771],[224,29746,29748],{"className":29747},[776,777,778,779],[224,29749,20315],{"className":29750},[468,469,779],[224,29752,787],{"className":29753},[786],[224,29755,29757],{"className":29756},[759],[224,29758,29760],{"className":29759,"style":20409},[763],[224,29761],{},[224,29763,268],{"className":29764},[542],[224,29766,1435],{"className":29767},[468,469],[224,29769,440],{"className":29770},[549],[224,29772,29774],{"className":29773},[468],[224,29775,1716],{"className":29776},[2363,2364],[224,29778,29779,29782],{"style":29656},[224,29780],{"className":29781,"style":2108},[771],[224,29783,29785,29788,29791,29794,29797,29800,29803,29806,29809,29812,29815,29818],{"className":29784},[468],[224,29786],{"className":29787},[468],[224,29789],{"className":29790,"style":496},[473],[224,29792,278],{"className":29793},[500],[224,29795],{"className":29796,"style":496},[473],[224,29798,1705],{"className":29799,"style":1740},[468,469],[224,29801,239],{"className":29802},[542],[224,29804,1438],{"className":29805},[468],[224,29807,1716],{"className":29808},[549],[224,29810],{"className":29811,"style":496},[473],[224,29813,278],{"className":29814},[500],[224,29816],{"className":29817,"style":496},[473],[224,29819,25772],{"className":29820},[468],[224,29822,787],{"className":29823},[786],[224,29825,29827],{"className":29826},[759],[224,29828,29830],{"className":29829,"style":29672},[763],[224,29831],{},[11,29833,29834,29835,29863,29864,29892,29893,29985],{},"这就是非负性。",[224,29836,29838,29851],{"className":29837,"translate":388},[392],[224,29839,29841],{"className":29840},[396],[398,29842,29843],{"xmlns":400},[403,29844,29845,29849],{},[406,29846,29847],{},[409,29848,1705],{},[447,29850,1705],{"encoding":449},[224,29852,29854],{"className":29853,"ariaHidden":455},[454],[224,29855,29857,29860],{"className":29856},[459],[224,29858],{"className":29859,"style":2565},[463],[224,29861,1705],{"className":29862,"style":1740},[468,469]," 的凸性是关键：若 ",[224,29865,29867,29880],{"className":29866,"translate":388},[392],[224,29868,29870],{"className":29869},[396],[398,29871,29872],{"xmlns":400},[403,29873,29874,29878],{},[406,29875,29876],{},[409,29877,1705],{},[447,29879,1705],{"encoding":449},[224,29881,29883],{"className":29882,"ariaHidden":455},[454],[224,29884,29886,29889],{"className":29885},[459],[224,29887],{"className":29888,"style":2565},[463],[224,29890,1705],{"className":29891,"style":1740},[468,469]," 是凹函数，Jensen 方向反转，",[224,29894,29896,29918],{"className":29895,"translate":388},[392],[224,29897,29899],{"className":29898},[396],[398,29900,29901],{"xmlns":400},[403,29902,29903,29915],{},[406,29904,29905,29911,29913],{},[695,29906,29907,29909],{},[409,29908,20303],{},[409,29910,1705],{},[412,29912,2070],{},[419,29914,284],{},[447,29916,29917],{"encoding":449},"D_f\\le 0",[224,29919,29921,29976],{"className":29920,"ariaHidden":455},[454],[224,29922,29924,29927,29967,29970,29973],{"className":29923},[459],[224,29925],{"className":29926,"style":24063},[463],[224,29928,29930,29933],{"className":29929},[468],[224,29931,20303],{"className":29932,"style":2473},[468,469],[224,29934,29936],{"className":29935},[750],[224,29937,29939,29959],{"className":29938},[754,755],[224,29940,29942,29956],{"className":29941},[759],[224,29943,29945],{"className":29944,"style":20387},[763],[224,29946,29947,29950],{"style":20390},[224,29948],{"className":29949,"style":772},[771],[224,29951,29953],{"className":29952},[776,777,778,779],[224,29954,1705],{"className":29955,"style":1740},[468,469,779],[224,29957,787],{"className":29958},[786],[224,29960,29962],{"className":29961},[759],[224,29963,29965],{"className":29964,"style":20409},[763],[224,29966],{},[224,29968],{"className":29969,"style":496},[473],[224,29971,2070],{"className":29972},[500],[224,29974],{"className":29975,"style":496},[473],[224,29977,29979,29982],{"className":29978},[459],[224,29980],{"className":29981,"style":510},[463],[224,29983,284],{"className":29984},[468],"，那就不是「散度」了。",[2674,29987,29988,30139],{},[2677,29989,29990,30021,30022,3117],{},[378,29991,29992,30020],{},[224,29993,29995,30008],{"className":29994,"translate":388},[392],[224,29996,29998],{"className":29997},[396],[398,29999,30000],{"xmlns":400},[403,30001,30002,30006],{},[406,30003,30004],{},[409,30005,1705],{},[447,30007,1705],{"encoding":449},[224,30009,30011],{"className":30010,"ariaHidden":455},[454],[224,30012,30014,30017],{"className":30013},[459],[224,30015],{"className":30016,"style":2565},[463],[224,30018,1705],{"className":30019,"style":1740},[468,469]," 凸","：保证 ",[224,30023,30025,30057],{"className":30024,"translate":388},[392],[224,30026,30028],{"className":30027},[396],[398,30029,30030],{"xmlns":400},[403,30031,30032,30054],{},[406,30033,30034,30040,30042,30044,30046,30048,30050,30052],{},[695,30035,30036,30038],{},[409,30037,20303],{},[409,30039,1705],{},[412,30041,239],{"stretchy":435},[409,30043,16018],{},[409,30045,20312],{"mathvariant":678},[409,30047,20315],{},[412,30049,1716],{"stretchy":435},[412,30051,9766],{},[419,30053,284],{},[447,30055,30056],{"encoding":449},"D_f(P\\|Q)\\ge 0",[224,30058,30060,30130],{"className":30059,"ariaHidden":455},[454],[224,30061,30063,30066,30106,30109,30112,30115,30118,30121,30124,30127],{"className":30062},[459],[224,30064],{"className":30065,"style":20368},[463],[224,30067,30069,30072],{"className":30068},[468],[224,30070,20303],{"className":30071,"style":2473},[468,469],[224,30073,30075],{"className":30074},[750],[224,30076,30078,30098],{"className":30077},[754,755],[224,30079,30081,30095],{"className":30080},[759],[224,30082,30084],{"className":30083,"style":20387},[763],[224,30085,30086,30089],{"style":20390},[224,30087],{"className":30088,"style":772},[771],[224,30090,30092],{"className":30091},[776,777,778,779],[224,30093,1705],{"className":30094,"style":1740},[468,469,779],[224,30096,787],{"className":30097},[786],[224,30099,30101],{"className":30100},[759],[224,30102,30104],{"className":30103,"style":20409},[763],[224,30105],{},[224,30107,239],{"className":30108},[542],[224,30110,16018],{"className":30111,"style":653},[468,469],[224,30113,20312],{"className":30114},[468],[224,30116,20315],{"className":30117},[468,469],[224,30119,1716],{"className":30120},[549],[224,30122],{"className":30123,"style":496},[473],[224,30125,9766],{"className":30126},[500],[224,30128],{"className":30129,"style":496},[473],[224,30131,30133,30136],{"className":30132},[459],[224,30134],{"className":30135,"style":510},[463],[224,30137,284],{"className":30138},[468],[2677,30140,30141,30208,30209,30260,30261,30395,30396,3117],{},[378,30142,30143],{},[224,30144,30146,30169],{"className":30145,"translate":388},[392],[224,30147,30149],{"className":30148},[396],[398,30150,30151],{"xmlns":400},[403,30152,30153,30167],{},[406,30154,30155,30157,30159,30161,30163,30165],{},[409,30156,1705],{},[412,30158,239],{"stretchy":435},[419,30160,1438],{},[412,30162,1716],{"stretchy":435},[412,30164,278],{},[419,30166,284],{},[447,30168,20243],{"encoding":449},[224,30170,30172,30199],{"className":30171,"ariaHidden":455},[454],[224,30173,30175,30178,30181,30184,30187,30190,30193,30196],{"className":30174},[459],[224,30176],{"className":30177,"style":535},[463],[224,30179,1705],{"className":30180,"style":1740},[468,469],[224,30182,239],{"className":30183},[542],[224,30185,1438],{"className":30186},[468],[224,30188,1716],{"className":30189},[549],[224,30191],{"className":30192,"style":496},[473],[224,30194,278],{"className":30195},[500],[224,30197],{"className":30198,"style":496},[473],[224,30200,30202,30205],{"className":30201},[459],[224,30203],{"className":30204,"style":510},[463],[224,30206,284],{"className":30207},[468],"：保证当 ",[224,30210,30212,30230],{"className":30211,"translate":388},[392],[224,30213,30215],{"className":30214},[396],[398,30216,30217],{"xmlns":400},[403,30218,30219,30227],{},[406,30220,30221,30223,30225],{},[409,30222,16018],{},[412,30224,278],{},[409,30226,20315],{},[447,30228,30229],{"encoding":449},"P=Q",[224,30231,30233,30251],{"className":30232,"ariaHidden":455},[454],[224,30234,30236,30239,30242,30245,30248],{"className":30235},[459],[224,30237],{"className":30238,"style":649},[463],[224,30240,16018],{"className":30241,"style":653},[468,469],[224,30243],{"className":30244,"style":496},[473],[224,30246,278],{"className":30247},[500],[224,30249],{"className":30250,"style":496},[473],[224,30252,30254,30257],{"className":30253},[459],[224,30255],{"className":30256,"style":20637},[463],[224,30258,20315],{"className":30259},[468,469]," 时，",[224,30262,30264,30294],{"className":30263,"translate":388},[392],[224,30265,30267],{"className":30266},[396],[398,30268,30269],{"xmlns":400},[403,30270,30271,30291],{},[406,30272,30273,30287,30289],{},[20338,30274,30275,30281],{},[406,30276,30277,30279],{},[409,30278,13182],{},[409,30280,16018],{},[406,30282,30283,30285],{},[409,30284,13182],{},[409,30286,20315],{},[412,30288,278],{},[419,30290,1438],{},[447,30292,30293],{"encoding":449},"\\frac{dP}{dQ}=1",[224,30295,30297,30386],{"className":30296,"ariaHidden":455},[454],[224,30298,30300,30303,30377,30380,30383],{"className":30299},[459],[224,30301],{"className":30302,"style":22705},[463],[224,30304,30306,30309,30374],{"className":30305},[468],[224,30307],{"className":30308},[542,8479],[224,30310,30312],{"className":30311},[20338],[224,30313,30315,30366],{"className":30314},[754,755],[224,30316,30318,30363],{"className":30317},[759],[224,30319,30321,30338,30346],{"className":30320,"style":22724},[763],[224,30322,30323,30326],{"style":22566},[224,30324],{"className":30325,"style":2108},[771],[224,30327,30329],{"className":30328},[776,777,778,779],[224,30330,30332,30335],{"className":30331},[468,779],[224,30333,13182],{"className":30334},[468,469,779],[224,30336,20315],{"className":30337},[468,469,779],[224,30339,30340,30343],{"style":20546},[224,30341],{"className":30342,"style":2108},[771],[224,30344],{"className":30345,"style":20554},[20553],[224,30347,30348,30351],{"style":22589},[224,30349],{"className":30350,"style":2108},[771],[224,30352,30354],{"className":30353},[776,777,778,779],[224,30355,30357,30360],{"className":30356},[468,779],[224,30358,13182],{"className":30359},[468,469,779],[224,30361,16018],{"className":30362,"style":653},[468,469,779],[224,30364,787],{"className":30365},[786],[224,30367,30369],{"className":30368},[759],[224,30370,30372],{"className":30371,"style":22776},[763],[224,30373],{},[224,30375],{"className":30376},[549,8479],[224,30378],{"className":30379,"style":496},[473],[224,30381,278],{"className":30382},[500],[224,30384],{"className":30385,"style":496},[473],[224,30387,30389,30392],{"className":30388},[459],[224,30390],{"className":30391,"style":510},[463],[224,30393,1438],{"className":30394},[468],"，从而 ",[224,30397,30399,30431],{"className":30398,"translate":388},[392],[224,30400,30402],{"className":30401},[396],[398,30403,30404],{"xmlns":400},[403,30405,30406,30428],{},[406,30407,30408,30414,30416,30418,30420,30422,30424,30426],{},[695,30409,30410,30412],{},[409,30411,20303],{},[409,30413,1705],{},[412,30415,239],{"stretchy":435},[409,30417,16018],{},[409,30419,20312],{"mathvariant":678},[409,30421,16018],{},[412,30423,1716],{"stretchy":435},[412,30425,278],{},[419,30427,284],{},[447,30429,30430],{"encoding":449},"D_f(P\\|P)=0",[224,30432,30434,30504],{"className":30433,"ariaHidden":455},[454],[224,30435,30437,30440,30480,30483,30486,30489,30492,30495,30498,30501],{"className":30436},[459],[224,30438],{"className":30439,"style":20368},[463],[224,30441,30443,30446],{"className":30442},[468],[224,30444,20303],{"className":30445,"style":2473},[468,469],[224,30447,30449],{"className":30448},[750],[224,30450,30452,30472],{"className":30451},[754,755],[224,30453,30455,30469],{"className":30454},[759],[224,30456,30458],{"className":30457,"style":20387},[763],[224,30459,30460,30463],{"style":20390},[224,30461],{"className":30462,"style":772},[771],[224,30464,30466],{"className":30465},[776,777,778,779],[224,30467,1705],{"className":30468,"style":1740},[468,469,779],[224,30470,787],{"className":30471},[786],[224,30473,30475],{"className":30474},[759],[224,30476,30478],{"className":30477,"style":20409},[763],[224,30479],{},[224,30481,239],{"className":30482},[542],[224,30484,16018],{"className":30485,"style":653},[468,469],[224,30487,20312],{"className":30488},[468],[224,30490,16018],{"className":30491,"style":653},[468,469],[224,30493,1716],{"className":30494},[549],[224,30496],{"className":30497,"style":496},[473],[224,30499,278],{"className":30500},[500],[224,30502],{"className":30503,"style":496},[473],[224,30505,30507,30510],{"className":30506},[459],[224,30508],{"className":30509,"style":510},[463],[224,30511,284],{"className":30512},[468],[11,30514,30515],{},"因此 f-散度天然具有非负性和「相同分布散度为零」的性质。",[613,30517,30519],{"id":30518},"_22-凸性为何必需","2.2 凸性为何必需",[11,30521,30522,30523,30592,30593,30621],{},"有人会问：凸性是不是一个过强的技术条件？答案是否定的。若要求 ",[224,30524,30526,30543],{"className":30525,"translate":388},[392],[224,30527,30529],{"className":30528},[396],[398,30530,30531],{"xmlns":400},[403,30532,30533,30541],{},[406,30534,30535],{},[695,30536,30537,30539],{},[409,30538,20303],{},[409,30540,1705],{},[447,30542,24053],{"encoding":449},[224,30544,30546],{"className":30545,"ariaHidden":455},[454],[224,30547,30549,30552],{"className":30548},[459],[224,30550],{"className":30551,"style":24063},[463],[224,30553,30555,30558],{"className":30554},[468],[224,30556,20303],{"className":30557,"style":2473},[468,469],[224,30559,30561],{"className":30560},[750],[224,30562,30564,30584],{"className":30563},[754,755],[224,30565,30567,30581],{"className":30566},[759],[224,30568,30570],{"className":30569,"style":20387},[763],[224,30571,30572,30575],{"style":20390},[224,30573],{"className":30574,"style":772},[771],[224,30576,30578],{"className":30577},[776,777,778,779],[224,30579,1705],{"className":30580,"style":1740},[468,469,779],[224,30582,787],{"className":30583},[786],[224,30585,30587],{"className":30586},[759],[224,30588,30590],{"className":30589,"style":20409},[763],[224,30591],{}," 对一切「随机变换」（马尔可夫核）单调不增（即第 4.5 节的数据处理不等式），那么 ",[224,30594,30596,30609],{"className":30595,"translate":388},[392],[224,30597,30599],{"className":30598},[396],[398,30600,30601],{"xmlns":400},[403,30602,30603,30607],{},[406,30604,30605],{},[409,30606,1705],{},[447,30608,1705],{"encoding":449},[224,30610,30612],{"className":30611,"ariaHidden":455},[454],[224,30613,30615,30618],{"className":30614},[459],[224,30616],{"className":30617,"style":2565},[463],[224,30619,1705],{"className":30620,"style":1740},[468,469]," 必须凸。二元分布上就能看到这一点。",[21055,30623,30625],{"title":30624},"命题 2.1（Csiszár 刻画）",[11,30626,30627,30628,30823,30824,30865,30866,30894,30895,30923],{},"若数据处理不等式 ",[224,30629,30631,30682],{"className":30630,"translate":388},[392],[224,30632,30634],{"className":30633},[396],[398,30635,30636],{"xmlns":400},[403,30637,30638,30679],{},[406,30639,30640,30646,30648,30650,30653,30655,30657,30659,30661,30663,30669,30671,30673,30675,30677],{},[695,30641,30642,30644],{},[409,30643,20303],{},[409,30645,1705],{},[412,30647,239],{"stretchy":435},[409,30649,16018],{},[409,30651,30652],{},"K",[409,30654,20312],{"mathvariant":678},[409,30656,20315],{},[409,30658,30652],{},[412,30660,1716],{"stretchy":435},[412,30662,2070],{},[695,30664,30665,30667],{},[409,30666,20303],{},[409,30668,1705],{},[412,30670,239],{"stretchy":435},[409,30672,16018],{},[409,30674,20312],{"mathvariant":678},[409,30676,20315],{},[412,30678,1716],{"stretchy":435},[447,30680,30681],{"encoding":449},"D_f(PK\\|QK)\\le D_f(P\\|Q)",[224,30683,30685,30762],{"className":30684,"ariaHidden":455},[454],[224,30686,30688,30691,30731,30734,30737,30741,30744,30747,30750,30753,30756,30759],{"className":30687},[459],[224,30689],{"className":30690,"style":20368},[463],[224,30692,30694,30697],{"className":30693},[468],[224,30695,20303],{"className":30696,"style":2473},[468,469],[224,30698,30700],{"className":30699},[750],[224,30701,30703,30723],{"className":30702},[754,755],[224,30704,30706,30720],{"className":30705},[759],[224,30707,30709],{"className":30708,"style":20387},[763],[224,30710,30711,30714],{"style":20390},[224,30712],{"className":30713,"style":772},[771],[224,30715,30717],{"className":30716},[776,777,778,779],[224,30718,1705],{"className":30719,"style":1740},[468,469,779],[224,30721,787],{"className":30722},[786],[224,30724,30726],{"className":30725},[759],[224,30727,30729],{"className":30728,"style":20409},[763],[224,30730],{},[224,30732,239],{"className":30733},[542],[224,30735,16018],{"className":30736,"style":653},[468,469],[224,30738,30652],{"className":30739,"style":30740},[468,469],"margin-right:0.0715em;",[224,30742,20312],{"className":30743},[468],[224,30745,20315],{"className":30746},[468,469],[224,30748,30652],{"className":30749,"style":30740},[468,469],[224,30751,1716],{"className":30752},[549],[224,30754],{"className":30755,"style":496},[473],[224,30757,2070],{"className":30758},[500],[224,30760],{"className":30761,"style":496},[473],[224,30763,30765,30768,30808,30811,30814,30817,30820],{"className":30764},[459],[224,30766],{"className":30767,"style":20368},[463],[224,30769,30771,30774],{"className":30770},[468],[224,30772,20303],{"className":30773,"style":2473},[468,469],[224,30775,30777],{"className":30776},[750],[224,30778,30780,30800],{"className":30779},[754,755],[224,30781,30783,30797],{"className":30782},[759],[224,30784,30786],{"className":30785,"style":20387},[763],[224,30787,30788,30791],{"style":20390},[224,30789],{"className":30790,"style":772},[771],[224,30792,30794],{"className":30793},[776,777,778,779],[224,30795,1705],{"className":30796,"style":1740},[468,469,779],[224,30798,787],{"className":30799},[786],[224,30801,30803],{"className":30802},[759],[224,30804,30806],{"className":30805,"style":20409},[763],[224,30807],{},[224,30809,239],{"className":30810},[542],[224,30812,16018],{"className":30813,"style":653},[468,469],[224,30815,20312],{"className":30816},[468],[224,30818,20315],{"className":30819},[468,469],[224,30821,1716],{"className":30822},[549]," 对一切概率对 ",[224,30825,30827,30844],{"className":30826,"translate":388},[392],[224,30828,30830],{"className":30829},[396],[398,30831,30832],{"xmlns":400},[403,30833,30834,30842],{},[406,30835,30836,30838,30840],{},[409,30837,16018],{},[412,30839,1877],{"separator":455},[409,30841,20315],{},[447,30843,20627],{"encoding":449},[224,30845,30847],{"className":30846,"ariaHidden":455},[454],[224,30848,30850,30853,30856,30859,30862],{"className":30849},[459],[224,30851],{"className":30852,"style":20637},[463],[224,30854,16018],{"className":30855,"style":653},[468,469],[224,30857,1877],{"className":30858},[1917],[224,30860],{"className":30861,"style":724},[473],[224,30863,20315],{"className":30864},[468,469]," 与一切马尔可夫核 ",[224,30867,30869,30882],{"className":30868,"translate":388},[392],[224,30870,30872],{"className":30871},[396],[398,30873,30874],{"xmlns":400},[403,30875,30876,30880],{},[406,30877,30878],{},[409,30879,30652],{},[447,30881,30652],{"encoding":449},[224,30883,30885],{"className":30884,"ariaHidden":455},[454],[224,30886,30888,30891],{"className":30887},[459],[224,30889],{"className":30890,"style":649},[463],[224,30892,30652],{"className":30893,"style":30740},[468,469]," 成立，则 ",[224,30896,30898,30911],{"className":30897,"translate":388},[392],[224,30899,30901],{"className":30900},[396],[398,30902,30903],{"xmlns":400},[403,30904,30905,30909],{},[406,30906,30907],{},[409,30908,1705],{},[447,30910,1705],{"encoding":449},[224,30912,30914],{"className":30913,"ariaHidden":455},[454],[224,30915,30917,30920],{"className":30916},[459],[224,30918],{"className":30919,"style":2565},[463],[224,30921,1705],{"className":30922,"style":1740},[468,469]," 凸。",[11,30925,30926],{},[378,30927,30928],{},"证明（两步）。",[11,30930,30931,31017,31018,22325,31251,31280,31281,31356,31357,372,31555,31752,31753,31889],{},[378,30932,30933,30934,26478,30964,31016],{},"第一步：DPI ",[224,30935,30937,30951],{"className":30936,"translate":388},[392],[224,30938,30940],{"className":30939},[396],[398,30941,30942],{"xmlns":400},[403,30943,30944,30948],{},[406,30945,30946],{},[412,30947,9886],{},[447,30949,30950],{"encoding":449},"\\Rightarrow",[224,30952,30954],{"className":30953,"ariaHidden":455},[454],[224,30955,30957,30961],{"className":30956},[459],[224,30958],{"className":30959,"style":30960},[463],"height:0.3669em;",[224,30962,9886],{"className":30963},[500],[224,30965,30967,30989],{"className":30966,"translate":388},[392],[224,30968,30970],{"className":30969},[396],[398,30971,30972],{"xmlns":400},[403,30973,30974,30986],{},[406,30975,30976,30978,30980,30982,30984],{},[412,30977,239],{"stretchy":435},[409,30979,16018],{},[412,30981,1877],{"separator":455},[409,30983,20315],{},[412,30985,1716],{"stretchy":435},[447,30987,30988],{"encoding":449},"(P,Q)",[224,30990,30992],{"className":30991,"ariaHidden":455},[454],[224,30993,30995,30998,31001,31004,31007,31010,31013],{"className":30994},[459],[224,30996],{"className":30997,"style":535},[463],[224,30999,239],{"className":31000},[542],[224,31002,16018],{"className":31003,"style":653},[468,469],[224,31005,1877],{"className":31006},[1917],[224,31008],{"className":31009,"style":724},[473],[224,31011,20315],{"className":31012},[468,469],[224,31014,1716],{"className":31015},[549]," 联合凸。"," 设 ",[224,31019,31021,31063],{"className":31020,"translate":388},[392],[224,31022,31024],{"className":31023},[396],[398,31025,31026],{"xmlns":400},[403,31027,31028,31060],{},[406,31029,31030,31036,31038,31044,31046,31052,31054],{},[695,31031,31032,31034],{},[409,31033,16018],{},[419,31035,1438],{},[412,31037,1877],{"separator":455},[695,31039,31040,31042],{},[409,31041,16018],{},[419,31043,1447],{},[412,31045,1877],{"separator":455},[695,31047,31048,31050],{},[409,31049,20315],{},[419,31051,1438],{},[412,31053,1877],{"separator":455},[695,31055,31056,31058],{},[409,31057,20315],{},[419,31059,1447],{},[447,31061,31062],{"encoding":449},"P_1,P_2,Q_1,Q_2",[224,31064,31066],{"className":31065,"ariaHidden":455},[454],[224,31067,31069,31072,31113,31116,31119,31159,31162,31165,31205,31208,31211],{"className":31068},[459],[224,31070],{"className":31071,"style":20637},[463],[224,31073,31075,31078],{"className":31074},[468],[224,31076,16018],{"className":31077,"style":653},[468,469],[224,31079,31081],{"className":31080},[750],[224,31082,31084,31105],{"className":31083},[754,755],[224,31085,31087,31102],{"className":31086},[759],[224,31088,31090],{"className":31089,"style":1479},[763],[224,31091,31093,31096],{"style":31092},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[224,31094],{"className":31095,"style":772},[771],[224,31097,31099],{"className":31098},[776,777,778,779],[224,31100,1438],{"className":31101},[468,779],[224,31103,787],{"className":31104},[786],[224,31106,31108],{"className":31107},[759],[224,31109,31111],{"className":31110,"style":794},[763],[224,31112],{},[224,31114,1877],{"className":31115},[1917],[224,31117],{"className":31118,"style":724},[473],[224,31120,31122,31125],{"className":31121},[468],[224,31123,16018],{"className":31124,"style":653},[468,469],[224,31126,31128],{"className":31127},[750],[224,31129,31131,31151],{"className":31130},[754,755],[224,31132,31134,31148],{"className":31133},[759],[224,31135,31137],{"className":31136,"style":1479},[763],[224,31138,31139,31142],{"style":31092},[224,31140],{"className":31141,"style":772},[771],[224,31143,31145],{"className":31144},[776,777,778,779],[224,31146,1447],{"className":31147},[468,779],[224,31149,787],{"className":31150},[786],[224,31152,31154],{"className":31153},[759],[224,31155,31157],{"className":31156,"style":794},[763],[224,31158],{},[224,31160,1877],{"className":31161},[1917],[224,31163],{"className":31164,"style":724},[473],[224,31166,31168,31171],{"className":31167},[468],[224,31169,20315],{"className":31170},[468,469],[224,31172,31174],{"className":31173},[750],[224,31175,31177,31197],{"className":31176},[754,755],[224,31178,31180,31194],{"className":31179},[759],[224,31181,31183],{"className":31182,"style":1479},[763],[224,31184,31185,31188],{"style":767},[224,31186],{"className":31187,"style":772},[771],[224,31189,31191],{"className":31190},[776,777,778,779],[224,31192,1438],{"className":31193},[468,779],[224,31195,787],{"className":31196},[786],[224,31198,31200],{"className":31199},[759],[224,31201,31203],{"className":31202,"style":794},[763],[224,31204],{},[224,31206,1877],{"className":31207},[1917],[224,31209],{"className":31210,"style":724},[473],[224,31212,31214,31217],{"className":31213},[468],[224,31215,20315],{"className":31216},[468,469],[224,31218,31220],{"className":31219},[750],[224,31221,31223,31243],{"className":31222},[754,755],[224,31224,31226,31240],{"className":31225},[759],[224,31227,31229],{"className":31228,"style":1479},[763],[224,31230,31231,31234],{"style":767},[224,31232],{"className":31233,"style":772},[771],[224,31235,31237],{"className":31236},[776,777,778,779],[224,31238,1447],{"className":31239},[468,779],[224,31241,787],{"className":31242},[786],[224,31244,31246],{"className":31245},[759],[224,31247,31249],{"className":31248,"style":794},[763],[224,31250],{},[224,31252,31254,31268],{"className":31253,"translate":388},[392],[224,31255,31257],{"className":31256},[396],[398,31258,31259],{"xmlns":400},[403,31260,31261,31265],{},[406,31262,31263],{},[409,31264,20668],{"mathvariant":20667},[447,31266,31267],{"encoding":449},"\\mathcal X",[224,31269,31271],{"className":31270,"ariaHidden":455},[454],[224,31272,31274,31277],{"className":31273},[459],[224,31275],{"className":31276,"style":649},[463],[224,31278,20668],{"className":31279,"style":20694},[468,20693]," 上任给的四对分布，",[224,31282,31284,31311],{"className":31283,"translate":388},[392],[224,31285,31287],{"className":31286},[396],[398,31288,31289],{"xmlns":400},[403,31290,31291,31308],{},[406,31292,31293,31296,31298,31300,31302,31304,31306],{},[409,31294,31295],{},"λ",[412,31297,12248],{},[412,31299,239],{"stretchy":435},[419,31301,284],{},[412,31303,1877],{"separator":455},[419,31305,1438],{},[412,31307,1716],{"stretchy":435},[447,31309,31310],{"encoding":449},"\\lambda\\in(0,1)",[224,31312,31314,31332],{"className":31313,"ariaHidden":455},[454],[224,31315,31317,31320,31323,31326,31329],{"className":31316},[459],[224,31318],{"className":31319,"style":16682},[463],[224,31321,31295],{"className":31322},[468,469],[224,31324],{"className":31325,"style":496},[473],[224,31327,12248],{"className":31328},[500],[224,31330],{"className":31331,"style":496},[473],[224,31333,31335,31338,31341,31344,31347,31350,31353],{"className":31334},[459],[224,31336],{"className":31337,"style":535},[463],[224,31339,239],{"className":31340},[542],[224,31342,284],{"className":31343},[468],[224,31345,1877],{"className":31346},[1917],[224,31348],{"className":31349,"style":724},[473],[224,31351,1438],{"className":31352},[468],[224,31354,1716],{"className":31355},[549],"，记 ",[224,31358,31360,31402],{"className":31359,"translate":388},[392],[224,31361,31363],{"className":31362},[396],[398,31364,31365],{"xmlns":400},[403,31366,31367,31399],{},[406,31368,31369,31371,31373,31375,31381,31383,31385,31387,31389,31391,31393],{},[409,31370,16018],{},[412,31372,278],{},[409,31374,31295],{},[695,31376,31377,31379],{},[409,31378,16018],{},[419,31380,1438],{},[412,31382,319],{},[412,31384,239],{"stretchy":435},[419,31386,1438],{},[412,31388,5268],{},[409,31390,31295],{},[412,31392,1716],{"stretchy":435},[695,31394,31395,31397],{},[409,31396,16018],{},[419,31398,1447],{},[447,31400,31401],{"encoding":449},"P=\\lambda P_1+(1-\\lambda)P_2",[224,31403,31405,31423,31482,31503],{"className":31404,"ariaHidden":455},[454],[224,31406,31408,31411,31414,31417,31420],{"className":31407},[459],[224,31409],{"className":31410,"style":649},[463],[224,31412,16018],{"className":31413,"style":653},[468,469],[224,31415],{"className":31416,"style":496},[473],[224,31418,278],{"className":31419},[500],[224,31421],{"className":31422,"style":496},[473],[224,31424,31426,31430,31433,31473,31476,31479],{"className":31425},[459],[224,31427],{"className":31428,"style":31429},[463],"height:0.8444em;vertical-align:-0.15em;",[224,31431,31295],{"className":31432},[468,469],[224,31434,31436,31439],{"className":31435},[468],[224,31437,16018],{"className":31438,"style":653},[468,469],[224,31440,31442],{"className":31441},[750],[224,31443,31445,31465],{"className":31444},[754,755],[224,31446,31448,31462],{"className":31447},[759],[224,31449,31451],{"className":31450,"style":1479},[763],[224,31452,31453,31456],{"style":31092},[224,31454],{"className":31455,"style":772},[771],[224,31457,31459],{"className":31458},[776,777,778,779],[224,31460,1438],{"className":31461},[468,779],[224,31463,787],{"className":31464},[786],[224,31466,31468],{"className":31467},[759],[224,31469,31471],{"className":31470,"style":794},[763],[224,31472],{},[224,31474],{"className":31475,"style":474},[473],[224,31477,319],{"className":31478},[478],[224,31480],{"className":31481,"style":474},[473],[224,31483,31485,31488,31491,31494,31497,31500],{"className":31484},[459],[224,31486],{"className":31487,"style":535},[463],[224,31489,239],{"className":31490},[542],[224,31492,1438],{"className":31493},[468],[224,31495],{"className":31496,"style":474},[473],[224,31498,5268],{"className":31499},[478],[224,31501],{"className":31502,"style":474},[473],[224,31504,31506,31509,31512,31515],{"className":31505},[459],[224,31507],{"className":31508,"style":535},[463],[224,31510,31295],{"className":31511},[468,469],[224,31513,1716],{"className":31514},[549],[224,31516,31518,31521],{"className":31517},[468],[224,31519,16018],{"className":31520,"style":653},[468,469],[224,31522,31524],{"className":31523},[750],[224,31525,31527,31547],{"className":31526},[754,755],[224,31528,31530,31544],{"className":31529},[759],[224,31531,31533],{"className":31532,"style":1479},[763],[224,31534,31535,31538],{"style":31092},[224,31536],{"className":31537,"style":772},[771],[224,31539,31541],{"className":31540},[776,777,778,779],[224,31542,1447],{"className":31543},[468,779],[224,31545,787],{"className":31546},[786],[224,31548,31550],{"className":31549},[759],[224,31551,31553],{"className":31552,"style":794},[763],[224,31554],{},[224,31556,31558,31600],{"className":31557,"translate":388},[392],[224,31559,31561],{"className":31560},[396],[398,31562,31563],{"xmlns":400},[403,31564,31565,31597],{},[406,31566,31567,31569,31571,31573,31579,31581,31583,31585,31587,31589,31591],{},[409,31568,20315],{},[412,31570,278],{},[409,31572,31295],{},[695,31574,31575,31577],{},[409,31576,20315],{},[419,31578,1438],{},[412,31580,319],{},[412,31582,239],{"stretchy":435},[419,31584,1438],{},[412,31586,5268],{},[409,31588,31295],{},[412,31590,1716],{"stretchy":435},[695,31592,31593,31595],{},[409,31594,20315],{},[419,31596,1447],{},[447,31598,31599],{"encoding":449},"Q=\\lambda Q_1+(1-\\lambda)Q_2",[224,31601,31603,31621,31679,31700],{"className":31602,"ariaHidden":455},[454],[224,31604,31606,31609,31612,31615,31618],{"className":31605},[459],[224,31607],{"className":31608,"style":20637},[463],[224,31610,20315],{"className":31611},[468,469],[224,31613],{"className":31614,"style":496},[473],[224,31616,278],{"className":31617},[500],[224,31619],{"className":31620,"style":496},[473],[224,31622,31624,31627,31630,31670,31673,31676],{"className":31623},[459],[224,31625],{"className":31626,"style":2565},[463],[224,31628,31295],{"className":31629},[468,469],[224,31631,31633,31636],{"className":31632},[468],[224,31634,20315],{"className":31635},[468,469],[224,31637,31639],{"className":31638},[750],[224,31640,31642,31662],{"className":31641},[754,755],[224,31643,31645,31659],{"className":31644},[759],[224,31646,31648],{"className":31647,"style":1479},[763],[224,31649,31650,31653],{"style":767},[224,31651],{"className":31652,"style":772},[771],[224,31654,31656],{"className":31655},[776,777,778,779],[224,31657,1438],{"className":31658},[468,779],[224,31660,787],{"className":31661},[786],[224,31663,31665],{"className":31664},[759],[224,31666,31668],{"className":31667,"style":794},[763],[224,31669],{},[224,31671],{"className":31672,"style":474},[473],[224,31674,319],{"className":31675},[478],[224,31677],{"className":31678,"style":474},[473],[224,31680,31682,31685,31688,31691,31694,31697],{"className":31681},[459],[224,31683],{"className":31684,"style":535},[463],[224,31686,239],{"className":31687},[542],[224,31689,1438],{"className":31690},[468],[224,31692],{"className":31693,"style":474},[473],[224,31695,5268],{"className":31696},[478],[224,31698],{"className":31699,"style":474},[473],[224,31701,31703,31706,31709,31712],{"className":31702},[459],[224,31704],{"className":31705,"style":535},[463],[224,31707,31295],{"className":31708},[468,469],[224,31710,1716],{"className":31711},[549],[224,31713,31715,31718],{"className":31714},[468],[224,31716,20315],{"className":31717},[468,469],[224,31719,31721],{"className":31720},[750],[224,31722,31724,31744],{"className":31723},[754,755],[224,31725,31727,31741],{"className":31726},[759],[224,31728,31730],{"className":31729,"style":1479},[763],[224,31731,31732,31735],{"style":767},[224,31733],{"className":31734,"style":772},[771],[224,31736,31738],{"className":31737},[776,777,778,779],[224,31739,1447],{"className":31740},[468,779],[224,31742,787],{"className":31743},[786],[224,31745,31747],{"className":31746},[759],[224,31748,31750],{"className":31749,"style":794},[763],[224,31751],{},"。构造扩大一倍的样本空间 ",[224,31754,31756,31793],{"className":31755,"translate":388},[392],[224,31757,31759],{"className":31758},[396],[398,31760,31761],{"xmlns":400},[403,31762,31763,31790],{},[406,31764,31765,31773,31775,31777,31779,31781,31783,31785,31788],{},[31766,31767,31768,31770],"mover",{"accent":455},[409,31769,20668],{"mathvariant":20667},[412,31771,31772],{},"~",[412,31774,278],{},[412,31776,8110],{"stretchy":435},[419,31778,1438],{},[412,31780,1877],{"separator":455},[419,31782,1447],{},[412,31784,15077],{"stretchy":435},[412,31786,31787],{},"×",[409,31789,20668],{"mathvariant":20667},[447,31791,31792],{"encoding":449},"\\tilde{\\mathcal X}=\\{1,2\\}\\times\\mathcal X",[224,31794,31796,31847,31880],{"className":31795,"ariaHidden":455},[454],[224,31797,31799,31803,31838,31841,31844],{"className":31798},[459],[224,31800],{"className":31801,"style":31802},[463],"height:0.9202em;",[224,31804,31807],{"className":31805},[468,31806],"accent",[224,31808,31810],{"className":31809},[754],[224,31811,31813],{"className":31812},[759],[224,31814,31816,31824],{"className":31815,"style":31802},[763],[224,31817,31818,31821],{"style":2120},[224,31819],{"className":31820,"style":2108},[771],[224,31822,20668],{"className":31823,"style":20694},[468,20693],[224,31825,31827,31830],{"style":31826},"top:-3.6023em;",[224,31828],{"className":31829,"style":2108},[771],[224,31831,31835],{"className":31832,"style":31834},[31833],"accent-body","left:-0.1111em;",[224,31836,31772],{"className":31837},[468],[224,31839],{"className":31840,"style":496},[473],[224,31842,278],{"className":31843},[500],[224,31845],{"className":31846,"style":496},[473],[224,31848,31850,31853,31856,31859,31862,31865,31868,31871,31874,31877],{"className":31849},[459],[224,31851],{"className":31852,"style":535},[463],[224,31854,8110],{"className":31855},[542],[224,31857,1438],{"className":31858},[468],[224,31860,1877],{"className":31861},[1917],[224,31863],{"className":31864,"style":724},[473],[224,31866,1447],{"className":31867},[468],[224,31869,15077],{"className":31870},[549],[224,31872],{"className":31873,"style":474},[473],[224,31875,31787],{"className":31876},[478],[224,31878],{"className":31879,"style":474},[473],[224,31881,31883,31886],{"className":31882},[459],[224,31884],{"className":31885,"style":649},[463],[224,31887,20668],{"className":31888,"style":20694},[468,20693]," 及其上的两个分块分布：",[224,31891,31893],{"className":31892,"translate":388},[387],[224,31894,31896,32030],{"className":31895,"translate":388},[392],[224,31897,31899],{"className":31898},[396],[398,31900,31901],{"xmlns":400,"display":401},[403,31902,31903,32027],{},[406,31904,31905,31911,31913,31915,31917,31919,31921,31923,31925,31927,31933,31935,31941,31943,31945,31947,31949,31951,31957,31959,31961,31963,31965,31967,31969,31971,31973,31979,31981,31987,31989,31991,31993,31995,31997,32003,32005,32007,32009,32011,32017,32019,32021,32023,32025],{},[31766,31906,31907,31909],{"accent":455},[409,31908,16018],{},[412,31910,31772],{},[412,31912,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[412,31914,239],{"stretchy":435},[409,31916,1880],{},[412,31918,1877],{"separator":455},[409,31920,371],{},[412,31922,1716],{"stretchy":435},[412,31924,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[412,31926,278],{},[695,31928,31929,31931],{},[409,31930,13639],{},[409,31932,1880],{},[423,31934,682],{},[695,31936,31937,31939],{},[409,31938,16018],{},[409,31940,1880],{},[412,31942,239],{"stretchy":435},[409,31944,371],{},[412,31946,1716],{"stretchy":435},[412,31948,1877],{"separator":455},[473,31950],{"width":4247},[31766,31952,31953,31955],{"accent":455},[409,31954,20315],{},[412,31956,31772],{},[412,31958,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[412,31960,239],{"stretchy":435},[409,31962,1880],{},[412,31964,1877],{"separator":455},[409,31966,371],{},[412,31968,1716],{"stretchy":435},[412,31970,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[412,31972,278],{},[695,31974,31975,31977],{},[409,31976,13639],{},[409,31978,1880],{},[423,31980,682],{},[695,31982,31983,31985],{},[409,31984,20315],{},[409,31986,1880],{},[412,31988,239],{"stretchy":435},[409,31990,371],{},[412,31992,1716],{"stretchy":435},[412,31994,1877],{"separator":455},[473,31996],{"width":4247},[695,31998,31999,32001],{},[409,32000,13639],{},[419,32002,1438],{},[412,32004,278],{},[409,32006,31295],{},[412,32008,1877],{"separator":455},[423,32010,15054],{},[695,32012,32013,32015],{},[409,32014,13639],{},[419,32016,1447],{},[412,32018,278],{},[419,32020,1438],{},[412,32022,5268],{},[409,32024,31295],{},[409,32026,688],{"mathvariant":678},[447,32028,32029],{"encoding":449},"\\tilde P\\big((i,x)\\big)=w_i\\,P_i(x),\\qquad\n\\tilde Q\\big((i,x)\\big)=w_i\\,Q_i(x),\\qquad\nw_1=\\lambda,\\ w_2=1-\\lambda .",[224,32031,32033,32111,32301,32457,32524,32542],{"className":32032,"ariaHidden":455},[454],[224,32034,32036,32040,32072,32078,32081,32084,32087,32090,32093,32096,32102,32105,32108],{"className":32035},[459],[224,32037],{"className":32038,"style":32039},[463],"height:1.2702em;vertical-align:-0.35em;",[224,32041,32043],{"className":32042},[468,31806],[224,32044,32046],{"className":32045},[754],[224,32047,32049],{"className":32048},[759],[224,32050,32052,32060],{"className":32051,"style":31802},[763],[224,32053,32054,32057],{"style":2120},[224,32055],{"className":32056,"style":2108},[771],[224,32058,16018],{"className":32059,"style":653},[468,469],[224,32061,32062,32065],{"style":31826},[224,32063],{"className":32064,"style":2108},[771],[224,32066,32069],{"className":32067,"style":32068},[31833],"left:-0.1667em;",[224,32070,31772],{"className":32071},[468],[224,32073,32075],{"className":32074},[468],[224,32076,239],{"className":32077},[2363,2364],[224,32079,239],{"className":32080},[542],[224,32082,1880],{"className":32083},[468,469],[224,32085,1877],{"className":32086},[1917],[224,32088],{"className":32089,"style":724},[473],[224,32091,371],{"className":32092},[468,469],[224,32094,1716],{"className":32095},[549],[224,32097,32099],{"className":32098},[468],[224,32100,1716],{"className":32101},[2363,2364],[224,32103],{"className":32104,"style":496},[473],[224,32106,278],{"className":32107},[500],[224,32109],{"className":32110,"style":496},[473],[224,32112,32114,32117,32158,32161,32201,32204,32207,32210,32213,32216,32219,32262,32268,32271,32274,32277,32280,32283,32286,32292,32295,32298],{"className":32113},[459],[224,32115],{"className":32116,"style":32039},[463],[224,32118,32120,32123],{"className":32119},[468],[224,32121,13639],{"className":32122,"style":13754},[468,469],[224,32124,32126],{"className":32125},[750],[224,32127,32129,32150],{"className":32128},[754,755],[224,32130,32132,32147],{"className":32131},[759],[224,32133,32135],{"className":32134,"style":4075},[763],[224,32136,32138,32141],{"style":32137},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[224,32139],{"className":32140,"style":772},[771],[224,32142,32144],{"className":32143},[776,777,778,779],[224,32145,1880],{"className":32146},[468,469,779],[224,32148,787],{"className":32149},[786],[224,32151,32153],{"className":32152},[759],[224,32154,32156],{"className":32155,"style":794},[763],[224,32157],{},[224,32159],{"className":32160,"style":724},[473],[224,32162,32164,32167],{"className":32163},[468],[224,32165,16018],{"className":32166,"style":653},[468,469],[224,32168,32170],{"className":32169},[750],[224,32171,32173,32193],{"className":32172},[754,755],[224,32174,32176,32190],{"className":32175},[759],[224,32177,32179],{"className":32178,"style":4075},[763],[224,32180,32181,32184],{"style":31092},[224,32182],{"className":32183,"style":772},[771],[224,32185,32187],{"className":32186},[776,777,778,779],[224,32188,1880],{"className":32189},[468,469,779],[224,32191,787],{"className":32192},[786],[224,32194,32196],{"className":32195},[759],[224,32197,32199],{"className":32198,"style":794},[763],[224,32200],{},[224,32202,239],{"className":32203},[542],[224,32205,371],{"className":32206},[468,469],[224,32208,1716],{"className":32209},[549],[224,32211,1877],{"className":32212},[1917],[224,32214],{"className":32215,"style":4416},[473],[224,32217],{"className":32218,"style":724},[473],[224,32220,32222],{"className":32221},[468,31806],[224,32223,32225,32253],{"className":32224},[754,755],[224,32226,32228,32250],{"className":32227},[759],[224,32229,32231,32239],{"className":32230,"style":31802},[763],[224,32232,32233,32236],{"style":2120},[224,32234],{"className":32235,"style":2108},[771],[224,32237,20315],{"className":32238},[468,469],[224,32240,32241,32244],{"style":31826},[224,32242],{"className":32243,"style":2108},[771],[224,32245,32247],{"className":32246,"style":32068},[31833],[224,32248,31772],{"className":32249},[468],[224,32251,787],{"className":32252},[786],[224,32254,32256],{"className":32255},[759],[224,32257,32260],{"className":32258,"style":32259},[763],"height:0.1944em;",[224,32261],{},[224,32263,32265],{"className":32264},[468],[224,32266,239],{"className":32267},[2363,2364],[224,32269,239],{"className":32270},[542],[224,32272,1880],{"className":32273},[468,469],[224,32275,1877],{"className":32276},[1917],[224,32278],{"className":32279,"style":724},[473],[224,32281,371],{"className":32282},[468,469],[224,32284,1716],{"className":32285},[549],[224,32287,32289],{"className":32288},[468],[224,32290,1716],{"className":32291},[2363,2364],[224,32293],{"className":32294,"style":496},[473],[224,32296,278],{"className":32297},[500],[224,32299],{"className":32300,"style":496},[473],[224,32302,32304,32307,32347,32350,32390,32393,32396,32399,32402,32405,32408,32448,32451,32454],{"className":32303},[459],[224,32305],{"className":32306,"style":535},[463],[224,32308,32310,32313],{"className":32309},[468],[224,32311,13639],{"className":32312,"style":13754},[468,469],[224,32314,32316],{"className":32315},[750],[224,32317,32319,32339],{"className":32318},[754,755],[224,32320,32322,32336],{"className":32321},[759],[224,32323,32325],{"className":32324,"style":4075},[763],[224,32326,32327,32330],{"style":32137},[224,32328],{"className":32329,"style":772},[771],[224,32331,32333],{"className":32332},[776,777,778,779],[224,32334,1880],{"className":32335},[468,469,779],[224,32337,787],{"className":32338},[786],[224,32340,32342],{"className":32341},[759],[224,32343,32345],{"className":32344,"style":794},[763],[224,32346],{},[224,32348],{"className":32349,"style":724},[473],[224,32351,32353,32356],{"className":32352},[468],[224,32354,20315],{"className":32355},[468,469],[224,32357,32359],{"className":32358},[750],[224,32360,32362,32382],{"className":32361},[754,755],[224,32363,32365,32379],{"className":32364},[759],[224,32366,32368],{"className":32367,"style":4075},[763],[224,32369,32370,32373],{"style":767},[224,32371],{"className":32372,"style":772},[771],[224,32374,32376],{"className":32375},[776,777,778,779],[224,32377,1880],{"className":32378},[468,469,779],[224,32380,787],{"className":32381},[786],[224,32383,32385],{"className":32384},[759],[224,32386,32388],{"className":32387,"style":794},[763],[224,32389],{},[224,32391,239],{"className":32392},[542],[224,32394,371],{"className":32395},[468,469],[224,32397,1716],{"className":32398},[549],[224,32400,1877],{"className":32401},[1917],[224,32403],{"className":32404,"style":4416},[473],[224,32406],{"className":32407,"style":724},[473],[224,32409,32411,32414],{"className":32410},[468],[224,32412,13639],{"className":32413,"style":13754},[468,469],[224,32415,32417],{"className":32416},[750],[224,32418,32420,32440],{"className":32419},[754,755],[224,32421,32423,32437],{"className":32422},[759],[224,32424,32426],{"className":32425,"style":1479},[763],[224,32427,32428,32431],{"style":32137},[224,32429],{"className":32430,"style":772},[771],[224,32432,32434],{"className":32433},[776,777,778,779],[224,32435,1438],{"className":32436},[468,779],[224,32438,787],{"className":32439},[786],[224,32441,32443],{"className":32442},[759],[224,32444,32446],{"className":32445,"style":794},[763],[224,32447],{},[224,32449],{"className":32450,"style":496},[473],[224,32452,278],{"className":32453},[500],[224,32455],{"className":32456,"style":496},[473],[224,32458,32460,32463,32466,32469,32472,32475,32515,32518,32521],{"className":32459},[459],[224,32461],{"className":32462,"style":2565},[463],[224,32464,31295],{"className":32465},[468,469],[224,32467,1877],{"className":32468},[1917],[224,32470,15054],{"className":32471},[473],[224,32473],{"className":32474,"style":724},[473],[224,32476,32478,32481],{"className":32477},[468],[224,32479,13639],{"className":32480,"style":13754},[468,469],[224,32482,32484],{"className":32483},[750],[224,32485,32487,32507],{"className":32486},[754,755],[224,32488,32490,32504],{"className":32489},[759],[224,32491,32493],{"className":32492,"style":1479},[763],[224,32494,32495,32498],{"style":32137},[224,32496],{"className":32497,"style":772},[771],[224,32499,32501],{"className":32500},[776,777,778,779],[224,32502,1447],{"className":32503},[468,779],[224,32505,787],{"className":32506},[786],[224,32508,32510],{"className":32509},[759],[224,32511,32513],{"className":32512,"style":794},[763],[224,32514],{},[224,32516],{"className":32517,"style":496},[473],[224,32519,278],{"className":32520},[500],[224,32522],{"className":32523,"style":496},[473],[224,32525,32527,32530,32533,32536,32539],{"className":32526},[459],[224,32528],{"className":32529,"style":7047},[463],[224,32531,1438],{"className":32532},[468],[224,32534],{"className":32535,"style":474},[473],[224,32537,5268],{"className":32538},[478],[224,32540],{"className":32541,"style":474},[473],[224,32543,32545,32548,32551],{"className":32544},[459],[224,32546],{"className":32547,"style":7642},[463],[224,32549,31295],{"className":32550},[468,469],[224,32552,688],{"className":32553},[468],[11,32555,32556,32557,26478,32560,32661,32662,32690,32691,372,32779,32879,32880,33293,33294,33364],{},"取",[378,32558,32559],{},"确定性核",[224,32561,32563,32595],{"className":32562,"translate":388},[392],[224,32564,32566],{"className":32565},[396],[398,32567,32568],{"xmlns":400},[403,32569,32570,32592],{},[406,32571,32572,32574,32576,32578,32580,32582,32584,32586,32588,32590],{},[409,32573,30652],{},[412,32575,1071],{},[423,32577,682],{},[412,32579,239],{"stretchy":435},[409,32581,1880],{},[412,32583,1877],{"separator":455},[409,32585,371],{},[412,32587,1716],{"stretchy":435},[412,32589,28649],{},[409,32591,371],{},[447,32593,32594],{"encoding":449},"K:\\,(i,x)\\mapsto x",[224,32596,32598,32619,32652],{"className":32597,"ariaHidden":455},[454],[224,32599,32601,32604,32607,32610,32613,32616],{"className":32600},[459],[224,32602],{"className":32603,"style":649},[463],[224,32605,30652],{"className":32606,"style":30740},[468,469],[224,32608],{"className":32609,"style":496},[473],[224,32611,1071],{"className":32612},[500],[224,32614],{"className":32615,"style":724},[473],[224,32617],{"className":32618,"style":496},[473],[224,32620,32622,32625,32628,32631,32634,32637,32640,32643,32646,32649],{"className":32621},[459],[224,32623],{"className":32624,"style":535},[463],[224,32626,239],{"className":32627},[542],[224,32629,1880],{"className":32630},[468,469],[224,32632,1877],{"className":32633},[1917],[224,32635],{"className":32636,"style":724},[473],[224,32638,371],{"className":32639},[468,469],[224,32641,1716],{"className":32642},[549],[224,32644],{"className":32645,"style":496},[473],[224,32647,28649],{"className":32648},[500],[224,32650],{"className":32651,"style":496},[473],[224,32653,32655,32658],{"className":32654},[459],[224,32656],{"className":32657,"style":839},[463],[224,32659,371],{"className":32660},[468,469],"（\"擦掉\"分量下标，只保留 ",[224,32663,32665,32678],{"className":32664,"translate":388},[392],[224,32666,32668],{"className":32667},[396],[398,32669,32670],{"xmlns":400},[403,32671,32672,32676],{},[406,32673,32674],{},[409,32675,371],{},[447,32677,371],{"encoding":449},[224,32679,32681],{"className":32680,"ariaHidden":455},[454],[224,32682,32684,32687],{"className":32683},[459],[224,32685],{"className":32686,"style":839},[463],[224,32688,371],{"className":32689},[468,469],"）。由分块结构立刻算出 ",[224,32692,32694,32718],{"className":32693,"translate":388},[392],[224,32695,32697],{"className":32696},[396],[398,32698,32699],{"xmlns":400},[403,32700,32701,32715],{},[406,32702,32703,32709,32711,32713],{},[31766,32704,32705,32707],{"accent":455},[409,32706,16018],{},[412,32708,31772],{},[409,32710,30652],{},[412,32712,278],{},[409,32714,16018],{},[447,32716,32717],{"encoding":449},"\\tilde PK=P",[224,32719,32721,32770],{"className":32720,"ariaHidden":455},[454],[224,32722,32724,32727,32758,32761,32764,32767],{"className":32723},[459],[224,32725],{"className":32726,"style":31802},[463],[224,32728,32730],{"className":32729},[468,31806],[224,32731,32733],{"className":32732},[754],[224,32734,32736],{"className":32735},[759],[224,32737,32739,32747],{"className":32738,"style":31802},[763],[224,32740,32741,32744],{"style":2120},[224,32742],{"className":32743,"style":2108},[771],[224,32745,16018],{"className":32746,"style":653},[468,469],[224,32748,32749,32752],{"style":31826},[224,32750],{"className":32751,"style":2108},[771],[224,32753,32755],{"className":32754,"style":32068},[31833],[224,32756,31772],{"className":32757},[468],[224,32759,30652],{"className":32760,"style":30740},[468,469],[224,32762],{"className":32763,"style":496},[473],[224,32765,278],{"className":32766},[500],[224,32768],{"className":32769,"style":496},[473],[224,32771,32773,32776],{"className":32772},[459],[224,32774],{"className":32775,"style":649},[463],[224,32777,16018],{"className":32778,"style":653},[468,469],[224,32780,32782,32806],{"className":32781,"translate":388},[392],[224,32783,32785],{"className":32784},[396],[398,32786,32787],{"xmlns":400},[403,32788,32789,32803],{},[406,32790,32791,32797,32799,32801],{},[31766,32792,32793,32795],{"accent":455},[409,32794,20315],{},[412,32796,31772],{},[409,32798,30652],{},[412,32800,278],{},[409,32802,20315],{},[447,32804,32805],{"encoding":449},"\\tilde QK=Q",[224,32807,32809,32870],{"className":32808,"ariaHidden":455},[454],[224,32810,32812,32816,32858,32861,32864,32867],{"className":32811},[459],[224,32813],{"className":32814,"style":32815},[463],"height:1.1146em;vertical-align:-0.1944em;",[224,32817,32819],{"className":32818},[468,31806],[224,32820,32822,32850],{"className":32821},[754,755],[224,32823,32825,32847],{"className":32824},[759],[224,32826,32828,32836],{"className":32827,"style":31802},[763],[224,32829,32830,32833],{"style":2120},[224,32831],{"className":32832,"style":2108},[771],[224,32834,20315],{"className":32835},[468,469],[224,32837,32838,32841],{"style":31826},[224,32839],{"className":32840,"style":2108},[771],[224,32842,32844],{"className":32843,"style":32068},[31833],[224,32845,31772],{"className":32846},[468],[224,32848,787],{"className":32849},[786],[224,32851,32853],{"className":32852},[759],[224,32854,32856],{"className":32855,"style":32259},[763],[224,32857],{},[224,32859,30652],{"className":32860,"style":30740},[468,469],[224,32862],{"className":32863,"style":496},[473],[224,32865,278],{"className":32866},[500],[224,32868],{"className":32869,"style":496},[473],[224,32871,32873,32876],{"className":32872},[459],[224,32874],{"className":32875,"style":20637},[463],[224,32877,20315],{"className":32878},[468,469],"，且似然比是分块恒定的：",[224,32881,32883,32961],{"className":32882,"translate":388},[392],[224,32884,32886],{"className":32885},[396],[398,32887,32888],{"xmlns":400},[403,32889,32890,32958],{},[406,32891,32892,32916,32918,32920,32922,32924,32926,32928,32952,32954,32956],{},[5221,32893,32894],{"scriptlevel":284,"displaystyle":455},[20338,32895,32896,32906],{},[406,32897,32898,32900],{},[409,32899,13182],{},[31766,32901,32902,32904],{"accent":455},[409,32903,16018],{},[412,32905,31772],{},[406,32907,32908,32910],{},[409,32909,13182],{},[31766,32911,32912,32914],{"accent":455},[409,32913,20315],{},[412,32915,31772],{},[412,32917,239],{"stretchy":435},[409,32919,1880],{},[412,32921,1877],{"separator":455},[409,32923,371],{},[412,32925,1716],{"stretchy":435},[412,32927,278],{},[5221,32929,32930],{"scriptlevel":284,"displaystyle":455},[20338,32931,32932,32942],{},[406,32933,32934,32936],{},[409,32935,13182],{},[695,32937,32938,32940],{},[409,32939,16018],{},[409,32941,1880],{},[406,32943,32944,32946],{},[409,32945,13182],{},[695,32947,32948,32950],{},[409,32949,20315],{},[409,32951,1880],{},[412,32953,239],{"stretchy":435},[409,32955,371],{},[412,32957,1716],{"stretchy":435},[447,32959,32960],{"encoding":449},"\\dfrac{d\\tilde P}{d\\tilde Q}(i,x)=\\dfrac{dP_i}{dQ_i}(x)",[224,32962,32964,33136],{"className":32963,"ariaHidden":455},[454],[224,32965,32967,32971,33109,33112,33115,33118,33121,33124,33127,33130,33133],{"className":32966},[459],[224,32968],{"className":32969,"style":32970},[463],"height:2.6018em;vertical-align:-1.0046em;",[224,32972,32974,32977,33106],{"className":32973},[468],[224,32975],{"className":32976},[542,8479],[224,32978,32980],{"className":32979},[20338],[224,32981,32983,33097],{"className":32982},[754,755],[224,32984,32986,33094],{"className":32985},[759],[224,32987,32990,33044,33052],{"className":32988,"style":32989},[763],"height:1.5972em;",[224,32991,32993,32996],{"style":32992},"top:-2.1898em;",[224,32994],{"className":32995,"style":2108},[771],[224,32997,32999,33002],{"className":32998},[468],[224,33000,13182],{"className":33001},[468,469],[224,33003,33005],{"className":33004},[468,31806],[224,33006,33008,33036],{"className":33007},[754,755],[224,33009,33011,33033],{"className":33010},[759],[224,33012,33014,33022],{"className":33013,"style":31802},[763],[224,33015,33016,33019],{"style":2120},[224,33017],{"className":33018,"style":2108},[771],[224,33020,20315],{"className":33021},[468,469],[224,33023,33024,33027],{"style":31826},[224,33025],{"className":33026,"style":2108},[771],[224,33028,33030],{"className":33029,"style":32068},[31833],[224,33031,31772],{"className":33032},[468],[224,33034,787],{"className":33035},[786],[224,33037,33039],{"className":33038},[759],[224,33040,33042],{"className":33041,"style":32259},[763],[224,33043],{},[224,33045,33046,33049],{"style":20546},[224,33047],{"className":33048,"style":2108},[771],[224,33050],{"className":33051,"style":20554},[20553],[224,33053,33054,33057],{"style":20557},[224,33055],{"className":33056,"style":2108},[771],[224,33058,33060,33063],{"className":33059},[468],[224,33061,13182],{"className":33062},[468,469],[224,33064,33066],{"className":33065},[468,31806],[224,33067,33069],{"className":33068},[754],[224,33070,33072],{"className":33071},[759],[224,33073,33075,33083],{"className":33074,"style":31802},[763],[224,33076,33077,33080],{"style":2120},[224,33078],{"className":33079,"style":2108},[771],[224,33081,16018],{"className":33082,"style":653},[468,469],[224,33084,33085,33088],{"style":31826},[224,33086],{"className":33087,"style":2108},[771],[224,33089,33091],{"className":33090,"style":32068},[31833],[224,33092,31772],{"className":33093},[468],[224,33095,787],{"className":33096},[786],[224,33098,33100],{"className":33099},[759],[224,33101,33104],{"className":33102,"style":33103},[763],"height:1.0046em;",[224,33105],{},[224,33107],{"className":33108},[549,8479],[224,33110,239],{"className":33111},[542],[224,33113,1880],{"className":33114},[468,469],[224,33116,1877],{"className":33117},[1917],[224,33119],{"className":33120,"style":724},[473],[224,33122,371],{"className":33123},[468,469],[224,33125,1716],{"className":33126},[549],[224,33128],{"className":33129,"style":496},[473],[224,33131,278],{"className":33132},[500],[224,33134],{"className":33135,"style":496},[473],[224,33137,33139,33142,33284,33287,33290],{"className":33138},[459],[224,33140],{"className":33141,"style":20903},[463],[224,33143,33145,33148,33281],{"className":33144},[468],[224,33146],{"className":33147},[542,8479],[224,33149,33151],{"className":33150},[20338],[224,33152,33154,33273],{"className":33153},[754,755],[224,33155,33157,33270],{"className":33156},[759],[224,33158,33160,33211,33219],{"className":33159,"style":20528},[763],[224,33161,33162,33165],{"style":20531},[224,33163],{"className":33164,"style":2108},[771],[224,33166,33168,33171],{"className":33167},[468],[224,33169,13182],{"className":33170},[468,469],[224,33172,33174,33177],{"className":33173},[468],[224,33175,20315],{"className":33176},[468,469],[224,33178,33180],{"className":33179},[750],[224,33181,33183,33203],{"className":33182},[754,755],[224,33184,33186,33200],{"className":33185},[759],[224,33187,33189],{"className":33188,"style":4075},[763],[224,33190,33191,33194],{"style":767},[224,33192],{"className":33193,"style":772},[771],[224,33195,33197],{"className":33196},[776,777,778,779],[224,33198,1880],{"className":33199},[468,469,779],[224,33201,787],{"className":33202},[786],[224,33204,33206],{"className":33205},[759],[224,33207,33209],{"className":33208,"style":794},[763],[224,33210],{},[224,33212,33213,33216],{"style":20546},[224,33214],{"className":33215,"style":2108},[771],[224,33217],{"className":33218,"style":20554},[20553],[224,33220,33221,33224],{"style":20557},[224,33222],{"className":33223,"style":2108},[771],[224,33225,33227,33230],{"className":33226},[468],[224,33228,13182],{"className":33229},[468,469],[224,33231,33233,33236],{"className":33232},[468],[224,33234,16018],{"className":33235,"style":653},[468,469],[224,33237,33239],{"className":33238},[750],[224,33240,33242,33262],{"className":33241},[754,755],[224,33243,33245,33259],{"className":33244},[759],[224,33246,33248],{"className":33247,"style":4075},[763],[224,33249,33250,33253],{"style":31092},[224,33251],{"className":33252,"style":772},[771],[224,33254,33256],{"className":33255},[776,777,778,779],[224,33257,1880],{"className":33258},[468,469,779],[224,33260,787],{"className":33261},[786],[224,33263,33265],{"className":33264},[759],[224,33266,33268],{"className":33267,"style":794},[763],[224,33269],{},[224,33271,787],{"className":33272},[786],[224,33274,33276],{"className":33275},[759],[224,33277,33279],{"className":33278,"style":20579},[763],[224,33280],{},[224,33282],{"className":33283},[549,8479],[224,33285,239],{"className":33286},[542],[224,33288,371],{"className":33289},[468,469],[224,33291,1716],{"className":33292},[549],"（权重 ",[224,33295,33297,33315],{"className":33296,"translate":388},[392],[224,33298,33300],{"className":33299},[396],[398,33301,33302],{"xmlns":400},[403,33303,33304,33312],{},[406,33305,33306],{},[695,33307,33308,33310],{},[409,33309,13639],{},[409,33311,1880],{},[447,33313,33314],{"encoding":449},"w_i",[224,33316,33318],{"className":33317,"ariaHidden":455},[454],[224,33319,33321,33324],{"className":33320},[459],[224,33322],{"className":33323,"style":3625},[463],[224,33325,33327,33330],{"className":33326},[468],[224,33328,13639],{"className":33329,"style":13754},[468,469],[224,33331,33333],{"className":33332},[750],[224,33334,33336,33356],{"className":33335},[754,755],[224,33337,33339,33353],{"className":33338},[759],[224,33340,33342],{"className":33341,"style":4075},[763],[224,33343,33344,33347],{"style":32137},[224,33345],{"className":33346,"style":772},[771],[224,33348,33350],{"className":33349},[776,777,778,779],[224,33351,1880],{"className":33352},[468,469,779],[224,33354,787],{"className":33355},[786],[224,33357,33359],{"className":33358},[759],[224,33360,33362],{"className":33361,"style":794},[763],[224,33363],{}," 在分子分母中约去）。代入定义：",[224,33366,33368],{"className":33367,"translate":388},[387],[224,33369,33371,33514],{"className":33370,"translate":388},[392],[224,33372,33374],{"className":33373},[396],[398,33375,33376],{"xmlns":400,"display":401},[403,33377,33378,33511],{},[406,33379,33380,33386,33388,33394,33396,33402,33404,33406,33413,33419,33421,33427,33429,33435,33437,33443,33445,33447,33449,33455,33457,33463,33465,33471,33473,33475,33477,33479,33481,33483,33485,33491,33493,33499,33501,33507,33509],{},[695,33381,33382,33384],{},[409,33383,20303],{},[409,33385,1705],{},[412,33387,239],{"stretchy":435},[31766,33389,33390,33392],{"accent":455},[409,33391,16018],{},[412,33393,31772],{},[409,33395,20312],{"mathvariant":678},[31766,33397,33398,33400],{"accent":455},[409,33399,20315],{},[412,33401,31772],{},[412,33403,1716],{"stretchy":435},[412,33405,278],{},[1965,33407,33408,33411],{},[412,33409,33410],{},"∑",[409,33412,1880],{},[695,33414,33415,33417],{},[409,33416,13639],{},[409,33418,1880],{},[423,33420,682],{},[695,33422,33423,33425],{},[409,33424,20303],{},[409,33426,1705],{},[412,33428,239],{"stretchy":435},[695,33430,33431,33433],{},[409,33432,16018],{},[409,33434,1880],{},[409,33436,20312],{"mathvariant":678},[695,33438,33439,33441],{},[409,33440,20315],{},[409,33442,1880],{},[412,33444,1716],{"stretchy":435},[412,33446,278],{},[409,33448,31295],{},[695,33450,33451,33453],{},[409,33452,20303],{},[409,33454,1705],{},[412,33456,239],{"stretchy":435},[695,33458,33459,33461],{},[409,33460,16018],{},[419,33462,1438],{},[409,33464,20312],{"mathvariant":678},[695,33466,33467,33469],{},[409,33468,20315],{},[419,33470,1438],{},[412,33472,1716],{"stretchy":435},[412,33474,319],{},[412,33476,239],{"stretchy":435},[419,33478,1438],{},[412,33480,5268],{},[409,33482,31295],{},[412,33484,1716],{"stretchy":435},[695,33486,33487,33489],{},[409,33488,20303],{},[409,33490,1705],{},[412,33492,239],{"stretchy":435},[695,33494,33495,33497],{},[409,33496,16018],{},[419,33498,1447],{},[409,33500,20312],{"mathvariant":678},[695,33502,33503,33505],{},[409,33504,20315],{},[419,33506,1447],{},[412,33508,1716],{"stretchy":435},[409,33510,688],{"mathvariant":678},[447,33512,33513],{"encoding":449},"D_f(\\tilde P\\|\\tilde Q)\n=\\sum_i w_i\\,D_f(P_i\\|Q_i)\n=\\lambda D_f(P_1\\|Q_1)+(1-\\lambda)D_f(P_2\\|Q_2).",[224,33515,33517,33655,33896,34043,34064],{"className":33516,"ariaHidden":455},[454],[224,33518,33520,33524,33564,33567,33598,33601,33643,33646,33649,33652],{"className":33519},[459],[224,33521],{"className":33522,"style":33523},[463],"height:1.2063em;vertical-align:-0.2861em;",[224,33525,33527,33530],{"className":33526},[468],[224,33528,20303],{"className":33529,"style":2473},[468,469],[224,33531,33533],{"className":33532},[750],[224,33534,33536,33556],{"className":33535},[754,755],[224,33537,33539,33553],{"className":33538},[759],[224,33540,33542],{"className":33541,"style":20387},[763],[224,33543,33544,33547],{"style":20390},[224,33545],{"className":33546,"style":772},[771],[224,33548,33550],{"className":33549},[776,777,778,779],[224,33551,1705],{"className":33552,"style":1740},[468,469,779],[224,33554,787],{"className":33555},[786],[224,33557,33559],{"className":33558},[759],[224,33560,33562],{"className":33561,"style":20409},[763],[224,33563],{},[224,33565,239],{"className":33566},[542],[224,33568,33570],{"className":33569},[468,31806],[224,33571,33573],{"className":33572},[754],[224,33574,33576],{"className":33575},[759],[224,33577,33579,33587],{"className":33578,"style":31802},[763],[224,33580,33581,33584],{"style":2120},[224,33582],{"className":33583,"style":2108},[771],[224,33585,16018],{"className":33586,"style":653},[468,469],[224,33588,33589,33592],{"style":31826},[224,33590],{"className":33591,"style":2108},[771],[224,33593,33595],{"className":33594,"style":32068},[31833],[224,33596,31772],{"className":33597},[468],[224,33599,20312],{"className":33600},[468],[224,33602,33604],{"className":33603},[468,31806],[224,33605,33607,33635],{"className":33606},[754,755],[224,33608,33610,33632],{"className":33609},[759],[224,33611,33613,33621],{"className":33612,"style":31802},[763],[224,33614,33615,33618],{"style":2120},[224,33616],{"className":33617,"style":2108},[771],[224,33619,20315],{"className":33620},[468,469],[224,33622,33623,33626],{"style":31826},[224,33624],{"className":33625,"style":2108},[771],[224,33627,33629],{"className":33628,"style":32068},[31833],[224,33630,31772],{"className":33631},[468],[224,33633,787],{"className":33634},[786],[224,33636,33638],{"className":33637},[759],[224,33639,33641],{"className":33640,"style":32259},[763],[224,33642],{},[224,33644,1716],{"className":33645},[549],[224,33647],{"className":33648,"style":496},[473],[224,33650,278],{"className":33651},[500],[224,33653],{"className":33654,"style":496},[473],[224,33656,33658,33662,33712,33715,33755,33758,33798,33801,33841,33844,33884,33887,33890,33893],{"className":33657},[459],[224,33659],{"className":33660,"style":33661},[463],"height:2.3277em;vertical-align:-1.2777em;",[224,33663,33666],{"className":33664},[21531,33665],"op-limits",[224,33667,33669,33703],{"className":33668},[754,755],[224,33670,33672,33700],{"className":33671},[759],[224,33673,33676,33689],{"className":33674,"style":33675},[763],"height:1.05em;",[224,33677,33679,33683],{"style":33678},"top:-1.8723em;margin-left:0em;",[224,33680],{"className":33681,"style":33682},[771],"height:3.05em;",[224,33684,33686],{"className":33685},[776,777,778,779],[224,33687,1880],{"className":33688},[468,469,779],[224,33690,33692,33695],{"style":33691},"top:-3.05em;",[224,33693],{"className":33694,"style":33682},[771],[224,33696,33697],{},[224,33698,33410],{"className":33699},[21531,21532,21533],[224,33701,787],{"className":33702},[786],[224,33704,33706],{"className":33705},[759],[224,33707,33710],{"className":33708,"style":33709},[763],"height:1.2777em;",[224,33711],{},[224,33713],{"className":33714,"style":724},[473],[224,33716,33718,33721],{"className":33717},[468],[224,33719,13639],{"className":33720,"style":13754},[468,469],[224,33722,33724],{"className":33723},[750],[224,33725,33727,33747],{"className":33726},[754,755],[224,33728,33730,33744],{"className":33729},[759],[224,33731,33733],{"className":33732,"style":4075},[763],[224,33734,33735,33738],{"style":32137},[224,33736],{"className":33737,"style":772},[771],[224,33739,33741],{"className":33740},[776,777,778,779],[224,33742,1880],{"className":33743},[468,469,779],[224,33745,787],{"className":33746},[786],[224,33748,33750],{"className":33749},[759],[224,33751,33753],{"className":33752,"style":794},[763],[224,33754],{},[224,33756],{"className":33757,"style":724},[473],[224,33759,33761,33764],{"className":33760},[468],[224,33762,20303],{"className":33763,"style":2473},[468,469],[224,33765,33767],{"className":33766},[750],[224,33768,33770,33790],{"className":33769},[754,755],[224,33771,33773,33787],{"className":33772},[759],[224,33774,33776],{"className":33775,"style":20387},[763],[224,33777,33778,33781],{"style":20390},[224,33779],{"className":33780,"style":772},[771],[224,33782,33784],{"className":33783},[776,777,778,779],[224,33785,1705],{"className":33786,"style":1740},[468,469,779],[224,33788,787],{"className":33789},[786],[224,33791,33793],{"className":33792},[759],[224,33794,33796],{"className":33795,"style":20409},[763],[224,33797],{},[224,33799,239],{"className":33800},[542],[224,33802,33804,33807],{"className":33803},[468],[224,33805,16018],{"className":33806,"style":653},[468,469],[224,33808,33810],{"className":33809},[750],[224,33811,33813,33833],{"className":33812},[754,755],[224,33814,33816,33830],{"className":33815},[759],[224,33817,33819],{"className":33818,"style":4075},[763],[224,33820,33821,33824],{"style":31092},[224,33822],{"className":33823,"style":772},[771],[224,33825,33827],{"className":33826},[776,777,778,779],[224,33828,1880],{"className":33829},[468,469,779],[224,33831,787],{"className":33832},[786],[224,33834,33836],{"className":33835},[759],[224,33837,33839],{"className":33838,"style":794},[763],[224,33840],{},[224,33842,20312],{"className":33843},[468],[224,33845,33847,33850],{"className":33846},[468],[224,33848,20315],{"className":33849},[468,469],[224,33851,33853],{"className":33852},[750],[224,33854,33856,33876],{"className":33855},[754,755],[224,33857,33859,33873],{"className":33858},[759],[224,33860,33862],{"className":33861,"style":4075},[763],[224,33863,33864,33867],{"style":767},[224,33865],{"className":33866,"style":772},[771],[224,33868,33870],{"className":33869},[776,777,778,779],[224,33871,1880],{"className":33872},[468,469,779],[224,33874,787],{"className":33875},[786],[224,33877,33879],{"className":33878},[759],[224,33880,33882],{"className":33881,"style":794},[763],[224,33883],{},[224,33885,1716],{"className":33886},[549],[224,33888],{"className":33889,"style":496},[473],[224,33891,278],{"className":33892},[500],[224,33894],{"className":33895,"style":496},[473],[224,33897,33899,33902,33905,33945,33948,33988,33991,34031,34034,34037,34040],{"className":33898},[459],[224,33900],{"className":33901,"style":20368},[463],[224,33903,31295],{"className":33904},[468,469],[224,33906,33908,33911],{"className":33907},[468],[224,33909,20303],{"className":33910,"style":2473},[468,469],[224,33912,33914],{"className":33913},[750],[224,33915,33917,33937],{"className":33916},[754,755],[224,33918,33920,33934],{"className":33919},[759],[224,33921,33923],{"className":33922,"style":20387},[763],[224,33924,33925,33928],{"style":20390},[224,33926],{"className":33927,"style":772},[771],[224,33929,33931],{"className":33930},[776,777,778,779],[224,33932,1705],{"className":33933,"style":1740},[468,469,779],[224,33935,787],{"className":33936},[786],[224,33938,33940],{"className":33939},[759],[224,33941,33943],{"className":33942,"style":20409},[763],[224,33944],{},[224,33946,239],{"className":33947},[542],[224,33949,33951,33954],{"className":33950},[468],[224,33952,16018],{"className":33953,"style":653},[468,469],[224,33955,33957],{"className":33956},[750],[224,33958,33960,33980],{"className":33959},[754,755],[224,33961,33963,33977],{"className":33962},[759],[224,33964,33966],{"className":33965,"style":1479},[763],[224,33967,33968,33971],{"style":31092},[224,33969],{"className":33970,"style":772},[771],[224,33972,33974],{"className":33973},[776,777,778,779],[224,33975,1438],{"className":33976},[468,779],[224,33978,787],{"className":33979},[786],[224,33981,33983],{"className":33982},[759],[224,33984,33986],{"className":33985,"style":794},[763],[224,33987],{},[224,33989,20312],{"className":33990},[468],[224,33992,33994,33997],{"className":33993},[468],[224,33995,20315],{"className":33996},[468,469],[224,33998,34000],{"className":33999},[750],[224,34001,34003,34023],{"className":34002},[754,755],[224,34004,34006,34020],{"className":34005},[759],[224,34007,34009],{"className":34008,"style":1479},[763],[224,34010,34011,34014],{"style":767},[224,34012],{"className":34013,"style":772},[771],[224,34015,34017],{"className":34016},[776,777,778,779],[224,34018,1438],{"className":34019},[468,779],[224,34021,787],{"className":34022},[786],[224,34024,34026],{"className":34025},[759],[224,34027,34029],{"className":34028,"style":794},[763],[224,34030],{},[224,34032,1716],{"className":34033},[549],[224,34035],{"className":34036,"style":474},[473],[224,34038,319],{"className":34039},[478],[224,34041],{"className":34042,"style":474},[473],[224,34044,34046,34049,34052,34055,34058,34061],{"className":34045},[459],[224,34047],{"className":34048,"style":535},[463],[224,34050,239],{"className":34051},[542],[224,34053,1438],{"className":34054},[468],[224,34056],{"className":34057,"style":474},[473],[224,34059,5268],{"className":34060},[478],[224,34062],{"className":34063,"style":474},[473],[224,34065,34067,34070,34073,34076,34116,34119,34159,34162,34202,34205],{"className":34066},[459],[224,34068],{"className":34069,"style":20368},[463],[224,34071,31295],{"className":34072},[468,469],[224,34074,1716],{"className":34075},[549],[224,34077,34079,34082],{"className":34078},[468],[224,34080,20303],{"className":34081,"style":2473},[468,469],[224,34083,34085],{"className":34084},[750],[224,34086,34088,34108],{"className":34087},[754,755],[224,34089,34091,34105],{"className":34090},[759],[224,34092,34094],{"className":34093,"style":20387},[763],[224,34095,34096,34099],{"style":20390},[224,34097],{"className":34098,"style":772},[771],[224,34100,34102],{"className":34101},[776,777,778,779],[224,34103,1705],{"className":34104,"style":1740},[468,469,779],[224,34106,787],{"className":34107},[786],[224,34109,34111],{"className":34110},[759],[224,34112,34114],{"className":34113,"style":20409},[763],[224,34115],{},[224,34117,239],{"className":34118},[542],[224,34120,34122,34125],{"className":34121},[468],[224,34123,16018],{"className":34124,"style":653},[468,469],[224,34126,34128],{"className":34127},[750],[224,34129,34131,34151],{"className":34130},[754,755],[224,34132,34134,34148],{"className":34133},[759],[224,34135,34137],{"className":34136,"style":1479},[763],[224,34138,34139,34142],{"style":31092},[224,34140],{"className":34141,"style":772},[771],[224,34143,34145],{"className":34144},[776,777,778,779],[224,34146,1447],{"className":34147},[468,779],[224,34149,787],{"className":34150},[786],[224,34152,34154],{"className":34153},[759],[224,34155,34157],{"className":34156,"style":794},[763],[224,34158],{},[224,34160,20312],{"className":34161},[468],[224,34163,34165,34168],{"className":34164},[468],[224,34166,20315],{"className":34167},[468,469],[224,34169,34171],{"className":34170},[750],[224,34172,34174,34194],{"className":34173},[754,755],[224,34175,34177,34191],{"className":34176},[759],[224,34178,34180],{"className":34179,"style":1479},[763],[224,34181,34182,34185],{"style":767},[224,34183],{"className":34184,"style":772},[771],[224,34186,34188],{"className":34187},[776,777,778,779],[224,34189,1447],{"className":34190},[468,779],[224,34192,787],{"className":34193},[786],[224,34195,34197],{"className":34196},[759],[224,34198,34200],{"className":34199,"style":794},[763],[224,34201],{},[224,34203,1716],{"className":34204},[549],[224,34206,688],{"className":34207},[468],[11,34209,34210,34211,34339],{},"对 ",[224,34212,34214,34244],{"className":34213,"translate":388},[392],[224,34215,34217],{"className":34216},[396],[398,34218,34219],{"xmlns":400},[403,34220,34221,34241],{},[406,34222,34223,34225,34231,34233,34239],{},[412,34224,239],{"stretchy":435},[31766,34226,34227,34229],{"accent":455},[409,34228,16018],{},[412,34230,31772],{},[412,34232,1877],{"separator":455},[31766,34234,34235,34237],{"accent":455},[409,34236,20315],{},[412,34238,31772],{},[412,34240,1716],{"stretchy":435},[447,34242,34243],{"encoding":449},"(\\tilde P,\\tilde Q)",[224,34245,34247],{"className":34246,"ariaHidden":455},[454],[224,34248,34250,34254,34257,34288,34291,34294,34336],{"className":34249},[459],[224,34251],{"className":34252,"style":34253},[463],"height:1.1702em;vertical-align:-0.25em;",[224,34255,239],{"className":34256},[542],[224,34258,34260],{"className":34259},[468,31806],[224,34261,34263],{"className":34262},[754],[224,34264,34266],{"className":34265},[759],[224,34267,34269,34277],{"className":34268,"style":31802},[763],[224,34270,34271,34274],{"style":2120},[224,34272],{"className":34273,"style":2108},[771],[224,34275,16018],{"className":34276,"style":653},[468,469],[224,34278,34279,34282],{"style":31826},[224,34280],{"className":34281,"style":2108},[771],[224,34283,34285],{"className":34284,"style":32068},[31833],[224,34286,31772],{"className":34287},[468],[224,34289,1877],{"className":34290},[1917],[224,34292],{"className":34293,"style":724},[473],[224,34295,34297],{"className":34296},[468,31806],[224,34298,34300,34328],{"className":34299},[754,755],[224,34301,34303,34325],{"className":34302},[759],[224,34304,34306,34314],{"className":34305,"style":31802},[763],[224,34307,34308,34311],{"style":2120},[224,34309],{"className":34310,"style":2108},[771],[224,34312,20315],{"className":34313},[468,469],[224,34315,34316,34319],{"style":31826},[224,34317],{"className":34318,"style":2108},[771],[224,34320,34322],{"className":34321,"style":32068},[31833],[224,34323,31772],{"className":34324},[468],[224,34326,787],{"className":34327},[786],[224,34329,34331],{"className":34330},[759],[224,34332,34334],{"className":34333,"style":32259},[763],[224,34335],{},[224,34337,1716],{"className":34338},[549]," 使用 DPI，即得到联合凸性：",[224,34341,34343],{"className":34342,"translate":388},[387],[224,34344,34346,34526],{"className":34345,"translate":388},[392],[224,34347,34349],{"className":34348},[396],[398,34350,34351],{"xmlns":400,"display":401},[403,34352,34353,34523],{},[5209,34354,34355,34439],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,34356,34357,34363],{},[5218,34358,34359],{},[5221,34360,34361],{"scriptlevel":284,"displaystyle":455},[406,34362],{},[5218,34364,34365],{},[5221,34366,34367],{"scriptlevel":284,"displaystyle":455},[406,34368,34369,34371,34377,34379,34381,34387,34389,34391,34393,34395,34397,34399,34405,34407,34409,34411,34413,34419,34421,34423,34425,34427,34429,34431,34437],{},[406,34370],{},[695,34372,34373,34375],{},[409,34374,20303],{},[409,34376,1705],{},[412,34378,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,34380,31295],{},[695,34382,34383,34385],{},[409,34384,16018],{},[419,34386,1438],{},[412,34388,319],{},[412,34390,239],{"stretchy":435},[419,34392,1438],{},[412,34394,5268],{},[409,34396,31295],{},[412,34398,1716],{"stretchy":435},[695,34400,34401,34403],{},[409,34402,16018],{},[419,34404,1447],{},[423,34406,682],{},[412,34408,20312],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[423,34410,682],{},[409,34412,31295],{},[695,34414,34415,34417],{},[409,34416,20315],{},[419,34418,1438],{},[412,34420,319],{},[412,34422,239],{"stretchy":435},[419,34424,1438],{},[412,34426,5268],{},[409,34428,31295],{},[412,34430,1716],{"stretchy":435},[695,34432,34433,34435],{},[409,34434,20315],{},[419,34436,1447],{},[412,34438,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[5215,34440,34441,34447],{},[5218,34442,34443],{},[5221,34444,34445],{"scriptlevel":284,"displaystyle":455},[406,34446],{},[5218,34448,34449],{},[5221,34450,34451],{"scriptlevel":284,"displaystyle":455},[406,34452,34453,34455,34457,34459,34461,34467,34469,34475,34477,34483,34485,34487,34489,34491,34493,34495,34497,34503,34505,34511,34513,34519,34521],{},[406,34454],{},[473,34456],{"width":4247},[412,34458,2070],{},[409,34460,31295],{},[695,34462,34463,34465],{},[409,34464,20303],{},[409,34466,1705],{},[412,34468,239],{"stretchy":435},[695,34470,34471,34473],{},[409,34472,16018],{},[419,34474,1438],{},[409,34476,20312],{"mathvariant":678},[695,34478,34479,34481],{},[409,34480,20315],{},[419,34482,1438],{},[412,34484,1716],{"stretchy":435},[412,34486,319],{},[412,34488,239],{"stretchy":435},[419,34490,1438],{},[412,34492,5268],{},[409,34494,31295],{},[412,34496,1716],{"stretchy":435},[695,34498,34499,34501],{},[409,34500,20303],{},[409,34502,1705],{},[412,34504,239],{"stretchy":435},[695,34506,34507,34509],{},[409,34508,16018],{},[419,34510,1447],{},[409,34512,20312],{"mathvariant":678},[695,34514,34515,34517],{},[409,34516,20315],{},[419,34518,1447],{},[412,34520,1716],{"stretchy":435},[409,34522,688],{"mathvariant":678},[447,34524,34525],{"encoding":449},"\\begin{aligned}\n&D_f\\big(\\lambda P_1+(1-\\lambda)P_2\n\\,\\big\\|\\,\\lambda Q_1+(1-\\lambda)Q_2\\big)\\\\\n&\\qquad\\le \\lambda 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h43z",[224,34806,787],{"className":34807},[786],[224,34809,34811],{"className":34810},[759],[224,34812,34815],{"className":34813,"style":34814},[763],"height:0.35em;",[224,34816],{},[224,34818],{"className":34819,"style":724},[473],[224,34821,31295],{"className":34822},[468,469],[224,34824,34826,34829],{"className":34825},[468],[224,34827,20315],{"className":34828},[468,469],[224,34830,34832],{"className":34831},[750],[224,34833,34835,34855],{"className":34834},[754,755],[224,34836,34838,34852],{"className":34837},[759],[224,34839,34841],{"className":34840,"style":1479},[763],[224,34842,34843,34846],{"style":767},[224,34844],{"className":34845,"style":772},[771],[224,34847,34849],{"className":34848},[776,777,778,779],[224,34850,1438],{"className":34851},[468,779],[224,34853,787],{"className":34854},[786],[224,34856,34858],{"className":34857},[759],[224,34859,34861],{"className":34860,"style":794},[763],[224,34862],{},[224,34864],{"className":34865,"style":474},[473],[224,34867,319],{"className":34868},[478],[224,34870],{"className":34871,"style":474},[473],[224,34873,239],{"className":34874},[542],[224,34876,1438],{"className":34877},[468],[224,34879],{"className":34880,"style":474},[473],[224,34882,5268],{"className":34883},[478],[224,34885],{"className":34886,"style":474},[473],[224,34888,31295],{"className":34889},[468,469],[224,34891,1716],{"className":34892},[549],[224,34894,34896,34899],{"className":34895},[468],[224,34897,20315],{"className":34898},[468,469],[224,34900,34902],{"className":34901},[750],[224,34903,34905,34925],{"className":34904},[754,755],[224,34906,34908,34922],{"className":34907},[759],[224,34909,34911],{"className":34910,"style":1479},[763],[224,34912,34913,34916],{"style":767},[224,34914],{"className":34915,"style":772},[771],[224,34917,34919],{"className":34918},[776,777,778,779],[224,34920,1447],{"className":34921},[468,779],[224,34923,787],{"className":34924},[786],[224,34926,34928],{"className":34927},[759],[224,34929,34931],{"className":34930,"style":794},[763],[224,34932],{},[224,34934,34936],{"className":34935},[468],[224,34937,1716],{"className":34938},[2363,2364],[224,34940,34942,34945],{"style":34941},"top:-2.255em;",[224,34943],{"className":34944,"style":2108},[771],[224,34946,34948,34951,34954,34957,34960,34963,34966,35006,35009,35049,35052,35092,35095,35098,35101,35104,35107,35110,35113,35116,35119,35122,35125,35165,35168,35208,35211,35251,35254],{"className":34947},[468],[224,34949],{"className":34950},[468],[224,34952],{"className":34953,"style":4416},[473],[224,34955],{"className":34956,"style":496},[473],[224,34958,2070],{"className":34959},[500],[224,34961],{"className":34962,"style":496},[473],[224,34964,31295],{"className":34965},[468,469],[224,34967,34969,34972],{"className":34968},[468],[224,34970,20303],{"className":34971,"style":2473},[468,469],[224,34973,34975],{"className":34974},[750],[224,34976,34978,34998],{"className":34977},[754,755],[224,34979,34981,34995],{"className":34980},[759],[224,34982,34984],{"className":34983,"style":20387},[763],[224,34985,34986,34989],{"style":20390},[224,34987],{"className":34988,"style":772},[771],[224,34990,34992],{"className":34991},[776,777,778,779],[224,34993,1705],{"className":34994,"style":1740},[468,469,779],[224,34996,787],{"className":34997},[786],[224,34999,35001],{"className":35000},[759],[224,35002,35004],{"className":35003,"style":20409},[763],[224,35005],{},[224,35007,239],{"className":35008},[542],[224,35010,35012,35015],{"className":35011},[468],[224,35013,16018],{"className":35014,"style":653},[468,469],[224,35016,35018],{"className":35017},[750],[224,35019,35021,35041],{"className":35020},[754,755],[224,35022,35024,35038],{"className":35023},[759],[224,35025,35027],{"className":35026,"style":1479},[763],[224,35028,35029,35032],{"style":31092},[224,35030],{"className":35031,"style":772},[771],[224,35033,35035],{"className":35034},[776,777,778,779],[224,35036,1438],{"className":35037},[468,779],[224,35039,787],{"className":35040},[786],[224,35042,35044],{"className":35043},[759],[224,35045,35047],{"className":35046,"style":794},[763],[224,35048],{},[224,35050,20312],{"className":35051},[468],[224,35053,35055,35058],{"className":35054},[468],[224,35056,20315],{"className":35057},[468,469],[224,35059,35061],{"className":35060},[750],[224,35062,35064,35084],{"className":35063},[754,755],[224,35065,35067,35081],{"className":35066},[759],[224,35068,35070],{"className":35069,"style":1479},[763],[224,35071,35072,35075],{"style":767},[224,35073],{"className":35074,"style":772},[771],[224,35076,35078],{"className":35077},[776,777,778,779],[224,35079,1438],{"className":35080},[468,779],[224,35082,787],{"className":35083},[786],[224,35085,35087],{"className":35086},[759],[224,35088,35090],{"className":35089,"style":794},[763],[224,35091],{},[224,35093,1716],{"className":35094},[549],[224,35096],{"className":35097,"style":474},[473],[224,35099,319],{"className":35100},[478],[224,35102],{"className":35103,"style":474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",[224,35274,35276,35289],{"className":35275,"translate":388},[392],[224,35277,35279],{"className":35278},[396],[398,35280,35281],{"xmlns":400},[403,35282,35283,35287],{},[406,35284,35285],{},[412,35286,9886],{},[447,35288,30950],{"encoding":449},[224,35290,35292],{"className":35291,"ariaHidden":455},[454],[224,35293,35295,35298],{"className":35294},[459],[224,35296],{"className":35297,"style":30960},[463],[224,35299,9886],{"className":35300},[500],[224,35302,35304,35317],{"className":35303,"translate":388},[392],[224,35305,35307],{"className":35306},[396],[398,35308,35309],{"xmlns":400},[403,35310,35311,35315],{},[406,35312,35313],{},[409,35314,1705],{},[447,35316,1705],{"encoding":449},[224,35318,35320],{"className":35319,"ariaHidden":455},[454],[224,35321,35323,35326],{"className":35322},[459],[224,35324],{"className":35325,"style":2565},[463],[224,35327,1705],{"className":35328,"style":1740},[468,469]," 固定 ",[224,35331,35333,35355],{"className":35332,"translate":388},[392],[224,35334,35336],{"className":35335},[396],[398,35337,35338],{"xmlns":400},[403,35339,35340,35352],{},[406,35341,35342,35344,35346,35348,35350],{},[409,35343,1858],{},[412,35345,1877],{"separator":455},[409,35347,4130],{},[412,35349,9766],{},[419,35351,284],{},[447,35353,35354],{"encoding":449},"s,u\\ge0",[224,35356,35358,35385],{"className":35357,"ariaHidden":455},[454],[224,35359,35361,35364,35367,35370,35373,35376,35379,35382],{"className":35360},[459],[224,35362],{"className":35363,"style":2644},[463],[224,35365,1858],{"className":35366},[468,469],[224,35368,1877],{"className":35369},[1917],[224,35371],{"className":35372,"style":724},[473],[224,35374,4130],{"className":35375},[468,469],[224,35377],{"className":35378,"style":496},[473],[224,35380,9766],{"className":35381},[500],[224,35383],{"className":35384,"style":496},[473],[224,35386,35388,35391],{"className":35387},[459],[224,35389],{"className":35390,"style":510},[463],[224,35392,284],{"className":35393},[468],[224,35395,35397,35422],{"className":35396,"translate":388},[392],[224,35398,35400],{"className":35399},[396],[398,35401,35402],{"xmlns":400},[403,35403,35404,35420],{},[406,35405,35406,35408,35410,35412,35414,35416,35418],{},[409,35407,31295],{},[412,35409,12248],{},[412,35411,239],{"stretchy":435},[419,35413,284],{},[412,35415,1877],{"separator":455},[419,35417,1438],{},[412,35419,1716],{"stretchy":435},[447,35421,31310],{"encoding":449},[224,35423,35425,35443],{"className":35424,"ariaHidden":455},[454],[224,35426,35428,35431,35434,35437,35440],{"className":35427},[459],[224,35429],{"className":35430,"style":16682},[463],[224,35432,31295],{"className":35433},[468,469],[224,35435],{"className":35436,"style":496},[473],[224,35438,12248],{"className":35439},[500],[224,35441],{"className":35442,"style":496},[473],[224,35444,35446,35449,35452,35455,35458,35461,35464],{"className":35445},[459],[224,35447],{"className":35448,"style":535},[463],[224,35450,239],{"className":35451},[542],[224,35453,284],{"className":35454},[468],[224,35456,1877],{"className":35457},[1917],[224,35459],{"className":35460,"style":724},[473],[224,35462,1438],{"className":35463},[468],[224,35465,1716],{"className":35466},[549],"，记目标值 ",[224,35469,35471,35505],{"className":35470,"translate":388},[392],[224,35472,35474],{"className":35473},[396],[398,35475,35476],{"xmlns":400},[403,35477,35478,35502],{},[406,35479,35480,35482,35484,35486,35488,35490,35492,35494,35496,35498,35500],{},[409,35481,13639],{},[412,35483,278],{},[409,35485,31295],{},[409,35487,1858],{},[412,35489,319],{},[412,35491,239],{"stretchy":435},[419,35493,1438],{},[412,35495,5268],{},[409,35497,31295],{},[412,35499,1716],{"stretchy":435},[409,35501,4130],{},[447,35503,35504],{"encoding":449},"w=\\lambda s+(1-\\lambda)u",[224,35506,35508,35526,35547,35568],{"className":35507,"ariaHidden":455},[454],[224,35509,35511,35514,35517,35520,35523],{"className":35510},[459],[224,35512],{"className":35513,"style":839},[463],[224,35515,13639],{"className":35516,"style":13754},[468,469],[224,35518],{"className":35519,"style":496},[473],[224,35521,278],{"className":35522},[500],[224,35524],{"className":35525,"style":496},[473],[224,35527,35529,35532,35535,35538,35541,35544],{"className":35528},[459],[224,35530],{"className":35531,"style":4348},[463],[224,35533,31295],{"className":35534},[468,469],[224,35536,1858],{"className":35537},[468,469],[224,35539],{"className":35540,"style":474},[473],[224,35542,319],{"className":35543},[478],[224,35545],{"className":35546,"style":474},[473],[224,35548,35550,35553,35556,35559,35562,35565],{"className":35549},[459],[224,35551],{"className":35552,"style":535},[463],[224,35554,239],{"className":35555},[542],[224,35557,1438],{"className":35558},[468],[224,35560],{"className":35561,"style":474},[473],[224,35563,5268],{"className":35564},[478],[224,35566],{"className":35567,"style":474},[473],[224,35569,35571,35574,35577,35580],{"className":35570},[459],[224,35572],{"className":35573,"style":535},[463],[224,35575,31295],{"className":35576},[468,469],[224,35578,1716],{"className":35579},[549],[224,35581,4130],{"className":35582},[468,469],"。在两点空间 ",[224,35585,35587,35609],{"className":35586,"translate":388},[392],[224,35588,35590],{"className":35589},[396],[398,35591,35592],{"xmlns":400},[403,35593,35594,35606],{},[406,35595,35596,35598,35600,35602,35604],{},[412,35597,8110],{"stretchy":435},[419,35599,1438],{},[412,35601,1877],{"separator":455},[419,35603,1447],{},[412,35605,15077],{"stretchy":435},[447,35607,35608],{"encoding":449},"\\{1,2\\}",[224,35610,35612],{"className":35611,"ariaHidden":455},[454],[224,35613,35615,35618,35621,35624,35627,35630,35633],{"className":35614},[459],[224,35616],{"className":35617,"style":535},[463],[224,35619,8110],{"className":35620},[542],[224,35622,1438],{"className":35623},[468],[224,35625,1877],{"className":35626},[1917],[224,35628],{"className":35629,"style":724},[473],[224,35631,1447],{"className":35632},[468],[224,35634,15077],{"className":35635},[549]," 上取同一个底分布 ",[224,35638,35640,35673],{"className":35639,"translate":388},[392],[224,35641,35643],{"className":35642},[396],[398,35644,35645],{"xmlns":400},[403,35646,35647,35670],{},[406,35648,35649,35651,35653,35655,35658,35660,35662,35664,35666,35668],{},[409,35650,20315],{},[412,35652,278],{},[412,35654,239],{"stretchy":435},[409,35656,35657],{},"δ",[412,35659,1877],{"separator":455},[423,35661,682],{},[419,35663,1438],{},[412,35665,5268],{},[409,35667,35657],{},[412,35669,1716],{"stretchy":435},[447,35671,35672],{"encoding":449},"Q=(\\delta,\\,1-\\delta)",[224,35674,35676,35694,35728],{"className":35675,"ariaHidden":455},[454],[224,35677,35679,35682,35685,35688,35691],{"className":35678},[459],[224,35680],{"className":35681,"style":20637},[463],[224,35683,20315],{"className":35684},[468,469],[224,35686],{"className":35687,"style":496},[473],[224,35689,278],{"className":35690},[500],[224,35692],{"className":35693,"style":496},[473],[224,35695,35697,35700,35703,35707,35710,35713,35716,35719,35722,35725],{"className":35696},[459],[224,35698],{"className":35699,"style":535},[463],[224,35701,239],{"className":35702},[542],[224,35704,35657],{"className":35705,"style":35706},[468,469],"margin-right:0.0379em;",[224,35708,1877],{"className":35709},[1917],[224,35711],{"className":35712,"style":724},[473],[224,35714],{"className":35715,"style":724},[473],[224,35717,1438],{"className":35718},[468],[224,35720],{"className":35721,"style":474},[473],[224,35723,5268],{"className":35724},[478],[224,35726],{"className":35727,"style":474},[473],[224,35729,35731,35734,35737],{"className":35730},[459],[224,35732],{"className":35733,"style":535},[463],[224,35735,35657],{"className":35736,"style":35706},[468,469],[224,35738,1716],{"className":35739},[549],"（",[224,35742,35744,35770],{"className":35743,"translate":388},[392],[224,35745,35747],{"className":35746},[396],[398,35748,35749],{"xmlns":400},[403,35750,35751,35767],{},[406,35752,35753,35755,35757,35759,35761,35763,35765],{},[409,35754,35657],{},[412,35756,12248],{},[412,35758,239],{"stretchy":435},[419,35760,284],{},[412,35762,1877],{"separator":455},[419,35764,1438],{},[412,35766,1716],{"stretchy":435},[447,35768,35769],{"encoding":449},"\\delta\\in(0,1)",[224,35771,35773,35791],{"className":35772,"ariaHidden":455},[454],[224,35774,35776,35779,35782,35785,35788],{"className":35775},[459],[224,35777],{"className":35778,"style":16682},[463],[224,35780,35657],{"className":35781,"style":35706},[468,469],[224,35783],{"className":35784,"style":496},[473],[224,35786,12248],{"className":35787},[500],[224,35789],{"className":35790,"style":496},[473],[224,35792,35794,35797,35800,35803,35806,35809,35812],{"className":35793},[459],[224,35795],{"className":35796,"style":535},[463],[224,35798,239],{"className":35799},[542],[224,35801,284],{"className":35802},[468],[224,35804,1877],{"className":35805},[1917],[224,35807],{"className":35808,"style":724},[473],[224,35810,1438],{"className":35811},[468],[224,35813,1716],{"className":35814},[549]," 充分小），并取似然比在第一坐标上为常数的分布：",[224,35817,35819],{"className":35818,"translate":388},[387],[224,35820,35822,35896],{"className":35821,"translate":388},[392],[224,35823,35825],{"className":35824},[396],[398,35826,35827],{"xmlns":400,"display":401},[403,35828,35829,35893],{},[406,35830,35831,35837,35839,35841,35843,35845,35847,35849,35851,35853,35855,35857,35859,35861,35863,35869,35871,35873,35875,35877,35879,35881,35883,35885,35887,35889,35891],{},[695,35832,35833,35835],{},[409,35834,16018],{},[419,35836,1438],{},[412,35838,278],{},[412,35840,239],{"stretchy":435},[409,35842,1858],{},[409,35844,35657],{},[412,35846,1877],{"separator":455},[423,35848,425],{},[419,35850,1438],{},[412,35852,5268],{},[409,35854,1858],{},[409,35856,35657],{},[412,35858,1716],{"stretchy":435},[412,35860,1877],{"separator":455},[473,35862],{"width":4247},[695,35864,35865,35867],{},[409,35866,16018],{},[419,35868,1447],{},[412,35870,278],{},[412,35872,239],{"stretchy":435},[409,35874,4130],{},[409,35876,35657],{},[412,35878,1877],{"separator":455},[423,35880,425],{},[419,35882,1438],{},[412,35884,5268],{},[409,35886,4130],{},[409,35888,35657],{},[412,35890,1716],{"stretchy":435},[412,35892,1877],{"separator":455},[447,35894,35895],{"encoding":449},"P_1=(s\\delta,\\;1-s\\delta),\\qquad\nP_2=(u\\delta,\\;1-u\\delta),",[224,35897,35899,35955,35989,36059,36095],{"className":35898,"ariaHidden":455},[454],[224,35900,35902,35906,35946,35949,35952],{"className":35901},[459],[224,35903],{"className":35904,"style":35905},[463],"height:0.8333em;vertical-align:-0.15em;",[224,35907,35909,35912],{"className":35908},[468],[224,35910,16018],{"className":35911,"style":653},[468,469],[224,35913,35915],{"className":35914},[750],[224,35916,35918,35938],{"className":35917},[754,755],[224,35919,35921,35935],{"className":35920},[759],[224,35922,35924],{"className":35923,"style":1479},[763],[224,35925,35926,35929],{"style":31092},[224,35927],{"className":35928,"style":772},[771],[224,35930,35932],{"className":35931},[776,777,778,779],[224,35933,1438],{"className":35934},[468,779],[224,35936,787],{"className":35937},[786],[224,35939,35941],{"className":35940},[759],[224,35942,35944],{"className":35943,"style":794},[763],[224,35945],{},[224,35947],{"className":35948,"style":496},[473],[224,35950,278],{"className":35951},[500],[224,35953],{"className":35954,"style":496},[473],[224,35956,35958,35961,35964,35968,35971,35974,35977,35980,35983,35986],{"className":35957},[459],[224,35959],{"className":35960,"style":535},[463],[224,35962,239],{"className":35963},[542],[224,35965,35967],{"className":35966,"style":35706},[468,469],"sδ",[224,35969,1877],{"className":35970},[1917],[224,35972],{"className":35973,"style":496},[473],[224,35975],{"className":35976,"style":724},[473],[224,35978,1438],{"className":35979},[468],[224,35981],{"className":35982,"style":474},[473],[224,35984,5268],{"className":35985},[478],[224,35987],{"className":35988,"style":474},[473],[224,35990,35992,35995,35998,36001,36004,36007,36010,36050,36053,36056],{"className":35991},[459],[224,35993],{"className":35994,"style":535},[463],[224,35996,35967],{"className":35997,"style":35706},[468,469],[224,35999,1716],{"className":36000},[549],[224,36002,1877],{"className":36003},[1917],[224,36005],{"className":36006,"style":4416},[473],[224,36008],{"className":36009,"style":724},[473],[224,36011,36013,36016],{"className":36012},[468],[224,36014,16018],{"className":36015,"style":653},[468,469],[224,36017,36019],{"className":36018},[750],[224,36020,36022,36042],{"className":36021},[754,755],[224,36023,36025,36039],{"className":36024},[759],[224,36026,36028],{"className":36027,"style":1479},[763],[224,36029,36030,36033],{"style":31092},[224,36031],{"className":36032,"style":772},[771],[224,36034,36036],{"className":36035},[776,777,778,779],[224,36037,1447],{"className":36038},[468,779],[224,36040,787],{"className":36041},[786],[224,36043,36045],{"className":36044},[759],[224,36046,36048],{"className":36047,"style":794},[763],[224,36049],{},[224,36051],{"className":36052,"style":496},[473],[224,36054,278],{"className":36055},[500],[224,36057],{"className":36058,"style":496},[473],[224,36060,36062,36065,36068,36071,36074,36077,36080,36083,36086,36089,36092],{"className":36061},[459],[224,36063],{"className":36064,"style":535},[463],[224,36066,239],{"className":36067},[542],[224,36069,4130],{"className":36070},[468,469],[224,36072,35657],{"className":36073,"style":35706},[468,469],[224,36075,1877],{"className":36076},[1917],[224,36078],{"className":36079,"style":496},[473],[224,36081],{"className":36082,"style":724},[473],[224,36084,1438],{"className":36085},[468],[224,36087],{"className":36088,"style":474},[473],[224,36090,5268],{"className":36091},[478],[224,36093],{"className":36094,"style":474},[473],[224,36096,36098,36101,36104,36107,36110],{"className":36097},[459],[224,36099],{"className":36100,"style":535},[463],[224,36102,4130],{"className":36103},[468,469],[224,36105,35657],{"className":36106,"style":35706},[468,469],[224,36108,1716],{"className":36109},[549],[224,36111,1877],{"className":36112},[1917],[11,36114,35740,36115,36144,36145,36187,36188,22450,36216,36379,36380,36408,36409,7098],{},[224,36116,36118,36132],{"className":36117,"translate":388},[392],[224,36119,36121],{"className":36120},[396],[398,36122,36123],{"xmlns":400},[403,36124,36125,36129],{},[406,36126,36127],{},[409,36128,35657],{},[447,36130,36131],{"encoding":449},"\\delta",[224,36133,36135],{"className":36134,"ariaHidden":455},[454],[224,36136,36138,36141],{"className":36137},[459],[224,36139],{"className":36140,"style":7642},[463],[224,36142,35657],{"className":36143,"style":35706},[468,469]," 足够小时这都是合法概率分布；",[224,36146,36148,36166],{"className":36147,"translate":388},[392],[224,36149,36151],{"className":36150},[396],[398,36152,36153],{"xmlns":400},[403,36154,36155,36163],{},[406,36156,36157,36159,36161],{},[419,36158,284],{},[412,36160,1877],{"separator":455},[419,36162,1438],{},[447,36164,36165],{"encoding":449},"0,1",[224,36167,36169],{"className":36168,"ariaHidden":455},[454],[224,36170,36172,36175,36178,36181,36184],{"className":36171},[459],[224,36173],{"className":36174,"style":8688},[463],[224,36176,284],{"className":36177},[468],[224,36179,1877],{"className":36180},[1917],[224,36182],{"className":36183,"style":724},[473],[224,36185,1438],{"className":36186},[468]," 两点上的似然比分别为 ",[224,36189,36191,36204],{"className":36190,"translate":388},[392],[224,36192,36194],{"className":36193},[396],[398,36195,36196],{"xmlns":400},[403,36197,36198,36202],{},[406,36199,36200],{},[409,36201,1858],{},[447,36203,1858],{"encoding":449},[224,36205,36207],{"className":36206,"ariaHidden":455},[454],[224,36208,36210,36213],{"className":36209},[459],[224,36211],{"className":36212,"style":839},[463],[224,36214,1858],{"className":36215},[468,469],[224,36217,36219,36261],{"className":36218,"translate":388},[392],[224,36220,36222],{"className":36221},[396],[398,36223,36224],{"xmlns":400},[403,36225,36226,36258],{},[406,36227,36228,36230,36232,36234,36236,36238],{},[409,36229,24189],{},[412,36231,239],{"stretchy":435},[409,36233,1858],{},[412,36235,1716],{"stretchy":435},[412,36237,278],{},[20338,36239,36240,36250],{},[406,36241,36242,36244,36246,36248],{},[419,36243,1438],{},[412,36245,5268],{},[409,36247,1858],{},[409,36249,35657],{},[406,36251,36252,36254,36256],{},[419,36253,1438],{},[412,36255,5268],{},[409,36257,35657],{},[447,36259,36260],{"encoding":449},"g(s)=\\frac{1-s\\delta}{1-\\delta}",[224,36262,36264,36291],{"className":36263,"ariaHidden":455},[454],[224,36265,36267,36270,36273,36276,36279,36282,36285,36288],{"className":36266},[459],[224,36268],{"className":36269,"style":535},[463],[224,36271,24189],{"className":36272,"style":492},[468,469],[224,36274,239],{"className":36275},[542],[224,36277,1858],{"className":36278},[468,469],[224,36280,1716],{"className":36281},[549],[224,36283],{"className":36284,"style":496},[473],[224,36286,278],{"className":36287},[500],[224,36289],{"className":36290,"style":496},[473],[224,36292,36294,36298],{"className":36293},[459],[224,36295],{"className":36296,"style":36297},[463],"height:1.2834em;vertical-align:-0.4033em;",[224,36299,36301,36304,36376],{"className":36300},[468],[224,36302],{"className":36303},[542,8479],[224,36305,36307],{"className":36306},[20338],[224,36308,36310,36367],{"className":36309},[754,755],[224,36311,36313,36364],{"className":36312},[759],[224,36314,36316,36336,36344],{"className":36315,"style":22724},[763],[224,36317,36318,36321],{"style":22566},[224,36319],{"className":36320,"style":2108},[771],[224,36322,36324],{"className":36323},[776,777,778,779],[224,36325,36327,36330,36333],{"className":36326},[468,779],[224,36328,1438],{"className":36329},[468,779],[224,36331,5268],{"className":36332},[478,779],[224,36334,35657],{"className":36335,"style":35706},[468,469,779],[224,36337,36338,36341],{"style":20546},[224,36339],{"className":36340,"style":2108},[771],[224,36342],{"className":36343,"style":20554},[20553],[224,36345,36346,36349],{"style":22589},[224,36347],{"className":36348,"style":2108},[771],[224,36350,36352],{"className":36351},[776,777,778,779],[224,36353,36355,36358,36361],{"className":36354},[468,779],[224,36356,1438],{"className":36357},[468,779],[224,36359,5268],{"className":36360},[478,779],[224,36362,35967],{"className":36363,"style":35706},[468,469,779],[224,36365,787],{"className":36366},[786],[224,36368,36370],{"className":36369},[759],[224,36371,36374],{"className":36372,"style":36373},[763],"height:0.4033em;",[224,36375],{},[224,36377],{"className":36378},[549,8479],"）。于是一般公式给出：对任何第一坐标似然比为常值 ",[224,36381,36383,36396],{"className":36382,"translate":388},[392],[224,36384,36386],{"className":36385},[396],[398,36387,36388],{"xmlns":400},[403,36389,36390,36394],{},[406,36391,36392],{},[409,36393,2061],{},[447,36395,2061],{"encoding":449},[224,36397,36399],{"className":36398,"ariaHidden":455},[454],[224,36400,36402,36405],{"className":36401},[459],[224,36403],{"className":36404,"style":839},[463],[224,36406,2061],{"className":36407,"style":2473},[468,469]," 的分布 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+(1-\\delta)\\,\nf\\!\\left(\\frac{1-r\\delta}{1-\\delta}\\right).\n\\end{aligned}",[224,36592,36594],{"className":36593,"ariaHidden":455},[454],[224,36595,36597,36601],{"className":36596},[459],[224,36598],{"className":36599,"style":36600},[463],"height:3.91em;vertical-align:-1.705em;",[224,36602,36604],{"className":36603},[468],[224,36605,36607,36759],{"className":36606},[5209],[224,36608,36610],{"className":36609},[5331],[224,36611,36613,36751],{"className":36612},[754,755],[224,36614,36616,36748],{"className":36615},[759],[224,36617,36620,36739],{"className":36618,"style":36619},[763],"height:2.205em;",[224,36621,36623,36626],{"style":36622},"top:-4.805em;",[224,36624],{"className":36625,"style":21425},[771],[224,36627,36629,36669,36675,36678,36681,36684,36687,36690,36727,36730,36733],{"className":36628},[468],[224,36630,36632,36635],{"className":36631},[468],[224,36633,20303],{"className":36634,"style":2473},[468,469],[224,36636,36638],{"className":36637},[750],[224,36639,36641,36661],{"className":36640},[754,755],[224,36642,36644,36658],{"className":36643},[759],[224,36645,36647],{"className":36646,"style":20387},[763],[224,36648,36649,36652],{"style":20390},[224,36650],{"className":36651,"style":772},[771],[224,36653,36655],{"className":36654},[776,777,778,779],[224,36656,1705],{"className":36657,"style":1740},[468,469,779],[224,36659,787],{"className":36660},[786],[224,36662,36664],{"className":36663},[759],[224,36665,36667],{"className":36666,"style":20409},[763],[224,36668],{},[224,36670,36672],{"className":36671},[468],[224,36673,239],{"className":36674},[2363,2364],[224,36676,16018],{"className":36677,"style":653},[468,469],[224,36679,239],{"className":36680},[542],[224,36682,2061],{"className":36683,"style":2473},[468,469],[224,36685,1716],{"className":36686},[549],[224,36688],{"className":36689,"style":724},[473],[224,36691,36693],{"className":36692},[468],[224,36694,36696],{"className":36695},[2363,34778],[224,36697,36699,36719],{"className":36698},[754,755],[224,36700,36702,36716],{"className":36701},[759],[224,36703,36705],{"className":36704,"style":2285},[763],[224,36706,36707,36710],{"style":34790},[224,36708],{"className":36709,"style":34794},[771],[224,36711,36712],{"style":34797},[2155,36713,36714],{"xmlns":2157,"width":34800,"height":2004,"viewBox":34801},[2163,36715],{"d":34804},[224,36717,787],{"className":36718},[786],[224,36720,36722],{"className":36721},[759],[224,36723,36725],{"className":36724,"style":34814},[763],[224,36726],{},[224,36728],{"className":36729,"style":724},[473],[224,36731,20315],{"className":36732},[468,469],[224,36734,36736],{"className":36735},[468],[224,36737,1716],{"className":36738},[2363,2364],[224,36740,36742,36745],{"style":36741},"top:-2.695em;",[224,36743],{"className":36744,"style":21425},[771],[224,36746],{"className":36747},[468],[224,36749,787],{"className":36750},[786],[224,36752,36754],{"className":36753},[759],[224,36755,36757],{"className":36756,"style":29516},[763],[224,36758],{},[224,36760,36762],{"className":36761},[5399],[224,36763,36765,36979],{"className":36764},[754,755],[224,36766,36768,36976],{"className":36767},[759],[224,36769,36771,36809],{"className":36770,"style":36619},[763],[224,36772,36773,36776],{"style":36622},[224,36774],{"className":36775,"style":21425},[771],[224,36777,36779,36782,36785,36788,36791,36794,36797,36800,36803,36806],{"className":36778},[468],[224,36780],{"className":36781},[468],[224,36783],{"className":36784,"style":496},[473],[224,36786,278],{"className":36787},[500],[224,36789],{"className":36790,"style":496},[473],[224,36792,35657],{"className":36793,"style":35706},[468,469],[224,36795],{"className":36796,"style":724},[473],[224,36798,1705],{"className":36799,"style":1740},[468,469],[224,36801,239],{"className":36802},[542],[224,36804,2061],{"className":36805,"style":2473},[468,469],[224,36807,1716],{"className":36808},[549],[224,36810,36811,36814],{"style":36741},[224,36812],{"className":36813,"style":21425},[771],[224,36815,36817,36820,36823,36826,36829,36832,36835,36838,36841,36844,36847,36850,36853,36856,36859,36862,36865,36970,36973],{"className":36816},[468],[224,36818],{"className":36819},[468],[224,36821],{"className":36822,"style":4416},[473],[224,36824],{"className":36825,"style":474},[473],[224,36827,319],{"className":36828},[478],[224,36830],{"className":36831,"style":474},[473],[224,36833,239],{"className":36834},[542],[224,36836,1438],{"className":36837},[468],[224,36839],{"className":36840,"style":474},[473],[224,36842,5268],{"className":36843},[478],[224,36845],{"className":36846,"style":474},[473],[224,36848,35657],{"className":36849,"style":35706},[468,469],[224,36851,1716],{"className":36852},[549],[224,36854],{"className":36855,"style":724},[473],[224,36857,1705],{"className":36858,"style":1740},[468,469],[224,36860],{"className":36861,"style":21559},[473],[224,36863],{"className":36864,"style":724},[473],[224,36866,36868,36874,36964],{"className":36867},[2091],[224,36869,36871],{"clas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",[224,36991,36993,37011],{"className":36992,"translate":388},[392],[224,36994,36996],{"className":36995},[396],[398,36997,36998],{"xmlns":400},[403,36999,37000,37008],{},[406,37001,37002,37004,37006],{},[409,37003,2061],{},[412,37005,278],{},[409,37007,1858],{},[447,37009,37010],{"encoding":449},"r=s",[224,37012,37014,37032],{"className":37013,"ariaHidden":455},[454],[224,37015,37017,37020,37023,37026,37029],{"className":37016},[459],[224,37018],{"className":37019,"style":839},[463],[224,37021,2061],{"className":37022,"style":2473},[468,469],[224,37024],{"className":37025,"style":496},[473],[224,37027,278],{"className":37028},[500],[224,37030],{"className":37031,"style":496},[473],[224,37033,37035,37038],{"className":37034},[459],[224,37036],{"className":37037,"style":839},[463],[224,37039,1858],{"className":37040},[468,469],[224,37042,37044,37062],{"className":37043,"translate":388},[392],[224,37045,37047],{"className":37046},[396],[398,37048,37049],{"xmlns":400},[403,37050,37051,37059],{},[406,37052,37053,37055,37057],{},[409,37054,2061],{},[412,37056,278],{},[409,37058,4130],{},[447,37060,37061],{"encoding":449},"r=u",[224,37063,37065,37083],{"className":37064,"ariaHidden":455},[454],[224,37066,37068,37071,37074,37077,37080],{"className":37067},[459],[224,37069],{"className":37070,"style":839},[463],[224,37072,2061],{"className":37073,"style":2473},[468,469],[224,37075],{"className":37076,"style":496},[473],[224,37078,278],{"className":37079},[500],[224,37081],{"className":37082,"style":496},[473],[224,37084,37086,37089],{"className":37085},[459],[224,37087],{"className":37088,"style":839},[463],[224,37090,4130],{"className":37091},[468,469],"、以及混合后的 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",[224,37146,37148,37182],{"className":37147,"translate":388},[392],[224,37149,37151],{"className":37150},[396],[398,37152,37153],{"xmlns":400},[403,37154,37155,37179],{},[406,37156,37157,37159,37161,37163,37165,37167,37169,37171,37173,37175,37177],{},[409,37158,31295],{},[409,37160,1858],{},[412,37162,319],{},[412,37164,239],{"stretchy":435},[419,37166,1438],{},[412,37168,5268],{},[409,37170,31295],{},[412,37172,1716],{"stretchy":435},[409,37174,4130],{},[412,37176,278],{},[409,37178,13639],{},[447,37180,37181],{"encoding":449},"\\lambda 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下混合，比值线性）代入第一步的联合凸性：",[224,37294,37296],{"className":37295,"translate":388},[387],[224,37297,37299,37496],{"className":37298,"translate":388},[392],[224,37300,37302],{"className":37301},[396],[398,37303,37304],{"xmlns":400,"display":401},[403,37305,37306,37493],{},[5209,37307,37308,37360,37421],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,37309,37310,37316],{},[5218,37311,37312],{},[5221,37313,37314],{"scriptlevel":284,"displaystyle":455},[406,37315],{},[5218,37317,37318],{},[5221,37319,37320],{"scriptlevel":284,"displaystyle":455},[406,37321,37322,37324,37326,37328,37330,37332,37334,37336,37338,37340,37342,37344,37346,37348,37350,37352,37354,37356,37358],{},[406,37323],{},[409,37325,35657],{},[409,37327,1705],{},[412,37329,239],{"stretchy":435},[409,37331,13639],{},[412,37333,1716],{"stretchy":435},[412,37335,319],{},[412,37337,239],{"stretchy":435},[419,37339,1438],{},[412,37341,5268],{},[409,37343,35657],{},[412,37345,1716],{"stretchy":435},[409,37347,1705],{},[412,37349,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,37351,24189],{},[412,37353,239],{"stretchy":435},[409,37355,13639],{},[412,37357,1716],{"stretchy":435},[412,37359,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[5215,37361,37362,37368],{},[5218,37363,37364],{},[5221,37365,37366],{"scriptlevel":284,"displaystyle":455},[406,37367],{},[5218,37369,37370],{},[5221,37371,37372],{"scriptlevel":284,"displaystyle":455},[406,37373,37374,37376,37378,37380,37383,37385,37387,37389,37391,37393,37395,37397,37399,37401,37403,37405,37407,37409,37411,37413,37415,37417,37419],{},[406,37375],{},[412,37377,2070],{},[409,37379,31295],{},[412,37381,268],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},"1.8em",[409,37384,35657],{},[409,37386,1705],{},[412,37388,239],{"stretchy":435},[409,37390,1858],{},[412,37392,1716],{"stretchy":435},[412,37394,319],{},[412,37396,239],{"stretchy":435},[419,37398,1438],{},[412,37400,5268],{},[409,37402,35657],{},[412,37404,1716],{"stretchy":435},[409,37406,1705],{},[412,37408,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,37410,24189],{},[412,37412,239],{"stretchy":435},[409,37414,1858],{},[412,37416,1716],{"stretchy":435},[412,37418,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[412,37420,440],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[5215,37422,37423,37429],{},[5218,37424,37425],{},[5221,37426,37427],{"scriptlevel":284,"displaystyle":455},[406,37428],{},[5218,37430,37431],{},[5221,37432,37433],{"scriptlevel":284,"displaystyle":455},[406,37434,37435,37437,37439,37441,37443,37445,37447,37449,37451,37453,37455,37457,37459,37461,37463,37465,37467,37469,37471,37473,37475,37477,37479,37481,37483,37485,37487,37489,37491],{},[406,37436],{},[473,37438],{"width":4247},[412,37440,319],{},[412,37442,239],{"stretchy":435},[419,37444,1438],{},[412,37446,5268],{},[409,37448,31295],{},[412,37450,1716],{"stretchy":435},[412,37452,268],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[409,37454,35657],{},[409,37456,1705],{},[412,37458,239],{"stretchy":435},[409,37460,4130],{},[412,37462,1716],{"stretchy":435},[412,37464,319],{},[412,37466,239],{"stretchy":435},[419,37468,1438],{},[412,37470,5268],{},[409,37472,35657],{},[412,37474,1716],{"stretchy":435},[409,37476,1705],{},[412,37478,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,37480,24189],{},[412,37482,239],{"stretchy":435},[409,37484,4130],{},[412,37486,1716],{"stretchy":435},[412,37488,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[412,37490,440],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[412,37492,1877],{"separator":455},[447,37494,37495],{"encoding":449},"\\begin{aligned}\n&\\delta f(w)+(1-\\delta)f\\big(g(w)\\big)\\\\\n&\\le \\lambda\\Big[\\delta f(s)\n+(1-\\delta)f\\big(g(s)\\big)\\Big]\\\\\n&\\qquad+(1-\\lambda)\\Big[\\delta f(u)\n+(1-\\delta)f\\big(g(u)\\big)\\Big],\n\\end{aligned}",[224,37497,37499],{"className":37498,"ariaHidden":455},[454],[224,37500,37502,37506],{"className":37501},[459],[224,37503],{"className":37504,"style":37505},[463],"height:5.41em;vertical-align:-2.455em;",[224,37507,37509],{"className":37508},[468],[224,37510,37512,37565],{"className":37511},[5209],[224,37513,37515],{"className":37514},[5331],[224,37516,37518,37556],{"className":37517},[754,755],[224,37519,37521,37553],{"className":37520},[759],[224,37522,37525,37535,37544],{"className":37523,"style":37524},[763],"height:2.955em;",[224,37526,37528,37532],{"style":37527},"top:-5.255em;",[224,37529],{"className":37530,"style":37531},[771],"height:3.15em;",[224,37533],{"className":37534},[468],[224,37536,37538,37541],{"style":37537},"top:-3.445em;",[224,37539],{"className":37540,"style":37531},[771],[224,37542],{"className":37543},[468],[224,37545,37547,37550],{"style":37546},"top:-1.345em;",[224,37548],{"className":37549,"style":37531},[771],[224,37551],{"className":37552},[468],[224,37554,787],{"className":37555},[786],[224,37557,37559],{"className":37558},[759],[224,37560,37563],{"className":37561,"style":37562},[763],"height:2.455em;",[224,37564],{},[224,37566,37568],{"className":37567},[5399],[224,37569,37571,37901],{"className":37570},[754,755],[224,37572,37574,37898],{"className":37573},[759],[224,37575,37577,37660,37767],{"className":37576,"style":37524},[763],[224,37578,37579,37582],{"style":37527},[224,37580],{"className":37581,"style":37531},[771],[224,37583,37585,37588,37591,37594,37597,37600,37603,37606,37609,37612,37615,37618,37621,37624,37627,37630,37633,37636,37642,37645,37648,37651,37654],{"className":37584},[468],[224,37586],{"className":37587},[468],[224,37589,35657],{"className":37590,"style":35706},[468,469],[224,37592,1705],{"className":37593,"style":1740},[468,469],[224,37595,239],{"className":37596},[542],[224,37598,13639],{"className":37599,"style":13754},[468,469],[224,37601,1716],{"className":37602},[549],[224,37604],{"className":37605,"style":474},[473],[224,37607,319],{"className":37608},[478],[224,37610],{"className":37611,"style":474},[473],[224,37613,239],{"className":37614},[542],[224,37616,1438],{"className":37617},[468],[224,37619],{"className":37620,"style":474},[473],[224,37622,5268],{"className":37623},[478],[224,37625],{"className":37626,"style":474},[473],[224,37628,35657],{"className":37629,"style":35706},[468,469],[224,37631,1716],{"className":37632},[549],[224,37634,1705],{"className":37635,"style":1740},[468,469],[224,37637,37639],{"className":37638},[468],[224,37640,239],{"className":37641},[2363,2364],[224,37643,24189],{"className":37644,"style":492},[468,469],[224,37646,239],{"className":37647},[542],[224,37649,13639],{"className":37650,"style":13754},[468,469],[224,37652,1716],{"className":37653},[549],[224,37655,37657],{"className":37656},[468],[224,37658,1716],{"className":37659},[2363,2364],[224,37661,37662,37665],{"style":37537},[224,37663],{"className":37664,"style":37531},[771],[224,37666,37668,37671,37674,37677,37680,37683,37689,37692,37695,37698,37701,37704,37707,37710,37713,37716,37719,37722,37725,37728,37731,37734,37737,37743,37746,37749,37752,37755,37761],{"className":37667},[468],[224,37669],{"className":37670},[468],[224,37672],{"className":37673,"style":496},[473],[224,37675,2070],{"className":37676},[500],[224,37678],{"className":37679,"style":496},[473],[224,37681,31295],{"className":37682},[468,469],[224,37684,37686],{"className":37685},[468],[224,37687,268],{"className":37688},[2363,23751],[224,37690,35657],{"className":37691,"style":35706},[468,469],[224,37693,1705],{"className":37694,"style":1740},[468,469],[224,37696,239],{"className":37697},[542],[224,37699,1858],{"className":37700},[468,469],[224,37702,1716],{"className":37703},[549],[224,37705],{"className":37706,"style":474},[473],[224,37708,319],{"className":37709},[478],[224,37711],{"className":37712,"style":474},[473],[224,37714,239],{"className":37715},[542],[224,37717,1438],{"className":37718},[468],[224,37720],{"className":37721,"style":474},[473],[224,37723,5268],{"className":37724},[478],[224,37726],{"className":37727,"style":474},[473],[224,37729,35657],{"className":37730,"style":35706},[468,469],[224,37732,1716],{"className":37733},[549],[224,37735,1705],{"className":37736,"style":1740},[468,469],[224,37738,37740],{"className":37739},[468],[224,37741,239],{"className":37742},[2363,2364],[224,37744,24189],{"className":37745,"style":492},[468,469],[224,37747,239],{"className":37748},[542],[224,37750,1858],{"className":37751},[468,469],[224,37753,1716],{"className":37754},[549],[224,37756,37758],{"className":37757},[468],[224,37759,1716],{"className":37760},[2363,2364],[224,37762,37764],{"className":37763},[468],[224,37765,440],{"className":37766},[2363,23751],[224,37768,37769,37772],{"style":37546},[224,37770],{"className":37771,"style":37531},[771],[224,37773,37775,37778,37781,37784,37787,37790,37793,37796,37799,37802,378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224,38059,5268],{"className":38060},[478],[224,38062],{"className":38063,"style":474},[473],[224,38065,2061],{"className":38066,"style":2473},[468,469],[224,38068,35657],{"className":38069,"style":35706},[468,469],[224,38071,787],{"className":38072},[786],[224,38074,38076],{"className":38075},[759],[224,38077,38079],{"className":38078,"style":36958},[763],[224,38080],{},[224,38082],{"className":38083},[549,8479],"。注意 ",[224,38086,38088,38101],{"className":38087,"translate":388},[392],[224,38089,38091],{"className":38090},[396],[398,38092,38093],{"xmlns":400},[403,38094,38095,38099],{},[406,38096,38097],{},[409,38098,24189],{},[447,38100,24189],{"encoding":449},[224,38102,38104],{"className":38103,"ariaHidden":455},[454],[224,38105,38107,38110],{"className":38106},[459],[224,38108],{"className":38109,"style":806},[463],[224,38111,24189],{"className":38112,"style":492},[468,469]," 是仿射函数且 ",[224,38115,38117,38169],{"className":38116,"translate":388},[392],[224,38118,38120],{"className":38119},[396],[398,38121,38122],{"xmlns":400},[403,38123,38124,38166],{},[406,38125,38126,38128,38130,38132,38134,38136,38138,38140,38142,38144,38146,38148,38150,38152,38154,38156,38158,38160,38162,38164],{},[409,38127,24189],{},[412,38129,239],{"stretchy":435},[409,38131,13639],{},[412,38133,1716],{"stretchy":435},[412,38135,278],{},[409,38137,31295],{},[409,38139,24189],{},[412,38141,239],{"stretchy":435},[409,38143,1858],{},[412,38145,1716],{"stretchy":435},[412,38147,319],{},[412,38149,239],{"stretchy":435},[419,38151,1438],{},[412,38153,5268],{},[409,38155,31295],{},[412,38157,1716],{"stretchy":435},[409,38159,24189],{},[412,38161,239],{"stretchy":435},[409,38163,4130],{},[412,38165,1716],{"stretchy":435},[447,38167,38168],{"encoding":449},"g(w)=\\lambda g(s)+(1-\\lambda)g(u)",[224,38170,38172,38199,38229,38250],{"className":38171,"ariaHidden":455},[454],[224,38173,38175,38178,38181,38184,38187,38190,38193,38196],{"className":38174},[459],[224,38176],{"className":38177,"style":535},[463],[224,38179,24189],{"className":38180,"style":492},[468,469],[224,38182,239],{"className":38183},[542],[224,38185,13639],{"className":38186,"style":13754},[468,469],[224,38188,1716],{"className":38189},[549],[224,38191],{"className":38192,"style":496},[473],[224,38194,278],{"className":38195},[500],[224,38197],{"className":38198,"style":496},[473],[224,38200,38202,38205,38208,38211,38214,38217,38220,38223,38226],{"className":38201},[459],[224,38203],{"className":38204,"style":535},[463],[224,38206,31295],{"className":38207},[468,469],[224,38209,24189],{"className":38210,"style":492},[468,469],[224,38212,239],{"className":38213},[542],[224,38215,1858],{"className":38216},[468,469],[224,38218,1716],{"className":38219},[549],[224,38221],{"className":38222,"style":474},[473],[224,38224,319],{"className":38225},[478],[224,38227],{"className":38228,"style":474},[473],[224,38230,38232,38235,38238,38241,38244,38247],{"className":38231},[459],[224,38233],{"className":38234,"style":535},[463],[224,38236,239],{"className":38237},[542],[224,38239,1438],{"className":38240},[468],[224,38242],{"className":38243,"style":474},[473],[224,38245,5268],{"className":38246},[478],[224,38248],{"className":38249,"style":474},[473],[224,38251,38253,38256,38259,38262,38265,38268,38271],{"className":38252},[459],[224,38254],{"className":38255,"style":535},[463],[224,38257,31295],{"className":38258},[468,469],[224,38260,1716],{"className":38261},[549],[224,38263,24189],{"className":38264,"style":492},[468,469],[224,38266,239],{"className":38267},[542],[224,38269,4130],{"className":38270},[468,469],[224,38272,1716],{"className":38273},[549],"，用 ",[224,38276,38278,38291],{"className":38277,"translate":388},[392],[224,38279,38281],{"className":38280},[396],[398,38282,38283],{"xmlns":400},[403,38284,38285,38289],{},[406,38286,38287],{},[409,38288,1705],{},[447,38290,1705],{"encoding":449},[224,38292,38294],{"className":38293,"ariaHidden":455},[454],[224,38295,38297,38300],{"className":38296},[459],[224,38298],{"className":38299,"style":2565},[463],[224,38301,1705],{"className":38302,"style":1740},[468,469],[224,38304,38306,38319],{"className":38305,"translate":388},[392],[224,38307,38309],{"className":38308},[396],[398,38310,38311],{"xmlns":400},[403,38312,38313,38317],{},[406,38314,38315],{},[419,38316,1438],{},[447,38318,1438],{"encoding":449},[224,38320,38322],{"className":38321,"ariaHidden":455},[454],[224,38323,38325,38328],{"className":38324},[459],[224,38326],{"className":38327,"style":510},[463],[224,38329,1438],{"className":38330},[468]," 处的支撑线（凸函数在可微点处 ",[224,38333,38335,38401],{"className":38334,"translate":388},[392],[224,38336,38338],{"className":38337},[396],[398,38339,38340],{"xmlns":400},[403,38341,38342,38398],{},[406,38343,38344,38346,38348,38350,38352,38354,38356,38358,38360,38362,38364,38370,38372,38374,38376,38378,38380,38382,38384,38386,38392,38394,38396],{},[409,38345,1705],{},[412,38347,239],{"stretchy":435},[409,38349,1435],{},[412,38351,1716],{"stretchy":435},[412,38353,9766],{},[409,38355,1705],{},[412,38357,239],{"stretchy":435},[419,38359,1438],{},[412,38361,1716],{"stretchy":435},[412,38363,319],{},[3248,38365,38366,38368],{},[409,38367,1705],{},[412,38369,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,38371,239],{"stretchy":435},[419,38373,1438],{},[412,38375,1716],{"stretchy":435},[412,38377,239],{"stretchy":435},[409,38379,1435],{},[412,38381,5268],{},[419,38383,1438],{},[412,38385,1716],{"stretchy":435},[3248,38387,38388,38390],{},[409,38389,1705],{},[412,38391,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,38393,239],{"stretchy":435},[419,38395,1438],{},[412,38397,1716],{"stretchy":435},[447,38399,38400],{"encoding":449},"f(t)\\ge f(1)+f'(1)(t-1)f'(1)",[224,38402,38404,38431,38458,38520],{"className":38403,"ariaHidden":455},[454],[224,38405,38407,38410,38413,38416,38419,38422,38425,38428],{"className":38406},[459],[224,38408],{"className":38409,"style":535},[463],[224,38411,1705],{"className":38412,"style":1740},[468,469],[224,38414,239],{"className":38415},[542],[224,38417,1435],{"className":38418},[468,469],[224,38420,1716],{"className":38421},[549],[224,38423],{"className":38424,"style":496},[473],[224,38426,9766],{"className":38427},[500],[224,38429],{"className":38430,"style":496},[473],[224,38432,38434,38437,38440,38443,38446,38449,38452,38455],{"className":38433},[459],[224,38435],{"className":38436,"style":535},[463],[224,38438,1705],{"className":38439,"style":1740},[468,469],[224,38441,239],{"className":38442},[542],[224,38444,1438],{"className":38445},[468],[224,38447,1716],{"className":38448},[549],[224,38450],{"className":38451,"style":474},[473],[224,38453,319],{"className":38454},[478],[224,38456],{"className":38457,"style":474},[473],[224,38459,38461,38464,38496,38499,38502,38505,38508,38511,38514,38517],{"className":38460},[459],[224,38462],{"className":38463,"style":6966},[463],[224,38465,38467,38470],{"className":38466},[468],[224,38468,1705],{"className":38469,"style":1740},[468,469],[224,38471,38473],{"className":38472},[750],[224,38474,38476],{"className":38475},[754],[224,38477,38479],{"className":38478},[759],[224,38480,38482],{"className":38481,"style":6992},[763],[224,38483,38484,38487],{"style":3288},[224,38485],{"className":38486,"style":772},[771],[224,38488,38490],{"className":38489},[776,777,778,779],[224,38491,38493],{"className":38492},[468,779],[224,38494,6933],{"className":38495},[468,779],[224,38497,239],{"className":38498},[542],[224,38500,1438],{"className":38501},[468],[224,38503,1716],{"className":38504},[549],[224,38506,239],{"className":38507},[542],[224,38509,1435],{"className":38510},[468,469],[224,38512],{"className":38513,"style":474},[473],[224,38515,5268],{"className":38516},[478],[224,38518],{"className":38519,"style":474},[473],[224,38521,38523,38526,38529,38532,38564,38567,38570],{"className":38522},[459],[224,38524],{"className":38525,"style":6966},[463],[224,38527,1438],{"className":38528},[468],[224,38530,1716],{"className":38531},[549],[224,38533,38535,38538],{"className":38534},[468],[224,38536,1705],{"className":38537,"style":1740},[468,469],[224,38539,38541],{"className":38540},[750],[224,38542,38544],{"className":38543},[754],[224,38545,38547],{"className":38546},[759],[224,38548,38550],{"className":38549,"style":6992},[763],[224,38551,38552,38555],{"style":3288},[224,38553],{"className":38554,"style":772},[771],[224,38556,38558],{"className":38557},[776,777,778,779],[224,38559,38561],{"className":38560},[468,779],[224,38562,6933],{"className":38563},[468,779],[224,38565,239],{"className":38566},[542],[224,38568,1438],{"className":38569},[468],[224,38571,1716],{"className":38572},[549],"，此处只需次梯度 ",[224,38575,38577,38601],{"className":38576,"translate":388},[392],[224,38578,38580],{"className":38579},[396],[398,38581,38582],{"xmlns":400},[403,38583,38584,38598],{},[406,38585,38586,38592,38594,38596],{},[3248,38587,38588,38590],{},[409,38589,1705],{},[412,38591,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,38593,239],{"stretchy":435},[419,38595,1438],{},[412,38597,1716],{"stretchy":435},[447,38599,38600],{"encoding":449},"f'(1)",[224,38602,38604],{"className":38603,"ariaHidden":455},[454],[224,38605,38607,38610,38642,38645,38648],{"className":38606},[459],[224,38608],{"className":38609,"style":6966},[463],[224,38611,38613,38616],{"className":38612},[468],[224,38614,1705],{"className":38615,"style":1740},[468,469],[224,38617,38619],{"className":38618},[750],[224,38620,38622],{"className":38621},[754],[224,38623,38625],{"className":38624},[759],[224,38626,38628],{"className":38627,"style":6992},[763],[224,38629,38630,38633],{"style":3288},[224,38631],{"className":38632,"style":772},[771],[224,38634,38636],{"className":38635},[776,777,778,779],[224,38637,38639],{"className":38638},[468,779],[224,38640,6933],{"className":38641},[468,779],[224,38643,239],{"className":38644},[542],[224,38646,1438],{"className":38647},[468],[224,38649,1716],{"className":38650},[549],"，即 ",[224,38653,38655,38703],{"className":38654,"translate":388},[392],[224,38656,38658],{"className":38657},[396],[398,38659,38660],{"xmlns":400},[403,38661,38662,38700],{},[406,38663,38664,38666,38668,38670,38672,38674,38676,38678,38680,38682,38684,38690,38692,38694,38696,38698],{},[409,38665,1705],{},[412,38667,239],{"stretchy":435},[409,38669,1435],{},[412,38671,1716],{"stretchy":435},[412,38673,9766],{},[409,38675,1705],{},[412,38677,239],{"stretchy":435},[419,38679,1438],{},[412,38681,1716],{"stretchy":435},[412,38683,319],{},[695,38685,38686,38688],{},[409,38687,24189],{},[419,38689,1438],{},[412,38691,239],{"stretchy":435},[409,38693,1435],{},[412,38695,5268],{},[419,38697,1438],{},[412,38699,1716],{"stretchy":435},[447,38701,38702],{"encoding":449},"f(t)\\ge f(1)+g_1(t-1)",[224,38704,38706,38733,38760,38821],{"className":38705,"ariaHidden":455},[454],[224,38707,38709,38712,38715,38718,38721,38724,38727,38730],{"className":38708},[459],[224,38710],{"className":38711,"style":535},[463],[224,38713,1705],{"className":38714,"style":1740},[468,469],[224,38716,239],{"className":38717},[542],[224,38719,1435],{"className":38720},[468,469],[224,38722,1716],{"className":38723},[549],[224,38725],{"className":38726,"style":496},[473],[224,38728,9766],{"className":38729},[500],[224,38731],{"className":38732,"style":496},[473],[224,38734,38736,38739,38742,38745,38748,38751,38754,38757],{"className":38735},[459],[224,38737],{"className":38738,"style":535},[463],[224,38740,1705],{"className":38741,"style":1740},[468,469],[224,38743,239],{"className":38744},[542],[224,38746,1438],{"className":38747},[468],[224,38749,1716],{"className":38750},[549],[224,38752],{"className":38753,"style":474},[473],[224,38755,319],{"className":38756},[478],[224,38758],{"className":38759,"style":474},[473],[224,38761,38763,38766,38806,38809,38812,38815,38818],{"className":38762},[459],[224,38764],{"className":38765,"style":535},[463],[224,38767,38769,38772],{"className":38768},[468],[224,38770,24189],{"className":38771,"style":492},[468,469],[224,38773,38775],{"className":38774},[750],[224,38776,38778,38798],{"className":38777},[754,755],[224,38779,38781,38795],{"className":38780},[759],[224,38782,38784],{"className":38783,"style":1479},[763],[224,38785,38786,38789],{"style":3854},[224,38787],{"className":38788,"style":772},[771],[224,38790,38792],{"className":38791},[776,777,778,779],[224,38793,1438],{"className":38794},[468,779],[224,38796,787],{"className":38797},[786],[224,38799,38801],{"className":38800},[759],[224,38802,38804],{"className":38803,"style":794},[763],[224,38805],{},[224,38807,239],{"className":38808},[542],[224,38810,1435],{"className":38811},[468,469],[224,38813],{"className":38814,"style":474},[473],[224,38816,5268],{"className":38817},[478],[224,38819],{"className":38820,"style":474},[473],[224,38822,38824,38827,38830],{"className":38823},[459],[224,38825],{"className":38826,"style":535},[463],[224,38828,1438],{"className":38829},[468],[224,38831,1716],{"className":38832},[549]," 对任一次梯度 ",[224,38835,38837,38855],{"className":38836,"translate":388},[392],[224,38838,38840],{"className":38839},[396],[398,38841,38842],{"xmlns":400},[403,38843,38844,38852],{},[406,38845,38846],{},[695,38847,38848,38850],{},[409,38849,24189],{},[419,38851,1438],{},[447,38853,38854],{"encoding":449},"g_1",[224,38856,38858],{"className":38857,"ariaHidden":455},[454],[224,38859,38861,38864],{"className":38860},[459],[224,38862],{"className":38863,"style":806},[463],[224,38865,38867,38870],{"className":38866},[468],[224,38868,24189],{"className":38869,"style":492},[468,469],[224,38871,38873],{"className":38872},[750],[224,38874,38876,38896],{"className":38875},[754,755],[224,38877,38879,38893],{"className":38878},[759],[224,38880,38882],{"className":38881,"style":1479},[763],[224,38883,38884,38887],{"style":3854},[224,38885],{"className":38886,"style":772},[771],[224,38888,38890],{"className":38889},[776,777,778,779],[224,38891,1438],{"className":38892},[468,779],[224,38894,787],{"className":38895},[786],[224,38897,38899],{"className":38898},[759],[224,38900,38902],{"className":38901,"style":794},[763],[224,38903],{},"）处理非主项：对凸的 ",[224,38906,38908,38921],{"className":38907,"translate":388},[392],[224,38909,38911],{"className":38910},[396],[398,38912,38913],{"xmlns":400},[403,38914,38915,38919],{},[406,38916,38917],{},[409,38918,1705],{},[447,38920,1705],{"encoding":449},[224,38922,38924],{"className":38923,"ariaHidden":455},[454],[224,38925,38927,38930],{"className":38926},[459],[224,38928],{"className":38929,"style":2565},[463],[224,38931,1705],{"className":38932,"style":1740},[468,469],[224,38934,38936,38962],{"className":38935,"translate":388},[392],[224,38937,38939],{"className":38938},[396],[398,38940,38941],{"xmlns":400},[403,38942,38943,38959],{},[406,38944,38945,38947,38949,38951,38953,38955,38957],{},[409,38946,1705],{},[412,38948,239],{"stretchy":435},[409,38950,24189],{},[412,38952,239],{"stretchy":435},[409,38954,2061],{},[412,38956,1716],{"stretchy":435},[412,38958,1716],{"stretchy":435},[447,38960,38961],{"encoding":449},"f(g(r))",[224,38963,38965],{"className":38964,"ariaHidden":455},[454],[224,38966,38968,38971,38974,38977,38980,38983,38986],{"className":38967},[459],[224,38969],{"className":38970,"style":535},[463],[224,38972,1705],{"className":38973,"style":1740},[468,469],[224,38975,239],{"className":38976},[542],[224,38978,24189],{"className":38979,"style":492},[468,469],[224,38981,239],{"className":38982},[542],[224,38984,2061],{"className":38985,"style":2473},[468,469],[224,38987,38989],{"className":38988},[549],"))"," 中关于 ",[224,38992,38994,39007],{"className":38993,"translate":388},[392],[224,38995,38997],{"className":38996},[396],[398,38998,38999],{"xmlns":400},[403,39000,39001,39005],{},[406,39002,39003],{},[409,39004,2061],{},[447,39006,2061],{"encoding":449},[224,39008,39010],{"className":39009,"ariaHidden":455},[454],[224,39011,39013,39016],{"className":39012},[459],[224,39014],{"className":39015,"style":839},[463],[224,39017,2061],{"className":39018,"style":2473},[468,469]," 的\"非主项\"部分为",[224,39021,39023],{"className":39022,"translate":388},[387],[224,39024,39026,39231],{"className":39025,"translate":388},[392],[224,39027,39029],{"className":39028},[396],[398,39030,39031],{"xmlns":400,"display":401},[403,39032,39033,39228],{},[5209,39034,39035,39099,39174],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,39036,39037,39057],{},[5218,39038,39039],{},[5221,39040,39041],{"scriptlevel":284,"displaystyle":455},[406,39042,39043,39045,39047,39049,39051,39053,39055],{},[409,39044,1705],{},[412,39046,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,39048,24189],{},[412,39050,239],{"stretchy":435},[409,39052,2061],{},[412,39054,1716],{"stretchy":435},[412,39056,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[5218,39058,39059],{},[5221,39060,39061],{"scriptlevel":284,"displaystyle":455},[406,39062,39063,39065,39067,39069,39071,39073,39075,39077,39097],{},[406,39064],{},[412,39066,278],{},[409,39068,1705],{},[412,39070,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[419,39072,1438],{},[412,39074,319],{},[409,39076,35657],{},[5221,39078,39079],{"scriptlevel":284,"displaystyle":435},[20338,39080,39081,39089],{},[406,39082,39083,39085,39087],{},[419,39084,1438],{},[412,39086,5268],{},[409,39088,2061],{},[406,39090,39091,39093,39095],{},[419,39092,1438],{},[412,39094,5268],{},[409,39096,35657],{},[412,39098,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[5215,39100,39101,39107],{},[5218,39102,39103],{},[5221,39104,39105],{"scriptlevel":284,"displaystyle":455},[406,39106],{},[5218,39108,39109],{},[5221,39110,39111],{"scriptlevel":284,"displaystyle":455},[406,39112,39113,39115,39117,39119,39121,39123,39125,39127,39133,39135,39137,39139,39159,39161,39164,39166,39172],{},[406,39114],{},[412,39116,278],{},[409,39118,1705],{},[412,39120,239],{"stretchy":435},[419,39122,1438],{},[412,39124,1716],{"stretchy":435},[412,39126,319],{},[695,39128,39129,39131],{},[409,39130,24189],{},[419,39132,1438],{},[412,39134,23331],{},[409,39136,35657],{},[423,39138,682],{},[5221,39140,39141],{"scriptlevel":284,"displaystyle":435},[20338,39142,39143,39151],{},[406,39144,39145,39147,39149],{},[419,39146,1438],{},[412,39148,5268],{},[409,39150,2061],{},[406,39152,39153,39155,39157],{},[419,39154,1438],{},[412,39156,5268],{},[409,39158,35657],{},[412,39160,319],{},[409,39162,39163],{},"O",[412,39165,239],{"stretchy":435},[3248,39167,39168,39170],{},[409,39169,35657],{},[419,39171,1447],{},[412,39173,1716],{"stretchy":435},[5215,39175,39176,39182],{},[5218,39177,39178],{},[5221,39179,39180],{"scriptlevel":284,"displaystyle":455},[406,39181],{},[5218,39183,39184],{},[5221,39185,39186],{"scriptlevel":284,"displaystyle":455},[406,39187,39188,39190,39192,39198,39200,39202,39204,39206,39208,39210,39212,39214,39216,39218,39224,39226],{},[406,39189],{},[412,39191,278],{},[695,39193,39194,39196],{},[409,39195,24189],{},[419,39197,1438],{},[423,39199,682],{},[409,39201,35657],{},[412,39203,239],{"stretchy":435},[419,39205,1438],{},[412,39207,5268],{},[409,39209,2061],{},[412,39211,1716],{"stretchy":435},[412,39213,319],{},[409,39215,39163],{},[412,39217,239],{"stretchy":435},[3248,39219,39220,39222],{},[409,39221,35657],{},[419,39223,1447],{},[412,39225,1716],{"stretchy":435},[412,39227,1877],{"separator":455},[447,39229,39230],{"encoding":449},"\\begin{aligned}\nf\\big(g(r)\\big)\n&=f\\big(1+\\delta\\tfrac{1-r}{1-\\delta}\\big)\\\\\n&= f(1)+g_1\\cdot\\delta\\,\\tfrac{1-r}{1-\\delta}\n+O(\\delta^2)\\\\\n&= 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项在 ",[224,39984,39986,39999],{"className":39985,"translate":388},[392],[224,39987,39989],{"className":39988},[396],[398,39990,39991],{"xmlns":400},[403,39992,39993,39997],{},[406,39994,39995],{},[409,39996,2061],{},[447,39998,2061],{"encoding":449},[224,40000,40002],{"className":40001,"ariaHidden":455},[454],[224,40003,40005,40008],{"className":40004},[459],[224,40006],{"className":40007,"style":839},[463],[224,40009,2061],{"className":40010,"style":2473},[468,469]," 变化时一致有界，因为 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连续）。代回不等式：两边的主项（",[224,40145,40147,40169],{"className":40146,"translate":388},[392],[224,40148,40150],{"className":40149},[396],[398,40151,40152],{"xmlns":400},[403,40153,40154,40166],{},[406,40155,40156,40158,40160,40162,40164],{},[409,40157,35657],{},[409,40159,1705],{},[412,40161,239],{"stretchy":435},[412,40163,23331],{},[412,40165,1716],{"stretchy":435},[447,40167,40168],{"encoding":449},"\\delta f(\\cdot)",[224,40170,40172],{"className":40171,"ariaHidden":455},[454],[224,40173,40175,40178,40181,40184,40187,40190],{"className":40174},[459],[224,40176],{"className":40177,"style":535},[463],[224,40179,35657],{"className":40180,"style":35706},[468,469],[224,40182,1705],{"className":40183,"style":1740},[468,469],[224,40185,239],{"className":40186},[542],[224,40188,23331],{"className":40189},[468],[224,40191,1716],{"className":40192},[549]," 部分）正是 ",[224,40195,40197,40245],{"className":40196,"translate":388},[392],[224,40198,40200],{"className":40199},[396],[398,40201,40202],{"xmlns":400},[403,40203,40204,40242],{},[406,40205,40206,40208,40210,40212,40214,40216,40218,40220,40222,40224,40226,40228,40230,40232,40234,40236,40238,40240],{},[409,40207,35657],{},[412,40209,268],{"stretchy":435},[409,40211,31295],{},[409,40213,1705],{},[412,40215,239],{"stretchy":435},[409,40217,1858],{},[412,40219,1716],{"stretchy":435},[412,40221,319],{},[412,40223,239],{"stretchy":435},[419,40225,1438],{},[412,40227,5268],{},[409,40229,31295],{},[412,40231,1716],{"stretchy":435},[409,40233,1705],{},[412,40235,239],{"stretchy":435},[409,40237,4130],{},[412,40239,1716],{"stretchy":435},[412,40241,440],{"stretchy":435},[447,40243,40244],{"encoding":449},"\\delta[\\lambda f(s)+(1-\\lambda)f(u)]",[224,40246,40248,40284,40305],{"className":40247,"ariaHidden":455},[454],[224,40249,40251,40254,40257,40260,40263,40266,40269,40272,40275,40278,40281],{"className":40250},[459],[224,40252],{"className":40253,"style":535},[463],[224,40255,35657],{"className":40256,"style":35706},[468,469],[224,40258,268],{"className":40259},[542],[224,40261,31295],{"className":40262},[468,469],[224,40264,1705],{"className":40265,"style":1740},[468,469],[224,40267,239],{"className":40268},[542],[224,40270,1858],{"className":40271},[468,469],[224,40273,1716],{"className":40274},[549],[224,40276],{"className":40277,"style":474},[473],[224,40279,319],{"className":40280},[478],[224,40282],{"className":40283,"style":474},[473],[224,40285,40287,40290,40293,40296,40299,40302],{"className":40286},[459],[224,40288],{"className":40289,"style":535},[463],[224,40291,239],{"className":40292},[542],[224,40294,1438],{"className":40295},[468],[224,40297],{"className":40298,"style":474},[473],[224,40300,5268],{"className":40301},[478],[224,40303],{"className":40304,"style":474},[473],[224,40306,40308,40311,40314,40317,40320,40323,40326],{"className":40307},[459],[224,40309],{"className":40310,"style":535},[463],[224,40312,31295],{"className":40313},[468,469],[224,40315,1716],{"className":40316},[549],[224,40318,1705],{"className":40319,"style":1740},[468,469],[224,40321,239],{"className":40322},[542],[224,40324,4130],{"className":40325},[468,469],[224,40327,40329],{"className":40328},[549],")]",[224,40331,40333,40355],{"className":40332,"translate":388},[392],[224,40334,40336],{"className":40335},[396],[398,40337,40338],{"xmlns":400},[403,40339,40340,40352],{},[406,40341,40342,40344,40346,40348,40350],{},[409,40343,35657],{},[409,40345,1705],{},[412,40347,239],{"stretchy":435},[409,40349,13639],{},[412,40351,1716],{"stretchy":435},[447,40353,40354],{"encoding":449},"\\delta f(w)",[224,40356,40358],{"className":40357,"ariaHidden":455},[454],[224,40359,40361,40364,40367,40370,40373,40376],{"className":40360},[459],[224,40362],{"className":40363,"style":535},[463],[224,40365,35657],{"className":40366,"style":35706},[468,469],[224,40368,1705],{"className":40369,"style":1740},[468,469],[224,40371,239],{"className":40372},[542],[224,40374,13639],{"className":40375,"style":13754},[468,469],[224,40377,1716],{"className":40378},[549],"；非主项部分",[224,40381,40383],{"className":40382,"translate":388},[387],[224,40384,40386,40472],{"className":40385,"translate":388},[392],[224,40387,40389],{"className":40388},[396],[398,40390,40391],{"xmlns":400,"display":401},[403,40392,40393,40469],{},[406,40394,40395,40397,40399,40401,40403,40405,40407,40413,40415,40417,40419,40421,40423,40425,40427,40429,40431,40433,40435,40437,40439,40441,40443,40445,40447,40449,40451,40453,40455,40457,40459,40461,40463,40465,40467],{},[412,40396,239],{"stretchy":435},[419,40398,1438],{},[412,40400,5268],{},[409,40402,35657],{},[412,40404,1716],{"stretchy":435},[423,40406,682],{},[695,40408,40409,40411],{},[409,40410,24189],{},[419,40412,1438],{},[409,40414,35657],{},[412,40416,268],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,40418,31295],{},[412,40420,239],{"stretchy":435},[419,40422,1438],{},[412,40424,5268],{},[409,40426,1858],{},[412,40428,1716],{"stretchy":435},[412,40430,319],{},[412,40432,239],{"stretchy":435},[419,40434,1438],{},[412,40436,5268],{},[409,40438,31295],{},[412,40440,1716],{"stretchy":435},[412,40442,239],{"stretchy":435},[419,40444,1438],{},[412,40446,5268],{},[409,40448,4130],{},[412,40450,1716],{"stretchy":435},[412,40452,5268],{},[412,40454,239],{"stretchy":435},[419,40456,1438],{},[412,40458,5268],{},[409,40460,13639],{},[412,40462,1716],{"stretchy":435},[412,40464,440],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[412,40466,278],{},[419,40468,284],{},[447,40470,40471],{"encoding":449},"(1-\\delta)\\,g_1\\delta\\big[\\lambda(1-s)\n+(1-\\lambda)(1-u)-(1-w)\\big]=0",[224,40473,40475,40496,40578,40599,40620,40647,40668,40689,40716],{"className":40474,"ariaHidden":455},[454],[224,40476,40478,40481,40484,40487,40490,40493],{"className":40477},[459],[224,40479],{"className":40480,"style":535},[463],[224,40482,239],{"className":40483},[542],[224,40485,1438],{"className":40486},[468],[224,40488],{"className":40489,"style":474},[473],[224,40491,5268],{"className":40492},[478],[224,40494],{"className":40495,"style":474},[473],[224,40497,40499,40502,40505,40508,40511,40551,40554,40560,40563,40566,40569,40572,40575],{"className":40498},[459],[224,40500],{"className":40501,"style":2663},[463],[224,40503,35657],{"className":40504,"style":35706},[468,469],[224,40506,1716],{"className":40507},[549],[224,40509],{"className":40510,"style":724},[473],[224,40512,40514,40517],{"className":40513},[468],[224,40515,24189],{"className":40516,"style":492},[468,469],[224,40518,40520],{"className":40519},[750],[224,40521,40523,40543],{"className":40522},[754,755],[224,40524,40526,40540],{"className":40525},[759],[224,40527,40529],{"className":40528,"style":1479},[763],[224,40530,40531,40534],{"style":3854},[224,40532],{"className":40533,"style":772},[771],[224,40535,40537],{"className":40536},[776,777,778,779],[224,40538,1438],{"className":40539},[468,779],[224,40541,787],{"className":40542},[786],[224,40544,40546],{"className":40545},[759],[224,40547,40549],{"className":40548,"style":794},[763],[224,40550],{},[224,40552,35657],{"className":40553,"style":35706},[468,469],[224,40555,40557],{"className":40556},[468],[224,40558,268],{"className":40559},[2363,2364],[224,40561,31295],{"className":40562},[468,469],[224,40564,239],{"className":40565},[542],[224,40567,1438],{"className":40568},[468],[224,40570],{"className":40571,"style":474},[473],[224,40573,5268],{"className":40574},[478],[224,40576],{"className":40577,"style":474},[473],[224,40579,40581,40584,40587,40590,40593,40596],{"className":40580},[459],[224,40582],{"className":40583,"style":535},[463],[224,40585,1858],{"className":40586},[468,469],[224,40588,1716],{"className":40589},[549],[224,40591],{"className":40592,"style":474},[473],[224,40594,319],{"className":40595},[478],[224,40597],{"className":40598,"style":474},[473],[224,40600,40602,40605,40608,40611,40614,40617],{"className":40601},[459],[224,40603],{"className":40604,"style":535},[463],[224,40606,239],{"className":40607},[542],[224,40609,1438],{"className":40610},[468],[224,40612],{"className":40613,"style":474},[473],[224,40615,5268],{"className":40616},[478],[224,40618],{"className":40619,"style":474},[473],[224,40621,40623,40626,40629,40632,40635,40638,40641,40644],{"className":40622},[459],[224,40624],{"className":40625,"style":535},[463],[224,40627,31295],{"className":40628},[468,469],[224,40630,1716],{"className":40631},[549],[224,40633,239],{"className":40634},[542],[224,40636,1438],{"className":40637},[468],[224,40639],{"className":40640,"style":474},[473],[224,40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0，因为 ",[224,40729,40731,40764],{"className":40730,"translate":388},[392],[224,40732,40734],{"className":40733},[396],[398,40735,40736],{"xmlns":400},[403,40737,40738,40762],{},[406,40739,40740,40742,40744,40746,40748,40750,40752,40754,40756,40758,40760],{},[409,40741,13639],{},[412,40743,278],{},[409,40745,31295],{},[409,40747,1858],{},[412,40749,319],{},[412,40751,239],{"stretchy":435},[419,40753,1438],{},[412,40755,5268],{},[409,40757,31295],{},[412,40759,1716],{"stretchy":435},[409,40761,4130],{},[447,40763,35504],{"encoding":449},[224,40765,40767,40785,40806,40827],{"className":40766,"ariaHidden":455},[454],[224,40768,40770,40773,40776,40779,40782],{"className":40769},[459],[224,40771],{"className":40772,"style":839},[463],[224,40774,13639],{"className":40775,"style":13754},[468,469],[224,40777],{"className":40778,"style":496},[473],[224,40780,278],{"className":40781},[500],[224,40783],{"className":40784,"style":496},[473],[224,40786,40788,40791,40794,40797,40800,40803],{"className":40787},[459],[224,40789],{"className":40790,"style":4348},[463],[224,40792,31295],{"className":40793},[468,469],[224,40795,1858],{"className":40796},[468,469],[224,40798],{"className":40799,"style":474},[473],[224,40801,319],{"className":40802},[478],[224,40804],{"className":40805,"style":474},[473],[224,40807,40809,40812,40815,40818,40821,40824],{"className":40808},[459],[224,40810],{"className":40811,"style":535},[463],[224,40813,239],{"className":40814},[542],[224,40816,1438],{"className":40817},[468],[224,40819],{"className":40820,"style":474},[473],[224,40822,5268],{"className":40823},[478],[224,40825],{"className":40826,"style":474},[473],[224,40828,40830,40833,40836,40839],{"className":40829},[459],[224,40831],{"className":40832,"style":535},[463],[224,40834,31295],{"className":40835},[468,469],[224,40837,1716],{"className":40838},[549],[224,40840,4130],{"className":40841},[468,469],"）。于是不等式两边同除以 ",[224,40844,40846,40864],{"className":40845,"translate":388},[392],[224,40847,40849],{"className":40848},[396],[398,40850,40851],{"xmlns":400},[403,40852,40853,40861],{},[406,40854,40855,40857,40859],{},[409,40856,35657],{},[412,40858,325],{},[419,40860,284],{},[447,40862,40863],{"encoding":449},"\\delta>0",[224,40865,40867,40885],{"className":40866,"ariaHidden":455},[454],[224,40868,40870,40873,40876,40879,40882],{"className":40869},[459],[224,40871],{"className":40872,"style":16682},[463],[224,40874,35657],{"className":40875,"style":35706},[468,469],[224,40877],{"className":40878,"style":496},[473],[224,40880,325],{"className":40881},[500],[224,40883],{"className":40884,"style":496},[473],[224,40886,40888,40891],{"className":40887},[459],[224,40889],{"className":40890,"style":510},[463],[224,40892,284],{"className":40893},[468],"，令 ",[224,40896,40898,40920],{"className":40897,"translate":388},[392],[224,40899,40901],{"className":40900},[396],[398,40902,40903],{"xmlns":400},[403,40904,40905,40917],{},[406,40906,40907,40909,40911],{},[409,40908,35657],{},[412,40910,1144],{},[3248,40912,40913,40915],{},[419,40914,284],{},[412,40916,319],{},[447,40918,40919],{"encoding":449},"\\delta\\to0^+",[224,40921,40923,40941],{"className":40922,"ariaHidden":455},[454],[224,40924,40926,40929,40932,40935,40938],{"className":40925},[459],[224,40927],{"className":40928,"style":7642},[463],[224,40930,35657],{"className":40931,"style":35706},[468,469],[224,40933],{"className":40934,"style":496},[473],[224,40936,1144],{"className":40937},[500],[224,40939],{"className":40940,"style":496},[473],[224,40942,40944,40948],{"className":40943},[459],[224,40945],{"className":40946,"style":40947},[463],"height:0.7713em;",[224,40949,40951,40954],{"className":40950},[468],[224,40952,284],{"className":40953},[468],[224,40955,40957],{"className":40956},[750],[224,40958,40960],{"className":40959},[754],[224,40961,40963],{"className":40962},[759],[224,40964,40966],{"className":40965,"style":40947},[763],[224,40967,40968,40971],{"style":3288},[224,40969],{"className":40970,"style":772},[771],[224,40972,40974],{"className":40973},[776,777,778,779],[224,40975,319],{"className":40976},[478,779],"，得",[224,40979,40981],{"className":40980,"translate":388},[387],[224,40982,40984,41054],{"className":40983,"translate":388},[392],[224,40985,40987],{"className":40986},[396],[398,40988,40989],{"xmlns":400,"display":401},[403,40990,40991,41051],{},[406,40992,40993,40995,40997,40999,41001,41003,41005,41007,41009,41011,41013,41015,41017,41019,41021,41023,41025,41027,41029,41031,41033,41035,41037,41039,41041,41043,41045,41047,41049],{},[409,40994,1705],{},[412,40996,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,40998,31295],{},[409,41000,1858],{},[412,41002,319],{},[412,41004,239],{"stretchy":435},[419,41006,1438],{},[412,41008,5268],{},[409,41010,31295],{},[412,41012,1716],{"stretchy":435},[409,41014,4130],{},[412,41016,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[412,41018,2070],{},[409,41020,31295],{},[409,41022,1705],{},[412,41024,239],{"stretchy":435},[409,41026,1858],{},[412,41028,1716],{"stretchy":435},[412,41030,319],{},[412,41032,239],{"stretchy":435},[419,41034,1438],{},[412,41036,5268],{},[409,41038,31295],{},[412,41040,1716],{"stretchy":435},[409,41042,1705],{},[412,41044,239],{"stretchy":435},[409,41046,4130],{},[412,41048,1716],{"stretchy":435},[412,41050,1877],{"separator":455},[447,41052,41053],{"encoding":449},"f\\big(\\lambda s+(1-\\lambda)u\\big)\n\\le\\lambda f(s)+(1-\\lambda)f(u),",[224,41055,41057,41087,41108,41138,41168,41189],{"className":41056,"ariaHidden":455},[454],[224,41058,41060,41063,41066,41072,41075,41078,41081,41084],{"className":41059},[459],[224,41061],{"className":41062,"style":2663},[463],[224,41064,1705],{"className":41065,"style":1740},[468,469],[224,41067,41069],{"className":41068},[468],[224,41070,239],{"className":41071},[2363,2364],[224,41073,31295],{"className":41074},[468,469],[224,41076,1858],{"className":41077},[468,469],[224,41079],{"className":41080,"style":474},[473],[224,41082,319],{"className":41083},[478],[224,41085],{"className":41086,"style":474},[473],[224,41088,41090,41093,41096,41099,41102,41105],{"className":41089},[459],[224,41091],{"className":41092,"style":535},[463],[224,41094,239],{"className":41095},[542],[224,41097,1438],{"className":41098},[468],[224,41100],{"className":41101,"style":474},[473],[224,41103,5268],{"className":41104},[478],[224,41106],{"className":41107,"style":474},[473],[224,41109,41111,41114,41117,41120,41123,41129,41132,41135],{"className":41110},[459],[224,41112],{"className":41113,"style":2663},[463],[224,41115,31295],{"className":41116},[468,469],[224,41118,1716],{"className":41119},[549],[224,41121,4130],{"className":41122},[468,469],[224,41124,41126],{"className":41125},[468],[224,41127,1716],{"className":41128},[2363,2364],[224,41130],{"className":41131,"style":496},[473],[224,41133,2070],{"className":41134},[500],[224,41136],{"className":41137,"style":496},[473],[224,41139,41141,41144,41147,41150,41153,41156,41159,41162,41165],{"className":41140},[459],[224,41142],{"className":41143,"style":535},[463],[224,41145,31295],{"className":41146},[468,469],[224,41148,1705],{"className":41149,"style":1740},[468,469],[224,41151,239],{"className":41152},[542],[224,41154,1858],{"className":41155},[468,469],[224,41157,1716],{"className":41158},[549],[224,41160],{"className":41161,"style":474},[473],[224,41163,319],{"className":41164},[478],[224,41166],{"className":41167,"style":474},[473],[224,41169,41171,41174,41177,41180,41183,41186],{"className":41170},[459],[224,41172],{"className":41173,"style":535},[463],[224,41175,239],{"className":41176},[542],[224,41178,1438],{"className":41179},[468],[224,41181],{"className":41182,"style":474},[473],[224,41184,5268],{"className":41185},[478],[224,41187],{"className":41188,"style":474},[473],[224,41190,41192,41195,41198,41201,41204,41207,41210,41213],{"className":41191},[459],[224,41193],{"className":41194,"style":535},[463],[224,41196,31295],{"className":41197},[468,469],[224,41199,1716],{"className":41200},[549],[224,41202,1705],{"className":41203,"style":1740},[468,469],[224,41205,239],{"className":41206},[542],[224,41208,4130],{"className":41209},[468,469],[224,41211,1716],{"className":41212},[549],[224,41214,1877],{"className":41215},[1917],[11,41217,41218,41219,30923],{},"即 ",[224,41220,41222,41235],{"className":41221,"translate":388},[392],[224,41223,41225],{"className":41224},[396],[398,41226,41227],{"xmlns":400},[403,41228,41229,41233],{},[406,41230,41231],{},[409,41232,1705],{},[447,41234,1705],{"encoding":449},[224,41236,41238],{"className":41237,"ariaHidden":455},[454],[224,41239,41241,41244],{"className":41240},[459],[224,41242],{"className":41243,"style":2565},[463],[224,41245,1705],{"className":41246,"style":1740},[468,469],[11,41248,41249,41250,41253,41254,41282,41283,41313,41314,41342],{},"所以\"凸\"与\"数据处理\"是同一件事的两面。这也是 f-散度族成为\"一切满足数据处理不等式的最一般散度\"的原因：它不是一堆例子的偶然汇总，而是一个有刻画定理支撑的完整族。换句话说，Csiszár–Ali–Silvey 定理给出一个完整的",[378,41251,41252],{},"刻画","：凸函数 ",[224,41255,41257,41270],{"className":41256,"translate":388},[392],[224,41258,41260],{"className":41259},[396],[398,41261,41262],{"xmlns":400},[403,41263,41264,41268],{},[406,41265,41266],{},[409,41267,1705],{},[447,41269,1705],{"encoding":449},[224,41271,41273],{"className":41272,"ariaHidden":455},[454],[224,41274,41276,41279],{"className":41275},[459],[224,41277],{"className":41278,"style":2565},[463],[224,41280,1705],{"className":41281,"style":1740},[468,469],"（模去仿射项）",[224,41284,41286,41301],{"className":41285,"translate":388},[392],[224,41287,41289],{"className":41288},[396],[398,41290,41291],{"xmlns":400},[403,41292,41293,41298],{},[406,41294,41295],{},[412,41296,41297],{},"↔",[447,41299,41300],{"encoding":449},"\\leftrightarrow",[224,41302,41304],{"className":41303,"ariaHidden":455},[454],[224,41305,41307,41310],{"className":41306},[459],[224,41308],{"className":41309,"style":30960},[463],[224,41311,41297],{"className":41312},[500]," 联合凸泛函 ",[224,41315,41317,41330],{"className":41316,"translate":388},[392],[224,41318,41320],{"className":41319},[396],[398,41321,41322],{"xmlns":400},[403,41323,41324,41328],{},[406,41325,41326],{},[412,41327,41297],{},[447,41329,41300],{"encoding":449},[224,41331,41333],{"className":41332,"ariaHidden":455},[454],[224,41334,41336,41339],{"className":41335},[459],[224,41337],{"className":41338,"style":30960},[463],[224,41340,41297],{"className":41341},[500]," 满足数据处理不等式的分布差异度量，三者一一对应。",[608,41344,41346],{"id":41345},"_3-常见散度","3. 常见散度",[11,41348,41349,41350,41378],{},"不同的 ",[224,41351,41353,41366],{"className":41352,"translate":388},[392],[224,41354,41356],{"className":41355},[396],[398,41357,41358],{"xmlns":400},[403,41359,41360,41364],{},[406,41361,41362],{},[409,41363,1705],{},[447,41365,1705],{"encoding":449},[224,41367,41369],{"className":41368,"ariaHidden":455},[454],[224,41370,41372,41375],{"className":41371},[459],[224,41373],{"className":41374,"style":2565},[463],[224,41376,1705],{"className":41377,"style":1740},[468,469]," 对应不同的散度。KL、JS、全变差、卡方、Hellinger、Wasserstein、MMD 等散度，表面上都是「衡量两个分布差多少」，但它们的根本区别可以归结为几个维度：",[11,41380,41381],{},"它们对「差异」的定义方式不同，对「支撑集不重叠」的处理不同，对「几何\u002F度量结构」的利用不同，因而导出的拓扑、梯度行为和优化性质也不同。",[11,41383,41384],{},"可以把不同散度的区别归纳为五个核心问题：",[9997,41386,41387,41390,41393,41396,41399],{},[2677,41388,41389],{},"差异定义在什么量上？是密度比值、密度差、累积分布差，还是搬运质量？",[2677,41391,41392],{},"是否对称？",[2677,41394,41395],{},"是否有界？",[2677,41397,41398],{},"是否利用样本空间的几何结构？",[2677,41400,41401],{},"诱导的拓扑强弱如何？",[11,41403,41404],{},"这五点决定了散度在理论性质和实际优化中的行为差异。下面逐个把常用散度「代入模板」并验算。",[613,41406,41408],{"id":41407},"_31-kl-散度","3.1 KL 散度",[11,41410,41411],{},"取：",[224,41413,41415],{"className":41414,"translate":388},[387],[224,41416,41418,41447],{"className":41417,"translate":388},[392],[224,41419,41421],{"className":41420},[396],[398,41422,41423],{"xmlns":400,"display":401},[403,41424,41425,41445],{},[406,41426,41427,41429,41431,41433,41435,41437,41439,41441,41443],{},[409,41428,1705],{},[412,41430,239],{"stretchy":435},[409,41432,1435],{},[412,41434,1716],{"stretchy":435},[412,41436,278],{},[409,41438,1435],{},[409,41440,24131],{},[412,41442,24134],{},[409,41444,1435],{},[447,41446,24139],{"encoding":449},[224,41448,41450,41477],{"className":41449,"ariaHidden":455},[454],[224,41451,41453,41456,41459,41462,41465,41468,41471,41474],{"className":41452},[459],[224,41454],{"className":41455,"style":535},[463],[224,41457,1705],{"className":41458,"style":1740},[468,469],[224,41460,239],{"className":41461},[542],[224,41463,1435],{"className":41464},[468,469],[224,41466,1716],{"className":41467},[549],[224,41469],{"className":41470,"style":496},[473],[224,41472,278],{"className":41473},[500],[224,41475],{"className":41476,"style":496},[473],[224,41478,41480,41483,41486,41489,41494,41497],{"className":41479},[459],[224,41481],{"className":41482,"style":2565},[463],[224,41484,1435],{"className":41485},[468,469],[224,41487],{"className":41488,"style":724},[473],[224,41490,24185,41492],{"className":41491},[21531],[224,41493,24189],{"style":24188},[224,41495],{"className":41496,"style":724},[473],[224,41498,1435],{"className":41499},[468,469],[11,41501,41502,41503,41536,41537,7098,41602,41735],{},"（下面 ",[224,41504,41506,41522],{"className":41505,"translate":388},[392],[224,41507,41509],{"className":41508},[396],[398,41510,41511],{"xmlns":400},[403,41512,41513,41519],{},[406,41514,41515,41517],{},[409,41516,24131],{},[412,41518,24134],{},[447,41520,41521],{"encoding":449},"\\log",[224,41523,41525],{"className":41524,"ariaHidden":455},[454],[224,41526,41528,41531],{"className":41527},[459],[224,41529],{"className":41530,"style":2565},[463],[224,41532,24185,41534],{"className":41533},[21531],[224,41535,24189],{"style":24188}," 均为自然对数。）则 ",[224,41538,41540,41563],{"className":41539,"translate":388},[392],[224,41541,41543],{"className":41542},[396],[398,41544,41545],{"xmlns":400},[403,41546,41547,41561],{},[406,41548,41549,41551,41553,41555,41557,41559],{},[409,41550,1705],{},[412,41552,239],{"stretchy":435},[419,41554,1438],{},[412,41556,1716],{"stretchy":435},[412,41558,278],{},[419,41560,284],{},[447,41562,20243],{"encoding":449},[224,41564,41566,41593],{"className":41565,"ariaHidden":455},[454],[224,41567,41569,41572,41575,41578,41581,41584,41587,41590],{"className":41568},[459],[224,41570],{"className":41571,"style":535},[463],[224,41573,1705],{"className":41574,"style":1740},[468,469],[224,41576,239],{"className":41577},[542],[224,41579,1438],{"className":41580},[468],[224,41582,1716],{"className":41583},[549],[224,41585],{"className":41586,"style":496},[473],[224,41588,278],{"className":41589},[500],[224,41591],{"className":41592,"style":496},[473],[224,41594,41596,41599],{"className":41595},[459],[224,41597],{"className":41598,"style":510},[463],[224,41600,284],{"className":41601},[468],[224,41603,41605,41645],{"className":41604,"translate":388},[392],[224,41606,41608],{"className":41607},[396],[398,41609,41610],{"xmlns":400},[403,41611,41612,41642],{},[406,41613,41614,41624,41626,41628,41630,41632,41634,41636,41638,41640],{},[3248,41615,41616,41618],{},[409,41617,1705],{},[406,41619,41620,41622],{},[412,41621,6933],{"mathvariant":678},[412,41623,6933],{"mathvariant":678},[412,41625,239],{"stretchy":435},[409,41627,1435],{},[412,41629,1716],{"stretchy":435},[412,41631,278],{},[419,41633,1438],{},[409,41635,12413],{"mathvariant":678},[409,41637,1435],{},[412,41639,325],{},[419,41641,284],{},[447,41643,41644],{"encoding":449},"f''(t)=1\u002Ft>0",[224,41646,41648,41705,41726],{"className":41647,"ariaHidden":455},[454],[224,41649,41651,41654,41687,41690,41693,41696,41699,41702],{"className":41650},[459],[224,41652],{"className":41653,"style":6966},[463],[224,41655,41657,41660],{"className":41656},[468],[224,41658,1705],{"className":41659,"style":1740},[468,469],[224,41661,41663],{"className":41662},[750],[224,41664,41666],{"className":41665},[754],[224,41667,41669],{"className":41668},[759],[224,41670,41672],{"className":41671,"style":6992},[763],[224,41673,41674,41677],{"style":3288},[224,41675],{"className":41676,"style":772},[771],[224,41678,41680],{"className":41679},[776,777,778,779],[224,41681,41683],{"className":41682},[468,779],[224,41684,41686],{"className":41685},[468,779],"′′",[224,41688,239],{"className":41689},[542],[224,41691,1435],{"className":41692},[468,469],[224,41694,1716],{"className":41695},[549],[224,41697],{"className":41698,"style":496},[473],[224,41700,278],{"className":41701},[500],[224,41703],{"className":41704,"style":496},[473],[224,41706,41708,41711,41714,41717,41720,41723],{"className":41707},[459],[224,41709],{"className":41710,"style":535},[463],[224,41712,28555],{"className":41713},[468],[224,41715,1435],{"className":41716},[468,469],[224,41718],{"className":41719,"style":496},[473],[224,41721,325],{"className":41722},[500],[224,41724],{"className":41725,"style":496},[473],[224,41727,41729,41732],{"className":41728},[459],[224,41730],{"className":41731,"style":510},[463],[224,41733,284],{"className":41734},[468]," 凸。代入模板：",[224,41737,41739],{"className":41738,"translate":388},[387],[224,41740,41742,41845],{"className":41741,"translate":388},[392],[224,41743,41745],{"className":41744},[396],[398,41746,41747],{"xmlns":400,"display":401},[403,41748,41749,41842],{},[406,41750,41751,41757,41759,41761,41763,41765,41767,41769,41771,41773,41775,41781,41783,41785,41791,41793,41795,41797,41799,41801,41803,41805,41807,41813,41815,41817,41819,41821,41832,41834,41836,41838,41840],{},[695,41752,41753,41755],{},[409,41754,20303],{},[409,41756,1705],{},[412,41758,239],{"stretchy":435},[409,41760,16018],{},[409,41762,20312],{"mathvariant":678},[409,41764,20315],{},[412,41766,1716],{"stretchy":435},[412,41768,278],{},[412,41770,21338],{},[409,41772,2023],{},[412,41774,23331],{},[20338,41776,41777,41779],{},[409,41778,11],{},[409,41780,2023],{},[409,41782,24131],{},[412,41784,24134],{},[20338,41786,41787,41789],{},[409,41788,11],{},[409,41790,2023],{},[423,41792,682],{},[409,41794,13182],{},[409,41796,371],{},[412,41798,278],{},[412,41800,21338],{},[409,41802,11],{},[409,41804,24131],{},[412,41806,24134],{},[20338,41808,41809,41811],{},[409,41810,11],{},[409,41812,2023],{},[423,41814,682],{},[409,41816,13182],{},[409,41818,371],{},[412,41820,278],{},[695,41822,41823,41825],{},[409,41824,20303],{},[406,41826,41827,41829],{},[409,41828,30652],{},[409,41830,41831],{},"L",[412,41833,239],{"stretchy":435},[409,41835,16018],{},[409,41837,20312],{"mathvariant":678},[409,41839,20315],{},[412,41841,1716],{"stretchy":435},[447,41843,41844],{"encoding":449},"D_f(P\\|Q)\n=\\int q\\cdot \\frac{p}{q}\\log\\frac{p}{q}\\,dx\n=\\int p\\log\\frac{p}{q}\\,dx\n=D_{KL}(P\\|Q)",[224,41846,41848,41918,41942,42102,42209],{"className":41847,"ariaHidden":455},[454],[224,41849,41851,41854,41894,41897,41900,41903,41906,41909,41912,41915],{"className":41850},[459],[224,41852],{"className":41853,"style":20368},[463],[224,41855,41857,41860],{"className":41856},[468],[224,41858,20303],{"className":41859,"style":2473},[468,469],[224,41861,41863],{"className":41862},[750],[224,41864,41866,41886],{"className":41865},[754,755],[224,41867,41869,41883],{"className":41868},[759],[224,41870,41872],{"className":41871,"style":20387},[763],[224,41873,41874,41877],{"style":20390},[224,41875],{"className":41876,"style":772},[771],[224,41878,41880],{"className":41879},[776,777,778,779],[224,41881,1705],{"className":41882,"style":1740},[468,469,779],[224,41884,787],{"className":41885},[786],[224,41887,41889],{"className":41888},[759],[224,41890,41892],{"className":41891,"style":20409},[763],[224,41893],{},[224,41895,239],{"className":41896},[542],[224,41898,16018],{"className":41899,"style":653},[468,469],[224,41901,20312],{"className":41902},[468],[224,41904,20315],{"className":41905},[468,469],[224,41907,1716],{"className":41908},[549],[224,41910],{"className":41911,"style":496},[473],[224,41913,278],{"className":41914},[500],[224,41916],{"className":41917,"style":496},[473],[224,41919,41921,41924,41927,41930,41933,41936,41939],{"className":41920},[459],[224,41922],{"className":41923,"style":25785},[463],[224,41925,21338],{"className":41926,"style":21534},[21531,21532,21533],[224,41928],{"className":41929,"style":724},[473],[224,41931,2023],{"className":41932,"style":492},[468,469],[224,41934],{"className":41935,"style":474},[473],[224,41937,23331],{"className":41938},[478],[224,41940],{"className":41941,"style":474},[473],[224,41943,41945,41949,42011,42014,42019,42022,42084,42087,42090,42093,42096,42099],{"className":41944},[459],[224,41946],{"className":41947,"style":41948},[463],"height:1.988em;vertical-align:-0.8804em;",[224,41950,41952,41955,42008],{"className":41951},[468],[224,41953],{"className":41954},[542,8479],[224,41956,41958],{"className":41957},[20338],[224,41959,41961,42000],{"className":41960},[754,755],[224,41962,41964,41997],{"className":41963},[759],[224,41965,41967,41978,41986],{"className":41966,"style":23586},[763],[224,41968,41969,41972],{"style":20531},[224,41970],{"className":41971,"style":2108},[771],[224,41973,41975],{"className":41974},[468],[224,41976,2023],{"className":41977,"style":492},[468,469],[224,41979,41980,41983],{"style":20546},[224,41981],{"className":41982,"style":2108},[771],[224,41984],{"className":41985,"style":20554},[20553],[224,41987,41988,41991],{"style":20557},[224,41989],{"className":41990,"style":2108},[771],[224,41992,41994],{"className":41993},[468],[224,41995,11],{"className":41996},[468,469],[224,41998,787],{"className":41999},[786],[224,42001,42003],{"className":42002},[759],[224,42004,42006],{"className":42005,"style":20579},[763],[224,42007],{},[224,42009],{"className":42010},[549,8479],[224,42012],{"className":42013,"style":724},[473],[224,42015,24185,42017],{"className":42016},[21531],[224,42018,24189],{"style":24188},[224,42020],{"className":42021,"style":724},[473],[224,42023,42025,42028,42081],{"className":42024},[468],[224,42026],{"className":42027},[542,8479],[224,42029,42031],{"className":42030},[20338],[224,42032,42034,42073],{"className":42033},[754,755],[224,42035,42037,42070],{"className":42036},[759],[224,42038,42040,42051,42059],{"className":42039,"style":23586},[763],[224,42041,42042,42045],{"style":20531},[224,42043],{"className":42044,"style":2108},[771],[224,42046,42048],{"className":42047},[468],[224,42049,2023],{"className":42050,"style":492},[468,469],[224,42052,42053,42056],{"style":20546},[224,42054],{"className":42055,"style":2108},[771],[224,42057],{"className":42058,"style":20554},[20553],[224,42060,42061,42064],{"style":20557},[224,42062],{"className":42063,"style":2108},[771],[224,42065,42067],{"className":42066},[468],[224,42068,11],{"className":42069},[468,469],[224,42071,787],{"className":42072},[786],[224,42074,42076],{"className":42075},[759],[224,42077,42079],{"className":42078,"style":20579},[763],[224,42080],{},[224,42082],{"className":42083},[549,8479],[224,42085],{"className":42086,"style":724},[473],[224,42088,13182],{"className":42089},[468,469],[224,42091,371],{"className":42092},[468,469],[224,42094],{"className":42095,"style":496},[473],[224,42097,278],{"className":42098},[500],[224,42100],{"className":42101,"style":496},[473],[224,42103,42105,42109,42112,42115,42118,42121,42126,42129,42191,42194,42197,42200,42203,42206],{"className":42104},[459],[224,42106],{"className":42107,"style":42108},[463],"height:2.2404em;vertical-align:-0.8804em;",[224,42110,21338],{"className":42111,"style":21534},[21531,21532,21533],[224,42113],{"className":42114,"style":724},[473],[224,42116,11],{"className":42117},[468,469],[224,42119],{"className":42120,"style":724},[473],[224,42122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KL 散度 ",[224,42280,42282,42314],{"className":42281,"translate":388},[392],[224,42283,42285],{"className":42284},[396],[398,42286,42287],{"xmlns":400},[403,42288,42289,42311],{},[406,42290,42291,42301,42303,42305,42307,42309],{},[695,42292,42293,42295],{},[409,42294,20303],{},[406,42296,42297,42299],{},[409,42298,30652],{},[409,42300,41831],{},[412,42302,239],{"stretchy":435},[409,42304,20315],{},[409,42306,20312],{"mathvariant":678},[409,42308,16018],{},[412,42310,1716],{"stretchy":435},[447,42312,42313],{"encoding":449},"D_{KL}(Q\\|P)",[224,42315,42317],{"className":42316,"ariaHidden":455},[454],[224,42318,42320,42323,42369,42372,42375,42378,42381],{"className":42319},[459],[224,42321],{"className":42322,"style":535},[463],[224,42324,42326,42329],{"className":42325},[468],[224,42327,20303],{"className":42328,"style":2473},[468,469],[224,42330,42332],{"className":42331},[750],[224,42333,42335,42361],{"className":42334},[754,755],[224,42336,42338,42358],{"className":42337},[759],[224,42339,42341],{"className":42340,"style":764},[763],[224,42342,42343,42346],{"style":20390},[224,42344],{"className":42345,"style":772},[771],[224,42347,42349],{"className":42348},[776,777,778,779],[224,42350,42352,42355],{"className":42351},[468,779],[224,42353,30652],{"className":42354,"style":30740},[468,469,779],[224,42356,41831],{"className":42357},[468,469,779],[224,42359,787],{"className":42360},[786],[224,42362,42364],{"className":42363},[759],[224,42365,42367],{"className":42366,"style":794},[763],[224,42368],{},[224,42370,239],{"className":42371},[542],[224,42373,20315],{"className":42374},[468,469],[224,42376,20312],{"className":42377},[468],[224,42379,16018],{"className":42380,"style":653},[468,469],[224,42382,1716],{"className":42383},[549]," 对应斜共轭 ",[224,42386,42388,42414],{"className":42387,"translate":388},[392],[224,42389,42391],{"className":42390},[396],[398,42392,42393],{"xmlns":400},[403,42394,42395,42411],{},[406,42396,42397,42399,42401,42403,42405,42407,42409],{},[409,42398,1435],{},[409,42400,1705],{},[412,42402,239],{"stretchy":435},[419,42404,1438],{},[409,42406,12413],{"mathvariant":678},[409,42408,1435],{},[412,42410,1716],{"stretchy":435},[447,42412,42413],{"encoding":449},"t f(1\u002Ft)",[224,42415,42417],{"className":42416,"ariaHidden":455},[454],[224,42418,42420,42423,42426,42429,42432,42435,42438],{"className":42419},[459],[224,42421],{"className":42422,"style":535},[463],[224,42424,1435],{"className":42425},[468,469],[224,42427,1705],{"className":42428,"style":1740},[468,469],[224,42430,239],{"className":42431},[542],[224,42433,28555],{"className":42434},[468],[224,42436,1435],{"className":42437},[468,469],[224,42439,1716],{"className":42440},[549],[224,42442,42444],{"className":42443,"translate":388},[387],[224,42445,42447,42569],{"className":42446,"translate":388},[392],[224,42448,42450],{"className":42449},[396],[398,42451,42452],{"xmlns":400,"display":401},[403,42453,42454,42566],{},[5209,42455,42456,42490],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,42457,42458,42472],{},[5218,42459,42460],{},[5221,42461,42462],{"scriptlevel":284,"displaystyle":455},[406,42463,42464,42466,42468,42470],{},[409,42465,1705],{},[412,42467,239],{"stretchy":435},[409,42469,1435],{},[412,42471,1716],{"stretchy":435},[5218,42473,42474],{},[5221,42475,42476],{"scriptlevel":284,"displaystyle":455},[406,42477,42478,42480,42482,42484,42486,42488],{},[406,42479],{},[412,42481,278],{},[409,42483,1435],{},[409,42485,24131],{},[412,42487,24134],{},[409,42489,1435],{},[5215,42491,42492,42498],{},[5218,42493,42494],{},[5221,42495,42496],{"scriptlevel":284,"displaystyle":455},[406,42497],{},[5218,42499,42500],{},[5221,42501,42502],{"scriptlevel":284,"displaystyle":455},[406,42503,42504,42506,42508,42510,42526,42528,42530,42532,42534,42536,42538,42544,42546,42548,42554,42556,42558,42560,42562,42564],{},[406,42505],{},[412,42507,24315],{},[423,42509,15054],{},[27527,42511,42512,42514,42524],{},[409,42513,1705],{},[406,42515,42516,42518,42520,42522],{},[409,42517,1858],{},[409,42519,7630],{},[409,42521,1861],{},[409,42523,13639],{},[412,42525,27543],{"lspace":5213,"rspace":5213},[412,42527,239],{"stretchy":435},[409,42529,1435],{},[412,42531,1716],{"stretchy":435},[412,42533,278],{},[409,42535,1435],{},[412,42537,23331],{},[20338,42539,42540,42542],{},[419,42541,1438],{},[409,42543,1435],{},[409,42545,24131],{},[412,42547,24134],{},[20338,42549,42550,42552],{},[419,42551,1438],{},[409,42553,1435],{},[412,42555,278],{},[412,42557,5268],{},[409,42559,24131],{},[412,42561,24134],{},[409,42563,1435],{},[412,42565,1877],{"separator":455},[447,42567,42568],{"encoding":449},"\\begin{aligned}\nf(t)&=t\\log t\\\\[2pt]\n&\\Longrightarrow\\ f^{*}_{skew}(t)\n=t\\cdot\\frac1t\\log\\frac1t=-\\log t,\n\\end{aligned}",[224,42570,42572],{"className":42571,"ariaHidden":455},[454],[224,42573,42575,42579],{"className":42574},[459],[224,42576],{"className":42577,"style":42578},[463],"height:3.7074em;vertical-align:-1.6037em;",[224,42580,42582],{"className":42581},[468],[224,42583,42585,42641],{"className":42584},[5209],[224,42586,42588],{"className":42587},[5331],[224,42589,42591,42632],{"className":42590},[754,755],[224,42592,42594,42629],{"className":42593},[759],[224,42595,42598,42620],{"className":42596,"style":42597},[763],"height:2.1037em;",[224,42599,42601,42605],{"style":42600},"top:-4.5852em;",[224,42602],{"className":42603,"style":42604},[771],"height:3.3214em;",[224,42606,42608,42611,42614,42617],{"className":42607},[468],[224,42609,1705],{"className":42610,"style":1740},[468,469],[224,42612,239],{"className":42613},[542],[224,42615,1435],{"className":42616},[468,469],[224,42618,1716],{"className":42619},[549],[224,42621,42623,42626],{"style":42622},"top:-2.4037em;",[224,42624],{"className":42625,"style":42604},[771],[224,42627],{"className":42628},[468],[224,42630,787],{"className":42631},[786],[224,42633,42635],{"className":42634},[759],[224,42636,42639],{"className":42637,"style":42638},[763],"height:1.6037em;",[224,42640],{},[224,42642,42644],{"className":42643},[5399],[224,42645,42647,42977],{"className":42646},[754,755],[224,42648,42650,42974],{"className":42649},[759],[224,42651,42653,42690],{"className":42652,"style":42597},[763],[224,42654,42655,42658],{"style":42600},[224,42656],{"className":42657,"style":42604},[771],[224,42659,42661,42664,42667,42670,42673,42676,42679,42684,42687],{"className":42660},[468],[224,42662],{"className":42663},[468],[224,42665],{"className":42666,"style":496},[473],[224,42668,278],{"className":42669},[500],[224,42671],{"className":42672,"style":496},[473],[224,42674,1435],{"className":42675},[468,469],[224,42677],{"className":42678,"style":724},[473],[224,42680,24185,42682],{"className":42681},[21531],[224,42683,24189],{"style":24188},[224,42685],{"className":42686,"style":724},[473],[224,42688,1435],{"className":42689},[468,469],[224,42691,42692,42695],{"style":42622},[224,42693],{"className":42694,"style":42604},[771],[224,42696,42698,42701,42704,42707,42710,42713,42779,42782,42785,42788,42791,42794,42797,42800,42803,42806,42809,42872,42875,42880,42883,42945,42948,42951,42954,42957,42960,42965,42968,42971],{"className":42697},[468],[224,42699],{"className":42700},[468],[224,42702],{"className":42703,"style":496},[473],[224,42705,24315],{"className":42706},[500],[224,42708,15054],{"className":42709},[473],[224,42711],{"className":42712,"style":496},[473],[224,42714,42716,42719],{"className":42715},[468],[224,42717,1705],{"className":42718,"style":1740},[468,469],[224,42720,42722],{"className":42721},[750],[224,42723,42725,42771],{"className":42724},[754,755],[224,42726,42728,42768],{"className":42727},[759],[224,42729,42731,42754],{"className":42730,"style":28217},[763],[224,42732,42733,42736],{"style":28220},[224,42734],{"className":42735,"style":772},[771],[224,42737,42739],{"className":42738},[776,777,778,779],[224,42740,42742,42745,42748,42751],{"className":42741},[468,779],[224,42743,1858],{"className":42744},[468,469,779],[224,42746,7630],{"className":42747,"style":7646},[468,469,779],[224,42749,1861],{"className":42750},[468,469,779],[224,42752,13639],{"className":42753,"style":13754},[468,469,779],[224,42755,42756,42759],{"style":5074},[224,42757],{"className":42758,"style":772},[771],[224,42760,42762],{"className":42761},[776,777,778,779],[224,42763,42765],{"className":42764},[468,779],[224,42766,27543],{"className":42767},[468,779],[224,42769,787],{"className":42770},[786],[224,42772,42774],{"className":42773},[759],[224,42775,42777],{"className":42776,"style":28265},[763],[224,42778],{},[224,42780,239],{"className":42781},[542],[224,42783,1435],{"className":42784},[468,469],[224,42786,1716],{"className":42787},[549],[224,42789],{"className":42790,"style":496},[473],[224,42792,278],{"className":42793},[500],[224,42795],{"className":42796,"style":496},[473],[224,42798,1435],{"className":42799},[468,469],[224,42801],{"className":42802,"style":474},[473],[224,42804,23331],{"className":42805},[478],[224,42807],{"className":42808,"style":474},[473],[224,42810,42812,42815,42869],{"className":42811},[468],[224,42813],{"className":42814},[542,8479],[224,42816,42818],{"className":42817},[20338],[224,42819,42821,42861],{"className":42820},[754,755],[224,42822,42824,42858],{"className":42823},[759],[224,42825,42828,42839,42847],{"className":42826,"style":42827},[763],"height:1.3214em;",[224,42829,42830,42833],{"style":20531},[224,42831],{"className":42832,"style":2108},[771],[224,42834,42836],{"className":42835},[468],[224,42837,1435],{"className":42838},[468,469],[224,42840,42841,42844],{"style":20546},[224,42842],{"className":42843,"style":2108},[771],[224,42845],{"className":42846,"style":20554},[20553],[224,42848,42849,42852],{"style":20557},[224,42850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散度。"," 取 ",[224,43285,43287,43325],{"className":43286,"translate":388},[392],[224,43288,43290],{"className":43289},[396],[398,43291,43292],{"xmlns":400},[403,43293,43294,43322],{},[406,43295,43296,43298,43300,43302,43304,43306,43308,43310,43312,43314,43316,43318,43320],{},[409,43297,1705],{},[412,43299,239],{"stretchy":435},[409,43301,1435],{},[412,43303,1716],{"stretchy":435},[412,43305,278],{},[412,43307,239],{"stretchy":435},[409,43309,1435],{},[412,43311,5268],{},[419,43313,1438],{},[412,43315,1716],{"stretchy":435},[409,43317,24131],{},[412,43319,24134],{},[409,43321,1435],{},[447,43323,43324],{"encoding":449},"f(t)=(t-1)\\log t",[224,43326,43328,43355,43376],{"className":43327,"ariaHidden":455},[454],[224,43329,43331,43334,43337,43340,43343,43346,43349,43352],{"className":43330},[459],[224,43332],{"className":43333,"style":535},[463],[224,43335,1705],{"className":43336,"style":1740},[468,469],[224,43338,239],{"className":43339},[542],[224,43341,1435],{"className":43342},[468,469],[224,43344,1716],{"className":43345},[549],[224,43347],{"className":43348,"style":496},[473],[224,43350,278],{"className":43351},[500],[224,43353],{"className":43354,"style":496},[473],[224,43356,43358,43361,43364,43367,43370,43373],{"className":43357},[459],[224,43359],{"className":43360,"style":535},[463],[224,43362,239],{"className":43363},[542],[224,43365,1435],{"className":43366},[468,469],[224,43368],{"className":43369,"style":474},[473],[224,43371,5268],{"className":43372},[478],[224,43374],{"className":43375,"style":474},[473],[224,43377,43379,43382,43385,43388,43391,43396,43399],{"className":43378},[459],[224,43380],{"className":43381,"style":535},[463],[224,43383,1438],{"className":43384},[468],[224,43386,1716],{"className":43387},[549],[224,43389],{"className":43390,"style":724},[473],[224,43392,24185,43394],{"className":43393},[21531],[224,43395,24189],{"style":24188},[224,43397],{"className":43398,"style":724},[473],[224,43400,1435],{"className":43401},[468,469],"（它满足 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f(1\u002Ft)=f(t)",[224,43509,43511,43544],{"className":43510,"ariaHidden":455},[454],[224,43512,43514,43517,43520,43523,43526,43529,43532,43535,43538,43541],{"className":43513},[459],[224,43515],{"className":43516,"style":535},[463],[224,43518,1435],{"className":43519},[468,469],[224,43521,1705],{"className":43522,"style":1740},[468,469],[224,43524,239],{"className":43525},[542],[224,43527,28555],{"className":43528},[468],[224,43530,1435],{"className":43531},[468,469],[224,43533,1716],{"className":43534},[549],[224,43536],{"className":43537,"style":496},[473],[224,43539,278],{"className":43540},[500],[224,43542],{"className":43543,"style":496},[473],[224,43545,43547,43550,43553,43556,43559],{"className":43546},[459],[224,43548],{"className":43549,"style":535},[463],[224,43551,1705],{"className":43552,"style":1740},[468,469],[224,43554,239],{"className":43555},[542],[224,43557,1435],{"className":43558},[468,469],[224,43560,1716],{"className":43561},[549],"），则",[224,43564,43566],{"className":43565,"translate":388},[387],[224,43567,43569,43697],{"className":43568,"translate":388},[392],[224,43570,43572],{"className":43571},[396],[398,43573,43574],{"xmlns":400,"display":401},[403,43575,43576,43694],{},[406,43577,43578,43584,43586,43588,43590,43592,43594,43596,43598,43600,43602,43604,43606,43608,43610,43612,43618,43620,43622,43624,43627,43629,43639,43641,43643,43645,43647,43649,43651,43661,43663,43665,43667,43669,43671,43673,43675,43682,43684,43686,43688,43690,43692],{},[695,43579,43580,43582],{},[409,43581,20303],{},[409,43583,1705],{},[412,43585,239],{"stretchy":435},[409,43587,16018],{},[409,43589,20312],{"mathvariant":678},[409,43591,20315],{},[412,43593,1716],{"stretchy":435},[412,43595,278],{},[412,43597,21338],{},[412,43599,239],{"stretchy":435},[409,43601,11],{},[412,43603,5268],{},[409,43605,2023],{},[412,43607,1716],{"stretchy":435},[409,43609,24131],{},[412,43611,24134],{},[20338,43613,43614,43616],{},[409,43615,11],{},[409,43617,2023],{},[423,43619,682],{},[409,43621,13182],{},[409,43623,371],{},[473,43625],{"linebreak":43626},"newline",[412,43628,278],{},[695,43630,43631,43633],{},[409,43632,20303],{},[406,43634,43635,43637],{},[409,43636,30652],{},[409,43638,41831],{},[412,43640,239],{"stretchy":435},[409,43642,16018],{},[409,43644,20312],{"mathvariant":678},[409,43646,20315],{},[412,43648,1716],{"stretchy":435},[412,43650,319],{},[695,43652,43653,43655],{},[409,43654,20303],{},[406,43656,43657,43659],{},[409,43658,30652],{},[409,43660,41831],{},[412,43662,239],{"stretchy":435},[409,43664,20315],{},[409,43666,20312],{"mathvariant":678},[409,43668,16018],{},[412,43670,1716],{"stretchy":435},[412,43672,278],{},[412,43674,1071],{},[695,43676,43677,43679],{},[409,43678,20303],{},[409,43680,43681],{},"J",[412,43683,239],{"stretchy":435},[409,43685,16018],{},[412,43687,1877],{"separator":455},[409,43689,20315],{},[412,43691,1716],{"stretchy":435},[412,43693,1877],{"separator":455},[447,43695,43696],{"encoding":449},"D_f(P\\|Q)=\\int (p-q)\\log\\frac{p}{q}\\,dx \\\\\n=D_{KL}(P\\|Q)+D_{KL}(Q\\|P)=:D_J(P,Q),",[224,43698,43700,43770,43794,43888,43891,43903,43979,44056],{"className":43699,"ariaHidden":455},[454],[224,43701,43703,43706,43746,43749,43752,43755,43758,43761,43764,43767],{"className":43702},[459],[224,43704],{"className":43705,"style":20368},[463],[224,43707,43709,43712],{"className":43708},[468],[224,43710,20303],{"className":43711,"style":2473},[468,469],[224,43713,43715],{"className":43714},[750],[224,43716,43718,43738],{"className":43717},[754,755],[224,43719,43721,43735],{"className":43720},[759],[224,43722,43724],{"className":43723,"style":20387},[763],[224,43725,43726,43729],{"style":20390},[224,43727],{"className":43728,"style":772},[771],[224,43730,43732],{"className":43731},[776,777,778,779],[224,43733,1705],{"className":43734,"style":1740},[468,469,779],[224,43736,787],{"className":43737},[786],[224,43739,43741],{"className":43740},[759],[224,43742,43744],{"className":43743,"style":20409},[763],[224,43745],{},[224,43747,239],{"className":43748},[542],[224,43750,16018],{"className":43751,"style":653},[468,469],[224,43753,20312],{"className":43754},[468],[224,43756,20315],{"className":43757},[468,469],[224,43759,1716],{"className":43760},[549],[224,43762],{"className":43763,"style":496},[473],[224,43765,278],{"className":43766},[500],[224,43768],{"className":43769,"style":496},[473],[224,43771,43773,43776,43779,43782,43785,43788,43791],{"className":43772},[459],[224,43774],{"className":43775,"style":25785},[463],[224,43777,21338],{"className":43778,"style":21534},[21531,21532,21533],[224,43780,239],{"className":43781},[542],[224,43783,11],{"className":43784},[468,469],[224,43786],{"className":43787,"style":474},[473],[224,43789,5268],{"className":43790},[478],[224,43792],{"className":43793,"style":474},[473],[224,43795,43797,43800,43803,43806,43809,43814,43817,43879,43882,43885],{"className":43796},[459],[224,43798],{"className":43799,"style":41948},[463],[224,43801,2023],{"className":43802,"style":492},[468,469],[224,43804,1716],{"className":43805},[549],[224,43807],{"className":43808,"style":724},[473],[224,43810,24185,43812],{"className":43811},[21531],[224,43813,24189],{"style":24188},[224,43815],{"className":43816,"style":724},[473],[224,43818,43820,43823,43876],{"className":43819},[468],[224,43821],{"className":43822},[542,8479],[224,43824,43826],{"className":43825},[20338],[224,43827,43829,43868],{"className":43828},[754,755],[224,43830,43832,43865],{"className":43831},[759],[224,43833,43835,43846,43854],{"className":43834,"style":23586},[763],[224,43836,43837,43840],{"style":20531},[224,43838],{"className":43839,"style":2108},[771],[224,43841,43843],{"className":43842},[468],[224,43844,2023],{"className":43845,"style":492},[468,469],[224,43847,43848,43851],{"style":20546},[224,43849],{"className":43850,"style":2108},[771],[224,43852],{"className":43853,"style":20554},[20553],[224,43855,43856,43859],{"style":20557},[224,43857],{"className":43858,"style":2108},[771],[224,43860,43862],{"className":43861},[468],[224,43863,11],{"className":43864},[468,469],[224,43866,787],{"className":43867},[786],[224,43869,43871],{"className":43870},[759],[224,43872,43874],{"className":43873,"style":20579},[763],[224,43875],{},[224,43877],{"className":43878},[549,8479],[224,43880],{"className":43881,"style":724},[473],[224,43883,13182],{"className":43884},[468,469],[224,43886,371],{"className":43887},[468,469],[224,43889],{"className":43890},[473,43626],[224,43892,43894,43897,43900],{"className":43893},[459],[224,43895],{"className":43896,"style":30960},[463],[224,43898,278],{"className":43899},[500],[224,43901],{"className":43902,"style":496},[473],[224,43904,43906,43909,43955,43958,43961,43964,43967,43970,43973,43976],{"className":43905},[459],[224,43907],{"className":43908,"style":535},[463],[224,43910,43912,43915],{"className":43911},[468],[224,43913,20303],{"className":43914,"style":2473},[468,469],[224,43916,43918],{"className":43917},[750],[224,43919,43921,43947],{"className":43920},[754,755],[224,43922,43924,43944],{"className":43923},[759],[224,43925,43927],{"className":43926,"style":764},[763],[224,43928,43929,43932],{"style":20390},[224,43930],{"className":43931,"style":772},[771],[224,43933,43935],{"className":43934},[776,777,778,779],[224,43936,43938,43941],{"className":43937},[468,779],[224,43939,30652],{"className":43940,"style":30740},[468,469,779],[224,43942,41831],{"className":43943},[468,469,779],[224,43945,787],{"className":43946},[786],[224,43948,43950],{"className":43949},[759],[224,43951,43953],{"className":43952,"style":794},[763],[224,43954],{},[224,43956,239],{"className":43957},[542],[224,43959,16018],{"className":43960,"style":653},[468,469],[224,43962,20312],{"className":43963},[468],[224,43965,20315],{"className":43966},[468,469],[224,43968,1716],{"className":43969},[549],[224,43971],{"className":43972,"style":474},[473],[224,43974,319],{"className":43975},[478],[224,43977],{"className":43978,"style":474},[473],[224,43980,43982,43985,44031,44034,44037,44040,44043,44046,44049,44053],{"className":43981},[459],[224,43983],{"className":43984,"style":535},[463],[224,43986,43988,43991],{"className":43987},[468],[224,43989,20303],{"className":43990,"style":2473},[468,469],[224,43992,43994],{"className":43993},[750],[224,43995,43997,44023],{"className":43996},[754,755],[224,43998,44000,44020],{"className":43999},[759],[224,44001,44003],{"className":44002,"style":764},[763],[224,44004,44005,44008],{"style":20390},[224,44006],{"className":44007,"style":772},[771],[224,44009,44011],{"className":44010},[776,777,778,779],[224,44012,44014,44017],{"className":44013},[468,779],[224,44015,30652],{"className":44016,"style":30740},[468,469,779],[224,44018,41831],{"className":44019},[468,469,779],[224,44021,787],{"className":44022},[786],[224,44024,44026],{"className":44025},[759],[224,44027,44029],{"className":44028,"style":794},[763],[224,44030],{},[224,44032,239],{"className":44033},[542],[224,44035,20315],{"className":44036},[468,469],[224,44038,20312],{"className":44039},[468],[224,44041,16018],{"className":44042,"style":653},[468,469],[224,44044,1716],{"className":44045},[549],[224,44047],{"className":44048,"style":496},[473],[224,44050,44052],{"className":44051},[500],"=:",[224,44054],{"className":44055,"style":496},[473],[224,44057,44059,44062,44103,44106,44109,44112,44115,44118,44121],{"className":44058},[459],[224,44060],{"className":44061,"style":535},[463],[224,44063,44065,44068],{"className":44064},[468],[224,44066,20303],{"className":44067,"style":2473},[468,469],[224,44069,44071],{"className":44070},[750],[224,44072,44074,44095],{"className":44073},[754,755],[224,44075,44077,44092],{"className":44076},[759],[224,44078,44080],{"className":44079,"style":764},[763],[224,44081,44082,44085],{"style":20390},[224,44083],{"className":44084,"style":772},[771],[224,44086,44088],{"className":44087},[776,777,778,779],[224,44089,43681],{"className":44090,"style":44091},[468,469,779],"margin-right:0.0962em;",[224,44093,787],{"className":44094},[786],[224,44096,44098],{"className":44097},[759],[224,44099,44101],{"className":44100,"style":794},[763],[224,44102],{},[224,44104,239],{"className":44105},[542],[224,44107,16018],{"className":44108,"style":653},[468,469],[224,44110,1877],{"className":44111},[1917],[224,44113],{"className":44114,"style":724},[473],[224,44116,20315],{"className":44117},[468,469],[224,44119,1716],{"className":44120},[549],[224,44122,1877],{"className":44123},[1917],[11,44125,44126],{},"这是 KL 的两向之和，也是一个人为构造的对称 f-散度。",[11,44128,44129,44132,44133,44136,44137,44165,44166,44194,44195,44223,44224,44256,44257,44362,44363,44391,44392,44420,44421,44424],{},[378,44130,44131],{},"意义。"," KL 是最基础的散度，等价于信息论中的",[378,44134,44135],{},"相对熵","：若真实分布是 ",[224,44138,44140,44153],{"className":44139,"translate":388},[392],[224,44141,44143],{"className":44142},[396],[398,44144,44145],{"xmlns":400},[403,44146,44147,44151],{},[406,44148,44149],{},[409,44150,16018],{},[447,44152,16018],{"encoding":449},[224,44154,44156],{"className":44155,"ariaHidden":455},[454],[224,44157,44159,44162],{"className":44158},[459],[224,44160],{"className":44161,"style":649},[463],[224,44163,16018],{"className":44164,"style":653},[468,469],"、而用 ",[224,44167,44169,44182],{"className":44168,"translate":388},[392],[224,44170,44172],{"className":44171},[396],[398,44173,44174],{"xmlns":400},[403,44175,44176,44180],{},[406,44177,44178],{},[409,44179,20315],{},[447,44181,20315],{"encoding":449},[224,44183,44185],{"className":44184,"ariaHidden":455},[454],[224,44186,44188,44191],{"className":44187},[459],[224,44189],{"className":44190,"style":20637},[463],[224,44192,20315],{"className":44193},[468,469]," 去编码来自 ",[224,44196,44198,44211],{"className":44197,"translate":388},[392],[224,44199,44201],{"className":44200},[396],[398,44202,44203],{"xmlns":400},[403,44204,44205,44209],{},[406,44206,44207],{},[409,44208,16018],{},[447,44210,16018],{"encoding":449},[224,44212,44214],{"className":44213,"ariaHidden":455},[454],[224,44215,44217,44220],{"className":44216},[459],[224,44218],{"className":44219,"style":649},[463],[224,44221,16018],{"className":44222,"style":653},[468,469]," 的样本，那么 ",[224,44225,44227,44242],{"className":44226,"translate":388},[392],[224,44228,44230],{"className":44229},[396],[398,44231,44232],{"xmlns":400},[403,44233,44234,44240],{},[406,44235,44236,44238],{},[409,44237,24131],{},[412,44239,24134],{},[447,44241,41521],{"encoding":449},[224,44243,44245],{"className":44244,"ariaHidden":455},[454],[224,44246,44248,44251],{"className":44247},[459],[224,44249],{"className":44250,"style":2565},[463],[224,44252,24185,44254],{"className":44253},[21531],[224,44255,24189],{"style":24188}," 以 2 为底时 ",[224,44258,44260,44292],{"className":44259,"translate":388},[392],[224,44261,44263],{"className":44262},[396],[398,44264,44265],{"xmlns":400},[403,44266,44267,44289],{},[406,44268,44269,44279,44281,44283,44285,44287],{},[695,44270,44271,44273],{},[409,44272,20303],{},[406,44274,44275,44277],{},[409,44276,30652],{},[409,44278,41831],{},[412,44280,239],{"stretchy":435},[409,44282,16018],{},[409,44284,20312],{"mathvariant":678},[409,44286,20315],{},[412,44288,1716],{"stretchy":435},[447,44290,44291],{"encoding":449},"D_{KL}(P\\|Q)",[224,44293,44295],{"className":44294,"ariaHidden":455},[454],[224,44296,44298,44301,44347,44350,44353,44356,44359],{"className":44297},[459],[224,44299],{"className":44300,"style":535},[463],[224,44302,44304,44307],{"className":44303},[468],[224,44305,20303],{"className":44306,"style":2473},[468,469],[224,44308,44310],{"className":44309},[750],[224,44311,44313,44339],{"className":44312},[754,755],[224,44314,44316,44336],{"className":44315},[759],[224,44317,44319],{"className":44318,"style":764},[763],[224,44320,44321,44324],{"style":20390},[224,44322],{"className":44323,"style":772},[771],[224,44325,44327],{"className":44326},[776,777,778,779],[224,44328,44330,44333],{"className":44329},[468,779],[224,44331,30652],{"className":44332,"style":30740},[468,469,779],[224,44334,41831],{"className":44335},[468,469,779],[224,44337,787],{"className":44338},[786],[224,44340,44342],{"className":44341},[759],[224,44343,44345],{"className":44344,"style":794},[763],[224,44346],{},[224,44348,239],{"className":44349},[542],[224,44351,16018],{"className":44352,"style":653},[468,469],[224,44354,20312],{"className":44355},[468],[224,44357,20315],{"className":44358},[468,469],[224,44360,1716],{"className":44361},[549]," 正是用 ",[224,44364,44366,44379],{"className":44365,"translate":388},[392],[224,44367,44369],{"className":44368},[396],[398,44370,44371],{"xmlns":400},[403,44372,44373,44377],{},[406,44374,44375],{},[409,44376,20315],{},[447,44378,20315],{"encoding":449},[224,44380,44382],{"className":44381,"ariaHidden":455},[454],[224,44383,44385,44388],{"className":44384},[459],[224,44386],{"className":44387,"style":20637},[463],[224,44389,20315],{"className":44390},[468,469]," 编码比用 ",[224,44393,44395,44408],{"className":44394,"translate":388},[392],[224,44396,44398],{"className":44397},[396],[398,44399,44400],{"xmlns":400},[403,44401,44402,44406],{},[406,44403,44404],{},[409,44405,16018],{},[447,44407,16018],{"encoding":449},[224,44409,44411],{"className":44410,"ariaHidden":455},[454],[224,44412,44414,44417],{"className":44413},[459],[224,44415],{"className":44416,"style":649},[463],[224,44418,16018],{"className":44419,"style":653},[468,469]," 编码",[378,44422,44423],{},"多付出的比特数","，即",[224,44426,44428],{"className":44427,"translate":388},[387],[224,44429,44431,44573],{"className":44430,"translate":388},[392],[224,44432,44434],{"className":44433},[396],[398,44435,44436],{"xmlns":400,"display":401},[403,44437,44438,44570],{},[5209,44439,44440,44513],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,44441,44442,44468],{},[5218,44443,44444],{},[5221,44445,44446],{"scriptlevel":284,"displaystyle":455},[406,44447,44448,44458,44460,44462,44464,44466],{},[695,44449,44450,44452],{},[409,44451,20303],{},[406,44453,44454,44456],{},[409,44455,30652],{},[409,44457,41831],{},[412,44459,239],{"stretchy":435},[409,44461,16018],{},[409,44463,20312],{"mathvariant":678},[409,44465,20315],{},[412,44467,1716],{"stretchy":435},[5218,44469,44470],{},[5221,44471,44472],{"scriptlevel":284,"displaystyle":455},[406,44473,44474,44476,44478],{},[406,44475],{},[412,44477,278],{},[1965,44479,44480,44510],{},[1965,44481,44482,44508],{},[406,44483,44484,44490,44492,44494,44496,44498,44500,44502,44504,44506],{},[695,44485,44486,44488],{},[409,44487,20324],{"mathvariant":12251},[409,44489,16018],{},[412,44491,268],{"stretchy":435},[412,44493,5268],{},[409,44495,24131],{},[412,44497,24134],{},[409,44499,2023],{},[412,44501,239],{"stretchy":435},[409,44503,20668],{},[412,44505,1716],{"stretchy":435},[412,44507,440],{"stretchy":435},[412,44509,1987],{"stretchy":455},[423,44511,44512],{},"交叉熵",[5215,44514,44515,44521],{},[5218,44516,44517],{},[5221,44518,44519],{"scriptlevel":284,"displaystyle":455},[406,44520],{},[5218,44522,44523],{},[5221,44524,44525],{"scriptlevel":284,"displaystyle":455},[406,44526,44527,44529,44531,44533,44568],{},[406,44528],{},[473,44530],{"width":4247},[412,44532,5268],{},[1965,44534,44535,44565],{},[1965,44536,44537,44563],{},[406,44538,44539,44545,44547,44549,44551,44553,44555,44557,44559,44561],{},[695,44540,44541,44543],{},[409,44542,20324],{"mathvariant":12251},[409,44544,16018],{},[412,44546,268],{"stretchy":435},[412,44548,5268],{},[409,44550,24131],{},[412,44552,24134],{},[409,44554,11],{},[412,44556,239],{"stretchy":435},[409,44558,20668],{},[412,44560,1716],{"stretchy":435},[412,44562,440],{"stretchy":435},[412,44564,1987],{"stretchy":455},[423,44566,44567],{},"熵",[409,44569,688],{"mathvariant":678},[447,44571,44572],{"encoding":449},"\\begin{aligned}\nD_{KL}(P\\|Q)\n&=\\underbrace{\\mathbb E_P[-\\log q(X)]}\n_{\\text{交叉熵}}\\\\[2pt]\n&\\qquad-\\underbrace{\\mathbb E_P[-\\log p(X)]}\n_{\\text{熵}} .\n\\end{aligned}",[224,44574,44576],{"className":44575,"ariaHidden":455},[454],[224,44577,44579,44583],{"className":44578},[459],[224,44580],{"className":44581,"style":44582},[463],"height:5.1327em;vertical-align:-2.3163em;",[224,44584,44586],{"className":44585},[468],[224,44587,44589,44693],{"className":44588},[5209],[224,44590,44592],{"className":44591},[5331],[224,44593,44595,44684],{"className":44594},[754,755],[224,44596,44598,44681],{"className":44597},[759],[224,44599,44602,44672],{"className":44600,"style":44601},[763],"height:2.8163em;",[224,44603,44605,44608],{"style":44604},"top:-4.9763em;",[224,44606],{"className":44607,"style":2108},[771],[224,44609,44611,44657,44660,44663,44666,44669],{"className":44610},[468],[224,44612,44614,44617],{"className":44613},[468],[224,44615,20303],{"className":44616,"style":2473},[468,469],[224,44618,44620],{"className":44619},[750],[224,44621,44623,44649],{"className":44622},[754,755],[224,44624,44626,44646],{"className":44625},[759],[224,44627,44629],{"className":44628,"style":764},[763],[224,44630,44631,44634],{"style":20390},[224,44632],{"className":44633,"style":772},[771],[224,44635,44637],{"className":44636},[776,777,778,779],[224,44638,44640,44643],{"className":44639},[468,779],[224,44641,30652],{"className":44642,"style":30740},[468,469,779],[224,44644,41831],{"className":44645},[468,469,779],[224,44647,787],{"className":44648},[786],[224,44650,44652],{"className":44651},[759],[224,44653,44655],{"className":44654,"style":794},[763],[224,44656],{},[224,44658,239],{"className":44659},[542],[224,44661,16018],{"className":44662,"style":653},[468,469],[224,44664,20312],{"className":44665},[468],[224,44667,20315],{"className":44668},[468,469],[224,44670,1716],{"className":44671},[549],[224,44673,44675,44678],{"style":44674},"top:-2.26em;",[224,44676],{"className":44677,"style":2108},[771],[224,44679],{"className":44680},[468],[224,44682,787],{"className":44683},[786],[224,44685,44687],{"className":44686},[759],[224,44688,44691],{"className":44689,"style":44690},[763],"height:2.3163em;",[224,44692],{},[224,44694,44696],{"className":44695},[5399],[224,44697,44699,45107],{"className":44698},[754,755],[224,44700,44702,45104],{"className":44701},[759],[224,44703,44705,44900],{"className":44704,"style":44601},[763],[224,44706,44707,44710],{"style":44604},[224,44708],{"className":44709,"style":2108},[771],[224,44711,44713,44716,44719,44722,44725],{"className":44712},[468],[224,44714],{"className":44715},[468],[224,44717],{"className":44718,"style":496},[473],[224,44720,278],{"className":44721},[500],[224,44723],{"className":44724,"style":496},[473],[224,44726,44728],{"className":44727},[2091,1965],[224,44729,44731,44892],{"className":44730},[754,755],[224,44732,44734,44889],{"className":44733},[759],[224,44735,44737,44754],{"className":44736,"style":2101},[763],[224,44738,44739,44742],{"style":16143},[224,44740],{"className":44741,"style":2108},[771],[224,44743,44745],{"className":44744},[776,777,778,779],[224,44746,44748],{"className":44747},[468,779],[224,44749,44751],{"className":44750},[468,11373,779],[224,44752,44512],{"className":44753},[468,16159,779],[224,44755,44756,44759],{"style":2120},[224,44757],{"className":44758,"style":2108},[771],[224,44760,44762],{"className":44761},[2091,1965],[224,44763,44765,44881],{"className":44764},[754,755],[224,44766,44768,44878],{"className":44767},[759],[224,44769,44771,44801],{"className":44770,"style":2101},[763],[224,44772,44774,44777],{"className":44773,"style":2140},[2139],[224,44775],{"className":44776,"style":2108},[771],[224,44778,44780,44787,44794],{"className":44779,"style":2148},[2147],[224,44781,44783],{"className":44782,"style":2153},[2152],[2155,44784,44785],{"xmlns":2157,"width":2158,"height":2159,"viewBox":2160,"preserveAspectRatio":2161},[2163,44786],{"d":2165},[224,44788,44790],{"className":44789,"style":2153},[2169],[2155,44791,44792],{"xmlns":2157,"width":2158,"height":2159,"viewBox":2160,"preserveAspectRatio":2172},[2163,44793],{"d":2175},[224,44795,44797],{"className":44796,"style":2153},[2179],[2155,44798,44799],{"xmlns":2157,"width":2158,"height":2159,"viewBox":2160,"preserveAspectRatio":2182},[2163,44800],{"d":2185},[224,44802,44803,44806],{"style":2120},[224,44804],{"className":44805,"style":2108},[771],[224,44807,44809,44849,44852,44855,44858,44863,44866,44869,44872,44875],{"className":44808},[468],[224,44810,44812,44815],{"className":44811},[468],[224,44813,20324],{"className":44814},[468,12389],[224,44816,44818],{"className":44817},[750],[224,44819,44821,44841],{"className":44820},[754,755],[224,44822,44824,44838],{"className":44823},[759],[224,44825,44827],{"className":44826,"style":764},[763],[224,44828,44829,44832],{"style":767},[224,44830],{"className":44831,"style":772},[771],[224,44833,44835],{"className":44834},[776,777,778,779],[224,44836,16018],{"className":44837,"style":653},[468,469,779],[224,44839,787],{"className":44840},[786],[224,44842,44844],{"className":44843},[759],[224,44845,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Stein 引理里第二类错误的最优指数，也是 AIC\u002FBIC 这类模型选择的基石；在机器学习中它是最大似然训练、变分推断与知识蒸馏的目标。它的不对称性带来两种相反的行为：",[224,45119,45121,45152],{"className":45120,"translate":388},[392],[224,45122,45124],{"className":45123},[396],[398,45125,45126],{"xmlns":400},[403,45127,45128,45150],{},[406,45129,45130,45140,45142,45144,45146,45148],{},[695,45131,45132,45134],{},[409,45133,20303],{},[406,45135,45136,45138],{},[409,45137,30652],{},[409,45139,41831],{},[412,45141,239],{"stretchy":435},[409,45143,16018],{},[409,45145,20312],{"mathvariant":678},[409,45147,20315],{},[412,45149,1716],{"stretchy":435},[447,45151,44291],{"encoding":449},[224,45153,45155],{"className":45154,"ariaHidden":455},[454],[224,45156,45158,45161,45207,45210,45213,45216,45219],{"className":45157},[459],[224,45159],{"className":45160,"style":535},[463],[224,45162,45164,45167],{"className":45163},[468],[224,45165,20303],{"className":45166,"style":2473},[468,469],[224,45168,45170],{"className":45169},[750],[224,45171,45173,45199],{"className":45172},[754,755],[224,45174,45176,45196],{"className":45175},[759],[224,45177,45179],{"className":45178,"style":764},[763],[224,45180,45181,45184],{"style":20390},[224,45182],{"className":45183,"style":772},[771],[224,45185,45187],{"className":45186},[776,777,778,779],[224,45188,45190,45193],{"className":45189},[468,779],[224,45191,30652],{"className":45192,"style":30740},[468,469,779],[224,45194,41831],{"className":45195},[468,469,779],[224,45197,787],{"className":45198},[786],[224,45200,45202],{"className":45201},[759],[224,45203,45205],{"className":45204,"style":794},[763],[224,45206],{},[224,45208,239],{"className":45209},[542],[224,45211,16018],{"className":45212,"style":653},[468,469],[224,45214,20312],{"className":45215},[468],[224,45217,20315],{"className":45218},[468,469],[224,45220,1716],{"className":45221},[549]," 要求 ",[224,45224,45226,45239],{"className":45225,"translate":388},[392],[224,45227,45229],{"className":45228},[396],[398,45230,45231],{"xmlns":400},[403,45232,45233,45237],{},[406,45234,45235],{},[409,45236,20315],{},[447,45238,20315],{"encoding":449},[224,45240,45242],{"className":45241,"ariaHidden":455},[454],[224,45243,45245,45248],{"className":45244},[459],[224,45246],{"className":45247,"style":20637},[463],[224,45249,20315],{"className":45250},[468,469],[224,45252,45254,45267],{"className":45253,"translate":388},[392],[224,45255,45257],{"className":45256},[396],[398,45258,45259],{"xmlns":400},[403,45260,45261,45265],{},[406,45262,45263],{},[409,45264,16018],{},[447,45266,16018],{"encoding":449},[224,45268,45270],{"className":45269,"ariaHidden":455},[454],[224,45271,45273,45276],{"className":45272},[459],[224,45274],{"className":45275,"style":649},[463],[224,45277,16018],{"className":45278,"style":653},[468,469]," 有质量处不为零（",[378,45281,45282],{},"零回避","，mass-covering，倾向覆盖 ",[224,45285,45287,45300],{"className":45286,"translate":388},[392],[224,45288,45290],{"className":45289},[396],[398,45291,45292],{"xmlns":400},[403,45293,45294,45298],{},[406,45295,45296],{},[409,45297,16018],{},[447,45299,16018],{"encoding":449},[224,45301,45303],{"className":45302,"ariaHidden":455},[454],[224,45304,45306,45309],{"className":45305},[459],[224,45307],{"className":45308,"style":649},[463],[224,45310,16018],{"className":45311,"style":653},[468,469]," 的每一个模式），而 ",[224,45314,45316,45347],{"className":45315,"translate":388},[392],[224,45317,45319],{"className":45318},[396],[398,45320,45321],{"xmlns":400},[403,45322,45323,45345],{},[406,45324,45325,45335,45337,45339,45341,45343],{},[695,45326,45327,45329],{},[409,45328,20303],{},[406,45330,45331,45333],{},[409,45332,30652],{},[409,45334,41831],{},[412,45336,239],{"stretchy":435},[409,45338,20315],{},[409,45340,20312],{"mathvariant":678},[409,45342,16018],{},[412,45344,1716],{"stretchy":435},[447,45346,42313],{"encoding":449},[224,45348,45350],{"className":45349,"ariaHidden":455},[454],[224,45351,45353,45356,45402,45405,45408,45411,45414],{"className":45352},[459],[224,45354],{"className":45355,"style":535},[463],[224,45357,45359,45362],{"className":45358},[468],[224,45360,20303],{"className":45361,"style":2473},[468,469],[224,45363,45365],{"className":45364},[750],[224,45366,45368,45394],{"className":45367},[754,755],[224,45369,45371,45391],{"className":45370},[759],[224,45372,45374],{"className":45373,"style":764},[763],[224,45375,45376,45379],{"style":20390},[224,45377],{"className":45378,"style":772},[771],[224,45380,45382],{"className":45381},[776,777,778,779],[224,45383,45385,45388],{"className":45384},[468,779],[224,45386,30652],{"className":45387,"style":30740},[468,469,779],[224,45389,41831],{"className":45390},[468,469,779],[224,45392,787],{"className":45393},[786],[224,45395,45397],{"className":45396},[759],[224,45398,45400],{"className":45399,"style":794},[763],[224,45401],{},[224,45403,239],{"className":45404},[542],[224,45406,20315],{"className":45407},[468,469],[224,45409,20312],{"className":45410},[468],[224,45412,16018],{"className":45413,"style":653},[468,469],[224,45415,1716],{"className":45416},[549]," 会把 ",[224,45419,45421,45434],{"className":45420,"translate":388},[392],[224,45422,45424],{"className":45423},[396],[398,45425,45426],{"xmlns":400},[403,45427,45428,45432],{},[406,45429,45430],{},[409,45431,20315],{},[447,45433,20315],{"encoding":449},[224,45435,45437],{"className":45436,"ariaHidden":455},[454],[224,45438,45440,45443],{"className":45439},[459],[224,45441],{"className":45442,"style":20637},[463],[224,45444,20315],{"className":45445},[468,469],[224,45447,45449,45462],{"className":45448,"translate":388},[392],[224,45450,45452],{"className":45451},[396],[398,45453,45454],{"xmlns":400},[403,45455,45456,45460],{},[406,45457,45458],{},[409,45459,16018],{},[447,45461,16018],{"encoding":449},[224,45463,45465],{"className":45464,"ariaHidden":455},[454],[224,45466,45468,45471],{"className":45467},[459],[224,45469],{"className":45470,"style":649},[463],[224,45472,16018],{"className":45473,"style":653},[468,469]," 为零处压到零（",[378,45476,45477],{},"零强制","，mode-seeking，倾向锁定单一模式）。GAN 的模式崩塌与变分推断的均值场近似，本质都由这一不对称性决定。",[613,45480,45482],{"id":45481},"_32-js-散度","3.2 JS 散度",[11,45484,45485,45486,12570],{},"先回顾 JS 散度的定义，令 ",[224,45487,45489,45522],{"className":45488,"translate":388},[392],[224,45490,45492],{"className":45491},[396],[398,45493,45494],{"xmlns":400},[403,45495,45496,45519],{},[406,45497,45498,45501,45503,45509,45511,45513,45515,45517],{},[409,45499,45500],{},"M",[412,45502,278],{},[20338,45504,45505,45507],{},[419,45506,1438],{},[419,45508,1447],{},[412,45510,239],{"stretchy":435},[409,45512,16018],{},[412,45514,319],{},[409,45516,20315],{},[412,45518,1716],{"stretchy":435},[447,45520,45521],{"encoding":449},"M=\\frac12(P+Q)",[224,45523,45525,45544,45633],{"className":45524,"ariaHidden":455},[454],[224,45526,45528,45531,45535,45538,45541],{"className":45527},[459],[224,45529],{"className":45530,"style":649},[463],[224,45532,45500],{"className":45533,"style":45534},[468,469],"margin-right:0.109em;",[224,45536],{"className":45537,"style":496},[473],[224,45539,278],{"className":45540},[500],[224,45542],{"className":45543,"style":496},[473],[224,45545,45547,45550,45618,45621,45624,45627,45630],{"className":45546},[459],[224,45548],{"className":45549,"style":22544},[463],[224,45551,45553,45556,45615],{"className":45552},[468],[224,45554],{"className":45555},[542,8479],[224,45557,45559],{"className":45558},[20338],[224,45560,45562,45607],{"className":45561},[754,755],[224,45563,45565,45604],{"className":45564},[759],[224,45566,45568,45582,45590],{"className":45567,"style":22563},[763],[224,45569,45570,45573],{"style":22566},[224,45571],{"className":45572,"style":2108},[771],[224,45574,45576],{"className":45575},[776,777,778,779],[224,45577,45579],{"className":45578},[468,779],[224,45580,1447],{"className":45581},[468,779],[224,45583,45584,45587],{"style":20546},[224,45585],{"className":45586,"style":2108},[771],[224,45588],{"className":45589,"style":20554},[20553],[224,45591,45592,45595],{"style":22589},[224,45593],{"className":45594,"style":2108},[771],[224,45596,45598],{"className":45597},[776,777,778,779],[224,45599,45601],{"className":45600},[468,779],[224,45602,1438],{"className":45603},[468,779],[224,45605,787],{"className":45606},[786],[224,45608,45610],{"className":45609},[759],[224,45611,45613],{"className":45612,"style":22611},[763],[224,45614],{},[224,45616],{"className":45617},[549,8479],[224,45619,239],{"className":45620},[542],[224,45622,16018],{"className":45623,"style":653},[468,469],[224,45625],{"className":45626,"style":474},[473],[224,45628,319],{"className":45629},[478],[224,45631],{"className":45632,"style":474},[473],[224,45634,45636,45639,45642],{"className":45635},[459],[224,45637],{"className":45638,"style":535},[463],[224,45640,20315],{"className":45641},[468,469],[224,45643,1716],{"className":45644},[549],[224,45646,45648],{"className":45647,"translate":388},[387],[224,45649,45651,45749],{"className":45650,"translate":388},[392],[224,45652,45654],{"className":45653},[396],[398,45655,45656],{"xmlns":400,"display":401},[403,45657,45658,45746],{},[406,45659,45660,45670,45672,45674,45676,45678,45680,45682,45688,45698,45700,45712,45714,45720,45730,45732,45744],{},[695,45661,45662,45664],{},[409,45663,20303],{},[406,45665,45666,45668],{},[409,45667,43681],{},[409,45669,6924],{},[412,45671,239],{"stretchy":435},[409,45673,16018],{},[409,45675,20312],{"mathvariant":678},[409,45677,20315],{},[412,45679,1716],{"stretchy":435},[412,45681,278],{},[20338,45683,45684,45686],{},[419,45685,1438],{},[419,45687,1447],{},[695,45689,45690,45692],{},[409,45691,20303],{},[406,45693,45694,45696],{},[409,45695,30652],{},[409,45697,41831],{},[423,45699,21353],{},[406,45701,45702,45704,45706,45708,45710],{},[412,45703,239],{"fence":455},[409,45705,16018],{},[409,45707,20312],{"mathvariant":678},[409,45709,45500],{},[412,45711,1716],{"fence":455},[412,45713,319],{},[20338,45715,45716,45718],{},[419,45717,1438],{},[419,45719,1447],{},[695,45721,45722,45724],{},[409,45723,20303],{},[406,45725,45726,45728],{},[409,45727,30652],{},[409,45729,41831],{},[423,45731,21353],{},[406,45733,45734,45736,45738,45740,45742],{},[412,45735,239],{"fence":455},[409,45737,20315],{},[409,45739,20312],{"mathvariant":678},[409,45741,45500],{},[412,45743,1716],{"fence":455},[409,45745,688],{"mathvariant":678},[447,45747,45748],{"encoding":449},"D_{JS}(P\\|Q)=\\frac12 D_{KL}\\!\\left(P\\|M\\right)\n+\\frac12 D_{KL}\\!\\left(Q\\|M\\right).",[224,45750,45752,45828,45976],{"className":45751,"ariaHidden":455},[454],[224,45753,45755,45758,45804,45807,45810,45813,45816,45819,45822,45825],{"className":45754},[459],[224,45756],{"className":45757,"style":535},[463],[224,45759,45761,45764],{"className":45760},[468],[224,45762,20303],{"className":45763,"style":2473},[468,469],[224,45765,45767],{"className":45766},[750],[224,45768,45770,45796],{"className":45769},[754,755],[224,45771,45773,45793],{"className":45772},[759],[224,45774,45776],{"className":45775,"style":764},[763],[224,45777,45778,45781],{"style":20390},[224,45779],{"className":45780,"style":772},[771],[224,45782,45784],{"className":45783},[776,777,778,779],[224,45785,45787,45790],{"className":45786},[468,779],[224,45788,43681],{"className":45789,"style":44091},[468,469,779],[224,45791,6924],{"className":45792,"style":6970},[468,469,779],[224,45794,787],{"className":45795},[786],[224,45797,45799],{"className":45798},[759],[224,45800,45802],{"className":45801,"style":794},[763],[224,45803],{},[224,45805,239],{"className":45806},[542],[224,45808,16018],{"className":45809,"style":653},[468,469],[224,45811,20312],{"className":45812},[468],[224,45814,20315],{"className":45815},[468,469],[224,45817,1716],{"className":45818},[549],[224,45820],{"className":45821,"style":496},[473],[224,45823,278],{"className":45824},[500],[224,45826],{"className":45827,"style":496},[473],[224,45829,45831,45835,45897,45943,45946,45949,45967,45970,45973],{"className":45830},[459],[224,45832],{"className":45833,"style":45834},[463],"height:2.0074em;vertical-align:-0.686em;",[224,45836,45838,45841,45894],{"className":45837},[468],[224,45839],{"className":45840},[542,8479],[224,45842,45844],{"className":45843},[20338],[224,45845,45847,45886],{"className":45846},[754,755],[224,45848,45850,45883],{"className":45849},[759],[224,45851,45853,45864,45872],{"className":45852,"style":42827},[763],[224,45854,45855,45858],{"style":20531},[224,45856],{"className":45857,"style":2108},[771],[224,45859,45861],{"className":45860},[468],[224,45862,1447],{"className":45863},[468],[224,45865,45866,45869],{"style":20546},[224,45867],{"className":45868,"style":2108},[771],[224,45870],{"className":45871,"style":20554},[20553],[224,45873,45874,45877],{"style":20557},[224,45875],{"className":45876,"style":2108},[771],[224,45878,45880],{"className":45879},[468],[224,45881,1438],{"className":45882},[468],[224,45884,787],{"className":45885},[786],[224,45887,45889],{"className":45888},[759],[224,45890,45892],{"className":45891,"style":23809},[763],[224,45893],{},[224,45895],{"className":45896},[549,8479],[224,45898,45900,45903],{"className":45899},[468],[224,45901,20303],{"className":45902,"style":2473},[468,469],[224,45904,45906],{"className":45905},[750],[224,45907,45909,45935],{"className":45908},[754,755],[224,45910,45912,45932],{"className":45911},[759],[224,45913,45915],{"className":45914,"style":764},[763],[224,45916,45917,45920],{"style":20390},[224,45918],{"className":45919,"style":772},[771],[224,45921,45923],{"className":45922},[776,777,778,779],[224,45924,45926,45929],{"className":45925},[468,779],[224,45927,30652],{"className":45928,"style":30740},[468,469,779],[224,45930,41831],{"className":45931},[468,469,779],[224,45933,787],{"className":45934},[786],[224,45936,45938],{"className":45937},[759],[224,45939,45941],{"className":45940,"style":794},[763],[224,45942],{},[224,45944],{"className":45945,"style":21559},[473],[224,45947],{"className":45948,"style":724},[473],[224,45950,45952,45955,45958,45961,45964],{"className":45951},[2091],[224,45953,239],{"className":45954,"style":8281},[542,8280],[224,45956,16018],{"className":45957,"style":653},[468,469],[224,45959,20312],{"className":45960},[468],[224,45962,45500],{"className":45963,"style":45534},[468,469],[224,45965,1716],{"className":45966,"style":8281},[549,8280],[224,45968],{"className":45969,"style":474},[473],[224,45971,319],{"className":45972},[478],[224,45974],{"className":45975,"style":474},[473],[224,45977,45979,45982,46044,46090,46093,46096,46114,46117],{"className":45978},[459],[224,45980],{"className":45981,"style":45834},[463],[224,45983,45985,45988,46041],{"className":45984},[468],[224,45986],{"className":45987},[542,8479],[224,45989,45991],{"className":45990},[20338],[224,45992,45994,46033],{"className":45993},[754,755],[224,45995,45997,46030],{"className":45996},[759],[224,45998,46000,46011,46019],{"className":45999,"style":42827},[763],[224,46001,46002,46005],{"style":20531},[224,46003],{"className":46004,"style":2108},[771],[224,46006,46008],{"className":46007},[468],[224,46009,1447],{"className":46010},[468],[224,46012,46013,46016],{"style":20546},[224,46014],{"className":46015,"style":2108},[771],[224,46017],{"className":46018,"style":20554},[20553],[224,46020,46021,46024],{"style":20557},[224,46022],{"className":46023,"style":2108},[771],[224,46025,46027],{"className":46026},[468],[224,46028,1438],{"className":46029},[468],[224,4603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f-散度。"," 这一点很容易被误解。我们直接构造 ",[224,46127,46129,46142],{"className":46128,"translate":388},[392],[224,46130,46132],{"className":46131},[396],[398,46133,46134],{"xmlns":400},[403,46135,46136,46140],{},[406,46137,46138],{},[409,46139,1705],{},[447,46141,1705],{"encoding":449},[224,46143,46145],{"className":46144,"ariaHidden":455},[454],[224,46146,46148,46151],{"className":46147},[459],[224,46149],{"className":46150,"style":2565},[463],[224,46152,1705],{"className":46153,"style":1740},[468,469],"：代入 ",[224,46156,46158,46178],{"className":46157,"translate":388},[392],[224,46159,46161],{"className":46160},[396],[398,46162,46163],{"xmlns":400},[403,46164,46165,46175],{},[406,46166,46167,46169,46171,46173],{},[409,46168,11],{},[412,46170,278],{},[409,46172,1435],{},[409,46174,2023],{},[447,46176,46177],{"encoding":449},"p=tq",[224,46179,46181,46199],{"className":46180,"ariaHidden":455},[454],[224,46182,46184,46187,46190,46193,46196],{"className":46183},[459],[224,46185],{"className":46186,"style":806},[463],[224,46188,11],{"className":46189},[468,469],[224,46191],{"className":46192,"style":496},[473],[224,46194,278],{"className":46195},[500],[224,46197],{"className":46198,"style":496},[473],[224,46200,46202,46206],{"className":46201},[459],[224,46203],{"className":46204,"style":46205},[463],"height:0.8095em;vertical-align:-0.1944em;",[224,46207,46209],{"className":46208,"style":492},[468,469],"tq","，于是 ",[224,46212,46214,46227],{"className":46213,"translate":388},[392],[224,46215,46217],{"className":46216},[396],[398,46218,46219],{"xmlns":400},[403,46220,46221,46225],{},[406,46222,46223],{},[409,46224,45500],{},[447,46226,45500],{"encoding":449},[224,46228,46230],{"className":46229,"ariaHidden":455},[454],[224,46231,46233,46236],{"className":46232},[459],[224,46234],{"className":46235,"style":649},[463],[224,46237,45500],{"className":46238,"style":45534},[468,469]," 的密度为 ",[224,46241,46243,46277],{"className":46242,"translate":388},[392],[224,46244,46246],{"className":46245},[396],[398,46247,46248],{"xmlns":400},[403,46249,46250,46274],{},[406,46251,46252,46254,46256],{},[409,46253,13177],{},[412,46255,278],{},[20338,46257,46258,46272],{},[406,46259,46260,46262,46264,46266,46268,46270],{},[412,46261,239],{"stretchy":435},[409,46263,1435],{},[412,46265,319],{},[419,46267,1438],{},[412,46269,1716],{"stretchy":435},[409,46271,2023],{},[419,46273,1447],{},[447,46275,46276],{"encoding":449},"m=\\frac{(t+1)q}{2}",[224,46278,46280,46298],{"className":46279,"ariaHidden":455},[454],[224,46281,46283,46286,46289,46292,46295],{"className":46282},[459],[224,46284],{"className":46285,"style":839},[463],[224,46287,13177],{"className":46288},[468,469],[224,46290],{"className":46291,"style":496},[473],[224,46293,278],{"className":46294},[500],[224,46296],{"className":46297,"style":496},[473],[224,46299,46301,46305],{"className":46300},[459],[224,46302],{"className":46303,"style":46304},[463],"height:1.355em;vertical-align:-0.345em;",[224,46306,46308,46311,46387],{"className":46307},[468],[224,46309],{"className":46310},[542,8479],[224,46312,46314],{"className":46313},[20338],[224,46315,46317,46379],{"className":46316},[754,755],[224,46318,46320,46376],{"className":46319},[759],[224,46321,46324,46338,46346],{"className":46322,"style":46323},[763],"height:1.01em;",[224,46325,46326,46329],{"style":22566},[224,46327],{"className":46328,"style":2108},[771],[224,46330,46332],{"className":46331},[776,777,778,779],[224,46333,46335],{"className":46334},[468,779],[224,46336,1447],{"className":46337},[468,779],[224,46339,46340,46343],{"style":20546},[224,46341],{"className":46342,"style":2108},[771],[224,46344],{"className":46345,"style":20554},[20553],[224,46347,46349,46352],{"style":46348},"top:-3.485em;",[224,46350],{"className":46351,"style":2108},[771],[224,46353,46355],{"className":46354},[776,777,778,779],[224,46356,46358,46361,46364,46367,46370,46373],{"className":46357},[468,779],[224,46359,239],{"className":46360},[542,779],[224,46362,1435],{"className":46363},[468,469,779],[224,46365,319],{"className":46366},[478,779],[224,46368,1438],{"className":46369},[468,779],[224,46371,1716],{"className":46372},[549,779],[224,46374,2023],{"className":46375,"style":492},[468,469,779],[224,46377,787],{"className":46378},[786],[224,46380,46382],{"className":46381},[759],[224,46383,46385],{"className":46384,"style":22611},[763],[224,46386],{},[224,46388],{"className":46389},[549,8479],"，逐项算：",[224,46392,46394],{"className":46393,"translate":388},[387],[224,46395,46397,46459],{"className":46396,"translate":388},[392],[224,46398,46400],{"className":46399},[396],[398,46401,46402],{"xmlns":400,"display":401},[403,46403,46404,46456],{},[406,46405,46406,46412,46414,46430,46432,46434,46440,46442,46454],{},[20338,46407,46408,46410],{},[409,46409,11],{},[409,46411,13177],{},[412,46413,278],{},[20338,46415,46416,46422],{},[406,46417,46418,46420],{},[419,46419,1447],{},[409,46421,1435],{},[406,46423,46424,46426,46428],{},[409,46425,1435],{},[412,46427,319],{},[419,46429,1438],{},[412,46431,1877],{"separator":455},[473,46433],{"width":4247},[20338,46435,46436,46438],{},[409,46437,2023],{},[409,46439,13177],{},[412,46441,278],{},[20338,46443,46444,46446],{},[419,46445,1447],{},[406,46447,46448,46450,46452],{},[409,46449,1435],{},[412,46451,319],{},[419,46453,1438],{},[412,46455,1877],{"separator":455},[447,46457,46458],{"encoding":449},"\\frac{p}{m}=\\frac{2t}{t+1},\\qquad\n\\frac{q}{m}=\\frac{2}{t+1},",[224,46460,46462,46540,46704],{"className":46461,"ariaHidden":455},[454],[224,46463,46465,46469,46531,46534,46537],{"className":46464},[459],[224,46466],{"className":46467,"style":46468},[463],"height:1.7936em;vertical-align:-0.686em;",[224,46470,46472,46475,46528],{"className":46471},[468],[224,46473],{"className":46474},[542,8479],[224,46476,46478],{"className":46477},[20338],[224,46479,46481,46520],{"className":46480},[754,755],[224,46482,46484,46517],{"className":46483},[759],[224,46485,46487,46498,46506],{"className":46486,"style":23586},[763],[224,46488,46489,46492],{"style":20531},[224,46490],{"className":46491,"style":2108},[771],[224,46493,46495],{"className":46494},[468],[224,46496,13177],{"className":46497},[468,469],[224,46499,46500,46503],{"style":20546},[224,46501],{"className":46502,"style":2108},[771],[224,46504],{"className":46505,"style":20554},[20553],[224,46507,46508,46511],{"style":20557},[224,46509],{"className":46510,"style":2108},[771],[224,46512,46514],{"className":46513},[468],[224,46515,11],{"className":46516},[468,469],[224,46518,787],{"className":46519},[786],[224,46521,46523],{"className":46522},[759],[224,46524,46526],{"className":46525,"style":23809},[763],[224,46527],{},[224,46529],{"className":46530},[549,8479],[224,46532],{"className":46533,"style":496},[473],[224,46535,278],{"className":46536},[500],[224,46538],{"className":46539,"style":496},[473],[224,46541,46543,46547,46624,46627,46630,46633,46695,46698,46701],{"className":46542},[459],[224,46544],{"className":46545,"style":46546},[463],"height:2.0908em;vertical-align:-0.7693em;",[224,46548,46550,46553,46621],{"className":46549},[468],[224,46551],{"className":46552},[542,8479],[224,46554,46556],{"className":46555},[20338],[224,46557,46559,46613],{"className":46558},[754,755],[224,46560,46562,46610],{"className":46561},[759],[224,46563,46565,46588,46596],{"className":46564,"style":42827},[763],[224,46566,46567,46570],{"style":20531},[224,46568],{"className":46569,"style":2108},[771],[224,46571,46573,46576,46579,46582,46585],{"className":46572},[468],[224,46574,1435],{"className":46575},[468,469],[224,46577],{"className":46578,"style":474},[473],[224,46580,319],{"className":46581},[478],[224,46583],{"className":46584,"style":474},[473],[224,46586,1438],{"className":46587},[468],[224,46589,46590,46593],{"style":20546},[224,46591],{"className":46592,"style":2108},[771],[224,46594],{"className":46595,"style":20554},[20553],[224,46597,46598,46601],{"style":20557},[224,46599],{"className":46600,"style":2108},[771],[224,46602,46604,46607],{"className":46603},[468],[224,46605,1447],{"className":46606},[468],[224,46608,1435],{"className":46609},[468,469],[224,46611,787],{"className":46612},[786],[224,46614,46616],{"className":46615},[759],[224,46617,46619],{"className":46618,"style":36958},[763],[224,46620],{},[224,46622],{"className":46623},[549,8479],[224,46625,1877],{"className":46626},[1917],[224,46628],{"className":46629,"style":4416},[473],[224,46631],{"className":46632,"style":724},[473],[224,46634,46636,46639,46692],{"className":46635},[468],[224,46637],{"className":46638},[542,8479],[224,46640,46642],{"className":46641},[20338],[224,46643,46645,46684],{"className":46644},[754,755],[224,46646,46648,46681],{"className":46647},[759],[224,46649,46651,46662,46670],{"className":46650,"style":23586},[763],[224,46652,46653,46656],{"style":20531},[224,46654],{"className":46655,"style":2108},[771],[224,46657,46659],{"className":46658},[468],[224,46660,13177],{"className":46661},[468,469],[224,46663,46664,46667],{"style":20546},[224,46665],{"className":46666,"style":2108},[771],[224,46668],{"className":46669,"style":20554},[20553],[224,46671,46672,46675],{"style":20557},[224,46673],{"className":46674,"style":2108},[771],[224,46676,46678],{"className":46677},[468],[224,46679,2023],{"className":46680,"style":492},[468,469],[224,46682,787],{"className":46683},[786],[224,46685,46687],{"className":46686},[759],[224,46688,46690],{"className":46689,"style":23809},[763],[224,46691],{},[224,46693],{"className":46694},[549,8479],[224,46696],{"className":46697,"style":496},[473],[224,46699,278],{"className":46700},[500],[224,46702],{"className":46703,"style":496},[473],[224,46705,46707,46710,46784],{"className":46706},[459],[224,46708],{"className":46709,"style":46546},[463],[224,46711,46713,46716,46781],{"className":46712},[468],[224,46714],{"className":46715},[542,8479],[224,46717,46719],{"className":46718},[20338],[224,46720,46722,46773],{"className":46721},[754,755],[224,46723,46725,46770],{"className":46724},[759],[224,46726,46728,46751,46759],{"className":46727,"style":42827},[763],[224,46729,46730,46733],{"style":20531},[224,46731],{"className":46732,"style":2108},[771],[224,46734,46736,46739,46742,46745,46748],{"className":46735},[468],[224,46737,1435],{"className":46738},[468,469],[224,46740],{"className":46741,"style":474},[473],[224,46743,319],{"className":46744},[478],[224,46746],{"className":46747,"style":474},[473],[224,46749,1438],{"className":46750},[468],[224,46752,46753,46756],{"style":20546},[224,46754],{"className":46755,"style":2108},[771],[224,46757],{"className":46758,"style":20554},[20553],[224,46760,46761,46764],{"style":20557},[224,46762],{"className":46763,"style":2108},[771],[224,46765,46767],{"className":46766},[468],[224,46768,1447],{"className":46769},[468],[224,46771,787],{"className":46772},[786],[224,46774,46776],{"className":46775},[759],[224,46777,46779],{"className":46778,"style":36958},[763],[224,46780],{},[224,46782],{"className":46783},[549,8479],[224,46785,1877],{"className":46786},[1917],[224,46788,46790],{"className":46789,"translate":388},[387],[224,46791,46793,46933],{"className":46792,"translate":388},[392],[224,46794,46796],{"className":46795},[396],[398,46797,46798],{"xmlns":400,"display":401},[403,46799,46800,46930],{},[5209,46801,46802,46882],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,46803,46804,46830],{},[5218,46805,46806],{},[5221,46807,46808],{"scriptlevel":284,"displaystyle":455},[406,46809,46810,46820,46822,46824,46826,46828],{},[695,46811,46812,46814],{},[409,46813,20303],{},[406,46815,46816,46818],{},[409,46817,43681],{},[409,46819,6924],{},[412,46821,239],{"stretchy":435},[409,46823,16018],{},[409,46825,20312],{"mathvariant":678},[409,46827,20315],{},[412,46829,1716],{"stretchy":435},[5218,46831,46832],{},[5221,46833,46834],{"scriptlevel":284,"displaystyle":455},[406,46835,46836,46838,46840,46846,46848,46850,46852,46854,46860,46862,46868,46870,46872,46874,46876],{},[406,46837],{},[412,46839,278],{},[20338,46841,46842,46844],{},[419,46843,1438],{},[419,46845,1447],{},[412,46847,21338],{},[409,46849,11],{},[409,46851,24131],{},[412,46853,24134],{},[20338,46855,46856,46858],{},[409,46857,11],{},[409,46859,13177],{},[412,46861,319],{},[20338,46863,46864,46866],{},[419,46865,1438],{},[419,46867,1447],{},[412,46869,21338],{},[409,46871,2023],{},[409,46873,24131],{},[412,46875,24134],{},[20338,46877,46878,46880],{},[409,46879,2023],{},[409,46881,13177],{},[5215,46883,46884,46890],{},[5218,46885,46886],{},[5221,46887,46888],{"scriptlevel":284,"displaystyle":455},[406,46889],{},[5218,46891,46892],{},[5221,46893,46894],{"scriptlevel":284,"displaystyle":455},[406,46895,46896,46898,46900,46902,46904,46906,46916,46918,46920,46922,46924,46926,46928],{},[406,46897],{},[412,46899,278],{},[412,46901,21338],{},[409,46903,2023],{},[423,46905,682],{},[695,46907,46908,46910],{},[409,46909,1705],{},[406,46911,46912,46914],{},[409,46913,43681],{},[409,46915,6924],{},[412,46917,239],{"stretchy":435},[409,46919,1435],{},[412,46921,1716],{"stretchy":435},[423,46923,682],{},[409,46925,13182],{},[409,46927,371],{},[412,46929,1877],{"separator":455},[447,46931,46932],{"encoding":449},"\\begin{aligned}\nD_{JS}(P\\|Q)\n&=\\frac12\\int p\\log\\frac{p}{m}\n+\\frac12\\int q\\log\\frac{q}{m}\\\\[2pt]\n&=\\int 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f_{JS}(1\u002Ft)=f_{JS}(t)",[224,49673,49675,49751],{"className":49674,"ariaHidden":455},[454],[224,49676,49678,49681,49684,49730,49733,49736,49739,49742,49745,49748],{"className":49677},[459],[224,49679],{"className":49680,"style":535},[463],[224,49682,1435],{"className":49683},[468,469],[224,49685,49687,49690],{"className":49686},[468],[224,49688,1705],{"className":49689,"style":1740},[468,469],[224,49691,49693],{"className":49692},[750],[224,49694,49696,49722],{"className":49695},[754,755],[224,49697,49699,49719],{"className":49698},[759],[224,49700,49702],{"className":49701,"style":764},[763],[224,49703,49704,49707],{"style":47439},[224,49705],{"className":49706,"style":772},[771],[224,49708,49710],{"className":49709},[776,777,778,779],[224,49711,49713,49716],{"className":49712},[468,779],[224,49714,43681],{"className":49715,"style":44091},[468,469,779],[224,49717,6924],{"className":49718,"style":6970},[468,469,779],[224,49720,787],{"className":49721},[786],[224,49723,49725],{"className":49724},[759],[224,49726,49728],{"className":49727,"style":794},[763],[224,49729],{},[224,49731,239],{"className":49732},[542],[224,49734,28555],{"className":49735},[468],[224,49737,1435],{"className":49738},[468,469],[224,49740,1716],{"className":49741},[549],[224,49743],{"className":49744,"style":496},[473],[224,49746,278],{"className":49747},[500],[224,49749],{"className":49750,"style":496},[473],[224,49752,49754,49757,49803,49806,49809],{"className":49753},[459],[224,49755],{"className":49756,"style":535},[463],[224,49758,49760,49763],{"className":49759},[468],[224,49761,1705],{"className":49762,"style":1740},[468,469],[224,49764,49766],{"className":49765},[750],[224,49767,49769,49795],{"className":49768},[754,755],[224,49770,49772,49792],{"className":49771},[759],[224,49773,49775],{"className":49774,"style":764},[763],[224,49776,49777,49780],{"style":47439},[224,49778],{"className":49779,"style":772},[771],[224,49781,49783],{"className":49782},[776,777,778,779],[224,49784,49786,49789],{"className":49785},[468,779],[224,49787,43681],{"className":49788,"style":44091},[468,469,779],[224,49790,6924],{"className":49791,"style":6970},[468,469,779],[224,49793,787],{"className":49794},[786],[224,49796,49798],{"className":49797},[759],[224,49799,49801],{"className":49800,"style":794},[763],[224,49802],{},[224,49804,239],{"className":49805},[542],[224,49807,1435],{"className":49808},[468,469],[224,49810,1716],{"className":49811},[549],"。因此当 ",[224,49814,49816,49833],{"className":49815,"translate":388},[392],[224,49817,49819],{"className":49818},[396],[398,49820,49821],{"xmlns":400},[403,49822,49823,49831],{},[406,49824,49825,49827,49829],{},[409,49826,16018],{},[412,49828,12984],{},[409,49830,20315],{},[447,49832,20786],{"encoding":449},[224,49834,49836,49854],{"className":49835,"ariaHidden":455},[454],[224,49837,49839,49842,49845,49848,49851],{"className":49838},[459],[224,49840],{"className":49841,"style":20796},[463],[224,49843,16018],{"className":49844,"style":653},[468,469],[224,49846],{"className":49847,"style":496},[473],[224,49849,12984],{"className":49850},[500],[224,49852],{"className":49853,"style":496},[473],[224,49855,49857,49860],{"className":49856},[459],[224,49858],{"className":49859,"style":20637},[463],[224,49861,20315],{"className":49862},[468,469]," 时",[224,49865,49867],{"className":49866,"translate":388},[387],[224,49868,49870,49998],{"className":49869,"translate":388},[392],[224,49871,49873],{"className":49872},[396],[398,49874,49875],{"xmlns":400,"display":401},[403,49876,49877,49995],{},[406,49878,49879,49889,49891,49893,49895,49897,49899,49901,49915,49917,49919,49921,49923,49925,49927,49933,49939,49941,49943,49945,49947,49949,49951,49957,49963,49965,49967,49969,49971,49973,49975,49977,49979,49981,49983,49985,49987,49989,49991,49993],{},[695,49880,49881,49883],{},[409,49882,20303],{},[406,49884,49885,49887],{},[409,49886,43681],{},[409,49888,6924],{},[412,49890,239],{"stretchy":435},[409,49892,16018],{},[409,49894,20312],{"mathvariant":678},[409,49896,20315],{},[412,49898,1716],{"stretchy":435},[412,49900,278],{},[695,49902,49903,49905],{},[409,49904,20303],{},[695,49906,49907,49909],{},[409,49908,1705],{},[406,49910,49911,49913],{},[409,49912,43681],{},[409,49914,6924],{},[412,49916,239],{"stretchy":435},[409,49918,16018],{},[409,49920,20312],{"mathvariant":678},[409,49922,20315],{},[412,49924,1716],{"stretchy":435},[412,49926,278],{},[20338,49928,49929,49931],{},[419,49930,1438],{},[419,49932,1447],{},[695,49934,49935,49937],{},[409,49936,20303],{},[409,49938,1705],{},[412,49940,239],{"stretchy":435},[409,49942,16018],{},[409,49944,20312],{"mathvariant":678},[409,49946,45500],{},[412,49948,1716],{"stretchy":435},[412,49950,319],{},[20338,49952,49953,49955],{},[419,49954,1438],{},[419,49956,1447],{},[695,49958,49959,49961],{},[409,49960,20303],{},[409,49962,1705],{},[412,49964,239],{"stretchy":435},[409,49966,20315],{},[409,49968,20312],{"mathvariant":678},[409,49970,45500],{},[412,49972,1716],{"stretchy":435},[412,49974,1877],{"separator":455},[473,49976],{"width":4247},[412,49978,239],{"stretchy":435},[409,49980,1705],{},[412,49982,278],{},[409,49984,1435],{},[409,49986,24131],{},[412,49988,24134],{},[409,49990,1435],{},[412,49992,1716],{"stretchy":435},[412,49994,1877],{"separator":455},[447,49996,49997],{"encoding":449},"D_{JS}(P\\|Q)=D_{f_{JS}}(P\\|Q)\n=\\frac12D_f(P\\|M)\n+\\frac12D_f(Q\\|M),\\qquad(f=t\\log t),",[224,49999,50001,50077,50197,50329,50476],{"className":50000,"ariaHidden":455},[454],[224,50002,50004,50007,50053,50056,50059,50062,50065,50068,50071,50074],{"className":50003},[459],[224,50005],{"className":50006,"style":535},[463],[224,50008,50010,50013],{"className":50009},[468],[224,50011,20303],{"className":50012,"style":2473},[468,469],[224,50014,50016],{"className":50015},[750],[224,50017,50019,50045],{"className":50018},[754,755],[224,50020,50022,50042],{"className":50021},[759],[224,50023,50025],{"className":50024,"style":764},[763],[224,50026,50027,50030],{"style":20390},[224,50028],{"className":50029,"style":772},[771],[224,50031,50033],{"className":50032},[776,777,778,779],[224,50034,50036,50039],{"className":50035},[468,779],[224,50037,43681],{"className":50038,"style":44091},[468,469,779],[224,50040,6924],{"className":50041,"style":6970},[468,469,779],[224,50043,787],{"className":50044},[786],[224,50046,50048],{"className":50047},[759],[224,50049,50051],{"className":50050,"style":794},[763],[224,50052],{},[224,50054,239],{"className":50055},[542],[224,50057,16018],{"className":50058,"style":653},[468,469],[224,50060,20312],{"className":50061},[468],[224,50063,20315],{"className":50064},[468,469],[224,50066,1716],{"className":50067},[549],[224,50069],{"className":50070,"style":496},[473],[224,50072,278],{"className":50073},[500],[224,50075],{"className":50076,"style":496},[473],[224,50078,50080,50083,50173,50176,50179,50182,50185,50188,50191,50194],{"className":50079},[459],[224,50081],{"className":50082,"style":20368},[463],[224,50084,50086,50089],{"className":50085},[468],[224,50087,20303],{"className":50088,"style":2473},[468,469],[224,50090,50092],{"className":50091},[750],[224,50093,50095,50165],{"className":50094},[754,755],[224,50096,50098,50162],{"className":50097},[759],[224,50099,50101],{"className":50100,"style":20387},[763],[224,50102,50103,50106],{"style":20390},[224,50104],{"className":50105,"style":772},[771],[224,50107,50109],{"className":50108},[776,777,778,779],[224,50110,50112],{"className":50111},[468,779],[224,50113,50115,50118],{"className":50114},[468,779],[224,50116,1705],{"className":50117,"style":1740},[468,469,779],[224,50119,50121],{"className":50120},[750],[224,50122,50124,50153],{"className":50123},[754,755],[224,50125,50127,50150],{"className":50126},[759],[224,50128,50130],{"className":50129,"style":26101},[763],[224,50131,50133,50137],{"style":50132},"top:-2.3567em;margin-left:-0.1076em;margin-right:0.0714em;",[224,50134],{"className":50135,"style":50136},[771],"height:2.5em;",[224,50138,50141],{"className":50139},[776,50140,2364,779],"reset-size3",[224,50142,50144,50147],{"className":50143},[468,779],[224,50145,43681],{"className":50146,"style":44091},[468,469,779],[224,50148,6924],{"className":50149,"style":6970},[468,469,779],[224,50151,787],{"className":50152},[786],[224,50154,50156],{"className":50155},[759],[224,50157,50160],{"className":50158,"style":50159},[763],"height:0.1433em;",[224,50161],{},[224,50163,787],{"className":50164},[786],[224,50166,50168],{"className":50167},[759],[224,50169,50171],{"className":50170,"style":20409},[763],[224,50172],{},[224,50174,239],{"className":50175},[542],[224,50177,16018],{"className":50178,"style":653},[468,469],[224,50180,20312],{"className":50181},[468],[224,50183,20315],{"className":50184},[468,469],[224,50186,1716],{"className":50187},[549],[224,50189],{"className":50190,"style":496},[473],[224,50192,278],{"className":50193},[500],[224,50195],{"className":50196,"style":496},[473],[224,50198,50200,50203,50265,50305,50308,50311,50314,50317,50320,50323,50326],{"className":50199},[459],[224,50201],{"className":50202,"style":45834},[463],[224,50204,50206,50209,50262],{"className":50205},[468],[224,50207],{"className":50208},[542,8479],[224,50210,50212],{"className":50211},[20338],[224,50213,50215,50254],{"className":50214},[754,755],[224,50216,50218,50251],{"className":50217},[759],[224,50219,50221,50232,50240],{"className":50220,"style":42827},[763],[224,50222,50223,50226],{"style":20531},[224,50224],{"className":50225,"style":2108},[771],[224,50227,50229],{"className":50228},[468],[224,50230,1447],{"className":50231},[468],[224,50233,50234,50237],{"style":20546},[224,50235],{"className":50236,"style":2108},[771],[224,50238],{"className":50239,"style":20554},[20553],[224,50241,50242,50245],{"style":20557},[224,50243],{"className":50244,"style":2108},[771],[224,50246,50248],{"className":50247},[468],[224,50249,1438],{"className":50250},[468],[224,50252,787],{"className":50253},[786],[224,50255,50257],{"className":50256},[759],[224,50258,50260],{"className":50259,"style":23809},[763],[224,50261],{},[224,50263],{"className":50264},[549,8479],[224,50266,50268,50271],{"className":50267},[468],[224,50269,20303],{"className":50270,"style":2473},[468,469],[224,50272,50274],{"className":50273},[750],[224,50275,50277,50297],{"className":50276},[754,755],[224,50278,50280,50294],{"className":50279},[759],[224,50281,50283],{"className":50282,"style":20387},[763],[224,50284,50285,50288],{"style":20390},[224,50286],{"className":50287,"style":772},[771],[224,50289,50291],{"className":50290},[776,777,778,779],[224,50292,1705],{"className":50293,"style":1740},[468,469,779],[224,50295,787],{"className":50296},[786],[224,50298,50300],{"className":50299},[759],[224,50301,50303],{"className":50302,"style":20409},[763],[224,50304],{},[224,50306,239],{"className":50307},[542],[224,50309,16018],{"className":50310,"style":653},[468,469],[224,50312,20312],{"className":50313},[468],[224,50315,45500],{"className":50316,"style":45534},[468,469],[224,50318,1716],{"className":50319},[549],[224,50321],{"className":50322,"style":474},[473],[224,50324,319],{"className":50325},[478],[224,50327],{"className":50328,"style":474},[473],[224,50330,50332,50335,50397,50437,50440,50443,50446,50449,50452,50455,50458,50461,50464,50467,50470,50473],{"className":50331},[459],[224,50333],{"className":50334,"style":45834},[463],[224,50336,50338,50341,50394],{"className":50337},[468],[224,50339],{"className":50340},[542,8479],[224,50342,50344],{"className":50343},[20338],[224,50345,50347,50386],{"className":50346},[754,755],[224,50348,50350,50383],{"className":50349},[759],[224,50351,50353,50364,50372],{"className":50352,"style":42827},[763],[224,50354,50355,50358],{"style":20531},[224,50356],{"className":50357,"style":2108},[771],[224,50359,50361],{"className":50360},[468],[224,50362,1447],{"className":50363},[468],[224,50365,50366,50369],{"style":20546},[224,50367],{"className":50368,"style":2108},[771],[224,50370],{"className":50371,"style":20554},[20553],[224,50373,50374,50377],{"style":20557},[224,50375],{"className":50376,"style":2108},[771],[224,50378,50380],{"className":50379},[468],[224,50381,1438],{"className":50382},[468],[224,50384,787],{"className":50385},[786],[224,50387,50389],{"className":50388},[759],[224,50390,50392],{"className":50391,"style":23809},[763],[224,50393],{},[224,50395],{"className":50396},[549,8479],[224,50398,50400,50403],{"className":50399},[468],[224,50401,20303],{"className":50402,"style":2473},[468,469],[224,50404,50406],{"className":50405},[750],[224,50407,50409,50429],{"className":50408},[754,755],[224,50410,50412,50426],{"className":50411},[759],[224,50413,50415],{"className":50414,"style":20387},[763],[224,50416,50417,50420],{"style":20390},[224,50418],{"className":50419,"style":772},[771],[224,50421,50423],{"className":50422},[776,777,778,779],[224,50424,1705],{"className":50425,"style":1740},[468,469,779],[224,50427,787],{"className":50428},[786],[224,50430,50432],{"className":50431},[759],[224,50433,50435],{"className":50434,"style":20409},[763],[224,50436],{},[224,50438,239],{"className":50439},[542],[224,50441,20315],{"className":50442},[468,469],[224,50444,20312],{"className":50445},[468],[224,50447,45500],{"className":50448,"style":45534},[468,469],[224,50450,1716],{"className":50451},[549],[224,50453,1877],{"className":50454},[1917],[224,50456],{"className":50457,"style":4416},[473],[224,50459],{"className":50460,"style":724},[473],[224,50462,239],{"className":50463},[542],[224,50465,1705],{"className":50466,"style":1740},[468,469],[224,50468],{"className":50469,"style":496},[473],[224,50471,278],{"className":50472},[500],[224,50474],{"className":50475,"style":496},[473],[224,50477,50479,50482,50485,50488,50493,50496,50499,50502],{"className":50478},[459],[224,50480],{"className":50481,"style":535},[463],[224,50483,1435],{"className":50484},[468,469],[224,50486],{"className":50487,"style":724},[473],[224,50489,24185,50491],{"className":50490},[21531],[224,50492,24189],{"style":24188},[224,50494],{"className":50495,"style":724},[473],[224,50497,1435],{"className":50498},[468,469],[224,50500,1716],{"className":50501},[549],[224,50503,1877],{"className":50504},[1917],[11,50506,50507,50508,50536,50537,50606,50607,50742],{},"两种写法完全一致：JS 既是「两个 f-散度（相对 ",[224,50509,50511,50524],{"className":50510,"translate":388},[392],[224,50512,50514],{"className":50513},[396],[398,50515,50516],{"xmlns":400},[403,50517,50518,50522],{},[406,50519,50520],{},[409,50521,45500],{},[447,50523,45500],{"encoding":449},[224,50525,50527],{"className":50526,"ariaHidden":455},[454],[224,50528,50530,50533],{"className":50529},[459],[224,50531],{"className":50532,"style":649},[463],[224,50534,45500],{"className":50535,"style":45534},[468,469],"）的平均」，也是「一个对称 f-散度」。原文「不是单一 ",[224,50538,50540,50557],{"className":50539,"translate":388},[392],[224,50541,50543],{"className":50542},[396],[398,50544,50545],{"xmlns":400},[403,50546,50547,50555],{},[406,50548,50549],{},[695,50550,50551,50553],{},[409,50552,20303],{},[409,50554,1705],{},[447,50556,24053],{"encoding":449},[224,50558,50560],{"className":50559,"ariaHidden":455},[454],[224,50561,50563,50566],{"className":50562},[459],[224,50564],{"className":50565,"style":24063},[463],[224,50567,50569,50572],{"className":50568},[468],[224,50570,20303],{"className":50571,"style":2473},[468,469],[224,50573,50575],{"className":50574},[750],[224,50576,50578,50598],{"className":50577},[754,755],[224,50579,50581,50595],{"className":50580},[759],[224,50582,50584],{"className":50583,"style":20387},[763],[224,50585,50586,50589],{"style":20390},[224,50587],{"className":50588,"style":772},[771],[224,50590,50592],{"className":50591},[776,777,778,779],[224,50593,1705],{"className":50594,"style":1740},[468,469,779],[224,50596,787],{"className":50597},[786],[224,50599,50601],{"className":50600},[759],[224,50602,50604],{"className":50603,"style":20409},[763],[224,50605],{},"」的说法可修正为：",[378,50608,50609,50610,50660,50661,50741],{},"在 ",[224,50611,50613,50630],{"className":50612,"translate":388},[392],[224,50614,50616],{"className":50615},[396],[398,50617,50618],{"xmlns":400},[403,50619,50620,50628],{},[406,50621,50622,50624,50626],{},[409,50623,16018],{},[412,50625,12984],{},[409,50627,20315],{},[447,50629,20786],{"encoding":449},[224,50631,50633,50651],{"className":50632,"ariaHidden":455},[454],[224,50634,50636,50639,50642,50645,50648],{"className":50635},[459],[224,50637],{"className":50638,"style":20796},[463],[224,50640,16018],{"className":50641,"style":653},[468,469],[224,50643],{"className":50644,"style":496},[473],[224,50646,12984],{"className":50647},[500],[224,50649],{"className":50650,"style":496},[473],[224,50652,50654,50657],{"className":50653},[459],[224,50655],{"className":50656,"style":20637},[463],[224,50658,20315],{"className":50659},[468,469]," 时它是单一 ",[224,50662,50664,50686],{"className":50663,"translate":388},[392],[224,50665,50667],{"className":50666},[396],[398,50668,50669],{"xmlns":400},[403,50670,50671,50683],{},[406,50672,50673],{},[695,50674,50675,50677],{},[409,50676,1705],{},[406,50678,50679,50681],{},[409,50680,43681],{},[409,50682,6924],{},[447,50684,50685],{"encoding":449},"f_{JS}",[224,50687,50689],{"className":50688,"ariaHidden":455},[454],[224,50690,50692,50695],{"className":50691},[459],[224,50693],{"className":50694,"style":2565},[463],[224,50696,50698,50701],{"className":50697},[468],[224,50699,1705],{"className":50700,"style":1740},[468,469],[224,50702,50704],{"className":50703},[750],[224,50705,50707,50733],{"className":50706},[754,755],[224,50708,50710,50730],{"className":50709},[759],[224,50711,50713],{"className":50712,"style":764},[763],[224,50714,50715,50718],{"style":47439},[224,50716],{"className":50717,"style":772},[771],[224,50719,50721],{"className":50720},[776,777,778,779],[224,50722,50724,50727],{"className":50723},[468,779],[224,50725,43681],{"className":50726,"style":44091},[468,469,779],[224,50728,6924],{"className":50729,"style":6970},[468,469,779],[224,50731,787],{"className":50732},[786],[224,50734,50736],{"className":50735},[759],[224,50737,50739],{"className":50738,"style":794},[763],[224,50740],{}," 的 f-散度","；一般情形则用下面的双变量形式。",[11,50744,50745,50748,50749,50790,50791,50906,50907,50910],{},[378,50746,50747],{},"一般情形的双变量形式."," 当 ",[224,50750,50752,50769],{"className":50751,"translate":388},[392],[224,50753,50755],{"className":50754},[396],[398,50756,50757],{"xmlns":400},[403,50758,50759,50767],{},[406,50760,50761,50763,50765],{},[409,50762,16018],{},[412,50764,1877],{"separator":455},[409,50766,20315],{},[447,50768,20627],{"encoding":449},[224,50770,50772],{"className":50771,"ariaHidden":455},[454],[224,50773,50775,50778,50781,50784,50787],{"className":50774},[459],[224,50776],{"className":50777,"style":20637},[463],[224,50779,16018],{"className":50780,"style":653},[468,469],[224,50782,1877],{"className":50783},[1917],[224,50785],{"className":50786,"style":724},[473],[224,50788,20315],{"className":50789},[468,469]," 支撑集可能分离时，",[224,50792,50794,50830],{"className":50793,"translate":388},[392],[224,50795,50797],{"className":50796},[396],[398,50798,50799],{"xmlns":400},[403,50800,50801,50827],{},[406,50802,50803,50805,50807,50809,50819,50821,50823,50825],{},[412,50804,239],{"stretchy":435},[409,50806,1435],{},[412,50808,1877],{"separator":455},[695,50810,50811,50813],{},[409,50812,1705],{},[406,50814,50815,50817],{},[409,50816,43681],{},[409,50818,6924],{},[412,50820,239],{"stretchy":435},[409,50822,1435],{},[412,50824,1716],{"stretchy":435},[412,50826,1716],{"stretchy":435},[447,50828,50829],{"encoding":449},"(t,f_{JS}(t))",[224,50831,50833],{"className":50832,"ariaHidden":455},[454],[224,50834,50836,50839,50842,50845,50848,50851,50897,50900,50903],{"className":50835},[459],[224,50837],{"className":50838,"style":535},[463],[224,50840,239],{"className":50841},[542],[224,50843,1435],{"className":50844},[468,469],[224,50846,1877],{"className":50847},[1917],[224,50849],{"className":50850,"style":724},[473],[224,50852,50854,50857],{"className":50853},[468],[224,50855,1705],{"className":50856,"style":1740},[468,469],[224,50858,50860],{"className":50859},[750],[224,50861,50863,50889],{"className":50862},[754,755],[224,50864,50866,50886],{"className":50865},[759],[224,50867,50869],{"className":50868,"style":764},[763],[224,50870,50871,50874],{"style":47439},[224,50872],{"className":50873,"style":772},[771],[224,50875,50877],{"className":50876},[776,777,778,779],[224,50878,50880,50883],{"className":50879},[468,779],[224,50881,43681],{"className":50882,"style":44091},[468,469,779],[224,50884,6924],{"className":50885,"style":6970},[468,469,779],[224,50887,787],{"className":50888},[786],[224,50890,50892],{"className":50891},[759],[224,50893,50895],{"className":50894,"style":794},[763],[224,50896],{},[224,50898,239],{"className":50899},[542],[224,50901,1435],{"className":50902},[468,469],[224,50904,38989],{"className":50905},[549]," 的写法会漏掉奇异项。此时把 JS 写成 Csiszár 的",[378,50908,50909],{},"双变量（齐次、联合凸）泛函","最稳妥：",[224,50912,50914],{"className":50913,"translate":388},[387],[224,50915,50917,50973],{"className":50916,"translate":388},[392],[224,50918,50920],{"className":50919},[396],[398,50921,50922],{"xmlns":400,"display":401},[403,50923,50924,50970],{},[406,50925,50926,50936,50938,50940,50942,50944,50946,50948,50950,50952,50954,50956,50958,50960,50962,50964,50966,50968],{},[695,50927,50928,50930],{},[409,50929,20303],{},[406,50931,50932,50934],{},[409,50933,43681],{},[409,50935,6924],{},[412,50937,239],{"stretchy":435},[409,50939,16018],{},[409,50941,20312],{"mathvariant":678},[409,50943,20315],{},[412,50945,1716],{"stretchy":435},[412,50947,278],{},[412,50949,21338],{},[409,50951,704],{},[412,50953,239],{"stretchy":435},[409,50955,11],{},[412,50957,1877],{"separator":455},[409,50959,2023],{},[412,50961,1716],{"stretchy":435},[423,50963,682],{},[409,50965,13182],{},[409,50967,22405],{},[412,50969,1877],{"separator":455},[447,50971,50972],{"encoding":449},"D_{JS}(P\\|Q)=\\int 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联合凸且一次齐次。在 ",[224,51591,51593,51611],{"className":51592,"translate":388},[392],[224,51594,51596],{"className":51595},[396],[398,51597,51598],{"xmlns":400},[403,51599,51600,51608],{},[406,51601,51602,51604,51606],{},[409,51603,2023],{},[412,51605,325],{},[419,51607,284],{},[447,51609,51610],{"encoding":449},"q>0",[224,51612,51614,51632],{"className":51613,"ariaHidden":455},[454],[224,51615,51617,51620,51623,51626,51629],{"className":51616},[459],[224,51618],{"className":51619,"style":488},[463],[224,51621,2023],{"className":51622,"style":492},[468,469],[224,51624],{"className":51625,"style":496},[473],[224,51627,325],{"className":51628},[500],[224,51630],{"className":51631,"style":496},[473],[224,51633,51635,51638],{"className":51634},[459],[224,51636],{"className":51637,"style":510},[463],[224,51639,284],{"className":51640},[468]," 处 ",[224,51643,51645,51693],{"className":51644,"translate":388},[392],[224,51646,51648],{"className":51647},[396],[398,51649,51650],{"xmlns":400},[403,51651,51652,51690],{},[406,51653,51654,51656,51658,51660,51662,51664,51666,51668,51670,51672,51674,51684,51686,51688],{},[409,51655,704],{},[412,51657,239],{"stretchy":435},[409,51659,1435],{},[409,51661,2023],{},[412,51663,1877],{"separator":455},[409,51665,2023],{},[412,51667,1716],{"stretchy":435},[412,51669,278],{},[409,51671,2023],{},[423,51673,682],{},[695,51675,51676,51678],{},[409,51677,1705],{},[406,51679,51680,51682],{},[409,51681,43681],{},[409,51683,6924],{},[412,51685,239],{"stretchy":435},[409,51687,1435],{},[412,51689,1716],{"stretchy":435},[447,51691,51692],{"encoding":449},"\\varphi(tq,q)=q\\,f_{JS}(t)",[224,51694,51696,51732],{"className":51695,"ariaHidden":455},[454],[224,51697,51699,51702,51705,51708,51711,51714,51717,51720,51723,51726,51729],{"className":51698},[459],[224,51700],{"className":51701,"style":535},[463],[224,51703,704],{"className":51704},[468,469],[224,51706,239],{"className":51707},[542],[224,51709,46209],{"className":51710,"style":492},[468,469],[224,51712,1877],{"className":51713},[1917],[224,51715],{"className":51716,"style":724},[473],[224,51718,2023],{"className":51719,"style":492},[468,469],[224,51721,1716],{"className":51722},[549],[224,51724],{"className":51725,"style":496},[473],[224,51727,278],{"className":51728},[500],[224,51730],{"className":51731,"style":496},[473],[224,51733,51735,51738,51741,51744,51790,51793,51796],{"className":51734},[459],[224,51736],{"className":51737,"style":535},[463],[224,51739,2023],{"className":51740,"style":492},[468,469],[224,51742],{"className":51743,"style":724},[473],[224,51745,51747,51750],{"className":51746},[468],[224,51748,1705],{"className":51749,"style":1740},[468,469],[224,51751,51753],{"className":51752},[750],[224,51754,51756,51782],{"className":51755},[754,755],[224,51757,51759,51779],{"className":51758},[759],[224,51760,51762],{"className":51761,"style":764},[763],[224,51763,51764,51767],{"style":47439},[224,51765],{"className":51766,"style":772},[771],[224,51768,51770],{"className":51769},[776,777,778,779],[224,51771,51773,51776],{"className":51772},[468,779],[224,51774,43681],{"className":51775,"style":44091},[468,469,779],[224,51777,6924],{"className":51778,"style":6970},[468,469,779],[224,51780,787],{"className":51781},[786],[224,51783,51785],{"className":51784},[759],[224,51786,51788],{"className":51787,"style":794},[763],[224,51789],{},[224,51791,239],{"className":51792},[542],[224,51794,1435],{"className":51795},[468,469],[224,51797,1716],{"className":51798},[549],"，两者一致；在 ",[224,51801,51803,51829],{"className":51802,"translate":388},[392],[224,51804,51806],{"className":51805},[396],[398,51807,51808],{"xmlns":400},[403,51809,51810,51826],{},[406,51811,51812,51814,51816,51818,51820,51822,51824],{},[409,51813,2023],{},[412,51815,278],{},[419,51817,284],{},[412,51819,1877],{"separator":455},[409,51821,11],{},[412,51823,325],{},[419,51825,284],{},[447,51827,51828],{"encoding":449},"q=0,p>0",[224,51830,51832,51850,51877],{"className":51831,"ariaHidden":455},[454],[224,51833,51835,51838,51841,51844,51847],{"className":51834},[459],[224,51836],{"className":51837,"style":806},[463],[224,51839,2023],{"className":51840,"style":492},[468,469],[224,51842],{"className":51843,"style":496},[473],[224,51845,278],{"className":51846},[500],[224,51848],{"className":51849,"style":496},[473],[224,51851,51853,51856,51859,51862,51865,51868,51871,51874],{"className":51852},[459],[224,51854],{"className":51855,"style":8688},[463],[224,51857,284],{"className":51858},[468],[224,51860,1877],{"className":51861},[1917],[224,51863],{"className":51864,"style":724},[473],[224,51866,11],{"className":51867},[468,469],[224,51869],{"className":51870,"style":496},[473],[224,51872,325],{"className":51873},[500],[224,51875],{"className":51876,"style":496},[473],[224,51878,51880,51883],{"className":51879},[459],[224,51881],{"className":51882,"style":510},[463],[224,51884,284],{"className":51885},[468]," 处奇异项 ",[224,51888,51890,51916],{"className":51889,"translate":388},[392],[224,51891,51893],{"className":51892},[396],[398,51894,51895],{"xmlns":400},[403,51896,51897,51913],{},[406,51898,51899,51905,51907,51909,51911],{},[20338,51900,51901,51903],{},[419,51902,1438],{},[419,51904,1447],{},[409,51906,11],{},[409,51908,24131],{},[412,51910,24134],{},[419,51912,1447],{},[447,51914,51915],{"encoding":449},"\\frac12 p\\log 2",[224,51917,51919],{"className":51918,"ariaHidden":455},[454],[224,51920,51922,51925,51993,51996,51999,52004,52007],{"className":51921},[459],[224,51923],{"className":51924,"style":22544},[463],[224,51926,51928,51931,51990],{"className":51927},[468],[224,51929],{"className":51930},[542,8479],[224,51932,51934],{"className":51933},[20338],[224,51935,51937,51982],{"className":51936},[754,755],[224,51938,51940,51979],{"className":51939},[759],[224,51941,51943,51957,51965],{"className":51942,"style":22563},[763],[224,51944,51945,51948],{"style":22566},[224,51946],{"className":51947,"style":2108},[771],[224,51949,51951],{"className":51950},[776,777,778,779],[224,51952,51954],{"className":51953},[468,779],[224,51955,1447],{"className":51956},[468,779],[224,51958,51959,51962],{"style":20546},[224,51960],{"className":51961,"style":2108},[771],[224,51963],{"className":51964,"style":20554},[20553],[224,51966,51967,51970],{"style":22589},[224,51968],{"className":51969,"style":2108},[771],[224,51971,51973],{"className":51972},[776,777,778,779],[224,51974,51976],{"className":51975},[468,779],[224,51977,1438],{"className":51978},[468,779],[224,51980,787],{"className":51981},[786],[224,51983,51985],{"className":51984},[759],[224,51986,51988],{"className":51987,"style":22611},[763],[224,51989],{},[224,51991],{"className":51992},[549,8479],[224,51994,11],{"className":51995},[468,469],[224,51997],{"className":51998,"style":724},[473],[224,52000,24185,52002],{"className":52001},[21531],[224,52003,24189],{"style":24188},[224,52005],{"className":52006,"style":724},[473],[224,52008,1447],{"className":52009},[468]," 会被保留下来（这正是 1.1 节讨论的「支撑不重叠」约定）。",[11,52012,52013,52016,52017,52096,52097,35740,52100,12708,52151,52384,52385,52413,52414,52442,52443,52630,52631,52726],{},[378,52014,52015],{},"有界性。"," 注意 ",[224,52018,52020,52041],{"className":52019,"translate":388},[392],[224,52021,52023],{"className":52022},[396],[398,52024,52025],{"xmlns":400},[403,52026,52027,52039],{},[406,52028,52029],{},[695,52030,52031,52033],{},[409,52032,1705],{},[406,52034,52035,52037],{},[409,52036,43681],{},[409,52038,6924],{},[447,52040,50685],{"encoding":449},[224,52042,52044],{"className":52043,"ariaHidden":455},[454],[224,52045,52047,52050],{"className":52046},[459],[224,52048],{"className":52049,"style":2565},[463],[224,52051,52053,52056],{"className":52052},[468],[224,52054,1705],{"className":52055,"style":1740},[468,469],[224,52057,52059],{"className":52058},[750],[224,52060,52062,52088],{"className":52061},[754,755],[224,52063,52065,52085],{"className":52064},[759],[224,52066,52068],{"className":52067,"style":764},[763],[224,52069,52070,52073],{"style":47439},[224,52071],{"className":52072,"style":772},[771],[224,52074,52076],{"className":52075},[776,777,778,779],[224,52077,52079,52082],{"className":52078},[468,779],[224,52080,43681],{"className":52081,"style":44091},[468,469,779],[224,52083,6924],{"className":52084,"style":6970},[468,469,779],[224,52086,787],{"className":52087},[786],[224,52089,52091],{"className":52090},[759],[224,52092,52094],{"className":52093,"style":794},[763],[224,52095],{}," 逐点",[378,52098,52099],{},"无界",[224,52101,52103,52121],{"className":52102,"translate":388},[392],[224,52104,52106],{"className":52105},[396],[398,52107,52108],{"xmlns":400},[403,52109,52110,52118],{},[406,52111,52112,52114,52116],{},[409,52113,1435],{},[412,52115,1144],{},[409,52117,21085],{"mathvariant":678},[447,52119,52120],{"encoding":449},"t\\to\\infty",[224,52122,52124,52142],{"className":52123,"ariaHidden":455},[454],[224,52125,52127,52130,52133,52136,52139],{"className":52126},[459],[224,52128],{"className":52129,"style":25353},[463],[224,52131,1435],{"className":52132},[468,469],[224,52134],{"className":52135,"style":496},[473],[224,52137,1144],{"className":52138},[500],[224,52140],{"className":52141,"style":496},[473],[224,52143,52145,52148],{"className":52144},[459],[224,52146],{"className":52147,"style":839},[463],[224,52149,21085],{"className":52150},[468],[224,52152,52154,52202],{"className":52153,"translate":388},[392],[224,52155,52157],{"className":52156},[396],[398,52158,52159],{"xmlns":400},[403,52160,52161,52199],{},[406,52162,52163,52173,52175,52177,52179,52181,52187,52189,52191,52193,52195,52197],{},[695,52164,52165,52167],{},[409,52166,1705],{},[406,52168,52169,52171],{},[409,52170,43681],{},[409,52172,6924],{},[412,52174,239],{"stretchy":435},[409,52176,1435],{},[412,52178,1716],{"stretchy":435},[412,52180,28834],{},[20338,52182,52183,52185],{},[419,52184,1438],{},[419,52186,1447],{},[409,52188,1435],{},[409,52190,24131],{},[412,52192,24134],{},[419,52194,1447],{},[412,52196,1144],{},[409,52198,21085],{"mathvariant":678},[447,52200,52201],{"encoding":449},"f_{JS}(t)\\sim\\frac12 t\\log 2\\to\\infty",[224,52203,52205,52275,52375],{"className":52204,"ariaHidden":455},[454],[224,52206,52208,52211,52257,52260,52263,52266,52269,52272],{"className":52207},[459],[224,52209],{"className":52210,"style":535},[463],[224,52212,52214,52217],{"className":52213},[468],[224,52215,1705],{"className":52216,"style":1740},[468,469],[224,52218,52220],{"className":52219},[750],[224,52221,52223,52249],{"className":52222},[754,755],[224,52224,52226,52246],{"className":52225},[759],[224,52227,52229],{"className":52228,"style":764},[763],[224,52230,52231,52234],{"style":47439},[224,52232],{"className":52233,"style":772},[771],[224,52235,52237],{"className":52236},[776,777,778,779],[224,52238,52240,52243],{"className":52239},[468,779],[224,52241,43681],{"className":52242,"style":44091},[468,469,779],[224,52244,6924],{"className":52245,"style":6970},[468,469,779],[224,52247,787],{"className":52248},[786],[224,52250,52252],{"className":52251},[759],[224,52253,52255],{"className":52254,"style":794},[763],[224,52256],{},[224,52258,239],{"className":52259},[542],[224,52261,1435],{"className":52262},[468,469],[224,52264,1716],{"className":52265},[549],[224,52267],{"className":52268,"style":496},[473],[224,52270,28834],{"className":52271},[500],[224,52273],{"className":52274,"style":496},[473],[224,52276,52278,52281,52349,52352,52355,52360,52363,52366,52369,52372],{"className":52277},[459],[224,52279],{"className":52280,"style":22544},[463],[224,52282,52284,52287,52346],{"className":52283},[468],[224,52285],{"className":52286},[542,8479],[224,52288,52290],{"className":52289},[20338],[224,52291,52293,52338],{"className":52292},[754,755],[224,52294,52296,52335],{"className":52295},[759],[224,52297,52299,52313,52321],{"className":52298,"style":22563},[763],[224,52300,52301,52304],{"style":22566},[224,52302],{"className":52303,"style":2108},[771],[224,52305,52307],{"className":52306},[776,777,778,779],[224,52308,52310],{"className":52309},[468,779],[224,52311,1447],{"className":52312},[468,779],[224,52314,52315,52318],{"style":20546},[224,52316],{"className":52317,"style":2108},[771],[224,52319],{"className":52320,"style":20554},[20553],[224,52322,52323,52326],{"style":22589},[224,52324],{"className":52325,"style":2108},[771],[224,52327,52329],{"className":52328},[776,777,778,779],[224,52330,52332],{"className":52331},[468,779],[224,52333,1438],{"className":52334},[468,779],[224,52336,787],{"className":52337},[786],[224,52339,52341],{"className":52340},[759],[224,52342,52344],{"className":52343,"style":22611},[763],[224,52345],{},[224,52347],{"className":52348},[549,8479],[224,52350,1435],{"className":52351},[468,469],[224,52353],{"className":52354,"style":724},[473],[224,52356,24185,52358],{"className":52357},[21531],[224,52359,24189],{"style":24188},[224,52361],{"className":52362,"style":724},[473],[224,52364,1447],{"className":52365},[468],[224,52367],{"className":52368,"style":496},[473],[224,52370,1144],{"className":52371},[500],[224,52373],{"className":52374,"style":496},[473],[224,52376,52378,52381],{"className":52377},[459],[224,52379],{"className":52380,"style":839},[463],[224,52382,21085],{"className":52383},[468],"），但积分有界，因为 ",[224,52386,52388,52401],{"className":52387,"translate":388},[392],[224,52389,52391],{"className":52390},[396],[398,52392,52393],{"xmlns":400},[403,52394,52395,52399],{},[406,52396,52397],{},[409,52398,1435],{},[447,52400,1435],{"encoding":449},[224,52402,52404],{"className":52403,"ariaHidden":455},[454],[224,52405,52407,52410],{"className":52406},[459],[224,52408],{"className":52409,"style":25353},[463],[224,52411,1435],{"className":52412},[468,469]," 大的地方 ",[224,52415,52417,52430],{"className":52416,"translate":388},[392],[224,52418,52420],{"className":52419},[396],[398,52421,52422],{"xmlns":400},[403,52423,52424,52428],{},[406,52425,52426],{},[409,52427,2023],{},[447,52429,2023],{"encoding":449},[224,52431,52433],{"className":52432,"ariaHidden":455},[454],[224,52434,52436,52439],{"className":52435},[459],[224,52437],{"className":52438,"style":806},[463],[224,52440,2023],{"className":52441,"style":492},[468,469]," 小、权重趋于 0。严格地用两 KL 形式：",[224,52444,52446,52504],{"className":52445,"translate":388},[392],[224,52447,52449],{"className":52448},[396],[398,52450,52451],{"xmlns":400},[403,52452,52453,52501],{},[406,52454,52455,52457,52459,52461,52463,52465,52467,52469,52471,52473,52475,52477,52483,52485,52487,52489,52491,52493,52495,52497,52499],{},[419,52456,1447],{},[409,52458,11],{},[409,52460,12413],{"mathvariant":678},[412,52462,239],{"stretchy":435},[409,52464,11],{},[412,52466,319],{},[409,52468,2023],{},[412,52470,1716],{"stretchy":435},[412,52472,2070],{},[419,52474,1447],{},[412,52476,9886],{},[406,52478,52479,52481],{},[409,52480,30652],{"mathvariant":678},[409,52482,41831],{"mathvariant":678},[412,52484,239],{"stretchy":435},[409,52486,16018],{},[409,52488,20312],{"mathvariant":678},[409,52490,45500],{},[412,52492,1716],{"stretchy":435},[412,52494,2070],{},[409,52496,24131],{},[412,52498,24134],{},[419,52500,1447],{},[447,52502,52503],{"encoding":449},"2p\u002F(p+q)\\le 2\\Rightarrow \\mathrm{KL}(P\\|M)\\le\\log2",[224,52505,52507,52537,52558,52576,52613],{"className":52506,"ariaHidden":455},[454],[224,52508,52510,52513,52516,52519,52522,52525,52528,52531,52534],{"className":52509},[459],[224,52511],{"className":52512,"style":535},[463],[224,52514,1447],{"className":52515},[468],[224,52517,11],{"className":52518},[468,469],[224,52520,12413],{"className":52521},[468],[224,52523,239],{"className":52524},[542],[224,52526,11],{"className":52527},[468,469],[224,52529],{"className":52530,"style":474},[473],[224,52532,319],{"className":52533},[478],[224,52535],{"className":52536,"style":474},[473],[224,52538,52540,52543,52546,52549,52552,52555],{"className":52539},[459],[224,52541],{"className":52542,"style":535},[463],[224,52544,2023],{"className":52545,"style":492},[468,469],[224,52547,1716],{"className":52548},[549],[224,52550],{"className":52551,"style":496},[473],[224,52553,2070],{"className":52554},[500],[224,52556],{"className":52557,"style":496},[473],[224,52559,52561,52564,52567,52570,52573],{"className":52560},[459],[224,52562],{"className":52563,"style":510},[463],[224,52565,1447],{"className":52566},[468],[224,52568],{"className":52569,"style":496},[473],[224,52571,9886],{"className":52572},[500],[224,52574],{"className":52575,"style":496},[473],[224,52577,52579,52582,52589,52592,52595,52598,52601,52604,52607,52610],{"className":52578},[459],[224,52580],{"className":52581,"style":535},[463],[224,52583,52585],{"className":52584},[468],[224,52586,52588],{"className":52587},[468,1906],"KL",[224,52590,239],{"className":52591},[542],[224,52593,16018],{"className":52594,"style":653},[468,469],[224,52596,20312],{"className":52597},[468],[224,52599,45500],{"className":52600,"style":45534},[468,469],[224,52602,1716],{"className":52603},[549],[224,52605],{"className":52606,"style":496},[473],[224,52608,2070],{"className":52609},[500],[224,52611],{"className":52612,"style":496},[473],[224,52614,52616,52619,52624,52627],{"className":52615},[459],[224,52617],{"className":52618,"style":2565},[463],[224,52620,24185,52622],{"className":52621},[21531],[224,52623,24189],{"style":24188},[224,52625],{"className":52626,"style":724},[473],[224,52628,1447],{"className":52629},[468],"，同理 ",[224,52632,52634,52670],{"className":52633,"translate":388},[392],[224,52635,52637],{"className":52636},[396],[398,52638,52639],{"xmlns":400},[403,52640,52641,52667],{},[406,52642,52643,52649,52651,52653,52655,52657,52659,52661,52663,52665],{},[406,52644,52645,52647],{},[409,52646,30652],{"mathvariant":678},[409,52648,41831],{"mathvariant":678},[412,52650,239],{"stretchy":435},[409,52652,20315],{},[409,52654,20312],{"mathvariant":678},[409,52656,45500],{},[412,52658,1716],{"stretchy":435},[412,52660,2070],{},[409,52662,24131],{},[412,52664,24134],{},[419,52666,1447],{},[447,52668,52669],{"encoding":449},"\\mathrm{KL}(Q\\|M)\\le\\log2",[224,52671,52673,52709],{"className":52672,"ariaHidden":455},[454],[224,52674,52676,52679,52685,52688,52691,52694,52697,52700,52703,52706],{"className":52675},[459],[224,52677],{"className":52678,"style":535},[463],[224,52680,52682],{"className":52681},[468],[224,52683,52588],{"className":52684},[468,1906],[224,52686,239],{"className":52687},[542],[224,52689,20315],{"className":52690},[468,469],[224,52692,20312],{"className":52693},[468],[224,52695,45500],{"className":52696,"style":45534},[468,469],[224,52698,1716],{"className":52699},[549],[224,52701],{"className":52702,"style":496},[473],[224,52704,2070],{"className":52705},[500],[224,52707],{"className":52708,"style":496},[473],[224,52710,52712,52715,52720,52723],{"className":52711},[459],[224,52713],{"className":52714,"style":2565},[463],[224,52716,24185,52718],{"className":52717},[21531],[224,52719,24189],{"style":24188},[224,52721],{"className":52722,"style":724},[473],[224,52724,1447],{"className":52725},[468],"，故",[224,52728,52730],{"className":52729,"translate":388},[387],[224,52731,52733,52891],{"className":52732,"translate":388},[392],[224,52734,52736],{"className":52735},[396],[398,52737,52738],{"xmlns":400,"display":401},[403,52739,52740,52888],{},[5209,52741,52742,52830],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,52743,52744,52774],{},[5218,52745,52746],{},[5221,52747,52748],{"scriptlevel":284,"displaystyle":455},[406,52749,52750,52752,52754,52764,52766,52768,52770,52772],{},[419,52751,284],{},[412,52753,2070],{},[695,52755,52756,52758],{},[409,52757,20303],{},[406,52759,52760,52762],{},[409,52761,43681],{},[409,52763,6924],{},[412,52765,239],{"stretchy":435},[409,52767,16018],{},[409,52769,20312],{"mathvariant":678},[409,52771,20315],{},[412,52773,1716],{"stretchy":435},[5218,52775,52776],{},[5221,52777,52778],{"scriptlevel":284,"displaystyle":455},[406,52779,52780,52782,52784,52790,52796,52798,52800,52802,52804,52806,52808,52814,52820,52822,52824,52826,52828],{},[406,52781],{},[412,52783,278],{},[20338,52785,52786,52788],{},[419,52787,1438],{},[419,52789,1447],{},[406,52791,52792,52794],{},[409,52793,30652],{"mathvariant":678},[409,52795,41831],{"mathvariant":678},[412,52797,239],{"stretchy":435},[409,52799,16018],{},[409,52801,20312],{"mathvariant":678},[409,52803,45500],{},[412,52805,1716],{"stretchy":435},[412,52807,319],{},[20338,52809,52810,52812],{},[419,52811,1438],{},[419,52813,1447],{},[406,52815,52816,52818],{},[409,52817,30652],{"mathvariant":678},[409,52819,41831],{"mathvariant":678},[412,52821,239],{"stretchy":435},[409,52823,20315],{},[409,52825,20312],{"mathvariant":678},[409,52827,45500],{},[412,52829,1716],{"stretchy":435},[5215,52831,52832,52838],{},[5218,52833,52834],{},[5221,52835,52836],{"scriptlevel":284,"displaystyle":455},[406,52837],{},[5218,52839,52840],{},[5221,52841,52842],{"scriptlevel":284,"displaystyle":455},[406,52843,52844,52846,52848,52854,52856,52858,52860,52862,52868,52870,52872,52874,52876,52878,52880,52882,52885],{},[406,52845],{},[412,52847,2070],{},[20338,52849,52850,52852],{},[419,52851,1438],{},[419,52853,1447],{},[409,52855,24131],{},[412,52857,24134],{},[419,52859,1447],{},[412,52861,319],{},[20338,52863,52864,52866],{},[419,52865,1438],{},[419,52867,1447],{},[409,52869,24131],{},[412,52871,24134],{},[419,52873,1447],{},[412,52875,278],{},[409,52877,24131],{},[412,52879,24134],{},[419,52881,1447],{},[412,52883,52884],{},"≈",[419,52886,52887],{},"0.693.",[447,52889,52890],{"encoding":449},"\\begin{aligned}\n0\\le D_{JS}(P\\|Q)\n&=\\frac12\\mathrm{KL}(P\\|M)\n+\\frac12\\mathrm{KL}(Q\\|M)\\\\[2pt]\n&\\le \\frac12\\log 2+\\frac12\\log 2\n=\\log 2\\approx 0.693 .\n\\end{aligned}",[224,52892,52894],{"className":52893,"ariaHidden":455},[454],[224,52895,52897,52901],{"className":52896},[459],[224,52898],{"className":52899,"style":52900},[463],"height:4.3149em;vertical-align:-1.9074em;",[224,52902,52904],{"className":52903},[468],[224,52905,52907,53022],{"className":52906},[5209],[224,52908,52910],{"className":52909},[5331],[224,52911,52913,53013],{"className":52912},[754,755],[224,52914,52916,53010],{"className":52915},[759],[224,52917,52920,53002],{"className":52918,"style":52919},[763],"height:2.4074em;",[224,52921,52923,52926],{"style":52922},"top:-4.4074em;",[224,52924],{"className":52925,"style":42604},[771],[224,52927,52929,52932,52935,52938,52941,52987,52990,52993,52996,52999],{"className":52928},[468],[224,52930,284],{"className":52931},[468],[224,52933],{"className":52934,"style":496},[473],[224,52936,2070],{"className":52937},[500],[224,52939],{"className":52940,"style":496},[473],[224,52942,52944,52947],{"className":52943},[468],[224,52945,20303],{"className":52946,"style":2473},[468,469],[224,52948,52950],{"className":52949},[750],[224,52951,52953,52979],{"className":52952},[754,755],[224,52954,52956,52976],{"className":52955},[759],[224,52957,52959],{"className":52958,"style":764},[763],[224,52960,52961,52964],{"style":20390},[224,52962],{"className":52963,"style":772},[771],[224,52965,52967],{"className":52966},[776,777,778,779],[224,52968,52970,52973],{"className":52969},[468,779],[224,52971,43681],{"className":52972,"style":44091},[468,469,779],[224,52974,6924],{"className":52975,"style":6970},[468,469,779],[224,52977,787],{"className":52978},[786],[224,52980,52982],{"className":52981},[759],[224,52983,52985],{"className":52984,"style":794},[763],[224,52986],{},[224,52988,239],{"className":52989},[542],[224,52991,16018],{"className":52992,"style":653},[468,469],[224,52994,20312],{"className":52995},[468],[224,52997,20315],{"className":52998},[468,469],[224,53000,1716],{"className":53001},[549],[224,53003,53004,53007],{"style":21950},[224,53005],{"className":53006,"style":42604},[771],[224,53008],{"className":53009},[468],[224,53011,787],{"className":53012},[786],[224,53014,53016],{"className":53015},[759],[224,53017,53020],{"className":53018,"style":53019},[763],"height:1.9074em;",[224,53021],{},[224,53023,53025],{"className":53024},[5399],[224,53026,53028,53445],{"className":53027},[754,755],[224,53029,53031,53442],{"className":53030},[759],[224,53032,53034,53229],{"className":53033,"style":52919},[763],[224,53035,53036,53039],{"style":52922},[224,53037],{"className":53038,"style":42604},[771],[224,53040,53042,53045,53048,53051,53054,53116,53122,53125,53128,53131,53134,53137,53140,53143,53146,53208,53214,53217,53220,53223,53226],{"className":53041},[468],[224,53043],{"className":53044},[468],[224,53046],{"className":53047,"style":496},[473],[224,53049,278],{"className":53050},[500],[224,53052],{"className":53053,"style":496},[473],[224,53055,53057,53060,53113],{"className":53056},[468],[224,53058],{"className":53059},[542,8479],[224,53061,53063],{"className":53062},[20338],[224,53064,53066,53105],{"className":53065},[754,755],[224,53067,53069,53102],{"className":53068},[759],[224,53070,53072,53083,53091],{"className":53071,"style":42827},[763],[224,53073,53074,53077],{"style":20531},[224,53075],{"className":53076,"style":2108},[771],[224,53078,53080],{"className":53079},[468],[224,53081,1447],{"className":53082},[468],[224,53084,53085,53088],{"style":20546},[224,53086],{"className":53087,"style":2108},[771],[224,53089],{"className":53090,"style":20554},[20553],[224,53092,53093,53096],{"style":20557},[224,53094],{"className":53095,"style":2108},[771],[224,53097,53099],{"className":53098},[468],[224,53100,1438],{"className":53101},[468],[224,53103,787],{"className":53104},[786],[224,53106,53108],{"className":53107},[759],[224,53109,53111],{"className":53110,"style":23809},[763],[224,53112],{},[224,53114],{"className":53115},[549,8479],[224,53117,53119],{"className":53118},[468],[224,53120,52588],{"className":53121},[468,1906],[224,53123,239],{"className":53124},[542],[224,53126,16018],{"className":53127,"style":653},[468,469],[224,53129,20312],{"className":53130},[468],[224,53132,45500],{"className":53133,"style":45534},[468,469],[224,53135,1716],{"className":53136},[549],[224,53138],{"className":53139,"style":474},[473],[224,53141,319],{"className":53142},[478],[224,53144],{"className":53145,"style":474},[473],[224,53147,53149,53152,53205],{"className":53148},[468],[224,53150],{"className":53151},[542,8479],[224,53153,53155],{"className":53154},[20338],[224,53156,53158,53197],{"className":53157},[754,755],[224,53159,53161,53194],{"className":53160},[759],[224,53162,53164,53175,53183],{"className":53163,"style":42827},[763],[224,53165,53166,53169],{"style":20531},[224,53167],{"className":53168,"style":2108},[771],[224,53170,53172],{"className":53171},[468],[224,53173,1447],{"className":53174},[468],[224,53176,53177,53180],{"style":20546},[224,53178],{"className":53179,"style":2108},[771],[224,53181],{"className":53182,"style":20554},[20553],[224,53184,53185,53188],{"style":20557},[224,53186],{"className":53187,"style":2108},[771],[224,53189,53191],{"className":53190},[468],[224,53192,1438],{"className":53193},[468],[224,53195,787],{"className":53196},[786],[224,53198,53200],{"className":53199},[759],[224,53201,53203],{"className":53202,"style":23809},[763],[224,53204],{},[224,53206],{"className":53207},[549,8479],[224,53209,53211],{"className":53210},[468],[224,53212,52588],{"className":53213},[468,1906],[224,53215,239],{"className":53216},[542],[224,53218,20315],{"className":53219},[468,469],[224,53221,20312],{"className":53222},[468],[224,53224,45500],{"className":53225,"style":45534},[468,469],[224,53227,1716],{"className":53228},[549],[224,53230,53231,53234],{"style":21950},[224,53232],{"className":53233,"style":42604},[771],[224,53235,53237,53240,53243,53246,53249,53311,53314,53319,53322,53325,53328,53331,53334,53396,53399,53404,53407,53410,53413,53416,53419,53424,53427,53430,53433,53436,53439],{"className":53236},[468],[224,53238],{"className":53239},[468],[224,53241],{"className":53242,"style":496},[473],[224,53244,2070],{"className":53245},[500],[224,53247],{"className":53248,"style":496},[473],[224,53250,53252,53255,53308],{"className":53251},[468],[224,53253],{"className":53254},[542,8479],[224,53256,53258],{"className":53257},[20338],[224,53259,53261,53300],{"className":53260},[754,755],[224,53262,53264,53297],{"className":53263},[759],[224,53265,53267,53278,53286],{"className":53266,"style":42827},[763],[224,53268,53269,53272],{"style":20531},[224,53270],{"className":53271,"style":2108},[771],[224,53273,53275],{"className":53274},[468],[224,53276,1447],{"className":53277},[468],[224,53279,53280,53283],{"style":20546},[224,53281],{"className":53282,"style":2108},[771],[224,53284],{"className":53285,"style":20554},[20553],[224,53287,53288,53291],{"style":20557},[224,53289],{"className":53290,"style":2108},[771],[224,53292,53294],{"className":53293},[468],[224,53295,1438],{"className":53296},[468],[224,53298,787],{"className":53299},[786],[224,53301,53303],{"className":53302},[759],[224,53304,53306],{"className":53305,"style":23809},[763],[224,53307],{},[224,53309],{"className":53310},[549,8479],[224,53312],{"className":53313,"style":724},[473],[224,53315,24185,53317],{"className":53316},[21531],[224,53318,24189],{"style":24188},[224,53320],{"className":53321,"style":724},[473],[224,53323,1447],{"className":53324},[468],[224,53326],{"className":53327,"style":474},[473],[224,53329,319],{"className":53330},[478],[224,53332],{"className":53333,"style":474},[473],[224,53335,53337,53340,53393],{"className":53336},[468],[224,53338],{"className":53339},[542,8479],[224,53341,53343],{"className":53342},[20338],[224,53344,53346,53385],{"className":53345},[754,755],[224,53347,53349,53382],{"className":53348},[759],[224,53350,53352,53363,53371],{"className":53351,"style":42827},[763],[224,53353,53354,53357],{"style":20531},[224,53355],{"className":53356,"style":2108},[771],[224,53358,53360],{"className":53359},[468],[224,53361,1447],{"className":53362},[468],[224,53364,53365,53368],{"style":20546},[224,53366],{"className":53367,"style":2108},[771],[224,53369],{"className":53370,"style":20554},[20553],[224,53372,53373,53376],{"style":20557},[224,53374],{"className":53375,"style":2108},[771],[224,53377,53379],{"className":53378},[468],[224,53380,1438],{"className":53381},[468],[224,53383,787],{"className":53384},[786],[224,53386,53388],{"className":53387},[759],[224,53389,53391],{"className":53390,"style":23809},[763],[224,53392],{},[224,53394],{"className":53395},[549,8479],[224,53397],{"className":53398,"style":724},[473],[224,53400,24185,53402],{"className":53401},[21531],[224,53403,24189],{"style":24188},[224,53405],{"className":53406,"style":724},[473],[224,53408,1447],{"className":53409},[468],[224,53411],{"className":53412,"style":496},[473],[224,53414,278],{"className":53415},[500],[224,53417],{"className":53418,"style":496},[473],[224,53420,24185,53422],{"className":53421},[21531],[224,53423,24189],{"style":24188},[224,53425],{"className":53426,"style":724},[473],[224,53428,1447],{"className":53429},[468],[224,53431],{"className":53432,"style":496},[473],[224,53434,52884],{"className":53435},[500],[224,53437],{"className":53438,"style":496},[473],[224,53440,52887],{"className":53441},[468],[224,53443,787],{"className":53444},[786],[224,53446,53448],{"className":53447},[759],[224,53449,53451],{"className":53450,"style":53019},[763],[224,53452],{},[11,53454,53455,53456,53507],{},"等号成立当且仅当 ",[224,53457,53459,53477],{"className":53458,"translate":388},[392],[224,53460,53462],{"className":53461},[396],[398,53463,53464],{"xmlns":400},[403,53465,53466,53474],{},[406,53467,53468,53470,53472],{},[409,53469,16018],{},[412,53471,1088],{},[409,53473,20315],{},[447,53475,53476],{"encoding":449},"P\\perp Q",[224,53478,53480,53498],{"className":53479,"ariaHidden":455},[454],[224,53481,53483,53486,53489,53492,53495],{"className":53482},[459],[224,53484],{"className":53485,"style":7642},[463],[224,53487,16018],{"className":53488,"style":653},[468,469],[224,53490],{"className":53491,"style":496},[473],[224,53493,1088],{"className":53494},[500],[224,53496],{"className":53497,"style":496},[473],[224,53499,53501,53504],{"className":53500},[459],[224,53502],{"className":53503,"style":20637},[463],[224,53505,20315],{"className":53506},[468,469],"（支撑集不重叠）。有界是 JS 相对 KL 的最大优点（梯度消失则是它最大的缺点，二者互为因果）。",[11,53509,53510,53512,53513,53541,53542,53575,53576,53604,53605,53633,53634,53662,53663,53691,53692,22450,53720,53748,53749,12570],{},[378,53511,44131],{}," JS 有一个很直观的信息论解释：先掷一枚公平硬币 ",[224,53514,53516,53529],{"className":53515,"translate":388},[392],[224,53517,53519],{"className":53518},[396],[398,53520,53521],{"xmlns":400},[403,53522,53523,53527],{},[406,53524,53525],{},[409,53526,12252],{},[447,53528,12252],{"encoding":449},[224,53530,53532],{"className":53531,"ariaHidden":455},[454],[224,53533,53535,53538],{"className":53534},[459],[224,53536],{"className":53537,"style":649},[463],[224,53539,12252],{"className":53540,"style":30740},[468,469],"（正反面各 ",[224,53543,53545,53563],{"className":53544,"translate":388},[392],[224,53546,53548],{"className":53547},[396],[398,53549,53550],{"xmlns":400},[403,53551,53552,53560],{},[406,53553,53554,53556,53558],{},[419,53555,1438],{},[409,53557,12413],{"mathvariant":678},[419,53559,1447],{},[447,53561,53562],{"encoding":449},"1\u002F2",[224,53564,53566],{"className":53565,"ariaHidden":455},[454],[224,53567,53569,53572],{"className":53568},[459],[224,53570],{"className":53571,"style":535},[463],[224,53573,53562],{"className":53574},[468],"），正面时从 ",[224,53577,53579,53592],{"className":53578,"translate":388},[392],[224,53580,53582],{"className":53581},[396],[398,53583,53584],{"xmlns":400},[403,53585,53586,53590],{},[406,53587,53588],{},[409,53589,16018],{},[447,53591,16018],{"encoding":449},[224,53593,53595],{"className":53594,"ariaHidden":455},[454],[224,53596,53598,53601],{"className":53597},[459],[224,53599],{"className":53600,"style":649},[463],[224,53602,16018],{"className":53603,"style":653},[468,469]," 取样、反面时从 ",[224,53606,53608,53621],{"className":53607,"translate":388},[392],[224,53609,53611],{"className":53610},[396],[398,53612,53613],{"xmlns":400},[403,53614,53615,53619],{},[406,53616,53617],{},[409,53618,20315],{},[447,53620,20315],{"encoding":449},[224,53622,53624],{"className":53623,"ariaHidden":455},[454],[224,53625,53627,53630],{"className":53626},[459],[224,53628],{"className":53629,"style":20637},[463],[224,53631,20315],{"className":53632},[468,469]," 取样，得到观测 ",[224,53635,53637,53650],{"className":53636,"translate":388},[392],[224,53638,53640],{"className":53639},[396],[398,53641,53642],{"xmlns":400},[403,53643,53644,53648],{},[406,53645,53646],{},[409,53647,20668],{},[447,53649,20668],{"encoding":449},[224,53651,53653],{"className":53652,"ariaHidden":455},[454],[224,53654,53656,53659],{"className":53655},[459],[224,53657],{"className":53658,"style":649},[463],[224,53660,20668],{"className":53661,"style":28852},[468,469],"；则 JS 正是「看一眼 ",[224,53664,53666,53679],{"className":53665,"translate":388},[392],[224,53667,53669],{"className":53668},[396],[398,53670,53671],{"xmlns":400},[403,53672,53673,53677],{},[406,53674,53675],{},[409,53676,20668],{},[447,53678,20668],{"encoding":449},[224,53680,53682],{"className":53681,"ariaHidden":455},[454],[224,53683,53685,53688],{"className":53684},[459],[224,53686],{"className":53687,"style":649},[463],[224,53689,20668],{"className":53690,"style":28852},[468,469]," 能否判断硬币正反」所蕴含的信息量，即 ",[224,53693,53695,53708],{"className":53694,"translate":388},[392],[224,53696,53698],{"className":53697},[396],[398,53699,53700],{"xmlns":400},[403,53701,53702,53706],{},[406,53703,53704],{},[409,53705,12252],{},[447,53707,12252],{"encoding":449},[224,53709,53711],{"className":53710,"ariaHidden":455},[454],[224,53712,53714,53717],{"className":53713},[459],[224,53715],{"className":53716,"style":649},[463],[224,53718,12252],{"className":53719,"style":30740},[468,469],[224,53721,53723,53736],{"className":53722,"translate":388},[392],[224,53724,53726],{"className":53725},[396],[398,53727,53728],{"xmlns":400},[403,53729,53730,53734],{},[406,53731,53732],{},[409,53733,20668],{},[447,53735,20668],{"encoding":449},[224,53737,53739],{"className":53738,"ariaHidden":455},[454],[224,53740,53742,53745],{"className":53741},[459],[224,53743],{"className":53744,"style":649},[463],[224,53746,20668],{"className":53747,"style":28852},[468,469]," 的",[378,53750,53751],{},"互信息",[224,53753,53755],{"className":53754,"translate":388},[387],[224,53756,53758,53837],{"className":53757,"translate":388},[392],[224,53759,53761],{"className":53760},[396],[398,53762,53763],{"xmlns":400,"display":401},[403,53764,53765,53834],{},[406,53766,53767,53777,53779,53781,53783,53785,53787,53789,53792,53794,53796,53799,53801,53803,53805,53808,53810,53812,53814,53816,53818,53820,53822,53824,53826,53828,53830,53832],{},[695,53768,53769,53771],{},[409,53770,20303],{},[406,53772,53773,53775],{},[409,53774,43681],{},[409,53776,6924],{},[412,53778,239],{"stretchy":435},[409,53780,16018],{},[409,53782,20312],{"mathvariant":678},[409,53784,20315],{},[412,53786,1716],{"stretchy":435},[412,53788,278],{},[409,53790,53791],{},"I",[412,53793,239],{"stretchy":435},[409,53795,12252],{},[412,53797,53798],{"separator":455},";",[409,53800,20668],{},[412,53802,1716],{"stretchy":435},[412,53804,278],{},[409,53806,53807],{},"H",[412,53809,239],{"stretchy":435},[409,53811,20668],{},[412,53813,1716],{"stretchy":435},[412,53815,5268],{},[409,53817,53807],{},[412,53819,239],{"stretchy":435},[409,53821,20668],{},[423,53823,682],{},[409,53825,1083],{"mathvariant":678},[423,53827,682],{},[409,53829,12252],{},[412,53831,1716],{"stretchy":435},[409,53833,688],{"mathvariant":678},[447,53835,53836],{"encoding":449},"D_{JS}(P\\|Q)=I(Z;X)=H(X)-H(X\\,|\\,Z).",[224,53838,53840,53916,53952,53980],{"className":53839,"ariaHidden":455},[454],[224,53841,53843,53846,53892,53895,53898,53901,53904,53907,53910,53913],{"className":53842},[459],[224,53844],{"className":53845,"style":535},[463],[224,53847,53849,53852],{"className":53848},[468],[224,53850,20303],{"className":53851,"style":2473},[468,469],[224,53853,53855],{"className":53854},[750],[224,53856,53858,53884],{"className":53857},[754,755],[224,53859,53861,53881],{"className":53860},[759],[224,53862,53864],{"className":53863,"style":764},[763],[224,53865,53866,53869],{"style":20390},[224,53867],{"className":53868,"style":772},[771],[224,53870,53872],{"className":53871},[776,777,778,779],[224,53873,53875,53878],{"className":53874},[468,779],[224,53876,43681],{"className":53877,"style":44091},[468,469,779],[224,53879,6924],{"className":53880,"style":6970},[468,469,779],[224,53882,787],{"className":53883},[786],[224,53885,53887],{"className":53886},[759],[224,53888,53890],{"className":53889,"style":794},[763],[224,53891],{},[224,53893,239],{"className":53894},[542],[224,53896,16018],{"className":53897,"style":653},[468,469],[224,53899,20312],{"className":53900},[468],[224,53902,20315],{"className":53903},[468,469],[224,53905,1716],{"className":53906},[549],[224,53908],{"className":53909,"style":496},[473],[224,53911,278],{"className":53912},[500],[224,53914],{"className":53915,"style":496},[473],[224,53917,53919,53922,53925,53928,53931,53934,53937,53940,53943,53946,53949],{"className":53918},[459],[224,53920],{"className":53921,"style":535},[463],[224,53923,53791],{"className":53924,"style":28852},[468,469],[224,53926,239],{"className":53927},[542],[224,53929,12252],{"className":53930,"style":30740},[468,469],[224,53932,53798],{"className":53933},[1917],[224,53935],{"className":53936,"style":724},[473],[224,53938,20668],{"className":53939,"style":28852},[468,469],[224,53941,1716],{"className":53942},[549],[224,53944],{"className":53945,"style":496},[473],[224,53947,278],{"className":53948},[500],[224,53950],{"className":53951,"style":496},[473],[224,53953,53955,53958,53962,53965,53968,53971,53974,53977],{"className":53954},[459],[224,53956],{"className":53957,"style":535},[463],[224,53959,53807],{"className":53960,"style":53961},[468,469],"margin-right:0.0813em;",[224,53963,239],{"className":53964},[542],[224,53966,20668],{"className":53967,"style":28852},[468,469],[224,53969,1716],{"className":53970},[549],[224,53972],{"className":53973,"style":474},[473],[224,53975,5268],{"className":53976},[478],[224,53978],{"className":53979,"style":474},[473],[224,53981,53983,53986,53989,53992,53995,53998,54001,54004,54007,54010],{"className":53982},[459],[224,53984],{"className":53985,"style":535},[463],[224,53987,53807],{"className":53988,"style":53961},[468,469],[224,53990,239],{"className":53991},[542],[224,53993,20668],{"className":53994,"style":28852},[468,469],[224,53996],{"className":53997,"style":724},[473],[224,53999,1083],{"className":54000},[468],[224,54002],{"className":54003,"style":724},[473],[224,54005,12252],{"className":54006,"style":30740},[468,469],[224,54008,1716],{"className":54009},[549],[224,54011,688],{"className":54012},[468],[11,54014,54015,54016,54044,54045,54077,54078,54206,54207,54248,54249,54387,54388,54429],{},"由于 ",[224,54017,54019,54032],{"className":54018,"translate":388},[392],[224,54020,54022],{"className":54021},[396],[398,54023,54024],{"xmlns":400},[403,54025,54026,54030],{},[406,54027,54028],{},[409,54029,12252],{},[447,54031,12252],{"encoding":449},[224,54033,54035],{"className":54034,"ariaHidden":455},[454],[224,54036,54038,54041],{"className":54037},[459],[224,54039],{"className":54040,"style":649},[463],[224,54042,12252],{"className":54043,"style":30740},[468,469]," 是概率 ",[224,54046,54048,54065],{"className":54047,"translate":388},[392],[224,54049,54051],{"className":54050},[396],[398,54052,54053],{"xmlns":400},[403,54054,54055,54063],{},[406,54056,54057,54059,54061],{},[419,54058,1438],{},[409,54060,12413],{"mathvariant":678},[419,54062,1447],{},[447,54064,53562],{"encoding":449},[224,54066,54068],{"className":54067,"ariaHidden":455},[454],[224,54069,54071,54074],{"className":54070},[459],[224,54072],{"className":54073,"style":535},[463],[224,54075,53562],{"className":54076},[468]," 的伯努利变量，",[224,54079,54081,54123],{"className":54080,"translate":388},[392],[224,54082,54084],{"className":54083},[396],[398,54085,54086],{"xmlns":400},[403,54087,54088,54120],{},[406,54089,54090,54092,54094,54096,54098,54100,54102,54104,54106,54108,54110,54112,54114,54116,54118],{},[409,54091,53791],{},[412,54093,239],{"stretchy":435},[409,54095,12252],{},[412,54097,53798],{"separator":455},[409,54099,20668],{},[412,54101,1716],{"stretchy":435},[412,54103,2070],{},[409,54105,53807],{},[412,54107,239],{"stretchy":435},[409,54109,12252],{},[412,54111,1716],{"stretchy":435},[412,54113,278],{},[409,54115,24131],{},[412,54117,24134],{},[419,54119,1447],{},[447,54121,54122],{"encoding":449},"I(Z;X)\\le H(Z)=\\log 2",[224,54124,54126,54162,54189],{"className":54125,"ariaHidden":455},[454],[224,54127,54129,54132,54135,54138,54141,54144,54147,54150,54153,54156,54159],{"className":54128},[459],[224,54130],{"className":54131,"style":535},[463],[224,54133,53791],{"className":54134,"style":28852},[468,469],[224,54136,239],{"className":54137},[542],[224,54139,12252],{"className":54140,"style":30740},[468,469],[224,54142,53798],{"className":54143},[1917],[224,54145],{"className":54146,"style":724},[473],[224,54148,20668],{"className":54149,"style":28852},[468,469],[224,54151,1716],{"className":54152},[549],[224,54154],{"className":54155,"style":496},[473],[224,54157,2070],{"className":54158},[500],[224,54160],{"className":54161,"style":496},[473],[224,54163,54165,54168,54171,54174,54177,54180,54183,54186],{"className":54164},[459],[224,54166],{"className":54167,"style":535},[463],[224,54169,53807],{"className":54170,"style":53961},[468,469],[224,54172,239],{"className":54173},[542],[224,54175,12252],{"className":54176,"style":30740},[468,469],[224,54178,1716],{"className":54179},[549],[224,54181],{"className":54182,"style":496},[473],[224,54184,278],{"className":54185},[500],[224,54187],{"className":54188,"style":496},[473],[224,54190,54192,54195,54200,54203],{"className":54191},[459],[224,54193],{"className":54194,"style":2565},[463],[224,54196,24185,54198],{"className":54197},[21531],[224,54199,24189],{"style":24188},[224,54201],{"className":54202,"style":724},[473],[224,54204,1447],{"className":54205},[468],"，有界性由此而来，",[224,54208,54210,54228],{"className":54209,"translate":388},[392],[224,54211,54213],{"className":54212},[396],[398,54214,54215],{"xmlns":400},[403,54216,54217,54225],{},[406,54218,54219,54221,54223],{},[409,54220,24131],{},[412,54222,24134],{},[419,54224,1447],{},[447,54226,54227],{"encoding":449},"\\log 2",[224,54229,54231],{"className":54230,"ariaHidden":455},[454],[224,54232,54234,54237,54242,54245],{"className":54233},[459],[224,54235],{"className":54236,"style":2565},[463],[224,54238,24185,54240],{"className":54239},[21531],[224,54241,24189],{"style":24188},[224,54243],{"className":54244,"style":724},[473],[224,54246,1447],{"className":54247},[468]," 恰好对应「最多只能获得 1 比特」这一直觉。JS 对称、有界且 ",[224,54250,54252,54277],{"className":54251,"translate":388},[392],[224,54253,54255],{"className":54254},[396],[398,54256,54257],{"xmlns":400},[403,54258,54259,54274],{},[406,54260,54261],{},[54262,54263,54264],"msqrt",{},[695,54265,54266,54268],{},[409,54267,20303],{},[406,54269,54270,54272],{},[409,54271,43681],{},[409,54273,6924],{},[447,54275,54276],{"encoding":449},"\\sqrt{D_{JS}}",[224,54278,54280],{"className":54279,"ariaHidden":455},[454],[224,54281,54283,54287],{"className":54282},[459],[224,54284],{"className":54285,"style":54286},[463],"height:1.04em;vertical-align:-0.1883em;",[224,54288,54291],{"className":54289},[468,54290],"sqrt",[224,54292,54294,54378],{"className":54293},[754,755],[224,54295,54297,54375],{"className":54296},[759],[224,54298,54301,54357],{"className":54299,"style":54300},[763],"height:0.8517em;",[224,54302,54304,54307],{"className":54303,"style":2120},[2139],[224,54305],{"className":54306,"style":2108},[771],[224,54308,54311],{"className":54309,"style":54310},[468],"padding-left:0.833em;",[224,54312,54314,54317],{"className":54313},[468],[224,54315,20303],{"className":54316,"style":2473},[468,469],[224,54318,54320],{"className":54319},[750],[224,54321,54323,54349],{"className":54322},[754,755],[224,54324,54326,54346],{"className":54325},[759],[224,54327,54329],{"className":54328,"style":764},[763],[224,54330,54331,54334],{"style":20390},[224,54332],{"className":54333,"style":772},[771],[224,54335,54337],{"className":54336},[776,777,778,779],[224,54338,54340,54343],{"className":54339},[468,779],[224,54341,43681],{"className":54342,"style":44091},[468,469,779],[224,54344,6924],{"className":54345,"style":6970},[468,469,779],[224,54347,787],{"className":54348},[786],[224,54350,54352],{"className":54351},[759],[224,54353,54355],{"className":54354,"style":794},[763],[224,54356],{},[224,54358,54360,54363],{"style":54359},"top:-2.8117em;",[224,54361],{"className":54362,"style":2108},[771],[224,54364,54368],{"className":54365,"style":54367},[54366],"hide-tail","min-width:0.853em;height:1.08em;",[2155,54369,54372],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},"1.08em","0 0 400000 1080",[2163,54373],{"d":54374},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[224,54376,787],{"className":54377},[786],[224,54379,54381],{"className":54380},[759],[224,54382,54385],{"className":54383,"style":54384},[763],"height:0.1883em;",[224,54386],{}," 仍是度量，因此在文本\u002F分布聚类、模型比较与原始 GAN 目标里被广泛使用；代价同样来自「有界」与上面这个混合结构——真差异越大 JS 越接近饱和的 ",[224,54389,54391,54409],{"className":54390,"translate":388},[392],[224,54392,54394],{"className":54393},[396],[398,54395,54396],{"xmlns":400},[403,54397,54398,54406],{},[406,54399,54400,54402,54404],{},[409,54401,24131],{},[412,54403,24134],{},[419,54405,1447],{},[447,54407,54408],{"encoding":449},"\\log2",[224,54410,54412],{"className":54411,"ariaHidden":455},[454],[224,54413,54415,54418,54423,54426],{"className":54414},[459],[224,54416],{"className":54417,"style":2565},[463],[224,54419,24185,54421],{"className":54420},[21531],[224,54422,24189],{"style":24188},[224,54424],{"className":54425,"style":724},[473],[224,54427,1447],{"className":54428},[468],"，梯度随之消失，这正是 WGAN 等替代方案的动机。",[613,54431,54433],{"id":54432},"_33-全变差","3.3 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且凸（绝对值是凸函数），代入：",[224,54673,54675],{"className":54674,"translate":388},[387],[224,54676,54678,54788],{"className":54677,"translate":388},[392],[224,54679,54681],{"className":54680},[396],[398,54682,54683],{"xmlns":400,"display":401},[403,54684,54685,54785],{},[406,54686,54687,54693,54695,54697,54699,54701,54703,54705,54711,54713,54715,54731,54733,54735,54737,54743,54745,54747,54749,54751,54753,54755,54757,54759,54761,54763,54773,54775,54777,54779,54781,54783],{},[695,54688,54689,54691],{},[409,54690,20303],{},[409,54692,1705],{},[412,54694,239],{"stretchy":435},[409,54696,16018],{},[409,54698,20312],{"mathvariant":678},[409,54700,20315],{},[412,54702,1716],{"stretchy":435},[412,54704,278],{},[20338,54706,54707,54709],{},[419,54708,1438],{},[419,54710,1447],{},[412,54712,21338],{},[409,54714,2023],{},[406,54716,54717,54719,54725,54727,54729],{},[412,54718,1083],{"fence":455},[20338,54720,54721,54723],{},[409,54722,11],{},[409,54724,2023],{},[412,54726,5268],{},[419,54728,1438],{},[412,54730,1083],{"fence":455},[409,54732,13182],{},[409,54734,371],{},[412,54736,278],{},[20338,54738,54739,54741],{},[419,54740,1438],{},[419,54742,1447],{},[412,54744,21338],{},[409,54746,1083],{"mathvariant":678},[409,54748,11],{},[412,54750,5268],{},[409,54752,2023],{},[409,54754,1083],{"mathvariant":678},[423,54756,682],{},[409,54758,13182],{},[409,54760,371],{},[412,54762,278],{},[695,54764,54765,54767],{},[409,54766,20303],{},[406,54768,54769,54771],{},[409,54770,637],{},[409,54772,16059],{},[412,54774,239],{"stretchy":435},[409,54776,16018],{},[409,54778,20312],{"mathvariant":678},[409,54780,20315],{},[412,54782,1716],{"stretchy":435},[409,54784,688],{"mathvariant":678},[447,54786,54787],{"encoding":449},"D_f(P\\|Q)\n=\\frac12\\int 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55272},[776,777,778,779],[224,55274,55276,55279],{"className":55275},[468,779],[224,55277,637],{"className":55278,"style":653},[468,469,779],[224,55280,16059],{"className":55281,"style":474},[468,469,779],[224,55283,787],{"className":55284},[786],[224,55286,55288],{"className":55287},[759],[224,55289,55291],{"className":55290,"style":794},[763],[224,55292],{},[224,55294,239],{"className":55295},[542],[224,55297,16018],{"className":55298,"style":653},[468,469],[224,55300,20312],{"className":55301},[468],[224,55303,20315],{"className":55304},[468,469],[224,55306,1716],{"className":55307},[549],[224,55309,688],{"className":55310},[468],[11,55312,55313,55314,55579,55580,55705],{},"一般测度上 ",[224,55315,55317,55390],{"className":55316,"translate":388},[392],[224,55318,55320],{"className":55319},[396],[398,55321,55322],{"xmlns":400},[403,55323,55324,55387],{},[406,55325,55326,55336,55338,55340,55342,55344,55346,55348,55365,55367,55369,55371,55373,55375,55377,55379,55381,55383,55385],{},[695,55327,55328,55330],{},[409,55329,20303],{},[406,55331,55332,55334],{},[409,55333,637],{},[409,55335,16059],{},[412,55337,239],{"stretchy":435},[409,55339,16018],{},[412,55341,1877],{"separator":455},[409,55343,20315],{},[412,55345,1716],{"stretchy":435},[412,55347,278],{},[695,55349,55350,55357],{},[406,55351,55352,55355],{},[409,55353,55354],{},"sup",[412,55356,24134],{},[406,55358,55359,55361,55363],{},[409,55360,1097],{},[412,55362,12248],{},[409,55364,1097],{"mathvariant":20667},[409,55366,1083],{"mathvariant":678},[409,55368,16018],{},[412,55370,239],{"stretchy":435},[409,55372,1097],{},[412,55374,1716],{"stretchy":435},[412,55376,5268],{},[409,55378,20315],{},[412,55380,239],{"stretchy":435},[409,55382,1097],{},[412,55384,1716],{"stretchy":435},[409,55386,1083],{"mathvariant":678},[447,55388,55389],{"encoding":449},"D_{TV}(P,Q)=\\sup_{A\\in\\mathcal A}|P(A)-Q(A)|",[224,55391,55393,55472,55558],{"className":55392,"ariaHidden":455},[454],[224,55394,55396,55399,55445,55448,55451,55454,55457,55460,55463,55466,55469],{"className":55395},[459],[224,55397],{"className":55398,"style":535},[463],[224,55400,55402,55405],{"className":55401},[468],[224,55403,20303],{"className":55404,"style":2473},[468,469],[224,55406,55408],{"className":55407},[750],[224,55409,55411,55437],{"className":55410},[754,755],[224,55412,55414,55434],{"className":55413},[759],[224,55415,55417],{"className":55416,"style":764},[763],[224,55418,55419,55422],{"style":20390},[224,55420],{"className":55421,"style":772},[771],[224,55423,55425],{"className":55424},[776,777,778,779],[224,55426,55428,55431],{"className":55427},[468,779],[224,55429,637],{"className":55430,"style":653},[468,469,779],[224,55432,16059],{"className":55433,"style":474},[468,469,779],[224,55435,787],{"className":55436},[786],[224,55438,55440],{"className":55439},[759],[224,55441,55443],{"className":55442,"style":794},[763],[224,55444],{},[224,55446,239],{"className":55447},[542],[224,55449,16018],{"className":55450,"style":653},[468,469],[224,55452,1877],{"className":55453},[1917],[224,55455],{"className":55456,"style":724},[473],[224,55458,20315],{"className":55459},[468,469],[224,55461,1716],{"className":55462},[549],[224,55464],{"className":55465,"style":496},[473],[224,55467,278],{"className":55468},[500],[224,55470],{"className":55471,"style":496},[473],[224,55473,55475,55479,55531,55534,55537,55540,55543,55546,55549,55552,55555],{"className":55474},[459],[224,55476],{"className":55477,"style":55478},[463],"height:1.0215em;vertical-align:-0.2715em;",[224,55480,55482,55485],{"className":55481},[21531],[224,55483,55354],{"className":55484},[21531],[224,55486,55488],{"className":55487},[750],[224,55489,55491,55522],{"className":55490},[754,755],[224,55492,55494,55519],{"className":55493},[759],[224,55495,55498],{"className":55496,"style":55497},[763],"height:0.2342em;",[224,55499,55501,55504],{"style":55500},"top:-2.4559em;margin-right:0.05em;",[224,55502],{"className":55503,"style":772},[771],[224,55505,55507],{"className":55506},[776,777,778,779],[224,55508,55510,55513,55516],{"className":55509},[468,779],[224,55511,1097],{"className":55512},[468,469,779],[224,55514,12248],{"className":55515},[500,779],[224,55517,1097],{"className":55518},[468,20693,779],[224,55520,787],{"className":55521},[786],[224,55523,55525],{"className":55524},[759],[224,55526,55529],{"className":55527,"style":55528},[763],"height:0.2715em;",[224,55530],{},[224,55532],{"className":55533,"style":724},[473],[224,55535,1083],{"className":55536},[468],[224,55538,16018],{"className":55539,"style":653},[468,469],[224,55541,239],{"className":55542},[542],[224,55544,1097],{"className":55545},[468,469],[224,55547,1716],{"className":55548},[549],[224,55550],{"className":55551,"style":474},[473],[224,55553,5268],{"className":55554},[478],[224,55556],{"className":55557,"style":474},[473],[224,55559,55561,55564,55567,55570,55573,55576],{"className":55560},[459],[224,55562],{"className":55563,"style":535},[463],[224,55565,20315],{"className":55566},[468,469],[224,55568,239],{"className":55569},[542],[224,55571,1097],{"className":55572},[468,469],[224,55574,1716],{"className":55575},[549],[224,55577,1083],{"className":55578},[468],"；两个表达式相等（这是全变差的对偶刻画）。注意 ",[224,55581,55583,55617],{"className":55582,"translate":388},[392],[224,55584,55586],{"className":55585},[396],[398,55587,55588],{"xmlns":400},[403,55589,55590,55614],{},[406,55591,55592,55602,55604,55606,55608,55610,55612],{},[695,55593,55594,55596],{},[409,55595,20303],{},[406,55597,55598,55600],{},[409,55599,637],{},[409,55601,16059],{},[412,55603,12248],{},[412,55605,268],{"stretchy":435},[419,55607,284],{},[412,55609,1877],{"separator":455},[419,55611,1438],{},[412,55613,440],{"stretchy":435},[447,55615,55616],{"encoding":449},"D_{TV}\\in[0,1]",[224,55618,55620,55681],{"className":55619,"ariaHidden":455},[454],[224,55621,55623,55626,55672,55675,55678],{"className":55622},[459],[224,55624],{"className":55625,"style":35905},[463],[224,55627,55629,55632],{"className":55628},[468],[224,55630,20303],{"className":55631,"style":2473},[468,469],[224,55633,55635],{"className":55634},[750],[224,55636,55638,55664],{"className":55637},[754,755],[224,55639,55641,55661],{"className":55640},[759],[224,55642,55644],{"className":55643,"style":764},[763],[224,55645,55646,55649],{"style":20390},[224,55647],{"className":55648,"style":772},[771],[224,55650,55652],{"className":55651},[776,777,778,779],[224,55653,55655,55658],{"className":55654},[468,779],[224,55656,637],{"className":55657,"style":653},[468,469,779],[224,55659,16059],{"className":55660,"style":474},[468,469,779],[224,55662,787],{"className":55663},[786],[224,55665,55667],{"className":55666},[759],[224,55668,55670],{"className":55669,"style":794},[763],[224,55671],{},[224,55673],{"className":55674,"style":496},[473],[224,55676,12248],{"className":55677},[500],[224,55679],{"className":55680,"style":496},[473],[224,55682,55684,55687,55690,55693,55696,55699,55702],{"className":55683},[459],[224,55685],{"className":55686,"style":535},[463],[224,55688,268],{"className":55689},[542],[224,55691,284],{"className":55692},[468],[224,55694,1877],{"className":55695},[1917],[224,55697],{"className":55698,"style":724},[473],[224,55700,1438],{"className":55701},[468],[224,55703,440],{"className":55704},[549],"，总有界。",[11,55707,55708,12570,55711,56097],{},[378,55709,55710],{},"斜共轭",[224,55712,55714,55778],{"className":55713,"translate":388},[392],[224,55715,55717],{"className":55716},[396],[398,55718,55719],{"xmlns":400},[403,55720,55721,55775],{},[406,55722,55723,55725,55727,55733,55735,55741,55743,55745,55747,55749,55755,55757,55759,55761,55763,55765,55767,55769,55771,55773],{},[409,55724,1435],{},[412,55726,23331],{},[20338,55728,55729,55731],{},[419,55730,1438],{},[419,55732,1447],{},[409,55734,1083],{"mathvariant":678},[20338,55736,55737,55739],{},[419,55738,1438],{},[409,55740,1435],{},[412,55742,5268],{},[419,55744,1438],{},[409,55746,1083],{"mathvariant":678},[412,55748,278],{},[20338,55750,55751,55753],{},[419,55752,1438],{},[419,55754,1447],{},[409,55756,1083],{"mathvariant":678},[419,55758,1438],{},[412,55760,5268],{},[409,55762,1435],{},[409,55764,1083],{"mathvariant":678},[412,55766,278],{},[409,55768,1705],{},[412,55770,239],{"stretchy":435},[409,55772,1435],{},[412,55774,1716],{"stretchy":435},[447,55776,55777],{"encoding":449},"t\\cdot\\frac12|\\frac1t-1|=\\frac12|1-t|=f(t)",[224,55779,55781,55799,55953,55971,56058,56079],{"className":55780,"ariaHidden":455},[454],[224,55782,55784,55787,55790,55793,55796],{"className":55783},[459],[224,55785],{"className":55786,"style":25353},[463],[224,55788,1435],{"className":55789},[468,469],[224,55791],{"className":55792,"style":474},[473],[224,55794,23331],{"className":55795},[478],[224,55797],{"className":55798,"style":474},[473],[224,55800,55802,55805,55873,55876,55944,55947,55950],{"className":55801},[459],[224,55803],{"className":55804,"style":22544},[463],[224,55806,55808,55811,55870],{"className":55807},[468],[224,55809],{"className":55810},[542,8479],[224,55812,55814],{"className":55813},[20338],[224,55815,55817,55862],{"className":55816},[754,755],[224,55818,55820,55859],{"className":55819},[759],[224,55821,55823,55837,55845],{"className":55822,"style":22563},[763],[224,55824,55825,55828],{"style":22566},[224,55826],{"className":55827,"style":2108},[771],[224,55829,55831],{"className":55830},[776,777,778,779],[224,55832,55834],{"className":55833},[468,779],[224,55835,1447],{"className":55836},[468,779],[224,55838,55839,55842],{"style":20546},[224,55840],{"className":55841,"style":2108},[771],[224,55843],{"className":55844,"style":20554},[20553],[224,55846,55847,55850],{"style":22589},[224,55848],{"className":55849,"style":2108},[771],[224,55851,55853],{"className":55852},[776,777,778,779],[224,55854,55856],{"className":55855},[468,779],[224,55857,1438],{"className":55858},[468,779],[224,55860,787],{"className":55861},[786],[224,55863,55865],{"className":55864},[759],[224,55866,55868],{"className":55867,"style":22611},[763],[224,55869],{},[224,55871],{"className":55872},[549,8479],[224,55874,1083],{"className":55875},[468],[224,55877,55879,55882,55941],{"className":55878},[468],[224,55880],{"className":55881},[542,8479],[224,55883,55885],{"className":55884},[20338],[224,55886,55888,55933],{"className":55887},[754,755],[224,55889,55891,55930],{"className":55890},[759],[224,55892,55894,55908,55916],{"className":55893,"style":22563},[763],[224,55895,55896,55899],{"style":22566},[224,55897],{"className":55898,"style":2108},[771],[224,55900,55902],{"className":55901},[776,777,778,779],[224,55903,55905],{"className":55904},[468,779],[224,55906,1435],{"className":55907},[468,469,779],[224,55909,55910,55913],{"style":20546},[224,55911],{"className":55912,"style":2108},[771],[224,55914],{"className":55915,"style":20554},[20553],[224,55917,55918,55921],{"style":22589},[224,55919],{"className":55920,"style":2108},[771],[224,55922,55924],{"className":55923},[776,777,778,779],[224,55925,55927],{"className":55926},[468,779],[224,55928,1438],{"className":55929},[468,779],[224,55931,787],{"className":55932},[786],[224,55934,55936],{"className":55935},[759],[224,55937,55939],{"className":55938,"style":22611},[763],[224,55940],{},[224,55942],{"className":55943},[549,8479],[224,55945],{"className":55946,"style":474},[473],[224,55948,5268],{"className":55949},[478],[224,55951],{"className":55952,"style":474},[473],[224,55954,55956,55959,55962,55965,55968],{"className":55955},[459],[224,55957],{"className":55958,"style":535},[463],[224,55960,54602],{"className":55961},[468],[224,55963],{"className":55964,"style":496},[473],[224,55966,278],{"className":55967},[500],[224,55969],{"className":55970,"style":496},[473],[224,55972,55974,55977,56045,56049,56052,56055],{"className":55973},[459],[224,55975],{"className":55976,"style":22544},[463],[224,55978,55980,55983,56042],{"className":55979},[468],[224,55981],{"className":55982},[542,8479],[224,55984,55986],{"className":55985},[20338],[224,55987,55989,56034],{"className":55988},[754,755],[224,55990,55992,56031],{"className":55991},[759],[224,55993,55995,56009,56017],{"className":55994,"style":22563},[763],[224,55996,55997,56000],{"style":22566},[224,55998],{"className":55999,"style":2108},[771],[224,56001,56003],{"className":56002},[776,777,778,779],[224,56004,56006],{"className":56005},[468,779],[224,56007,1447],{"className":56008},[468,779],[224,56010,56011,56014],{"style":20546},[224,56012],{"className":56013,"style":2108},[771],[224,56015],{"className":56016,"style":20554},[20553],[224,56018,56019,56022],{"style":22589},[224,56020],{"className":56021,"style":2108},[771],[224,56023,56025],{"className":56024},[776,777,778,779],[224,56026,56028],{"className":56027},[468,779],[224,56029,1438],{"className":56030},[468,779],[224,56032,787],{"className":56033},[786],[224,56035,56037],{"className":56036},[759],[224,56038,56040],{"className":56039,"style":22611},[763],[224,56041],{},[224,56043],{"className":56044},[549,8479],[224,56046,56048],{"className":56047},[468],"∣1",[224,56050],{"className":56051,"style":474},[473],[224,56053,5268],{"className":56054},[478],[224,56056],{"className":56057,"style":474},[473],[224,56059,56061,56064,56067,56070,56073,56076],{"className":56060},[459],[224,56062],{"className":56063,"style":535},[463],[224,56065,1435],{"className":56066},[468,469],[224,56068,1083],{"className":56069},[468],[224,56071],{"className":56072,"style":496},[473],[224,56074,278],{"className":56075},[500],[224,56077],{"className":56078,"style":496},[473],[224,56080,56082,56085,56088,56091,56094],{"className":56081},[459],[224,56083],{"className":56084,"style":535},[463],[224,56086,1705],{"className":56087,"style":1740},[468,469],[224,56089,239],{"className":56090},[542],[224,56092,1435],{"className":56093},[468,469],[224,56095,1716],{"className":56096},[549],"，所以 TV 是对称的。",[11,56099,56100,56102,56103,56106],{},[378,56101,44131],{}," 全变差是概率测度空间上真正的",[378,56104,56105],{},"度量","（对称、满足三角不等式），它等于两个分布在一件事上能给出的最大概率差：",[224,56108,56110],{"className":56109,"translate":388},[387],[224,56111,56113,56187],{"className":56112,"translate":388},[392],[224,56114,56116],{"className":56115},[396],[398,56117,56118],{"xmlns":400,"display":401},[403,56119,56120,56184],{},[406,56121,56122,56132,56134,56136,56138,56140,56142,56144,56160,56162,56164,56166,56168,56170,56172,56174,56176,56178,56180,56182],{},[695,56123,56124,56126],{},[409,56125,20303],{},[406,56127,56128,56130],{},[409,56129,637],{},[409,56131,16059],{},[412,56133,239],{"stretchy":435},[409,56135,16018],{},[412,56137,1877],{"separator":455},[409,56139,20315],{},[412,56141,1716],{"stretchy":435},[412,56143,278],{},[1965,56145,56146,56152],{},[406,56147,56148,56150],{},[409,56149,55354],{},[412,56151,24134],{},[406,56153,56154,56156,56158],{},[409,56155,1097],{},[412,56157,12248],{},[409,56159,1097],{"mathvariant":20667},[412,56161,1083],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,56163,16018],{},[412,56165,239],{"stretchy":435},[409,56167,1097],{},[412,56169,1716],{"stretchy":435},[412,56171,5268],{},[409,56173,20315],{},[412,56175,239],{"stretchy":435},[409,56177,1097],{},[412,56179,1716],{"stretchy":435},[412,56181,1083],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,56183,688],{"mathvariant":678},[447,56185,56186],{"encoding":449},"D_{TV}(P,Q)\n=\\sup_{A\\in\\mathcal A}\\big|P(A)-Q(A)\\big| .",[224,56188,56190,56269,56395],{"className":56189,"ariaHidden":455},[454],[224,56191,56193,56196,56242,56245,56248,56251,56254,56257,56260,56263,56266],{"className":56192},[459],[224,56194],{"className":56195,"style":535},[463],[224,56197,56199,56202],{"className":56198},[468],[224,56200,20303],{"className":56201,"style":2473},[468,469],[224,56203,56205],{"className":56204},[750],[224,56206,56208,56234],{"className":56207},[754,755],[224,56209,56211,56231],{"className":56210},[759],[224,56212,56214],{"className":56213,"style":764},[763],[224,56215,56216,56219],{"style":20390},[224,56217],{"className":56218,"style":772},[771],[224,56220,56222],{"className":56221},[776,777,778,779],[224,56223,56225,56228],{"className":56224},[468,779],[224,56226,637],{"className":56227,"style":653},[468,469,779],[224,56229,16059],{"className":56230,"style":474},[468,469,779],[224,56232,787],{"className":56233},[786],[224,56235,56237],{"className":56236},[759],[224,56238,56240],{"className":56239,"style":794},[763],[224,56241],{},[224,56243,239],{"className":56244},[542],[224,56246,16018],{"className":56247,"style":653},[468,469],[224,56249,1877],{"className":56250},[1917],[224,56252],{"className":56253,"style":724},[473],[224,56255,20315],{"className":56256},[468,469],[224,56258,1716],{"className":56259},[549],[224,56261],{"className":56262,"style":496},[473],[224,56264,278],{"className":56265},[500],[224,56267],{"className":56268,"style":496},[473],[224,56270,56272,56276,56331,56334,56374,56377,56380,56383,56386,56389,56392],{"className":56271},[459],[224,56273],{"className":56274,"style":56275},[463],"height:1.8161em;vertical-align:-0.9661em;",[224,56277,56279],{"className":56278},[21531,33665],[224,56280,56282,56322],{"className":56281},[754,755],[224,56283,56285,56319],{"className":56284},[759],[224,56286,56288,56309],{"className":56287,"style":839},[763],[224,56289,56291,56294],{"style":56290},"top:-2.1612em;margin-left:0em;",[224,56292],{"className":56293,"style":2108},[771],[224,56295,56297],{"className":56296},[776,777,778,779],[224,56298,56300,56303,56306],{"className":56299},[468,779],[224,56301,1097],{"className":56302},[468,469,779],[224,56304,12248],{"className":56305},[500,779],[224,56307,1097],{"className":56308},[468,20693,779],[224,56310,56311,56314],{"style":2120},[224,56312],{"className":56313,"style":2108},[771],[224,56315,56316],{},[224,56317,55354],{"className":56318},[21531],[224,56320,787],{"className":56321},[786],[224,56323,56325],{"className":56324},[759],[224,56326,56329],{"className":56327,"style":56328},[763],"height:0.9661em;",[224,56330],{},[224,56332],{"className":56333,"style":724},[473],[224,56335,56337],{"className":56336},[468],[224,56338,56340],{"className":56339},[2363,34778],[224,56341,56343,56366],{"className":56342},[754,755],[224,56344,56346,56363],{"className":56345},[759],[224,56347,56349],{"className":56348,"style":2285},[763],[224,56350,56351,56354],{"style":34790},[224,56352],{"className":56353,"style":34794},[771],[224,56355,56357],{"style":56356},"width:0.333em;height:1.2em;",[2155,56358,56360],{"xmlns":2157,"width":54973,"height":2004,"viewBox":56359},"0 0 333 1200",[2163,56361],{"d":56362},"M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z",[224,56364,787],{"className":56365},[786],[224,56367,56369],{"className":56368},[759],[224,56370,56372],{"className":56371,"style":34814},[763],[224,56373],{},[224,56375,16018],{"className":56376,"style":653},[468,469],[224,56378,239],{"className":56379},[542],[224,56381,1097],{"className":56382},[468,469],[224,56384,1716],{"className":56385},[549],[224,56387],{"className":56388,"style":474},[473],[224,56390,5268],{"className":56391},[478],[224,56393],{"className":56394,"style":474},[473],[224,56396,56398,56401,56404,56407,56410,56413,56450],{"className":56397},[459],[224,56399],{"className":56400,"style":2663},[463],[224,56402,20315],{"className":56403},[468,469],[224,56405,239],{"className":56406},[542],[224,56408,1097],{"className":56409},[468,469],[224,56411,1716],{"className":56412},[549],[224,56414,56416],{"className":56415},[468],[224,56417,56419],{"className":56418},[2363,34778],[224,56420,56422,56442],{"className":56421},[754,755],[224,56423,56425,56439],{"className":56424},[759],[224,56426,56428],{"className":56427,"style":2285},[763],[224,56429,56430,56433],{"style":34790},[224,56431],{"className":56432,"style":34794},[771],[224,56434,56435],{"style":56356},[2155,56436,56437],{"xmlns":2157,"width":54973,"height":2004,"viewBox":56359},[2163,56438],{"d":56362},[224,56440,787],{"className":56441},[786],[224,56443,56445],{"className":56444},[759],[224,56446,56448],{"className":56447,"style":34814},[763],[224,56449],{},[224,56451,688],{"className":56452},[468],[11,56454,56455,56456,22450,56484,56512,56513,56707,56708,56711,56712,56764,56765,57067],{},"它还有一个判决论解释：在等先验下用最优检验区分 ",[224,56457,56459,56472],{"className":56458,"translate":388},[392],[224,56460,56462],{"className":56461},[396],[398,56463,56464],{"xmlns":400},[403,56465,56466,56470],{},[406,56467,56468],{},[409,56469,16018],{},[447,56471,16018],{"encoding":449},[224,56473,56475],{"className":56474,"ariaHidden":455},[454],[224,56476,56478,56481],{"className":56477},[459],[224,56479],{"className":56480,"style":649},[463],[224,56482,16018],{"className":56483,"style":653},[468,469],[224,56485,56487,56500],{"className":56486,"translate":388},[392],[224,56488,56490],{"className":56489},[396],[398,56491,56492],{"xmlns":400},[403,56493,56494,56498],{},[406,56495,56496],{},[409,56497,20315],{},[447,56499,20315],{"encoding":449},[224,56501,56503],{"className":56502,"ariaHidden":455},[454],[224,56504,56506,56509],{"className":56505},[459],[224,56507],{"className":56508,"style":20637},[463],[224,56510,20315],{"className":56511},[468,469]," 时，最小平均错误率为 ",[224,56514,56516,56554],{"className":56515,"translate":388},[392],[224,56517,56519],{"className":56518},[396],[398,56520,56521],{"xmlns":400},[403,56522,56523,56551],{},[406,56524,56525,56533,56535,56537,56539,56549],{},[5221,56526,56527],{"scriptlevel":284,"displaystyle":435},[20338,56528,56529,56531],{},[419,56530,1438],{},[419,56532,1447],{},[412,56534,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[419,56536,1438],{},[412,56538,5268],{},[695,56540,56541,56543],{},[409,56542,20303],{},[406,56544,56545,56547],{},[409,56546,637],{},[409,56548,16059],{},[412,56550,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[447,56552,56553],{"encoding":449},"\\tfrac12\\big(1-D_{TV}\\big)",[224,56555,56557,56649],{"className":56556,"ariaHidden":455},[454],[224,56558,56560,56563,56631,56637,56640,56643,56646],{"className":56559},[459],[224,56561],{"className":56562,"style":2663},[463],[224,56564,56566,56569,56628],{"className":56565},[468],[224,56567],{"className":56568},[542,8479],[224,56570,56572],{"className":56571},[20338],[224,56573,56575,56620],{"className":56574},[754,755],[224,56576,56578,56617],{"className":56577},[759],[224,56579,56581,56595,56603],{"className":56580,"style":22563},[763],[224,56582,56583,56586],{"style":22566},[224,56584],{"className":56585,"style":2108},[771],[224,56587,56589],{"className":56588},[776,777,778,779],[224,56590,56592],{"className":56591},[468,779],[224,56593,1447],{"className":56594},[468,779],[224,56596,56597,56600],{"style":20546},[224,56598],{"className":56599,"style":2108},[771],[224,56601],{"className":56602,"style":20554},[20553],[224,56604,56605,56608],{"style":22589},[224,56606],{"className":56607,"style":2108},[771],[224,56609,56611],{"className":56610},[776,777,778,779],[224,56612,56614],{"className":56613},[468,779],[224,56615,1438],{"className":56616},[468,779],[224,56618,787],{"className":56619},[786],[224,56621,56623],{"className":56622},[759],[224,56624,56626],{"className":56625,"style":22611},[763],[224,56627],{},[224,56629],{"className":56630},[549,8479],[224,56632,56634],{"className":56633},[468],[224,56635,239],{"className":56636},[2363,2364],[224,56638,1438],{"className":56639},[468],[224,56641],{"className":56642,"style":474},[473],[224,56644,5268],{"className":56645},[478],[224,56647],{"className":56648,"style":474},[473],[224,56650,56652,56655,56701],{"className":56651},[459],[224,56653],{"className":56654,"style":2663},[463],[224,56656,56658,56661],{"className":56657},[468],[224,56659,20303],{"className":56660,"style":2473},[468,469],[224,56662,56664],{"className":56663},[750],[224,56665,56667,56693],{"className":56666},[754,755],[224,56668,56670,56690],{"className":56669},[759],[224,56671,56673],{"className":56672,"style":764},[763],[224,56674,56675,56678],{"style":20390},[224,56676],{"className":56677,"style":772},[771],[224,56679,56681],{"className":56680},[776,777,778,779],[224,56682,56684,56687],{"className":56683},[468,779],[224,56685,637],{"className":56686,"style":653},[468,469,779],[224,56688,16059],{"className":56689,"style":474},[468,469,779],[224,56691,787],{"className":56692},[786],[224,56694,56696],{"className":56695},[759],[224,56697,56699],{"className":56698,"style":794},[763],[224,56700],{},[224,56702,56704],{"className":56703},[468],[224,56705,1716],{"className":56706},[2363,2364],"，因此 TV 直接度量了",[378,56709,56710],{},"可分性","，「TV 小」就等于「几乎分不开」。TV 是这里所有 f-散度中最「弱」的标尺（总在 ",[224,56713,56715,56737],{"className":56714,"translate":388},[392],[224,56716,56718],{"className":56717},[396],[398,56719,56720],{"xmlns":400},[403,56721,56722,56734],{},[406,56723,56724,56726,56728,56730,56732],{},[412,56725,268],{"stretchy":435},[419,56727,284],{},[412,56729,1877],{"separator":455},[419,56731,1438],{},[412,56733,440],{"stretchy":435},[447,56735,56736],{"encoding":449},"[0,1]",[224,56738,56740],{"className":56739,"ariaHidden":455},[454],[224,56741,56743,56746,56749,56752,56755,56758,56761],{"className":56742},[459],[224,56744],{"className":56745,"style":535},[463],[224,56747,268],{"className":56748},[542],[224,56750,284],{"className":56751},[468],[224,56753,1877],{"className":56754},[1917],[224,56756],{"className":56757,"style":724},[473],[224,56759,1438],{"className":56760},[468],[224,56762,440],{"className":56763},[549]," 内），但按 Csiszár 定理它与 KL 诱导同一个拓扑，所以常常已经够用；在一维实直线上它退化为 Kolmogorov 距离 ",[224,56766,56768,56833],{"className":56767,"translate":388},[392],[224,56769,56771],{"className":56770},[396],[398,56772,56773],{"xmlns":400},[403,56774,56775,56830],{},[406,56776,56777,56787,56789,56799,56801,56808,56810,56812,56814,56816,56822,56824,56826,56828],{},[695,56778,56779,56781],{},[409,56780,20303],{},[406,56782,56783,56785],{},[409,56784,637],{},[409,56786,16059],{},[412,56788,278],{},[695,56790,56791,56797],{},[406,56792,56793,56795],{},[409,56794,55354],{},[412,56796,24134],{},[409,56798,371],{},[409,56800,1083],{"mathvariant":678},[695,56802,56803,56806],{},[409,56804,56805],{},"F",[409,56807,16018],{},[412,56809,239],{"stretchy":435},[409,56811,371],{},[412,56813,1716],{"stretchy":435},[412,56815,5268],{},[695,56817,56818,56820],{},[409,56819,56805],{},[409,56821,20315],{},[412,56823,239],{"stretchy":435},[409,56825,371],{},[412,56827,1716],{"stretchy":435},[409,56829,1083],{"mathvariant":678},[447,56831,56832],{"encoding":449},"D_{TV}=\\sup_x|F_P(x)-F_Q(x)|",[224,56834,56836,56897,57009],{"className":56835,"ariaHidden":455},[454],[224,56837,56839,56842,56888,56891,56894],{"className":56838},[459],[224,56840],{"className":56841,"style":35905},[463],[224,56843,56845,56848],{"className":56844},[468],[224,56846,20303],{"className":56847,"style":2473},[468,469],[224,56849,56851],{"className":56850},[750],[224,56852,56854,56880],{"className":56853},[754,755],[224,56855,56857,56877],{"className":56856},[759],[224,56858,56860],{"className":56859,"style":764},[763],[224,56861,56862,56865],{"style":20390},[224,56863],{"className":56864,"style":772},[771],[224,56866,56868],{"className":56867},[776,777,778,779],[224,56869,56871,56874],{"className":56870},[468,779],[224,56872,637],{"className":56873,"style":653},[468,469,779],[224,56875,16059],{"className":56876,"style":474},[468,469,779],[224,56878,787],{"className":56879},[786],[224,56881,56883],{"className":56882},[759],[224,56884,56886],{"className":56885,"style":794},[763],[224,56887],{},[224,56889],{"className":56890,"style":496},[473],[224,56892,278],{"className":56893},[500],[224,56895],{"className":56896,"style":496},[473],[224,56898,56900,56903,56945,56948,56951,56991,56994,56997,57000,57003,57006],{"className":56899},[459],[224,56901],{"className":56902,"style":535},[463],[224,56904,56906,56909],{"className":56905},[21531],[224,56907,55354],{"className":56908},[21531],[224,56910,56912],{"className":56911},[750],[224,56913,56915,56936],{"className":56914},[754,755],[224,56916,56918,56933],{"className":56917},[759],[224,56919,56922],{"className":56920,"style":56921},[763],"height:0.0573em;",[224,56923,56924,56927],{"style":55500},[224,56925],{"className":56926,"style":772},[771],[224,56928,56930],{"className":56929},[776,777,778,779],[224,56931,371],{"className":56932},[468,469,779],[224,56934,787],{"className":56935},[786],[224,56937,56939],{"className":56938},[759],[224,56940,56943],{"className":56941,"style":56942},[763],"height:0.2441em;",[224,56944],{},[224,56946],{"className":56947,"style":724},[473],[224,56949,1083],{"className":56950},[468],[224,56952,56954,56957],{"className":56953},[468],[224,56955,56805],{"className":56956,"style":653},[468,469],[224,56958,56960],{"className":56959},[750],[224,56961,56963,56983],{"className":56962},[754,755],[224,56964,56966,56980],{"className":56965},[759],[224,56967,56969],{"className":56968,"style":764},[763],[224,56970,56971,56974],{"style":31092},[224,56972],{"className":56973,"style":772},[771],[224,56975,56977],{"className":56976},[776,777,778,779],[224,56978,16018],{"className":56979,"style":653},[468,469,779],[224,56981,787],{"className":56982},[786],[224,56984,56986],{"className":56985},[759],[224,56987,56989],{"className":56988,"style":794},[763],[224,56990],{},[224,56992,239],{"className":56993},[542],[224,56995,371],{"className":56996},[468,469],[224,56998,1716],{"className":56999},[549],[224,57001],{"className":57002,"style":474},[473],[224,57004,5268],{"className":57005},[478],[224,57007],{"className":57008,"style":474},[473],[224,57010,57012,57015,57055,57058,57061,57064],{"className":57011},[459],[224,57013],{"className":57014,"style":20368},[463],[224,57016,57018,57021],{"className":57017},[468],[224,57019,56805],{"className":57020,"style":653},[468,469],[224,57022,57024],{"className":57023},[750],[224,57025,57027,57047],{"className":57026},[754,755],[224,57028,57030,57044],{"className":57029},[759],[224,57031,57033],{"className":57032,"style":764},[763],[224,57034,57035,57038],{"style":31092},[224,57036],{"className":57037,"style":772},[771],[224,57039,57041],{"className":57040},[776,777,778,779],[224,57042,20315],{"className":57043},[468,469,779],[224,57045,787],{"className":57046},[786],[224,57048,57050],{"className":57049},[759],[224,57051,57053],{"className":57052,"style":20409},[763],[224,57054],{},[224,57056,239],{"className":57057},[542],[224,57059,371],{"className":57060},[468,469],[224,57062,1716],{"className":57063},[549],[224,57065,1083],{"className":57066},[468],"。马尔可夫链的混合时间、耦合构造与统计距离（total variation distance）都以它为核心。",[613,57069,57071],{"id":57070},"_34-卡方散度","3.4 卡方散度",[11,57073,41411],{},[224,57075,57077],{"className":57076,"translate":388},[387],[224,57078,57080,57116],{"className":57079,"translate":388},[392],[224,57081,57083],{"className":57082},[396],[398,57084,57085],{"xmlns":400,"display":401},[403,57086,57087,57113],{},[406,57088,57089,57091,57093,57095,57097,57099,57101,57103,57105,57107],{},[409,57090,1705],{},[412,57092,239],{"stretchy":435},[409,57094,1435],{},[412,57096,1716],{"stretchy":435},[412,57098,278],{},[412,57100,239],{"stretchy":435},[409,57102,1435],{},[412,57104,5268],{},[419,57106,1438],{},[3248,57108,57109,57111],{},[412,57110,1716],{"stretchy":435},[419,57112,1447],{},[447,57114,57115],{"encoding":449},"f(t)=(t-1)^2",[224,57117,57119,57146,57167],{"className":57118,"ariaHidden":455},[454],[224,57120,57122,57125,57128,57131,57134,57137,57140,57143],{"className":57121},[459],[224,57123],{"className":57124,"style":535},[463],[224,57126,1705],{"className":57127,"style":1740},[468,469],[224,57129,239],{"className":57130},[542],[224,57132,1435],{"className":57133},[468,469],[224,57135,1716],{"className":57136},[549],[224,57138],{"className":57139,"style":496},[473],[224,57141,278],{"className":57142},[500],[224,57144],{"className":57145,"style":496},[473],[224,57147,57149,57152,57155,57158,57161,57164],{"className":57148},[459],[224,57150],{"className":57151,"style":535},[463],[224,57153,239],{"className":57154},[542],[224,57156,1435],{"className":57157},[468,469],[224,57159],{"className":57160,"style":474},[473],[224,57162,5268],{"className":57163},[478],[224,57165],{"className":57166,"style":474},[473],[224,57168,57170,57173,57176],{"className":57169},[459],[224,57171],{"className":57172,"style":13273},[463],[224,57174,1438],{"className":57175},[468],[224,57177,57179,57182],{"className":57178},[549],[224,57180,1716],{"className":57181},[549],[224,57183,57185],{"className":57184},[750],[224,57186,57188],{"className":57187},[754],[224,57189,57191],{"className":57190},[759],[224,57192,57194],{"className":57193,"style":5071},[763],[224,57195,57196,57199],{"style":5074},[224,57197],{"className":57198,"style":772},[771],[224,57200,57202],{"className":57201},[776,777,778,779],[224,57203,1447],{"className":57204},[468,779],[11,57206,54605,57207,57272],{},[224,57208,57210,57233],{"className":57209,"translate":388},[392],[224,57211,57213],{"className":57212},[396],[398,57214,57215],{"xmlns":400},[403,57216,57217,57231],{},[406,57218,57219,57221,57223,57225,57227,57229],{},[409,57220,1705],{},[412,57222,239],{"stretchy":435},[419,57224,1438],{},[412,57226,1716],{"stretchy":435},[412,57228,278],{},[419,57230,284],{},[447,57232,20243],{"encoding":449},[224,57234,57236,57263],{"className":57235,"ariaHidden":455},[454],[224,57237,57239,57242,57245,57248,57251,57254,57257,57260],{"className":57238},[459],[224,57240],{"className":57241,"style":535},[463],[224,57243,1705],{"className":57244,"style":1740},[468,469],[224,57246,239],{"className":57247},[542],[224,57249,1438],{"className":57250},[468],[224,57252,1716],{"className":57253},[549],[224,57255],{"className":57256,"style":496},[473],[224,57258,278],{"className":57259},[500],[224,57261],{"className":57262,"style":496},[473],[224,57264,57266,57269],{"className":57265},[459],[224,57267],{"className":57268,"style":510},[463],[224,57270,284],{"className":57271},[468],"、凸，代入：",[224,57274,57276],{"className":57275,"translate":388},[387],[224,57277,57279,57386],{"className":57278,"translate":388},[392],[224,57280,57282],{"className":57281},[396],[398,57283,57284],{"xmlns":400,"display":401},[403,57285,57286,57383],{},[406,57287,57288,57294,57296,57298,57300,57302,57304,57306,57308,57310,57330,57332,57334,57336,57338,57358,57360,57362,57364,57366,57373,57375,57377,57379,57381],{},[695,57289,57290,57292],{},[409,57291,20303],{},[409,57293,1705],{},[412,57295,239],{"stretchy":435},[409,57297,16018],{},[409,57299,20312],{"mathvariant":678},[409,57301,20315],{},[412,57303,1716],{"stretchy":435},[412,57305,278],{},[412,57307,21338],{},[409,57309,2023],{},[3248,57311,57312,57328],{},[406,57313,57314,57316,57322,57324,57326],{},[412,57315,239],{"fence":455},[20338,57317,57318,57320],{},[409,57319,11],{},[409,57321,2023],{},[412,57323,5268],{},[419,57325,1438],{},[412,57327,1716],{"fence":455},[419,57329,1447],{},[409,57331,13182],{},[409,57333,371],{},[412,57335,278],{},[412,57337,21338],{},[20338,57339,57340,57356],{},[406,57341,57342,57344,57346,57348,57350],{},[412,57343,239],{"stretchy":435},[409,57345,11],{},[412,57347,5268],{},[409,57349,2023],{},[3248,57351,57352,57354],{},[412,57353,1716],{"stretchy":435},[419,57355,1447],{},[409,57357,2023],{},[423,57359,682],{},[409,57361,13182],{},[409,57363,371],{},[412,57365,278],{},[3248,57367,57368,57371],{},[409,57369,57370],{},"χ",[419,57372,1447],{},[412,57374,239],{"stretchy":435},[409,57376,16018],{},[409,57378,20312],{"mathvariant":678},[409,57380,20315],{},[412,57382,1716],{"stretchy":435},[447,57384,57385],{"encoding":449},"D_f(P\\|Q)=\\int q\\left(\\frac{p}{q}-1\\right)^2dx\n=\\int \\frac{(p-q)^2}{q}\\,dx\n=\\chi^2(P\\|Q)",[224,57387,57389,57459,57613,57751],{"className":57388,"ariaHidden":455},[454],[224,57390,57392,57395,57435,57438,57441,57444,57447,57450,57453,57456],{"className":57391},[459],[224,57393],{"className":57394,"style":20368},[463],[224,57396,57398,57401],{"className":57397},[468],[224,57399,20303],{"className":57400,"style":2473},[468,469],[224,57402,57404],{"className":57403},[750],[224,57405,57407,57427],{"className":57406},[754,755],[224,57408,57410,57424],{"className":57409},[759],[224,57411,57413],{"className":57412,"style":20387},[763],[224,57414,57415,57418],{"style":20390},[224,57416],{"className":57417,"style":772},[771],[224,57419,57421],{"className":57420},[776,777,778,779],[224,57422,1705],{"className":57423,"style":1740},[468,469,779],[224,57425,787],{"className":57426},[786],[224,57428,57430],{"className":57429},[759],[224,57431,57433],{"className":57432,"style":20409},[763],[224,57434],{},[224,57436,239],{"className":57437},[542],[224,57439,16018],{"className":57440,"style":653},[468,469],[224,57442,20312],{"className":57443},[468],[224,57445,20315],{"className":57446},[468,469],[224,57448,1716],{"className":57449},[549],[224,57451],{"className":57452,"style":496},[473],[224,57454,278],{"className":57455},[500],[224,57457],{"className":57458,"style":496},[473],[224,57460,57462,57466,57469,57472,57475,57478,57595,57598,57601,57604,57607,57610],{"className":57461},[459],[224,57463],{"className":57464,"style":57465},[463],"height:2.604em;vertical-align:-0.95em;",[224,57467,21338],{"className":57468,"style":21534},[21531,21532,21533],[224,57470],{"className":57471,"style":724},[473],[224,57473,2023],{"className":57474,"style":492},[468,469],[224,57476],{"className":57477,"style":724},[473],[224,57479,57481,57570],{"className":57480},[2091],[224,57482,57484,57490,57552,57555,57558,57561,57564],{"className":57483},[2091],[224,57485,57487],{"className":57486,"style":8281},[542,8280],[224,57488,239],{"className":57489},[2363,778],[224,57491,57493,57496,57549],{"className":57492},[468],[224,57494],{"className":57495},[542,8479],[224,57497,57499],{"className":57498},[20338],[224,57500,57502,57541],{"className":57501},[754,755],[224,57503,57505,57538],{"className":57504},[759],[224,57506,57508,57519,57527],{"className":57507,"style":23586},[763],[224,57509,57510,57513],{"style":20531},[224,57511],{"className":57512,"style":2108},[771],[224,57514,57516],{"className":57515},[468],[224,57517,2023],{"className":57518,"style":492},[468,469],[224,57520,57521,57524],{"style":20546},[224,57522],{"className":57523,"style":2108},[771],[224,57525],{"className":57526,"style":20554},[20553],[224,57528,57529,57532],{"style":20557},[224,57530],{"className":57531,"style":2108},[771],[224,57533,57535],{"className":57534},[468],[224,57536,11],{"className":57537},[468,469],[224,57539,787],{"className":57540},[786],[224,57542,57544],{"className":57543},[759],[224,57545,57547],{"className":57546,"style":20579},[763],[224,57548],{},[224,57550],{"className":57551},[549,8479],[224,57553],{"className":57554,"style":474},[473],[224,57556,5268],{"className":57557},[478],[224,57559],{"className":57560,"style":474},[473],[224,57562,1438],{"className":57563},[468],[224,57565,57567],{"className":57566,"style":8281},[549,8280],[224,57568,1716],{"className":57569},[2363,778],[224,57571,57573],{"className":57572},[750],[224,57574,57576],{"className":57575},[754],[224,57577,57579],{"className":57578},[759],[224,57580,57583],{"className":57581,"style":57582},[763],"height:1.654em;",[224,57584,57586,57589],{"style":57585},"top:-3.9029em;margin-right:0.05em;",[224,57587],{"className":57588,"style":772},[771],[224,57590,57592],{"className":57591},[776,777,778,779],[224,57593,1447],{"className":57594},[468,779],[224,57596],{"className":57597,"style":724},[473],[224,57599,13182],{"className":57600},[468,469],[224,57602,371],{"className":57603},[468,469],[224,57605],{"className":57606,"style":496},[473],[224,57608,278],{"className":57609},[500],[224,57611],{"className":57612,"style":496},[473],[224,57614,57616,57620,57623,57626,57733,57736,57739,57742,57745,57748],{"className":57615},[459],[224,57617],{"className":57618,"style":57619},[463],"height:2.3715em;vertical-align:-0.8804em;",[224,57621,21338],{"className":57622,"style":21534},[21531,21532,21533],[224,57624],{"className":57625,"style":724},[473],[224,57627,57629,57632,57730],{"className":57628},[468],[224,57630],{"className":57631},[542,8479],[224,57633,57635],{"className":57634},[20338],[224,57636,57638,57722],{"className":57637},[754,755],[224,57639,57641,57719],{"className":57640},[759],[224,57642,57645,57656,57664],{"className":57643,"style":57644},[763],"height:1.4911em;",[224,57646,57647,57650],{"style":20531},[224,57648],{"className":57649,"style":2108},[771],[224,57651,57653],{"className":57652},[468],[224,57654,2023],{"className":57655,"style":492},[468,469],[224,57657,57658,57661],{"style":20546},[224,57659],{"className":57660,"style":2108},[771],[224,57662],{"className":57663,"style":20554},[20553],[224,57665,57666,57669],{"style":20557},[224,57667],{"className":57668,"style":2108},[771],[224,57670,57672,57675,57678,57681,57684,57687,57690],{"className":57671},[468],[224,57673,239],{"className":57674},[542],[224,57676,11],{"className":57677},[468,469],[224,57679],{"className":57680,"style":474},[473],[224,57682,5268],{"className":57683},[478],[224,57685],{"className":57686,"style":474},[473],[224,57688,2023],{"className":57689,"style":492},[468,469],[224,57691,57693,57696],{"className":57692},[549],[224,57694,1716],{"className":57695},[549],[224,57697,57699],{"className":57698},[750],[224,57700,57702],{"className":57701},[754],[224,57703,57705],{"className":57704},[759],[224,57706,57708],{"className":57707,"style":5671},[763],[224,57709,57710,57713],{"style":3288},[224,57711],{"className":57712,"style":772},[771],[224,57714,57716],{"className":57715},[776,777,778,779],[224,57717,1447],{"className":57718},[468,779],[224,57720,787],{"className":57721},[786],[224,57723,57725],{"className":57724},[759],[224,57726,57728],{"className":57727,"style":20579},[763],[224,57729],{},[224,57731],{"className":57732},[549,8479],[224,57734],{"className":57735,"style":724},[473],[224,57737,13182],{"className":57738},[468,469],[224,57740,371],{"className":57741},[468,469],[224,57743],{"className":57744,"style":496},[473],[224,57746,278],{"className":57747},[500],[224,57749],{"className":57750,"style":496},[473],[224,57752,57754,57757,57786,57789,57792,57795,57798],{"className":57753},[459],[224,57755],{"className":57756,"style":13273},[463],[224,57758,57760,57763],{"className":57759},[468],[224,57761,57370],{"className":57762},[468,469],[224,57764,57766],{"className":57765},[750],[224,57767,57769],{"className":57768},[754],[224,57770,57772],{"className":57771},[759],[224,57773,57775],{"className":57774,"style":5071},[763],[224,57776,57777,57780],{"style":5074},[224,57778],{"className":57779,"style":772},[771],[224,57781,57783],{"className":57782},[776,777,778,779],[224,57784,1447],{"className":57785},[468,779],[224,57787,239],{"className":57788},[542],[224,57790,16018],{"className":57791,"style":653},[468,469],[224,57793,20312],{"className":57794},[468],[224,57796,20315],{"className":57797},[468,469],[224,57799,1716],{"className":57800},[549],[11,57802,57803,57804,57807],{},"这是",[378,57805,57806],{},"皮尔逊（Pearson）卡方散度","。它非对称、无界。其斜共轭是",[224,57809,57811],{"className":57810,"translate":388},[387],[224,57812,57814,57890],{"className":57813,"translate":388},[392],[224,57815,57817],{"className":57816},[396],[398,57818,57819],{"xmlns":400,"display":401},[403,57820,57821,57887],{},[406,57822,57823,57825,57827,57829,57831,57833,57835,57837,57839,57841,57843,57863,57865,57885],{},[409,57824,1435],{},[423,57826,682],{},[409,57828,1705],{},[412,57830,239],{"stretchy":435},[419,57832,1438],{},[409,57834,12413],{"mathvariant":678},[409,57836,1435],{},[412,57838,1716],{"stretchy":435},[412,57840,278],{},[409,57842,1435],{},[3248,57844,57845,57861],{},[406,57846,57847,57849,57855,57857,57859],{},[412,57848,239],{"fence":455},[20338,57850,57851,57853],{},[419,57852,1438],{},[409,57854,1435],{},[412,57856,5268],{},[419,57858,1438],{},[412,57860,1716],{"fence":455},[419,57862,1447],{},[412,57864,278],{},[20338,57866,57867,57883],{},[406,57868,57869,57871,57873,57875,57877],{},[412,57870,239],{"stretchy":435},[409,57872,1435],{},[412,57874,5268],{},[419,57876,1438],{},[3248,57878,57879,57881],{},[412,57880,1716],{"stretchy":435},[419,57882,1447],{},[409,57884,1435],{},[412,57886,1877],{"separator":455},[447,57888,57889],{"encoding":449},"t\\,f(1\u002Ft)=t\\left(\\frac1t-1\\right)^2\n=\\frac{(t-1)^2}{t},",[224,57891,57893,57929,58065],{"className":57892,"ariaHidden":455},[454],[224,57894,57896,57899,57902,57905,57908,57911,57914,57917,57920,57923,57926],{"className":57895},[459],[224,57897],{"className":57898,"style":535},[463],[224,57900,1435],{"className":57901},[468,469],[224,57903],{"className":57904,"style":724},[473],[224,57906,1705],{"className":57907,"style":1740},[468,469],[224,57909,239],{"className":57910},[542],[224,57912,28555],{"className":57913},[468],[224,57915,1435],{"className":57916},[468,469],[224,57918,1716],{"className":57919},[549],[224,57921],{"className":57922,"style":496},[473],[224,57924,278],{"className":57925},[500],[224,57927],{"className":57928,"style":496},[473],[224,57930,57932,57935,57938,57941,58056,58059,58062],{"className":57931},[459],[224,57933],{"className":57934,"style":57465},[463],[224,57936,1435],{"className":57937},[468,469],[224,57939],{"className":57940,"style":724},[473],[224,57942,57944,58033],{"className":57943},[2091],[224,57945,57947,57953,58015,58018,58021,58024,58027],{"className":57946},[2091],[224,57948,57950],{"className":57949,"style":8281},[542,8280],[224,57951,239],{"className":57952},[2363,778],[224,57954,57956,57959,58012],{"className":57955},[468],[224,57957],{"className":57958},[542,8479],[224,57960,57962],{"classN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q)^2\\approx\\frac14\\chi^2",[224,58920,58922,58991,59081],{"className":58921,"ariaHidden":455},[454],[224,58923,58925,58929,58932,58935,58982,58985,58988],{"className":58924},[459],[224,58926],{"className":58927,"style":58928},[463],"height:1.1419em;vertical-align:-0.3369em;",[224,58930,21338],{"className":58931,"style":25626},[21531,21532,25625],[224,58933,239],{"className":58934},[542],[224,58936,58938],{"className":58937},[468,54290],[224,58939,58941,58973],{"className":58940},[754,755],[224,58942,58944,58970],{"className":58943},[759],[224,58945,58948,58957],{"className":58946,"style":58947},[763],"height:0.7031em;",[224,58949,58951,58954],{"className":58950,"style":2120},[2139],[224,58952],{"className":58953,"style":2108},[771],[224,58955,11],{"className":58956,"style":54310},[468,469],[224,58958,58960,58963],{"style":58959},"top:-2.6631em;",[224,58961],{"className":58962,"style":2108},[771],[224,58964,58966],{"className":58965,"style":54367},[54366],[2155,58967,58968],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,58969],{"d":54374},[224,58971,787],{"className":58972},[786],[224,58974,58976],{"className":58975},[759],[224,58977,58980],{"className":58978,"style":58979},[763],"height:0.3369em;",[224,58981],{},[224,58983],{"className":58984,"style":474},[473],[224,58986,5268],{"className":58987},[478],[224,58989],{"className":58990,"style":474},[473],[224,58992,58994,58998,59043,59072,59075,59078],{"className":58993},[459],[224,58995],{"className":58996,"style":58997},[463],"height:1.151em;vertical-align:-0.3369em;",[224,58999,59001],{"className":59000},[468,54290],[224,59002,59004,59035],{"className":59003},[754,755],[224,59005,59007,59032],{"className":59006},[759],[224,59008,59010,59020],{"className":59009,"style":58947},[763],[224,59011,59013,59016],{"className":59012,"style":2120},[2139],[224,59014],{"className":59015,"style":2108},[771],[224,59017,2023],{"className":59018,"style":59019},[468,469],"margin-right:0.0359em;padding-left:0.833em;",[224,59021,59022,59025],{"style":58959},[224,59023],{"className":59024,"style":2108},[771],[224,59026,59028],{"className":59027,"style":54367},[54366],[2155,59029,59030],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,59031],{"d":54374},[224,59033,787],{"className":59034},[786],[224,59036,59038],{"className":59037},[759],[224,59039,59041],{"className":59040,"style":58979},[763],[224,59042],{},[224,59044,59046,59049],{"className":59045},[549],[224,59047,1716],{"className":59048},[549],[224,59050,59052],{"className":59051},[750],[224,59053,59055],{"className":59054},[754],[224,59056,59058],{"className":59057},[759],[224,59059,59061],{"className":59060,"style":5671},[763],[224,59062,59063,59066],{"style":3288},[224,59064],{"className":59065,"style":772},[771],[224,59067,59069],{"className":59068},[776,777,778,779],[224,59070,1447],{"className":59071},[468,779],[224,59073],{"className":59074,"style":496},[473],[224,59076,52884],{"className":59077},[500],[224,59079],{"className":59080,"style":496},[473],[224,59082,59084,59087,59155],{"className":59083},[459],[224,59085],{"className":59086,"style":22544},[463],[224,59088,59090,59093,59152],{"className":59089},[468],[224,59091],{"className":59092},[542,8479],[224,59094,59096],{"className":59095},[20338],[224,59097,59099,59144],{"className":59098},[754,755],[224,59100,59102,59141],{"className":59101},[759],[224,59103,59105,59119,59127],{"className":59104,"style":22563},[763],[224,59106,59107,59110],{"style":22566},[224,59108],{"className":59109,"style":2108},[771],[224,59111,59113],{"className":59112},[776,777,778,779],[224,59114,59116],{"className":59115},[468,779],[224,59117,9409],{"className":59118},[468,779],[224,59120,59121,59124],{"style":20546},[224,59122],{"className":59123,"style":2108},[771],[224,59125],{"className":59126,"style":20554},[20553],[224,59128,59129,59132],{"style":22589},[224,59130],{"className":59131,"style":2108},[771],[224,59133,59135],{"className":59134},[776,777,778,779],[224,59136,59138],{"className":59137},[468,779],[224,59139,1438],{"className":59140},[468,779],[224,59142,787],{"className":59143},[786],[224,59145,59147],{"className":59146},[759],[224,59148,59150],{"className":59149,"style":22611},[763],[224,59151],{},[224,59153],{"className":59154},[549,8479],[224,59156,59158,59161],{"className":59157},[468],[224,59159,57370],{"className":59160},[468,469],[224,59162,59164],{"className":59163},[750],[224,59165,59167],{"className":59166},[754],[224,59168,59170],{"className":59169},[759],[224,59171,59173],{"className":59172,"style":5671},[763],[224,59174,59175,59178],{"style":3288},[224,59176],{"className":59177,"style":772},[771],[224,59179,59181],{"className":59180},[776,777,778,779],[224,59182,1447],{"className":59183},[468,779],"、Bhattacharyya 约定的 ",[224,59186,59188,59230],{"className":59187,"translate":388},[392],[224,59189,59191],{"className":59190},[396],[398,59192,59193],{"xmlns":400},[403,59194,59195,59227],{},[406,59196,59197,59203,59205,59207,59209,59211,59213,59215,59221],{},[3248,59198,59199,59201],{},[409,59200,53807],{},[419,59202,1447],{},[412,59204,278],{},[419,59206,1438],{},[412,59208,5268],{},[409,59210,9889],{},[409,59212,16021],{},[412,59214,52884],{},[20338,59216,59217,59219],{},[419,59218,1438],{},[419,59220,5430],{},[3248,59222,59223,59225],{},[409,59224,57370],{},[419,59226,1447],{},[447,59228,59229],{"encoding":449},"H^2=1-BC\\approx\\frac18\\chi^2",[224,59231,59233,59277,59295,59316],{"className":59232,"ariaHidden":455},[454],[224,59234,59236,59239,59268,59271,59274],{"className":59235},[459],[224,59237],{"className":59238,"style":5671},[463],[224,59240,59242,59245],{"className":59241},[468],[224,59243,53807],{"className":59244,"style":53961},[468,469],[224,59246,59248],{"className":59247},[750],[224,59249,59251],{"className":59250},[754],[224,59252,59254],{"className":59253},[759],[224,59255,59257],{"className":59256,"style":5671},[763],[224,59258,59259,59262],{"style":3288},[224,59260],{"className":59261,"style":772},[771],[224,59263,59265],{"className":59264},[776,777,778,779],[224,59266,1447],{"className":59267},[468,779],[224,59269],{"className":59270,"style":496},[473],[224,59272,278],{"className":59273},[500],[224,59275],{"className":59276,"style":496},[473],[224,59278,59280,59283,59286,59289,59292],{"className":59279},[459],[224,59281],{"className":59282,"style":7047},[463],[224,59284,1438],{"className":59285},[468],[224,59287],{"className":59288,"style":474},[473],[224,59290,5268],{"className":59291},[478],[224,59293],{"className":59294,"style":474},[473],[224,59296,59298,59301,59304,59307,59310,59313],{"className":59297},[459],[224,59299],{"className":59300,"style":649},[463],[224,59302,9889],{"className":59303,"style":9923},[468,469],[224,59305,16021],{"className":59306,"style":30740},[468,469],[224,59308],{"className":59309,"style":496},[473],[224,59311,52884],{"className":59312},[500],[224,59314],{"className":59315,"style":496},[473],[224,59317,59319,59322,59390],{"className":59318},[459],[224,59320],{"className":59321,"style":22544},[463],[224,59323,59325,59328,59387],{"className":59324},[468],[224,59326],{"className":59327},[542,8479],[224,59329,59331],{"className":59330},[20338],[224,59332,59334,59379],{"className":59333},[754,755],[224,59335,59337,59376],{"className":59336},[759],[224,59338,59340,59354,59362],{"className":59339,"style":22563},[763],[224,59341,59342,59345],{"style":22566},[224,59343],{"className":59344,"style":2108},[771],[224,59346,59348],{"className":59347},[776,777,778,779],[224,59349,59351],{"className":59350},[468,779],[224,59352,5430],{"className":59353},[468,779],[224,59355,59356,59359],{"style":20546},[224,59357],{"className":59358,"style":2108},[771],[224,59360],{"className":59361,"style":20554},[20553],[224,59363,59364,59367],{"style":22589},[224,59365],{"className":59366,"style":2108},[771],[224,59368,59370],{"className":59369},[776,777,778,779],[224,59371,59373],{"className":59372},[468,779],[224,59374,1438],{"className":59375},[468,779],[224,59377,787],{"className":59378},[786],[224,59380,59382],{"className":59381},[759],[224,59383,59385],{"className":59384,"style":22611},[763],[224,59386],{},[224,59388],{"className":59389},[549,8479],[224,59391,59393,59396],{"className":59392},[468],[224,59394,57370],{"className":59395},[468,469],[224,59397,59399],{"className":59398},[750],[224,59400,59402],{"className":59401},[754],[224,59403,59405],{"className":59404},[759],[224,59406,59408],{"className":59407,"style":5671},[763],[224,59409,59410,59413],{"style":3288},[224,59411],{"className":59412,"style":772},[771],[224,59414,59416],{"className":59415},[776,777,778,779],[224,59417,1447],{"className":59418},[468,779],"（见 4.6 节的局部展开），因此它是所有 f-散度在局部的「公因子」——只是在 ",[224,59421,59423,59451],{"className":59422,"translate":388},[392],[224,59424,59426],{"className":59425},[396],[398,59427,59428],{"xmlns":400},[403,59429,59430,59448],{},[406,59431,59432,59442,59444,59446],{},[3248,59433,59434,59436],{},[409,59435,1705],{},[406,59437,59438,59440],{},[412,59439,6933],{"mathvariant":678},[412,59441,6933],{"mathvariant":678},[412,59443,239],{"stretchy":435},[419,59445,1438],{},[412,59447,1716],{"stretchy":435},[447,59449,59450],{"encoding":449},"f''(1)",[224,59452,59454],{"className":59453,"ariaHidden":455},[454],[224,59455,59457,59460,59492,59495,59498],{"className":59456},[459],[224,59458],{"className":59459,"style":6966},[463],[224,59461,59463,59466],{"className":59462},[468],[224,59464,1705],{"className":59465,"style":1740},[468,469],[224,59467,59469],{"className":59468},[750],[224,59470,59472],{"className":59471},[754],[224,59473,59475],{"className":59474},[759],[224,59476,59478],{"className":59477,"style":6992},[763],[224,59479,59480,59483],{"style":3288},[224,59481],{"className":59482,"style":772},[771],[224,59484,59486],{"className":59485},[776,777,778,779],[224,59487,59489],{"className":59488},[468,779],[224,59490,41686],{"className":59491},[468,779],[224,59493,239],{"className":59494},[542],[224,59496,1438],{"className":59497},[468],[224,59499,1716],{"className":59500},[549]," 意义下的「单位」不同。",[11,59503,59504,59506,59507,59839,59840,59868,59869,7098,59897,59981,59982,60010,60011,60039,60040,60068,60069,60097,60098,60126,60127,60130,60131,60287,60288,60348],{},[378,59505,44131],{}," 卡方散度是 Pearson 拟合优度统计量 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的总体（无限样本）版本：把观测频率看作 ",[224,59841,59843,59856],{"className":59842,"translate":388},[392],[224,59844,59846],{"className":59845},[396],[398,59847,59848],{"xmlns":400},[403,59849,59850,59854],{},[406,59851,59852],{},[409,59853,16018],{},[447,59855,16018],{"encoding":449},[224,59857,59859],{"className":59858,"ariaHidden":455},[454],[224,59860,59862,59865],{"className":59861},[459],[224,59863],{"className":59864,"style":649},[463],[224,59866,16018],{"className":59867,"style":653},[468,469],"、期望频率看作 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就是「",[224,59983,59985,59998],{"className":59984,"translate":388},[392],[224,59986,59988],{"className":59987},[396],[398,59989,59990],{"xmlns":400},[403,59991,59992,59996],{},[406,59993,59994],{},[409,59995,16018],{},[447,59997,16018],{"encoding":449},[224,59999,60001],{"className":60000,"ariaHidden":455},[454],[224,60002,60004,60007],{"className":60003},[459],[224,60005],{"className":60006,"style":649},[463],[224,60008,16018],{"className":60009,"style":653},[468,469]," 是否来自 ",[224,60012,60014,60027],{"className":60013,"translate":388},[392],[224,60015,60017],{"className":60016},[396],[398,60018,60019],{"xmlns":400},[403,60020,60021,60025],{},[406,60022,60023],{},[409,60024,20315],{},[447,60026,20315],{"encoding":449},[224,60028,60030],{"className":60029,"ariaHidden":455},[454],[224,60031,60033,60036],{"className":60032},[459],[224,60034],{"className":60035,"style":20637},[463],[224,60037,20315],{"className":60038},[468,469],"」的标准化偏差，因此它是独立性检验、拟合优度检验与特征选择的自然尺度。它的分母 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是二次的，判别器梯度不会像 JS 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距离还给出了带权核的另一种「局部」相似度。",[613,60350,60352],{"id":60351},"_35-hellinger-平方","3.5 Hellinger 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athvariant":678},[419,60652,1447],{},[412,60654,325],{},[419,60656,284],{},[447,60658,60659],{"encoding":449},"f''(t)=\\frac{1}{2t^{3\u002F2}}>0",[224,60661,60663,60719,60841],{"className":60662,"ariaHidden":455},[454],[224,60664,60666,60669,60701,60704,60707,60710,60713,60716],{"className":60665},[459],[224,60667],{"className":60668,"style":6966},[463],[224,60670,60672,60675],{"className":60671},[468],[224,60673,1705],{"className":60674,"style":1740},[468,469],[224,60676,60678],{"className":60677},[750],[224,60679,60681],{"className":60680},[754],[224,60682,60684],{"className":60683},[759],[224,60685,60687],{"className":60686,"style":6992},[763],[224,60688,60689,60692],{"style":3288},[224,60690],{"className":60691,"style":772},[771],[224,60693,60695],{"className":60694},[776,777,778,779],[224,60696,60698],{"className":60697},[468,779],[224,60699,41686],{"className":60700},[468,779],[224,60702,239],{"className":60703},[542],[224,60705,1435],{"className":60706},[468,469],[224,60708,1716],{"className":60709},[549],[224,60711],{"className":60712,"style":496},[473],[224,60714,278],{"className":60715},[500],[224,60717],{"className":60718,"style":496},[473],[224,60720,60722,60726,60832,60835,60838],{"className":60721},[459],[224,60723],{"className":60724,"style":60725},[463],"height:1.2305em;vertical-align:-0.3854em;",[224,60727,60729,60732,60829],{"className":60728},[468],[224,60730],{"className":60731},[542,8479],[224,60733,60735],{"className":60734},[20338],[224,60736,60738,60820],{"className":60737},[754,755],[224,60739,60741,60817],{"className":60740},[759],[224,60742,60744,60795,60803],{"className":60743,"style":22563},[763],[224,60745,60747,60750],{"style":60746},"top:-2.6146em;",[224,60748],{"className":60749,"style":2108},[771],[224,60751,60753],{"className":60752},[776,777,778,779],[224,60754,60756,60759],{"className":60755},[468,779],[224,60757,1447],{"className":60758},[468,779],[224,60760,60762,60765],{"className":60761},[468,779],[224,60763,1435],{"className":60764},[468,469,779],[224,60766,60768],{"className":60767},[750],[224,60769,60771],{"className":60770},[754],[224,60772,60774],{"className":60773},[759],[224,60775,60778],{"className":60776,"style":60777},[763],"height:0.822em;",[224,60779,60781,60785],{"style":60780},"top:-2.822em;margin-right:0.0714em;",[224,60782],{"className":60783,"style":60784},[771],"height:2.5357em;",[224,60786,60788],{"className":60787},[776,50140,2364,779],[224,60789,60791],{"className":60790},[468,779],[224,60792,60794],{"className":60793},[468,779],"3\u002F2",[224,60796,60797,60800],{"style":20546},[224,60798],{"className":60799,"style":2108},[771],[224,60801],{"className":60802,"style":20554},[20553],[224,60804,60805,60808],{"style":22589},[224,60806],{"className":60807,"style":2108},[771],[224,60809,60811],{"className":60810},[776,777,778,779],[224,60812,60814],{"className":60813},[468,779],[224,60815,1438],{"className":60816},[468,779],[224,60818,787],{"className":60819},[786],[224,60821,60823],{"className":60822},[759],[224,60824,60827],{"className":60825,"style":60826},[763],"height:0.3854em;",[224,60828],{},[224,60830],{"className":60831},[549,8479],[224,60833],{"className":60834,"style":496},[473],[224,60836,325],{"className":60837},[500],[224,60839],{"className":60840,"style":496},[473],[224,60842,60844,60847],{"className":60843},[459],[224,60845],{"className":60846,"style":510},[463],[224,60848,284],{"className":60849},[468]," 凸，代入：",[224,60852,60854],{"className":60853,"translate":388},[387],[224,60855,60857,60947],{"className":60856,"translate":388},[392],[224,60858,60860],{"className":60859},[396],[398,60861,60862],{"xmlns":400,"display":401},[403,60863,60864,60944],{},[406,60865,60866,60872,60874,60876,60878,60880,60882,60884,60886,60888,60910,60912,60914,60916,60918,60920,60924,60926,60930,60936,60938,60940,60942],{},[695,60867,60868,60870],{},[409,60869,20303],{},[409,60871,1705],{},[412,60873,239],{"stretchy":435},[409,60875,16018],{},[409,60877,20312],{"mathvariant":678},[409,60879,20315],{},[412,60881,1716],{"stretchy":435},[412,60883,278],{},[412,60885,21338],{},[409,60887,2023],{},[3248,60889,60890,60908],{},[406,60891,60892,60894,60902,60904,60906],{},[412,60893,239],{"fence":455},[54262,60895,60896],{},[20338,60897,60898,60900],{},[409,60899,11],{},[409,60901,2023],{},[412,60903,5268],{},[419,60905,1438],{},[412,60907,1716],{"fence":455},[419,60909,1447],{},[409,60911,13182],{},[409,60913,371],{},[412,60915,278],{},[412,60917,21338],{},[412,60919,239],{"stretchy":435},[54262,60921,60922],{},[409,60923,11],{},[412,60925,5268],{},[54262,60927,60928],{},[409,60929,2023],{},[3248,60931,60932,60934],{},[412,60933,1716],{"stretchy":435},[419,60935,1447],{},[423,60937,682],{},[409,60939,13182],{},[409,60941,371],{},[409,60943,688],{"mathvariant":678},[447,60945,60946],{"encoding":449},"D_f(P\\|Q)\n=\\int q\\left(\\sqrt{\\frac{p}{q}}-1\\right)^2dx\n=\\int (\\sqrt{p}-\\sqrt{q})^2\\,dx .",[224,60948,60950,61020,61225,61295],{"className":60949,"ariaHidden":455},[454],[224,60951,60953,60956,60996,60999,61002,61005,61008,61011,61014,61017],{"className":60952},[459],[224,60954],{"className":60955,"style":20368},[463],[224,60957,60959,60962],{"className":60958},[468],[224,60960,20303],{"className":60961,"style":2473},[468,469],[224,60963,60965],{"className":60964},[750],[224,60966,60968,60988],{"className":60967},[754,755],[224,60969,60971,60985],{"className":60970},[759],[224,60972,60974],{"className":60973,"style":20387},[763],[224,60975,60976,60979],{"style":20390},[224,60977],{"className":60978,"style":772},[771],[224,60980,60982],{"className":60981},[776,777,778,779],[224,60983,1705],{"className":60984,"style":1740},[468,469,779],[224,60986,787],{"className":60987},[786],[224,60989,60991],{"className":60990},[759],[224,60992,60994],{"className":60993,"style":20409},[763],[224,60995],{},[224,60997,239],{"className":60998},[542],[224,61000,16018],{"className":61001,"style":653},[468,469],[224,61003,20312],{"className":61004},[468],[224,61006,20315],{"className":61007},[468,469],[224,61009,1716],{"className":61010},[549],[224,61012],{"className":61013,"style":496},[473],[224,61015,278],{"className":61016},[500],[224,61018],{"className":61019,"style":496},[473],[224,61021,61023,61027,61030,61033,61036,61039,61207,61210,61213,61216,61219,61222],{"className":61022},[459],[224,61024],{"className":61025,"style":61026},[463],"height:2.6466em;vertical-align:-0.9926em;",[224,61028,21338],{"className":61029,"style":21534},[21531,21532,21533],[224,61031],{"className":61032,"style":724},[473],[224,61034,2023],{"className":61035,"style":492},[468,469],[224,61037],{"className":61038,"style":724},[473],[224,61040,61042,61184],{"className":61041},[2091],[224,61043,61045,61051,61166,61169,61172,61175,61178],{"className":61044},[2091],[224,61046,61048],{"className":61047,"style":8281},[542,8280],[224,61049,239],{"className":61050},[2363,778],[224,61052,61054],{"className":61053},[468,54290],[224,61055,61057,61157],{"className":61056},[754,755],[224,61058,61060,61154],{"className":61059},[759],[224,61061,61064,61137],{"className":61062,"style":61063},[763],"height:1.4474em;",[224,61065,61068,61071],{"className":61066,"style":61067},[2139],"top:-4.4em;",[224,61069],{"className":61070,"style":54967},[771],[224,61072,61075],{"className":61073,"style":61074},[468],"padding-left:1em;",[224,61076,61078,61081,61134],{"className":61077},[468],[224,61079],{"className":61080},[542,8479],[224,61082,61084],{"className":61083},[20338],[224,61085,61087,61126],{"className":61086},[754,755],[224,61088,61090,61123],{"className":61089},[759],[224,61091,61093,61104,61112],{"className":61092,"style":23586},[763],[224,61094,61095,61098],{"style":20531},[224,61096],{"className":61097,"style":2108},[771],[224,61099,61101],{"className":61100},[468],[224,61102,2023],{"className":61103,"style":492},[468,469],[224,61105,61106,61109],{"style":20546},[224,61107],{"className":61108,"style":2108},[771],[224,61110],{"className":61111,"style":20554},[20553],[224,61113,61114,61117],{"style":20557},[224,61115],{"className":61116,"style":2108},[771],[224,61118,61120],{"className":61119},[468],[224,61121,11],{"className":61122},[468,469],[224,61124,787],{"className":61125},[786],[224,61127,61129],{"className":61128},[759],[224,61130,61132],{"className":61131,"style":20579},[763],[224,61133],{},[224,61135],{"className":61136},[549,8479],[224,61138,61140,61143],{"style":61139},"top:-3.4074em;",[224,61141],{"className":61142,"style":54967},[771],[224,61144,61147],{"className":61145,"style":61146},[54366],"min-width:1.02em;height:2.48em;",[2155,61148,61151],{"xmlns":2157,"width":2158,"height":61149,"viewBox":61150,"preserveAspectRatio":2161},"2.48em","0 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80\nh400000v40h-400000z",[224,61155,787],{"className":61156},[786],[224,61158,61160],{"className":61159},[759],[224,61161,61164],{"className":61162,"style":61163},[763],"height:0.9926em;",[224,61165],{},[224,61167],{"className":61168,"style":474},[473],[224,61170,5268],{"className":61171},[478],[224,61173],{"className":61174,"style":474},[473],[224,61176,1438],{"className":61177},[468],[224,61179,61181],{"className":61180,"style":8281},[549,8280],[224,61182,1716],{"className":61183},[2363,778],[224,61185,61187],{"className":61186},[750],[224,61188,61190],{"className":61189},[754],[224,61191,61193],{"className":61192},[759],[224,61194,61196],{"className":61195,"style":57582},[763],[224,61197,61198,61201],{"style":57585},[224,61199],{"className":61200,"style":772},[771],[224,61202,61204],{"className":61203},[776,777,778,779],[224,61205,1447],{"className":61206},[468,779],[224,61208],{"className":61209,"style":724},[473],[224,61211,13182],{"className":61212},[468,469],[224,61214,371],{"className":61215},[468,469],[224,61217],{"className":61218,"style":496},[473],[224,61220,278],{"className":61221},[500],[224,61223],{"className":61224,"style":496},[473],[224,61226,61228,61231,61234,61237,61286,61289,61292],{"className":61227},[459],[224,61229],{"className":61230,"style":25785},[463],[224,61232,21338],{"className":61233,"style":21534},[21531,21532,21533],[224,61235,239],{"className":61236},[542],[224,61238,61240],{"className":61239},[468,54290],[224,61241,61243,61277],{"className":61242},[754,755],[224,61244,61246,61274],{"className":61245},[759],[224,61247,61249,61261],{"className":61248,"style":6992},[763],[224,61250,61252,61255],{"className":61251,"style":2120},[2139],[224,61253],{"className":61254,"style":2108},[771],[224,61256,61258],{"className":61257,"style":54310},[468],[224,61259,11],{"className":61260},[468,469],[224,61262,61264,61267],{"style":61263},"top:-2.7119em;",[224,61265],{"className":61266,"style":2108},[771],[224,61268,61270],{"className":61269,"style":54367},[54366],[2155,61271,61272],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,61273],{"d":54374},[224,61275,787],{"className":61276},[786],[224,61278,61280],{"className":61279},[759],[224,61281,61284],{"className":61282,"style":61283},[763],"height:0.2881em;",[224,61285],{},[224,61287],{"className":61288,"style":474},[473],[224,61290,5268],{"className":61291},[478],[224,61293],{"className":61294,"style":474},[473],[224,61296,61298,61302,61349,61378,61381,61384,61387],{"className":61297},[459],[224,61299],{"className":61300,"style":61301},[463],"height:1.1522em;vertical-align:-0.2881em;",[224,61303,61305],{"className":61304},[468,54290],[224,61306,61308,61341],{"className":61307},[754,755],[224,61309,61311,61338],{"className":61310},[759],[224,61312,61314,61326],{"className":61313,"style":6992},[763],[224,61315,61317,61320],{"className":61316,"style":2120},[2139],[224,61318],{"className":61319,"style":2108},[771],[224,61321,61323],{"className":61322,"style":54310},[468],[224,61324,2023],{"className":61325,"style":492},[468,469],[224,61327,61328,61331],{"style":61263},[224,61329],{"className":61330,"style":2108},[771],[224,61332,61334],{"className":61333,"style":54367},[54366],[2155,61335,61336],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,61337],{"d":54374},[224,61339,787],{"className":61340},[786],[224,61342,61344],{"className":61343},[759],[224,61345,61347],{"className":61346,"style":61283},[763],[224,61348],{},[224,61350,61352,61355],{"className":61351},[549],[224,61353,1716],{"className":61354},[549],[224,61356,61358],{"className":61357},[750],[224,61359,61361],{"className":61360},[754],[224,61362,61364],{"className":61363},[759],[224,61365,61367],{"className":61366,"style":5071},[763],[224,61368,61369,61372],{"style":5074},[224,61370],{"className":61371,"style":772},[771],[224,61373,61375],{"className":61374},[776,777,778,779],[224,61376,1447],{"className":61377},[468,779],[224,61379],{"className":61380,"style":724},[473],[224,61382,13182],{"className":61383},[468,469],[224,61385,371],{"className":61386},[468,469],[224,61388,688],{"className":61389},[468],[11,61391,61392,61393,61551],{},"记 Bhattacharyya 系数 ",[224,61394,61396,61440],{"className":61395,"translate":388},[392],[224,61397,61399],{"className":61398},[396],[398,61400,61401],{"xmlns":400},[403,61402,61403,61437],{},[406,61404,61405,61407,61409,61411,61413,61415,61417,61419,61421,61423,61431,61433,61435],{},[409,61406,9889],{},[409,61408,16021],{},[412,61410,239],{"stretchy":435},[409,61412,16018],{},[412,61414,1877],{"separator":455},[409,61416,20315],{},[412,61418,1716],{"stretchy":435},[412,61420,278],{},[412,61422,21338],{},[54262,61424,61425],{},[406,61426,61427,61429],{},[409,61428,11],{},[409,61430,2023],{},[423,61432,682],{},[409,61434,13182],{},[409,61436,371],{},[447,61438,61439],{"encoding":449},"BC(P,Q)=\\int\\sqrt{pq}\\,dx",[224,61441,61443,61482],{"className":61442,"ariaHidden":455},[454],[224,61444,61446,61449,61452,61455,61458,61461,61464,61467,61470,61473,61476,61479],{"className":61445},[459],[224,61447],{"className":61448,"style":535},[463],[224,61450,9889],{"className":61451,"style":9923},[468,469],[224,61453,16021],{"className":61454,"style":30740},[468,469],[224,61456,239],{"className":61457},[542],[224,61459,16018],{"className":61460,"style":653},[468,469],[224,61462,1877],{"className":61463},[1917],[224,61465],{"className":61466,"style":724},[473],[224,61468,20315],{"className":61469},[468,469],[224,61471,1716],{"className":61472},[549],[224,61474],{"className":61475,"style":496},[473],[224,61477,278],{"className":61478},[500],[224,61480],{"className":61481,"style":496},[473],[224,61483,61485,61488,61491,61494,61542,61545,61548],{"className":61484},[459],[224,61486],{"className":61487,"style":58928},[463],[224,61489,21338],{"className":61490,"style":25626},[21531,21532,25625],[224,61492],{"className":61493,"style":724},[473],[224,61495,61497],{"className":61496},[468,54290],[224,61498,61500,61534],{"className":61499},[754,755],[224,61501,61503,61531],{"className":61502},[759],[224,61504,61506,61519],{"className":61505,"style":58947},[763],[224,61507,61509,61512],{"className":61508,"style":2120},[2139],[224,61510],{"className":61511,"style":2108},[771],[224,61513,61515],{"className":61514,"style":54310},[468],[224,61516,61518],{"className":61517,"style":492},[468,469],"pq",[224,61520,61521,61524],{"style":58959},[224,61522],{"className":61523,"style":2108},[771],[224,61525,61527],{"className":61526,"style":54367},[54366],[2155,61528,61529],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,61530],{"d":54374},[224,61532,787],{"className":61533},[786],[224,61535,61537],{"className":61536},[759],[224,61538,61540],{"className":61539,"style":58979},[763],[224,61541],{},[224,61543],{"className":61544,"style":724},[473],[224,61546,13182],{"className":61547},[468,469],[224,61549,371],{"className":61550},[468,469],"，则",[224,61553,61555],{"className":61554,"translate":388},[387],[224,61556,61558,61634],{"className":61557,"translate":388},[392],[224,61559,61561],{"className":61560},[396],[398,61562,61563],{"xmlns":400,"display":401},[403,61564,61565,61631],{},[406,61566,61567,61569,61571,61575,61577,61581,61587,61589,61591,61593,61595,61597,61599,61601,61603,61605,61613,61615,61617,61619,61621,61623,61625,61627,61629],{},[412,61568,21338],{},[412,61570,239],{"stretchy":435},[54262,61572,61573],{},[409,61574,11],{},[412,61576,5268],{},[54262,61578,61579],{},[409,61580,2023],{},[3248,61582,61583,61585],{},[412,61584,1716],{"stretchy":435},[419,61586,1447],{},[412,61588,278],{},[412,61590,21338],{},[409,61592,11],{},[412,61594,319],{},[412,61596,21338],{},[409,61598,2023],{},[412,61600,5268],{},[419,61602,1447],{},[412,61604,21338],{},[54262,61606,61607],{},[406,61608,61609,61611],{},[409,61610,11],{},[409,61612,2023],{},[412,61614,278],{},[419,61616,1447],{},[412,61618,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[419,61620,1438],{},[412,61622,5268],{},[409,61624,9889],{},[409,61626,16021],{},[412,61628,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,61630,688],{"mathvariant":678},[447,61632,61633],{"encoding":449},"\\int(\\sqrt p-\\sqrt q)^2\n=\\int p+\\int q-2\\int\\sqrt{pq}\n=2\\big(1-BC\\big).",[224,61635,61637,61702,61790,61814,61838,61912,61939],{"className":61636,"ariaHidden":455},[454],[224,61638,61640,61643,61646,61649,61693,61696,61699],{"className":61639},[459],[224,61641],{"className":61642,"style":25785},[463],[224,61644,21338],{"className":61645,"style":21534},[21531,21532,21533],[224,61647,239],{"className":61648},[542],[224,61650,61652],{"className":61651},[468,54290],[224,61653,61655,61685],{"className":61654},[754,755],[224,61656,61658,61682],{"className":61657},[759],[224,61659,61661,61670],{"className":61660,"style":6992},[763],[224,61662,61664,61667],{"className":61663,"style":2120},[2139],[224,61665],{"className":61666,"style":2108},[771],[224,61668,11],{"className":61669,"style":54310},[468,469],[224,61671,61672,61675],{"style":61263},[224,61673],{"className":61674,"style":2108},[771],[224,61676,61678],{"className":61677,"style":54367},[54366],[2155,61679,61680],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,61681],{"d":54374},[224,61683,787],{"className":61684},[786],[224,61686,61688],{"className":61687},[759],[224,61689,61691],{"className":61690,"style":61283},[763],[224,61692],{},[224,61694],{"className":61695,"style":474},[473],[224,61697,5268],{"className":61698},[478],[224,61700],{"className":61701,"style":474},[473],[224,61703,61705,61708,61752,61781,61784,61787],{"className":61704},[459],[224,61706],{"className":61707,"style":61301},[463],[224,61709,61711],{"className":61710},[468,54290],[224,61712,61714,61744],{"className":61713},[754,755],[224,61715,61717,61741],{"className":61716},[759],[224,61718,61720,61729],{"className":61719,"style":6992},[763],[224,61721,61723,61726],{"className":61722,"style":2120},[2139],[224,61724],{"className":61725,"style":2108},[771],[224,61727,2023],{"className":61728,"style":59019},[468,469],[224,61730,61731,61734],{"style":61263},[224,61732],{"className":61733,"style":2108},[771],[224,61735,61737],{"className":61736,"style":54367},[54366],[2155,61738,61739],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,61740],{"d":54374},[224,61742,787],{"className":61743},[786],[224,61745,61747],{"className":61746},[759],[224,61748,61750],{"className":61749,"style":61283},[763],[224,61751],{},[224,61753,61755,61758],{"className":61754},[549],[224,61756,1716],{"className":61757},[549],[224,61759,61761],{"className":61760},[750],[224,61762,61764],{"className":61763},[754],[224,61765,61767],{"className":61766},[759],[224,61768,61770],{"className":61769,"style":5071},[763],[224,61771,61772,61775],{"style":5074},[224,61773],{"className":61774,"style":772},[771],[224,61776,61778],{"className":61777},[776,777,778,779],[224,61779,1447],{"className":61780},[468,779],[224,61782],{"className":61783,"style":496},[473],[224,61785,278],{"className":61786},[500],[224,61788],{"className":61789,"style":496},[473],[224,61791,61793,61796,61799,61802,61805,61808,61811],{"className":61792},[459],[224,61794],{"className":61795,"style":25785},[463],[224,61797,21338],{"className":61798,"style":21534},[21531,21532,21533],[224,61800],{"className":61801,"style":724},[473],[224,61803,11],{"className":61804},[468,469],[224,61806],{"className":61807,"style":474},[473],[224,61809,319],{"className":61810},[478],[224,61812],{"className":61813,"style":474},[473],[224,61815,61817,61820,61823,61826,61829,61832,61835],{"className":61816},[459],[224,61818],{"className":61819,"style":25785},[463],[224,61821,21338],{"className":61822,"style":21534},[21531,21532,21533],[224,61824],{"className":61825,"style":724},[473],[224,61827,2023],{"className":61828,"style":492},[468,469],[224,61830],{"className":61831,"style":474},[473],[224,61833,5268],{"className":61834},[478],[224,61836],{"className":61837,"style":474},[473],[224,61839,61841,61844,61847,61850,61853,61856,61903,61906,61909],{"className":61840},[459],[224,61842],{"className":61843,"style":25785},[463],[224,61845,1447],{"className":61846},[468],[224,61848],{"className":61849,"style":724},[473],[224,61851,21338],{"className":61852,"style":21534},[21531,21532,21533],[224,61854],{"className":61855,"style":724},[473],[224,61857,61859],{"className":61858},[468,54290],[224,61860,61862,61895],{"className":61861},[754,755],[224,61863,61865,61892],{"className":61864},[759],[224,61866,61868,61880],{"className":61867,"style":6992},[763],[224,61869,61871,61874],{"className":61870,"style":2120},[2139],[224,61872],{"className":61873,"style":2108},[771],[224,61875,61877],{"className":61876,"style":54310},[468],[224,61878,61518],{"className":61879,"style":492},[468,469],[224,61881,61882,61885],{"style":61263},[224,61883],{"className":61884,"style":2108},[771],[224,61886,61888],{"className":61887,"style":54367},[54366],[2155,61889,61890],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,61891],{"d":54374},[224,61893,787],{"className":61894},[786],[224,61896,61898],{"className":61897},[759],[224,61899,61901],{"className":61900,"style":61283},[763],[224,61902],{},[224,61904],{"className":61905,"style":496},[473],[224,61907,278],{"className":61908},[500],[224,61910],{"className":61911,"style":496},[473],[224,61913,61915,61918,61921,61927,61930,61933,61936],{"className":61914},[459],[224,61916],{"className":61917,"style":2663},[463],[224,61919,1447],{"className":61920},[468],[224,61922,61924],{"className":61923},[468],[224,61925,239],{"className":61926},[2363,2364],[224,61928,1438],{"className":61929},[468],[224,61931],{"className":61932,"style":474},[473],[224,61934,5268],{"className":61935},[478],[224,61937],{"className":61938,"style":474},[473],[224,61940,61942,61945,61948,61951,61957],{"className":61941},[459],[224,61943],{"className":61944,"style":2663},[463],[224,61946,9889],{"className":61947,"style":9923},[468,469],[224,61949,16021],{"className":61950,"style":30740},[468,469],[224,61952,61954],{"className":61953},[468],[224,61955,1716],{"className":61956},[2363,2364],[224,61958,688],{"className":61959},[468],[11,61961,61962,61963,62022],{},"文献里对 ",[224,61964,61966,61984],{"className":61965,"translate":388},[392],[224,61967,61969],{"className":61968},[396],[398,61970,61971],{"xmlns":400},[403,61972,61973,61981],{},[406,61974,61975],{},[3248,61976,61977,61979],{},[409,61978,53807],{},[419,61980,1447],{},[447,61982,61983],{"encoding":449},"H^2",[224,61985,61987],{"className":61986,"ariaHidden":455},[454],[224,61988,61990,61993],{"className":61989},[459],[224,61991],{"className":61992,"style":5671},[463],[224,61994,61996,61999],{"className":61995},[468],[224,61997,53807],{"className":61998,"style":53961},[468,469],[224,62000,62002],{"className":62001},[750],[224,62003,62005],{"className":62004},[754],[224,62006,62008],{"className":62007},[759],[224,62009,62011],{"className":62010,"style":5671},[763],[224,62012,62013,62016],{"style":3288},[224,62014],{"className":62015,"style":772},[771],[224,62017,62019],{"className":62018},[776,777,778,779],[224,62020,1447],{"className":62021},[468,779]," 有两种约定：",[224,62024,62026],{"className":62025,"translate":388},[387],[224,62027,62029,62135],{"className":62028,"translate":388},[392],[224,62030,62032],{"className":62031},[396],[398,62033,62034],{"xmlns":400,"display":401},[403,62035,62036,62132],{},[5209,62037,62038,62073],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,62039,62040,62053],{},[5218,62041,62042],{},[5221,62043,62044],{"scriptlevel":284,"displaystyle":455},[27527,62045,62046,62048,62051],{},[409,62047,53807],{},[423,62049,62050],{},"Bhattacharyya",[419,62052,1447],{},[5218,62054,62055],{},[5221,62056,62057],{"scriptlevel":284,"displaystyle":455},[406,62058,62059,62061,62063,62065,62067,62069,62071],{},[406,62060],{},[412,62062,278],{},[419,62064,1438],{},[412,62066,5268],{},[409,62068,9889],{},[409,62070,16021],{},[412,62072,1877],{"separator":455},[5215,62074,62075,62094],{},[5218,62076,62077],{},[5221,62078,62079],{"scriptlevel":284,"displaystyle":455},[27527,62080,62081,62083,62092],{},[409,62082,53807],{},[406,62084,62085,62088,62090],{},[423,62086,62087],{},"this",[423,62089,682],{},[409,62091,1705],{},[419,62093,1447],{},[5218,62095,62096],{},[5221,62097,62098],{"scriptlevel":284,"displaystyle":455},[406,62099,62100,62102,62104,62106,62108,62110,62112,62114,62116,62118,62120,62122,62130],{},[406,62101],{},[412,62103,278],{},[419,62105,1447],{},[412,62107,239],{"stretchy":435},[419,62109,1438],{},[412,62111,5268],{},[409,62113,9889],{},[409,62115,16021],{},[412,62117,1716],{"stretchy":435},[412,62119,278],{},[419,62121,1447],{},[27527,62123,62124,62126,62128],{},[409,62125,53807],{},[423,62127,62050],{},[419,62129,1447],{},[409,62131,688],{"mathvariant":678},[447,62133,62134],{"encoding":449},"\\begin{aligned}\nH^2_{\\text{Bhattacharyya}}&=1-BC,\\\\[2pt]\nH^2_{\\text{this}\\,f}&=2(1-BC)\n=2H^2_{\\text{Bhattacharyya}} .\n\\end{aligned}",[224,62136,62138],{"className":62137,"ariaHidden":455},[454],[224,62139,62141,62145],{"className":62140},[459],[224,62142],{"className":62143,"style":62144},[463],"height:2.9713em;vertical-align:-1.2357em;",[224,62146,62148],{"className":62147},[468],[224,62149,62151,62317],{"className":62150},[5209],[224,62152,62154],{"className":62153},[5331],[224,62155,62157,62308],{"className":62156},[754,755],[224,62158,62160,62305],{"className":62159},[759],[224,62161,62164,62232],{"className":62162,"style":62163},[763],"height:1.7357em;",[224,62165,62167,62170],{"style":62166},"top:-3.8716em;",[224,62168],{"className":62169,"style":2108},[771],[224,62171,62173],{"className":62172},[468],[224,62174,62176,62179],{"className":62175},[468],[224,62177,53807],{"className":62178,"style":53961},[468,469],[224,62180,62182],{"className":62181},[750],[224,62183,62185,62223],{"className":62184},[754,755],[224,62186,62188,62220],{"className":62187},[759],[224,62189,62191,62209],{"className":62190,"style":5071},[763],[224,62192,62194,62197],{"style":62193},"top:-2.453em;margin-left:-0.0813em;margin-right:0.05em;",[224,62195],{"className":62196,"style":772},[771],[224,62198,62200],{"className":62199},[776,777,778,779],[224,62201,62203],{"className":62202},[468,779],[224,62204,62206],{"className":62205},[468,11373,779],[224,62207,62050],{"className":62208},[468,779],[224,62210,62211,62214],{"style":5074},[224,62212],{"className":62213,"style":772},[771],[224,62215,62217],{"className":62216},[776,777,778,779],[224,62218,1447],{"className":62219},[468,779],[224,62221,787],{"className":62222},[786],[224,62224,62226],{"className":62225},[759],[224,62227,62230],{"className":62228,"style":62229},[763],"height:0.3831em;",[224,62231],{},[224,62233,62235,62238],{"style":62234},"top:-2.1474em;",[224,62236],{"className":62237,"style":2108},[771],[224,62239,62241],{"className":62240},[468],[224,62242,62244,62247],{"className":62243},[468],[224,62245,53807],{"className":62246,"style":53961},[468,469],[224,62248,62250],{"className":62249},[750],[224,62251,62253,62297],{"className":62252},[754,755],[224,62254,62256,62294],{"className":62255},[759],[224,62257,62259,62283],{"className":62258,"style":5071},[763],[224,62260,62261,62264],{"style":62193},[224,62262],{"className":62263,"style":772},[771],[224,62265,62267],{"className":62266},[776,777,778,779],[224,62268,62270,62276,62280],{"className":62269},[468,779],[224,62271,62273],{"className":62272},[468,11373,779],[224,62274,62087],{"className":62275},[468,779],[224,62277],{"className":62278,"style":62279},[473,779],"margin-right:0.1952em;",[224,62281,1705],{"className":62282,"style":1740},[468,469,779],[224,62284,62285,62288],{"style":5074},[224,62286],{"className":62287,"style":772},[771],[224,62289,62291],{"className":62290},[776,777,778,779],[224,62292,1447],{"className":62293},[468,779],[224,62295,787],{"className":62296},[786],[224,62298,62300],{"className":62299},[759],[224,62301,62303],{"className":62302,"style":62229},[763],[224,62304],{},[224,62306,787],{"className":62307},[786],[224,62309,62311],{"className":62310},[759],[224,62312,62315],{"className":62313,"style":62314},[763],"height:1.2357em;",[224,62316],{},[224,62318,62320],{"className":62319},[5399],[224,62321,62323,62492],{"className":62322},[754,755],[224,62324,62326,62489],{"className":62325},[759],[224,62327,62329,62370],{"className":62328,"style":62163},[763],[224,62330,62331,62334],{"style":62166},[224,62332],{"className":62333,"style":2108},[771],[224,62335,62337,62340,62343,62346,62349,62352,62355,62358,62361,62364,62367],{"className":62336},[468],[224,62338],{"className":62339},[468],[224,62341],{"className":62342,"style":496},[473],[224,62344,278],{"className":62345},[500],[224,62347],{"className":62348,"style":496},[473],[224,62350,1438],{"className":62351},[468],[224,62353],{"className":62354,"style":474},[473],[224,62356,5268],{"className":62357},[478],[224,62359],{"className":62360,"style":474},[473],[224,62362,9889],{"className":62363,"style":9923},[468,469],[224,62365,16021],{"className":62366,"style":30740},[468,469],[224,62368,1877],{"className":62369},[1917],[224,62371,62372,62375],{"style":62234},[224,62373],{"className":62374,"style":2108},[771],[224,62376,62378,62381,62384,62387,62390,62393,62396,62399,62402,62405,62408,62411,62414,62417,62420,62423,62426,62429,62486],{"className":62377},[468],[224,62379],{"className":62380},[468],[224,62382],{"className":62383,"style":496},[473],[224,62385,278],{"className":62386},[500],[224,62388],{"className":62389,"style":496},[473],[224,62391,1447],{"className":62392},[468],[224,62394,239],{"className":62395},[542],[224,62397,1438],{"className":62398},[468],[224,62400],{"className":62401,"style":474},[473],[224,62403,5268],{"className":62404},[478],[224,62406],{"className":62407,"style":474},[473],[224,62409,9889],{"className":62410,"style":9923},[468,469],[224,62412,16021],{"className":62413,"style":30740},[468,469],[224,62415,1716],{"className":62416},[549],[224,62418],{"className":62419,"style":496},[473],[224,62421,278],{"className":62422},[500],[224,62424],{"className":62425,"style":496},[473],[224,62427,1447],{"className":62428},[468],[224,62430,62432,62435],{"className":62431},[468],[224,62433,53807],{"className":62434,"style":53961},[468,469],[224,62436,62438],{"className":62437},[750],[224,62439,62441,62478],{"className":62440},[754,755],[224,62442,62444,62475],{"className":62443},[759],[224,62445,62447,62464],{"className":62446,"style":5071},[763],[224,62448,62449,62452],{"style":62193},[224,62450],{"className":62451,"style":772},[771],[224,62453,62455],{"className":62454},[776,777,778,779],[224,62456,62458],{"className":62457},[468,779],[224,62459,62461],{"className":62460},[468,11373,779],[224,62462,62050],{"className":62463},[468,779],[224,62465,62466,62469],{"style":5074},[224,62467],{"className":62468,"style":772},[771],[224,62470,62472],{"className":62471},[776,777,778,779],[224,62473,1447],{"className":62474},[468,779],[224,62476,787],{"className":62477},[786],[224,62479,62481],{"className":62480},[759],[224,62482,62484],{"className":62483,"style":62229},[763],[224,62485],{},[224,62487,688],{"className":62488},[468],[224,62490,787],{"className":62491},[786],[224,62493,62495],{"className":62494},[759],[224,62496,62498],{"className":62497,"style":62314},[763],[224,62499],{},[11,62501,62502,62503,62678,62679,38084,63014,63072,63073,63116],{},"本文采用 ",[224,62504,62506,62544],{"className":62505,"translate":388},[392],[224,62507,62509],{"className":62508},[396],[398,62510,62511],{"xmlns":400},[403,62512,62513,62541],{},[406,62514,62515,62517,62519,62521,62523,62525,62527,62531,62533,62535],{},[409,62516,1705],{},[412,62518,239],{"stretchy":435},[409,62520,1435],{},[412,62522,1716],{"stretchy":435},[412,62524,278],{},[412,62526,239],{"stretchy":435},[54262,62528,62529],{},[409,62530,1435],{},[412,62532,5268],{},[419,62534,1438],{},[3248,62536,62537,62539],{},[412,62538,1716],{"stretchy":435},[419,62540,1447],{},[447,62542,62543],{"encoding":449},"f(t)=(\\sqrt t-1)^2",[224,62545,62547,62574,62640],{"className":62546,"ariaHidden":455},[454],[224,62548,62550,62553,62556,62559,62562,62565,62568,62571],{"className":62549},[459],[224,62551],{"className":62552,"style":535},[463],[224,62554,1705],{"className":62555,"style":1740},[468,469],[224,62557,239],{"className":62558},[542],[224,62560,1435],{"className":62561},[468,469],[224,62563,1716],{"className":62564},[549],[224,62566],{"className":62567,"style":496},[473],[224,62569,278],{"className":62570},[500],[224,62572],{"className":62573,"style":496},[473],[224,62575,62577,62581,62584,62631,62634,62637],{"className":62576},[459],[224,62578],{"className":62579,"style":62580},[463],"height:1.1425em;vertical-align:-0.25em;",[224,62582,239],{"className":62583},[542],[224,62585,62587],{"className":62586},[468,54290],[224,62588,62590,62622],{"className":62589},[754,755],[224,62591,62593,62619],{"className":62592},[759],[224,62594,62597,62606],{"className":62595,"style":62596},[763],"height:0.8925em;",[224,62598,62600,62603],{"className":62599,"style":2120},[2139],[224,62601],{"className":62602,"style":2108},[771],[224,62604,1435],{"className":62605,"style":54310},[468,469],[224,62607,62609,62612],{"style":62608},"top:-2.8525em;",[224,62610],{"className":62611,"style":2108},[771],[224,62613,62615],{"className":62614,"style":54367},[54366],[2155,62616,62617],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,62618],{"d":54374},[224,62620,787],{"className":62621},[786],[224,62623,62625],{"className":62624},[759],[224,62626,62629],{"className":62627,"style":62628},[763],"height:0.1475em;",[224,62630],{},[224,62632],{"className":62633,"style":474},[473],[224,62635,5268],{"className":62636},[478],[224,62638],{"className":62639,"style":474},[473],[224,62641,62643,62646,62649],{"className":62642},[459],[224,62644],{"className":62645,"style":39943},[463],[224,62647,1438],{"className":62648},[468],[224,62650,62652,62655],{"className":62651},[549],[224,62653,1716],{"className":62654},[549],[224,62656,62658],{"className":62657},[750],[224,62659,62661],{"className":62660},[754],[224,62662,62664],{"className":62663},[759],[224,62665,62667],{"className":62666,"style":5671},[763],[224,62668,62669,62672],{"style":3288},[224,62670],{"className":62671,"style":772},[771],[224,62673,62675],{"className":62674},[776,777,778,779],[224,62676,1447],{"className":62677},[468,779],"，故 ",[224,62680,62682,62734],{"className":62681,"translate":388},[392],[224,62683,62685],{"className":62684},[396],[398,62686,62687],{"xmlns":400},[403,62688,62689,62731],{},[406,62690,62691,62697,62699,62701,62709,62711,62713,62715,62719,62721,62725],{},[695,62692,62693,62695],{},[409,62694,20303],{},[409,62696,1705],{},[412,62698,278],{},[419,62700,1447],{},[27527,62702,62703,62705,62707],{},[409,62704,53807],{},[423,62706,62050],{},[419,62708,1447],{},[412,62710,278],{},[412,62712,21338],{},[412,62714,239],{"stretchy":435},[54262,62716,62717],{},[409,62718,11],{},[412,62720,5268],{},[54262,62722,62723],{},[409,62724,2023],{},[3248,62726,62727,62729],{},[412,62728,1716],{"stretchy":435},[419,62730,1447],{},[447,62732,62733],{"encoding":449},"D_f=2H^2_{\\text{Bhattacharyya}}=\\int(\\sqrt p-\\sqrt q)^2",[224,62735,62737,62792,62870,62935],{"className":62736,"ariaHidden":455},[454],[224,62738,62740,62743,62783,62786,62789],{"className":62739},[459],[224,62741],{"className":62742,"style":24063},[463],[224,62744,62746,62749],{"className":62745},[468],[224,62747,20303],{"className":62748,"style":2473},[468,469],[224,62750,62752],{"className":62751},[750],[224,62753,62755,62775],{"className":62754},[754,755],[224,62756,62758,62772],{"className":62757},[759],[224,62759,62761],{"className":62760,"style":20387},[763],[224,62762,62763,62766],{"style":20390},[224,62764],{"className":62765,"style":772},[771],[224,62767,62769],{"className":62768},[776,777,778,779],[224,62770,1705],{"className":62771,"style":1740},[468,469,779],[224,62773,787],{"className":62774},[786],[224,62776,62778],{"className":62777},[759],[224,62779,62781],{"className":62780,"style":20409},[763],[224,62782],{},[224,62784],{"className":62785,"style":496},[473],[224,62787,278],{"className":62788},[500],[224,62790],{"className":62791,"style":496},[473],[224,62793,62795,62799,62802,62861,62864,62867],{"className":62794},[459],[224,62796],{"className":62797,"style":62798},[463],"height:1.2333em;vertical-align:-0.4192em;",[224,62800,1447],{"className":62801},[468],[224,62803,62805,62808],{"className":62804},[468],[224,62806,53807],{"className":62807,"style":53961},[468,469],[224,62809,62811],{"className":62810},[750],[224,62812,62814,62852],{"className":62813},[754,755],[224,62815,62817,62849],{"className":62816},[759],[224,62818,62820,62838],{"className":62819,"style":5671},[763],[224,62821,62823,62826],{"style":62822},"top:-2.4169em;margin-left:-0.0813em;margin-right:0.05em;",[224,62824],{"className":62825,"style":772},[771],[224,62827,62829],{"className":62828},[776,777,778,779],[224,62830,62832],{"className":62831},[468,779],[224,62833,62835],{"className":62834},[468,11373,779],[224,62836,62050],{"className":62837},[468,779],[224,62839,62840,62843],{"style":3288},[224,62841],{"className":62842,"style":772},[771],[224,62844,62846],{"className":62845},[776,777,778,779],[224,62847,1447],{"className":62848},[468,779],[224,62850,787],{"className":62851},[786],[224,62853,62855],{"className":62854},[759],[224,62856,62859],{"className":62857,"style":62858},[763],"height:0.4192em;",[224,62860],{},[224,62862],{"className":62863,"style":496},[473],[224,62865,278],{"className":62866},[500],[224,62868],{"className":62869,"style":496},[473],[224,62871,62873,62876,62879,62882,62926,62929,62932],{"className":62872},[459],[224,62874],{"className":62875,"style":58928},[463],[224,62877,21338],{"className":62878,"style":25626},[21531,21532,25625],[224,62880,239],{"className":62881},[542],[224,62883,62885],{"className":62884},[468,54290],[224,62886,62888,62918],{"className":62887},[754,755],[224,62889,62891,62915],{"className":62890},[759],[224,62892,62894,62903],{"className":62893,"style":58947},[763],[224,62895,62897,62900],{"className":62896,"style":2120},[2139],[224,62898],{"className":62899,"style":2108},[771],[224,62901,11],{"className":62902,"style":54310},[468,469],[224,62904,62905,62908],{"style":58959},[224,62906],{"className":62907,"style":2108},[771],[224,62909,62911],{"className":62910,"style":54367},[54366],[2155,62912,62913],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,62914],{"d":54374},[224,62916,787],{"className":62917},[786],[224,62919,62921],{"className":62920},[759],[224,62922,62924],{"className":62923,"style":58979},[763],[224,62925],{},[224,62927],{"className":62928,"style":474},[473],[224,62930,5268],{"className":62931},[478],[224,62933],{"className":62934,"style":474},[473],[224,62936,62938,62941,62985],{"className":62937},[459],[224,62939],{"className":62940,"style":58997},[463],[224,62942,62944],{"className":62943},[468,54290],[224,62945,62947,62977],{"className":62946},[754,755],[224,62948,62950,62974],{"className":62949},[759],[224,62951,62953,62962],{"className":62952,"style":58947},[763],[224,62954,62956,62959],{"className":62955,"style":2120},[2139],[224,62957],{"className":62958,"style":2108},[771],[224,62960,2023],{"className":62961,"style":59019},[468,469],[224,62963,62964,62967],{"style":58959},[224,62965],{"className":62966,"style":2108},[771],[224,62968,62970],{"className":62969,"style":54367},[54366],[2155,62971,62972],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,62973],{"d":54374},[224,62975,787],{"className":62976},[786],[224,62978,62980],{"className":62979},[759],[224,62981,62983],{"className":62982,"style":58979},[763],[224,62984],{},[224,62986,62988,62991],{"className":62987},[549],[224,62989,1716],{"className":62990},[549],[224,62992,62994],{"className":62993},[750],[224,62995,62997],{"className":62996},[754],[224,62998,63000],{"className":62999},[759],[224,63001,63003],{"className":63002,"style":5671},[763],[224,63004,63005,63008],{"style":3288},[224,63006],{"className":63007,"style":772},[771],[224,63009,63011],{"className"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有界（",[224,63074,63076,63092],{"className":63075,"translate":388},[392],[224,63077,63079],{"className":63078},[396],[398,63080,63081],{"xmlns":400},[403,63082,63083,63089],{},[406,63084,63085,63087],{},[412,63086,2070],{},[419,63088,1447],{},[447,63090,63091],{"encoding":449},"\\le 2",[224,63093,63095,63107],{"className":63094,"ariaHidden":455},[454],[224,63096,63098,63101,63104],{"className":63097},[459],[224,63099],{"className":63100,"style":3204},[463],[224,63102,2070],{"className":63103},[500],[224,63105],{"className":63106,"style":496},[473],[224,63108,63110,63113],{"className":63109},[459],[224,63111],{"className":63112,"style":510},[463],[224,63114,1447],{"className":63115},[468],"）。对称性可直接验算：",[224,63118,63120],{"className":63119,"translate":388},[387],[224,63121,63123,63299],{"className":63122,"translate":388},[392],[224,63124,63126],{"className":63125},[396],[398,63127,63128],{"xmlns":400,"display":401},[403,63129,63130,63296],{},[5209,63131,63132,63190,63234],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,63133,63134,63156],{},[5218,63135,63136],{},[5221,63137,63138],{"scriptlevel":284,"displaystyle":455},[406,63139,63140,63142,63144,63146,63148,63150,63152,63154],{},[409,63141,1435],{},[423,63143,682],{},[409,63145,1705],{},[412,63147,239],{"stretchy":435},[419,63149,1438],{},[409,63151,12413],{"mathvariant":678},[409,63153,1435],{},[412,63155,1716],{"stretchy":435},[5218,63157,63158],{},[5221,63159,63160],{"scriptlevel":284,"displaystyle":455},[406,63161,63162,63164,63166,63168],{},[406,63163],{},[412,63165,278],{},[409,63167,1435],{},[3248,63169,63170,63188],{},[406,63171,63172,63174,63182,63184,63186],{},[412,63173,239],{"fence":455},[20338,63175,63176,63178],{},[419,63177,1438],{},[54262,63179,63180],{},[409,63181,1435],{},[412,63183,5268],{},[419,63185,1438],{},[412,63187,1716],{"fence":455},[419,63189,1447],{},[5215,63191,63192,63198],{},[5218,63193,63194],{},[5221,63195,63196],{"scriptlevel":284,"displaystyle":455},[406,63197],{},[5218,63199,63200],{},[5221,63201,63202],{"scriptlevel":284,"displaystyle":455},[406,63203,63204,63206,63208,63210,63212,63218,63220,63228,63230,63232],{},[406,63205],{},[412,63207,278],{},[409,63209,1435],{},[412,63211,239],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[20338,63213,63214,63216],{},[419,63215,1438],{},[409,63217,1435],{},[412,63219,5268],{},[20338,63221,63222,63224],{},[419,63223,1447],{},[54262,63225,63226],{},[409,63227,1435],{},[412,63229,319],{},[419,63231,1438],{},[412,63233,1716],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[5215,63235,63236,63242],{},[5218,63237,63238],{},[5221,63239,63240],{"scriptlevel":284,"displaystyle":455},[406,63241],{},[5218,63243,63244],{},[5221,63245,63246],{"scriptlevel":284,"displaystyle":455},[406,63247,63248,63250,63252,63254,63256,63258,63262,63264,63266,63268,63270,63274,63276,63278,63284,63286,63288,63290,63292,63294],{},[406,63249],{},[412,63251,278],{},[419,63253,1438],{},[412,63255,5268],{},[419,63257,1447],{},[54262,63259,63260],{},[409,63261,1435],{},[412,63263,319],{},[409,63265,1435],{},[412,63267,278],{},[412,63269,239],{"stretchy":435},[54262,63271,63272],{},[409,63273,1435],{},[412,63275,5268],{},[419,63277,1438],{},[3248,63279,63280,63282],{},[412,63281,1716],{"stretchy":435},[419,63283,1447],{},[412,63285,278],{},[409,63287,1705],{},[412,63289,239],{"stretchy":435},[409,63291,1435],{},[412,63293,1716],{"stretchy":435},[412,63295,1877],{"separator":455},[447,63297,63298],{"encoding":449},"\\begin{aligned}\nt\\,f(1\u002Ft)\n&=t\\left(\\frac{1}{\\sqrt t}-1\\right)^2\\\\\n&=t\\Big(\\frac1t-\\frac{2}{\\sqrt t}+1\\Big)\\\\[2pt]\n&=1-2\\sqrt t+t=(\\sqrt t-1)^2=f(t),\n\\end{aligned}",[224,63300,63302],{"className":63301,"ariaHidden":455},[454],[224,63303,63305,63309],{"className":63304},[459],[224,63306],{"className":63307,"style":63308},[463],"height:6.7569em;vertical-align:-3.1284em;",[224,63310,63312],{"className":63311},[468],[224,63313,63315,63389],{"className":63314},[5209],[224,63316,63318],{"className":63317},[5331],[224,63319,63321,63380],{"className":63320},[754,755],[224,63322,63324,63377],{"className":63323},[759],[224,63325,63328,63359,63368],{"className":63326,"style":63327},[763],"height:3.6284em;",[224,63329,63331,63335],{"style":63330},"top:-5.6284em;",[224,63332],{"className":63333,"style":63334},[771],"height:3.654em;",[224,63336,63338,63341,63344,63347,63350,63353,63356],{"className":63337},[468],[224,63339,1435],{"className":63340},[468,469],[224,63342],{"className":63343,"style":724},[473],[224,63345,1705],{"className":63346,"style":1740},[468,469],[224,63348,239],{"className":63349},[542],[224,63351,28555],{"className":63352},[468],[224,63354,1435],{"className":63355},[468,469],[224,63357,1716],{"className":63358},[549],[224,63360,63362,63365],{"style":63361},"top:-3.057em;",[224,63363],{"className":63364,"style":63334},[771],[224,63366],{"className":63367},[468],[224,63369,63371,63374],{"style":63370},"top:-0.8856em;",[224,63372],{"className":63373,"style":63334},[771],[224,63375],{"className":63376},[468],[224,63378,787],{"className":63379},[786],[224,63381,63383],{"className":63382},[759],[224,63384,63387],{"className":63385,"style":63386},[763],"height:3.1284em;",[224,63388],{},[224,63390,63392],{"className":63391},[5399],[224,63393,63395,64021],{"className":63394},[754,755],[224,63396,63398,64018],{"className":63397},[759],[224,63399,63401,63585,63806],{"className":63400,"style":63327},[763],[224,63402,63403,63406],{"style":63330},[224,63404],{"className":63405,"style":63334},[771],[224,63407,63409,63412,63415,63418,63421,63424,63427],{"className":63408},[468],[224,63410],{"className":63411},[468],[224,63413],{"className":63414,"style":496},[473],[224,63416,278],{"className":63417},[500],[224,63419],{"className":63420,"style":496},[473],[224,63422,1435],{"className":63423},[468,469],[224,63425],{"className":63426,"style":724},[473],[224,63428,63430,63562],{"className":63429},[2091],[224,63431,63433,63439,63544,63547,63550,63553,63556],{"className":63432},[2091],[224,63434,63436],{"className":63435,"style":8281},[542,8280],[224,63437,239],{"className":63438},[2363,778],[224,63440,63442,63445,63541],{"className":63441},[468],[224,63443],{"className":63444},[542,8479],[224,63446,63448],{"className":63447},[20338],[224,63449,63451,63532],{"className":63450},[754,755],[224,63452,63454,63529],{"className":63453},[759],[224,63455,63457,63510,63518],{"className":63456,"style":42827},[763],[224,63458,63460,63463],{"style":63459},"top:-2.2175em;",[224,63461],{"className":63462,"style":2108},[771],[224,63464,63466],{"className":63465},[468],[224,63467,63469],{"className":63468},[468,54290],[224,63470,63472,63502],{"className":63471},[754,755],[224,63473,63475,63499],{"className":63474},[759],[224,63476,63478,63487],{"className":63477,"style":62596},[763],[224,63479,63481,63484],{"className":63480,"style":2120},[2139],[224,63482],{"className":63483,"style":2108},[771],[224,63485,1435],{"className":63486,"style":54310},[468,469],[224,63488,63489,63492],{"style":62608},[224,63490],{"className":63491,"style":2108},[771],[224,63493,63495],{"className":63494,"style":54367},[54366],[2155,63496,63497],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,63498],{"d":54374},[224,63500,787],{"className":63501},[786],[224,63503,63505],{"className":63504},[759],[224,63506,63508],{"className":63507,"style":62628},[763],[224,63509],{},[224,63511,63512,63515],{"style":20546},[224,63513],{"className":63514,"style":2108},[771],[224,63516],{"className":63517,"style":20554},[20553],[224,63519,63520,63523],{"style":20557},[224,63521],{"className":63522,"style":2108},[771],[224,63524,63526],{"className":63525},[468],[224,63527,1438],{"className":63528},[468],[224,63530,787],{"className":63531},[786],[224,63533,63535],{"className":63534},[759],[224,63536,63539],{"className":63537,"style":63538},[763],"height:0.93em;",[224,63540],{},[224,63542],{"className":63543},[549,8479],[224,63545],{"className":63546,"style":474},[473],[224,63548,5268],{"className":63549},[478],[224,63551],{"className":63552,"style":474},[473],[224,63554,1438],{"className":63555},[468],[224,63557,63559],{"className":63558,"style":8281},[549,8280],[224,63560,1716],{"className":63561},[2363,778],[224,63563,63565],{"className":63564},[750],[224,63566,63568],{"className":63567},[754],[224,63569,63571],{"className":63570},[759],[224,63572,63574],{"className":63573,"style":57582},[763],[224,63575,63576,63579],{"style":57585},[224,63577],{"className":63578,"style":772},[771],[224,63580,63582],{"className":63581},[776,777,778,779],[224,63583,1447],{"className":63584},[468,779],[224,63586,63587,63590],{"style":63361},[224,63588],{"className":63589,"style":63334},[771],[224,63591,63593,63596,63599,63602,63605,63608,63614,63676,63679,63682,63685,63788,63791,63794,63797,63800],{"className":63592},[468],[224,63594],{"className":63595},[468],[224,63597],{"className":63598,"style":496},[473],[224,63600,278],{"className":63601},[500],[224,63603],{"className":63604,"style":496},[473],[224,63606,1435],{"className":63607},[468,469],[224,63609,63611],{"className":63610},[468],[224,63612,239],{"className":63613},[2363,23751],[224,63615,63617,63620,63673],{"className":63616},[468],[224,63618],{"className":63619},[542,8479],[224,63621,63623],{"className":63622},[20338],[224,63624,63626,63665],{"className":63625},[754,755],[224,63627,63629,63662],{"className":63628},[759],[224,63630,63632,63643,63651],{"className":63631,"style":42827},[763],[224,63633,63634,63637],{"style":20531},[224,63635],{"className":63636,"style":2108},[771],[224,63638,63640],{"className":63639},[468],[224,63641,1435],{"className":63642},[468,469],[224,63644,63645,63648],{"style":20546},[224,63646],{"className":63647,"style":2108},[771],[224,63649],{"className":63650,"style":20554},[20553],[224,63652,63653,63656],{"style":20557},[224,63654],{"className":63655,"style":2108},[771],[224,63657,63659],{"className":63658},[468],[224,63660,1438],{"className":63661},[468],[224,63663,787],{"className":63664},[786],[224,63666,63668],{"className":63667},[759],[224,63669,63671],{"className":63670,"style":23809},[763],[224,63672],{},[224,63674],{"className":63675},[549,8479],[224,63677],{"className":63678,"style":474},[473],[224,63680,5268],{"className":63681},[478],[224,63683],{"className":63684,"style":474},[473],[224,63686,63688,63691,63785],{"className":63687},[468],[224,63689],{"className":63690},[542,8479],[224,63692,63694],{"className":63693},[20338],[224,63695,63697,63777],{"className":63696},[754,755],[224,63698,63700,63774],{"className":63699},[759],[224,63701,63703,63755,63763],{"className":63702,"style":42827},[763],[224,63704,63705,63708],{"style":63459},[224,63706],{"className":63707,"style":2108},[771],[224,63709,63711],{"className":63710},[468],[224,63712,63714],{"className":63713},[468,54290],[224,63715,63717,63747],{"className":63716},[754,755],[224,63718,63720,63744],{"className":63719},[759],[224,63721,63723,63732],{"className":63722,"style":62596},[763],[224,63724,63726,63729],{"className":63725,"style":2120},[2139],[224,63727],{"className":63728,"style":2108},[771],[224,63730,1435],{"className":63731,"style":54310},[468,469],[224,63733,63734,63737],{"style":62608},[224,63735],{"className":63736,"style":2108},[771],[224,63738,63740],{"className":63739,"style":54367},[54366],[2155,63741,63742],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,63743],{"d":54374},[224,63745,787],{"className":63746},[786],[224,63748,63750],{"className":63749},[759],[224,63751,63753],{"className":63752,"style":62628},[763],[224,63754],{},[224,63756,63757,63760],{"style":20546},[224,63758],{"className":63759,"style":2108},[771],[224,63761],{"className":63762,"style":20554},[20553],[224,63764,63765,63768],{"style":20557},[224,63766],{"className":63767,"style":2108},[771],[224,63769,63771],{"className":63770},[468],[224,63772,1447],{"className":63773},[468],[224,63775,787],{"className":63776},[786],[224,63778,63780],{"className":63779},[759],[224,63781,63783],{"className":63782,"style":63538},[763],[224,63784],{},[224,63786],{"className":63787},[549,8479],[224,63789],{"className":63790,"style":474},[473],[224,63792,319],{"className":63793},[478],[224,63795],{"className":63796,"style":474},[473],[224,63798,1438],{"className":63799},[468],[224,63801,63803],{"className":63802},[468],[224,63804,1716],{"className":63805},[2363,23751],[224,63807,63808,63811],{"style":63370},[224,63809],{"className":63810,"style":63334},[771],[224,63812,63814,63817,63820,63823,63826,63829,63832,63835,63838,63841,63885,63888,63891,63894,63897,63900,63903,63906,63909,63953,63956,63959,63962,63965,63994,63997,64000,64003,64006,64009,64012,64015],{"className":63813},[468],[224,63815],{"className":63816},[468],[224,63818],{"className":63819,"style":496},[473],[224,63821,278],{"className":63822},[500],[224,63824],{"className":63825,"style":496},[473],[224,63827,1438],{"className":63828},[468],[224,63830],{"className":63831,"style":474},[473],[224,63833,5268],{"className":63834},[478],[224,63836],{"className":63837,"style":474},[473],[224,63839,1447],{"className":63840},[468],[224,63842,63844],{"className":63843},[468,54290],[224,63845,63847,63877],{"className":63846},[754,755],[224,63848,63850,63874],{"className":63849},[759],[224,63851,63853,63862],{"className":63852,"style":60451},[763],[224,63854,63856,63859],{"className":63855,"style":2120},[2139],[224,63857],{"className":63858,"style":2108},[771],[224,63860,1435],{"className":63861,"style":54310},[468,469],[224,63863,63864,63867],{"style":60466},[224,63865],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Hellinger",[224,64162,64164,64182],{"className":64163,"translate":388},[392],[224,64165,64167],{"className":64166},[396],[398,64168,64169],{"xmlns":400},[403,64170,64171,64179],{},[406,64172,64173],{},[3248,64174,64175,64177],{},[406,64176],{},[419,64178,1447],{},[447,64180,64181],{"encoding":449},"^2",[224,64183,64185],{"className":64184,"ariaHidden":455},[454],[224,64186,64188,64191],{"className":64187},[459],[224,64189],{"className":64190,"style":5671},[463],[224,64192,64194,64196],{"className":64193},[468],[224,64195],{},[224,64197,64199],{"className":64198},[750],[224,64200,64202],{"className":64201},[754],[224,64203,64205],{"className":64204},[759],[224,64206,64208],{"className":64207,"style":5671},[763],[224,64209,64210,64213],{"style":3288},[224,64211],{"className":64212,"style":772},[771],[224,64214,64216],{"className":64215},[776,777,778,779],[224,64217,1447],{"className":64218},[468,779]," 也满足对称判别准则（4.3 节）。",[11,64221,64222,64224,64225,64339,64340,64343,64344,64400,64401,64457,64458,64616,64617,64645],{},[378,64223,44131],{}," Hellinger 与 Bhattacharyya 系数 ",[224,64226,64228,64256],{"className":64227,"translate":388},[392],[224,64229,64231],{"className":64230},[396],[398,64232,64233],{"xmlns":400},[403,64234,64235,64253],{},[406,64236,64237,64239,64241,64243,64245],{},[409,64238,9889],{},[409,64240,16021],{},[412,64242,278],{},[412,64244,21338],{},[54262,64246,64247],{},[406,64248,64249,64251],{},[409,64250,11],{},[409,64252,2023],{},[447,64254,64255],{"encoding":449},"BC=\\int\\sqrt{pq}",[224,64257,64259,64280],{"className":64258,"ariaHidden":455},[454],[224,64260,64262,64265,64268,64271,64274,64277],{"className":64261},[459],[224,64263],{"className":64264,"style":649},[463],[224,64266,9889],{"className":64267,"style":9923},[468,469],[224,64269,16021],{"className":64270,"style":30740},[468,469],[224,64272],{"className":64273,"style":496},[473],[224,64275,278],{"className":64276},[500],[224,64278],{"className":64279,"style":496},[473],[224,64281,64283,64286,64289,64292],{"className":64282},[459],[224,64284],{"className":64285,"style":58928},[463],[224,64287,21338],{"className":64288,"style":25626},[21531,21532,25625],[224,64290],{"className":64291,"style":724},[473],[224,64293,64295],{"className":64294},[468,54290],[224,64296,64298,64331],{"className":64297},[754,755],[224,64299,64301,64328],{"className":64300},[759],[224,64302,64304,64316],{"className":64303,"style":58947},[763],[224,64305,64307,64310],{"className":64306,"style":2120},[2139],[224,64308],{"className":64309,"style":2108},[771],[224,64311,64313],{"className":64312,"style":54310},[468],[224,64314,61518],{"className":64315,"style":492},[468,469],[224,64317,64318,64321],{"style":58959},[224,64319],{"className":64320,"style":2108},[771],[224,64322,64324],{"className":64323,"style":54367},[54366],[2155,64325,64326],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,64327],{"d":54374},[224,64329,787],{"className":64330},[786],[224,64332,64334],{"className":64333},[759],[224,64335,64337],{"className":64336,"style":58979},[763],[224,64338],{}," 把「两个分布的",[378,64341,64342],{},"重叠程度","」量化：",[224,64345,64347,64367],{"className":64346,"translate":388},[392],[224,64348,64350],{"className":64349},[396],[398,64351,64352],{"xmlns":400},[403,64353,64354,64364],{},[406,64355,64356,64358,64360,64362],{},[409,64357,9889],{},[409,64359,16021],{},[412,64361,278],{},[419,64363,1438],{},[447,64365,64366],{"encoding":449},"BC=1",[224,64368,64370,64391],{"className":64369,"ariaHidden":455},[454],[224,64371,64373,64376,64379,64382,64385,64388],{"className":64372},[459],[224,64374],{"className":64375,"style":649},[463],[224,64377,9889],{"className":64378,"style":9923},[468,469],[224,64380,16021],{"className":64381,"style":30740},[468,469],[224,64383],{"className":64384,"style":496},[473],[224,64386,278],{"className":64387},[500],[224,64389],{"className":64390,"style":496},[473],[224,64392,64394,64397],{"className":64393},[459],[224,64395],{"className":64396,"style":510},[463],[224,64398,1438],{"className":64399},[468]," 表示完全重合，",[224,64402,64404,64424],{"className":64403,"translate":388},[392],[224,64405,64407],{"className":64406},[396],[398,64408,64409],{"xmlns":400},[403,64410,64411,64421],{},[406,64412,64413,64415,64417,64419],{},[409,64414,9889],{},[409,64416,16021],{},[412,64418,278],{},[419,64420,284],{},[447,64422,64423],{"encoding":449},"BC=0",[224,64425,64427,64448],{"className":64426,"ariaHidden":455},[454],[224,64428,64430,64433,64436,64439,64442,64445],{"className":64429},[459],[224,64431],{"className":64432,"style":649},[463],[224,64434,9889],{"className":64435,"style":9923},[468,469],[224,64437,16021],{"className":64438,"style":30740},[468,469],[224,64440],{"className":64441,"style":496},[473],[224,64443,278],{"className":64444},[500],[224,64446],{"className":64447,"style":496},[473],[224,64449,64451,64454],{"className":64450},[459],[224,64452],{"className":64453,"style":510},[463],[224,64455,284],{"className":64456},[468]," 表示支撑分离。由它可造出真正的距离 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",[224,65079,65081,65121],{"className":65080,"translate":388},[392],[224,65082,65084],{"className":65083},[396],[398,65085,65086],{"xmlns":400},[403,65087,65088,65118],{},[406,65089,65090,65092,65094,65106,65108,65110,65112,65114,65116],{},[409,65091,15],{},[412,65093,278],{},[20338,65095,65096,65104],{},[406,65097,65098,65100,65102],{},[419,65099,1438],{},[412,65101,319],{},[409,65103,64666],{},[419,65105,1447],{},[412,65107,12248],{},[412,65109,239],{"stretchy":435},[419,65111,284],{},[412,65113,1877],{"separator":455},[419,65115,1438],{},[412,65117,1716],{"stretchy":435},[447,65119,65120],{"encoding":449},"a=\\frac{1+\\alpha}{2}\\in(0,1)",[224,65122,65124,65142,65231],{"className":65123,"ariaHidden":455},[454],[224,65125,65127,65130,65133,65136,65139],{"className":65126},[459],[224,65128],{"className":65129,"style":839},[463],[224,65131,15],{"className":65132},[468,469],[224,65134],{"className":65135,"style":496},[473],[224,65137,278],{"className":65138},[500],[224,65140],{"className":65141,"style":496},[473],[224,65143,65145,65148,65222,65225,65228],{"className":65144},[459],[224,65146],{"className":65147,"style":22544},[463],[224,65149,65151,65154,65219],{"className":65150},[468],[224,65152],{"className":65153},[542,8479],[224,65155,65157],{"className":65156},[20338],[224,65158,65160,65211],{"className":65159},[754,755],[224,65161,65163,65208],{"className":65162},[759],[224,65164,65166,65180,65188],{"className":65165,"style":22563},[763],[224,65167,65168,65171],{"style":22566},[224,65169],{"className":65170,"style":2108},[771],[224,65172,65174],{"className":65173},[776,777,778,779],[224,65175,65177],{"className":65176},[468,779],[224,65178,1447],{"className":65179},[468,779],[224,65181,65182,65185],{"style":20546},[224,65183],{"className":65184,"style":2108},[771],[224,65186],{"className":65187,"style":20554},[20553],[224,65189,65190,65193],{"style":22589},[224,65191],{"className":65192,"style":2108},[771],[224,65194,65196],{"className":65195},[776,777,778,779],[224,65197,65199,65202,65205],{"className":65198},[468,779],[224,65200,1438],{"className":65201},[468,779],[224,65203,319],{"className":65204},[478,779],[224,65206,64666],{"className":65207,"style":64696},[468,469,779],[224,65209,787],{"className":65210},[786],[224,65212,65214],{"className":65213},[759],[224,65215,65217],{"className":65216,"style":22611},[763],[224,65218],{},[224,65220],{"className":65221},[549,8479],[224,65223],{"className":65224,"style":496},[473],[224,65226,12248],{"className":65227},[500],[224,65229],{"className":65230,"style":496},[473],[224,65232,65234,65237,65240,65243,65246,65249,65252],{"className":65233},[459],[224,65235],{"className":65236,"style":535},[463],[224,65238,239],{"className":65239},[542],[224,65241,284],{"className":65242},[468],[224,65244,1877],{"className":65245},[1917],[224,65247],{"className":65248,"style":724},[473],[224,65250,1438],{"className":65251},[468],[224,65253,1716],{"className":65254},[549],"（于是 ",[224,65257,65259,65307],{"className":65258,"translate":388},[392],[224,65260,65262],{"className":65261},[396],[398,65263,65264],{"xmlns":400},[403,65265,65266,65304],{},[406,65267,65268,65284,65286],{},[20338,65269,65270,65272],{},[419,65271,9409],{},[406,65273,65274,65276,65278],{},[419,65275,1438],{},[412,65277,5268],{},[3248,65279,65280,65282],{},[409,65281,64666],{},[419,65283,1447],{},[412,65285,278],{},[20338,65287,65288,65290],{},[419,65289,1438],{},[406,65291,65292,65294,65296,65298,65300,65302],{},[409,65293,15],{},[412,65295,239],{"stretchy":435},[419,65297,1438],{},[412,65299,5268],{},[409,65301,15],{},[412,65303,1716],{"stretchy":435},[447,65305,65306],{"encoding":449},"\\frac{4}{1-\\alpha^2}=\\frac{1}{a(1-a)}",[224,65308,65310,65428],{"className":65309,"ariaHidden":455},[454],[224,65311,65313,65317,65419,65422,65425],{"className":65312},[459],[224,65314],{"className":65315,"style":65316},[463],"height:1.2484em;vertical-align:-0.4033em;",[224,65318,65320,65323,65416],{"className":65319},[468],[224,65321],{"className":65322},[542,8479],[224,65324,65326],{"className":65325},[20338],[224,65327,65329,65408],{"className":65328},[754,755],[224,65330,65332,65405],{"className":65331},[759],[224,65333,65335,65383,65391],{"className":65334,"style":22563},[763],[224,65336,65337,65340],{"style":22566},[224,65338],{"className":65339,"style":2108},[771],[224,65341,65343],{"className":65342},[776,777,778,779],[224,65344,65346,65349,65352],{"className":65345},[468,779],[224,65347,1438],{"className":65348},[468,779],[224,65350,5268],{"className":65351},[478,779],[224,65353,65355,65358],{"className":65354},[468,779],[224,65356,64666],{"className":65357,"style":64696},[468,469,779],[224,65359,65361],{"className":65360},[750],[224,65362,65364],{"className":65363},[754],[224,65365,65367],{"className":65366},[759],[224,65368,65371],{"className":65369,"style":65370},[763],"height:0.7463em;",[224,65372,65374,65377],{"style":65373},"top:-2.786em;margin-right:0.0714em;",[224,65375],{"className":65376,"style":50136},[771],[224,65378,65380],{"className":65379},[776,50140,2364,779],[224,65381,1447],{"className":65382},[468,779],[224,65384,65385,65388],{"style":20546},[224,65386],{"className":65387,"style":2108},[771],[224,65389],{"className":65390,"style":20554},[20553],[224,65392,65393,65396],{"style":22589},[224,65394],{"className":65395,"style":2108},[771],[224,65397,65399],{"className":65398},[776,777,778,779],[224,65400,65402],{"className":65401},[468,779],[224,65403,9409],{"className":65404},[468,779],[224,65406,787],{"className":65407},[786],[224,65409,65411],{"className":65410},[759],[224,65412,65414],{"className":65413,"style":36373},[763],[224,65415],{},[224,65417],{"className":65418},[549,8479],[224,65420],{"className":65421,"style":496},[473],[224,65423,278],{"className":65424},[500],[224,65426],{"className":65427,"style":496},[473],[224,65429,65431,65435],{"className":65430},[459],[224,65432],{"className":65433,"style":65434},[463],"height:1.3651em;vertical-align:-0.52em;",[224,65436,65438,65441,65516],{"className":65437},[468],[224,65439],{"className":65440},[542,8479],[224,65442,65444],{"className":65443},[20338],[224,65445,65447,65507],{"className":65446},[754,755],[224,65448,65450,65504],{"className":65449},[759],[224,65451,65453,65482,65490],{"className":65452,"style":22563},[763],[224,65454,65455,65458],{"style":22566},[224,65456],{"className":65457,"style":2108},[771],[224,65459,65461],{"className":65460},[776,777,778,779],[224,65462,65464,65467,65470,65473,65476,65479],{"className":65463},[468,779],[224,65465,15],{"className":65466},[468,469,779],[224,65468,239],{"className":65469},[542,779],[224,65471,1438],{"className":65472},[468,779],[224,65474,5268],{"className":65475},[478,779],[224,65477,15],{"className":65478},[468,469,779],[224,65480,1716],{"className":65481},[549,779],[224,65483,65484,65487],{"style":20546},[224,65485],{"className":65486,"style":2108},[771],[224,65488],{"className":65489,"style":20554},[20553],[224,65491,65492,65495],{"style":22589},[224,65493],{"className":65494,"style":2108},[771],[224,65496,65498],{"className":65497},[776,777,778,779],[224,65499,65501],{"className":65500},[468,779],[224,65502,1438],{"className":65503},[468,779],[224,65505,787],{"className":65506},[786],[224,65508,65510],{"className":65509},[759],[224,65511,65514],{"className":65512,"style":65513},[763],"height:0.52em;",[224,65515],{},[224,65517],{"className":65518},[549,8479],"）：",[224,65521,65523],{"className":65522,"translate":388},[387],[224,65524,65526,65612],{"className":65525,"translate":388},[392],[224,65527,65529],{"className":65528},[396],[398,65530,65531],{"xmlns":400,"display":401},[403,65532,65533,65609],{},[406,65534,65535,65541,65543,65545,65547,65549,65551,65553,65571,65607],{},[695,65536,65537,65539],{},[409,65538,20303],{},[409,65540,64666],{},[412,65542,239],{"stretchy":435},[409,65544,16018],{},[409,65546,20312],{"mathvariant":678},[409,65548,20315],{},[412,65550,1716],{"stretchy":435},[412,65552,278],{},[20338,65554,65555,65557],{},[419,65556,1438],{},[406,65558,65559,65561,65563,65565,65567,65569],{},[409,65560,15],{},[412,65562,239],{"stretchy":435},[419,65564,1438],{},[412,65566,5268],{},[409,65568,15],{},[412,65570,1716],{"stretchy":435},[406,65572,65573,65575,65577,65579,65581,65587,65599,65601,65603,65605],{},[412,65574,239],{"fence":455},[419,65576,1438],{},[412,65578,5268],{},[412,65580,21338],{},[3248,65582,65583,65585],{},[409,65584,11],{},[409,65586,15],{},[3248,65588,65589,65591],{},[409,65590,2023],{},[406,65592,65593,65595,65597],{},[419,65594,1438],{},[412,65596,5268],{},[409,65598,15],{},[423,65600,682],{},[409,65602,13182],{},[409,65604,371],{},[412,65606,1716],{"fence":455},[409,65608,688],{"mathvariant":678},[447,65610,65611],{"encoding":449},"D_\\alpha(P\\|Q)=\\frac{1}{a(1-a)}\n\\left(1-\\int p^{a}q^{1-a}\\,dx\\right).",[224,65613,65615,65685],{"className":65614,"ariaHidden":455},[454],[224,65616,65618,65621,65661,65664,65667,65670,65673,65676,65679,65682],{"className":65617},[459],[224,65619],{"className":65620,"style":535},[463],[224,65622,65624,65627],{"className":65623},[468],[224,65625,20303],{"className":65626,"style":2473},[468,469],[224,65628,65630],{"className":65629},[750],[224,65631,65633,65653],{"className":65632},[754,755],[224,65634,65636,65650],{"className":65635},[759],[224,65637,65639],{"className":65638,"style":64841},[763],[224,65640,65641,65644],{"style":20390},[224,65642],{"className":65643,"style":772},[771],[224,65645,65647],{"className":65646},[776,777,778,779],[224,65648,64666],{"className":65649,"style":64696},[468,469,779],[224,65651,787],{"className":65652},[786],[224,65654,65656],{"className":65655},[759],[224,65657,65659],{"className":65658,"style":794},[763],[224,65660],{},[224,65662,239],{"className":65663},[542],[224,65665,16018],{"className":65666,"style":653},[468,469],[224,65668,20312],{"className":65669},[468],[224,65671,20315],{"className":65672},[468,469],[224,65674,1716],{"className":65675},[549],[224,65677],{"className":65678,"style":496},[473],[224,65680,278],{"className":65681},[500],[224,65683],{"className":65684,"style":496},[473],[224,65686,65688,65691,65774,65777,65890,65893],{"className":65687},[459],[224,65689],{"className":65690,"style":20442},[463],[224,65692,65694,65697,65771],{"className":65693},[468],[224,65695],{"className":65696},[542,8479],[224,65698,65700],{"className":65699},[20338],[224,65701,65703,65763],{"className":65702},[754,755],[224,65704,65706,65760],{"className":65705},[759],[224,65707,65709,65741,65749],{"className":65708,"style":42827},[763],[224,65710,65711,65714],{"style":20531},[224,65712],{"className":65713,"style":2108},[771],[224,65715,65717,65720,65723,65726,65729,65732,65735,65738],{"className":65716},[468],[224,65718,15],{"className":65719},[468,469],[224,65721,239],{"className":65722},[542],[224,65724,1438],{"className":65725},[468],[224,65727],{"className":65728,"style":474},[473],[224,65730,5268],{"className":65731},[478],[224,65733],{"className":65734,"style":474},[473],[224,65736,15],{"className":65737},[468,469],[224,65739,1716],{"className":65740},[549],[224,65742,65743,65746],{"style":20546},[224,65744],{"className":65745,"style":2108},[771],[224,65747],{"className":65748,"style":20554},[20553],[224,65750,65751,65754],{"style":20557},[224,65752],{"className":65753,"style":2108},[771],[224,65755,65757],{"className":65756},[468],[224,65758,1438],{"className":65759},[468],[224,65761,787],{"className":65762},[786],[224,65764,65766],{"className":65765},[759],[224,65767,65769],{"className":65768,"style":21648},[763],[224,65770],{},[224,65772],{"className":65773},[549,8479],[224,65775],{"className":65776,"style":724},[473],[224,65778,65780,65786,65789,65792,65795,65798,65801,65804,65837,65875,65878,65881,65884],{"className":65779},[2091],[224,65781,65783],{"className":65782,"style":8281},[542,8280],[224,65784,239],{"className":65785},[2363,778],[224,65787,1438],{"className":65788},[468],[224,65790],{"className":65791,"style":474},[473],[224,65793,5268],{"className":65794},[478],[224,65796],{"className":65797,"style":474},[473],[224,65799,21338],{"className":65800,"style":21534},[21531,21532,21533],[224,65802],{"className":65803,"style":724},[473],[224,65805,65807,65810],{"className":65806},[468],[224,65808,11],{"className":65809},[468,469],[224,65811,65813],{"className":65812},[750],[224,65814,65816],{"className":65815},[754],[224,65817,65819],{"className":65818},[759],[224,65820,65823],{"className":65821,"style":65822},[763],"height:0.7144em;",[224,65824,65825,65828],{"style":5074},[224,65826],{"className":65827,"style":772},[771],[224,65829,65831],{"className":65830},[776,777,778,779],[224,65832,65834],{"className":65833},[468,779],[224,65835,15],{"className":65836},[468,469,779],[224,65838,65840,65843],{"className":65839},[468],[224,65841,2023],{"className":65842,"style":492},[468,469],[224,65844,65846],{"className":65845},[750],[224,65847,65849],{"className":65848},[754],[224,65850,65852],{"className":65851},[759],[224,65853,65855],{"className":65854,"style":5071},[763],[224,65856,65857,65860],{"style":5074},[224,65858],{"className":65859,"style":772},[771],[224,65861,65863],{"className":65862},[776,777,778,779],[224,65864,65866,65869,65872],{"className":65865},[468,779],[224,65867,1438],{"className":65868},[468,779],[224,65870,5268],{"className":65871},[478,779],[224,65873,15],{"className":65874},[468,469,779],[224,65876],{"className":65877,"style":724},[473],[224,65879,13182],{"className":65880},[468,469],[224,65882,371],{"className":65883},[468,469],[224,65885,65887],{"className":65886,"style":8281},[549,8280],[224,65888,1716],{"className":65889},[2363,778],[224,65891],{"className":65892,"style":724},[473],[224,65894,688],{"className":65895},[468],[11,65897,65898,65899,65902,65903,12708,65958,12570],{},"这正是 Csiszár \u002F Amari 的 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是 Hellinger 型散度。两个端点极限要小心：",[224,66432,66434,66452],{"className":66433,"translate":388},[392],[224,66435,66437],{"className":66436},[396],[398,66438,66439],{"xmlns":400},[403,66440,66441,66449],{},[406,66442,66443],{},[695,66444,66445,66447],{},[409,66446,1705],{},[409,66448,64666],{},[447,66450,66451],{"encoding":449},"f_\\alpha",[224,66453,66455],{"className":66454,"ariaHidden":455},[454],[224,66456,66458,66461],{"className":66457},[459],[224,66459],{"className":66460,"style":2565},[463],[224,66462,66464,66467],{"className":66463},[468],[224,66465,1705],{"className":66466,"style":1740},[468,469],[224,66468,66470],{"className":66469},[750],[224,66471,66473,66493],{"className":66472},[754,755],[224,66474,66476,66490],{"className":66475},[759],[224,66477,66479],{"className":66478,"style":64841},[763],[224,66480,66481,66484],{"style":47439},[224,66482],{"className":66483,"style":772},[771],[224,66485,66487],{"className":66486},[776,777,778,779],[224,66488,64666],{"className":66489,"style":64696},[468,469,779],[224,66491,787],{"className":66492},[786],[224,66494,66496],{"className":66495},[759],[224,66497,66499],{"className":66498,"style":794},[763],[224,66500],{},[224,66502,66504,66525],{"className":66503,"translate":388},[392],[224,66505,66507],{"className":66506},[396],[398,66508,66509],{"xmlns":400},[403,66510,66511,66522],{},[406,66512,66513,66515,66517,66520],{},[409,66514,64666],{},[412,66516,1144],{},[412,66518,66519],{},"±",[419,66521,1438],{},[447,66523,66524],{"encoding":449},"\\alpha\\to\\pm1",[224,66526,66528,66546],{"className":66527,"ariaHidden":455},[454],[224,66529,66531,66534,66537,66540,66543],{"className":66530},[459],[224,66532],{"className":66533,"style":839},[463],[224,66535,64666],{"className":66536,"style":64696},[468,469],[224,66538],{"className":66539,"style":496},[473],[224,66541,1144],{"className":66542},[500],[224,66544],{"className":66545,"style":496},[473],[224,66547,66549,66552,66555],{"className":66548},[459],[224,66550],{"className":66551,"style":7047},[463],[224,66553,66519],{"className":66554},[468],[224,66556,1438],{"className":66557},[468]," 时逐点发散，但",[378,66560,66561],{},"散度值","收敛到 KL：",[224,66564,66566],{"className":66565,"translate":388},[387],[224,66567,66569,66698],{"className":66568,"translate":388},[392],[224,66570,66572],{"className":66571},[396],[398,66573,66574],{"xmlns":400,"display":401},[403,66575,66576,66695],{},[406,66577,66578,66595,66601,66603,66605,66607,66609,66611,66613,66623,66625,66627,66629,66631,66633,66635,66637,66655,66661,66663,66665,66667,66669,66671,66673,66683,66685,66687,66689,66691,66693],{},[1965,66579,66580,66587],{},[406,66581,66582,66585],{},[409,66583,66584],{},"lim",[412,66586,24134],{},[406,66588,66589,66591,66593],{},[409,66590,64666],{},[412,66592,1144],{},[419,66594,1438],{},[695,66596,66597,66599],{},[409,66598,20303],{},[409,66600,64666],{},[412,66602,239],{"stretchy":435},[409,66604,16018],{},[409,66606,20312],{"mathvariant":678},[409,66608,20315],{},[412,66610,1716],{"stretchy":435},[412,66612,278],{},[695,66614,66615,66617],{},[409,66616,20303],{},[406,66618,66619,66621],{},[409,66620,30652],{},[409,66622,41831],{},[412,66624,239],{"stretchy":435},[409,66626,16018],{},[409,66628,20312],{"mathvariant":678},[409,66630,20315],{},[412,66632,1716],{"stretchy":435},[412,66634,1877],{"separator":455},[473,66636],{"width":4247},[1965,66638,66639,66645],{},[406,66640,66641,66643],{},[409,66642,66584],{},[412,66644,24134],{},[406,66646,66647,66649,66651,66653],{},[409,66648,64666],{},[412,66650,1144],{},[412,66652,5268],{},[419,66654,1438],{},[695,66656,66657,66659],{},[409,66658,20303],{},[409,66660,64666],{},[412,66662,239],{"stretchy":435},[409,66664,16018],{},[409,66666,20312],{"mathvariant":678},[409,66668,20315],{},[412,66670,1716],{"stretchy":435},[412,66672,278],{},[695,66674,66675,66677],{},[409,66676,20303],{},[406,66678,66679,66681],{},[409,66680,30652],{},[409,66682,41831],{},[412,66684,239],{"stretchy":435},[409,66686,20315],{},[409,66688,20312],{"mathvariant":678},[409,66690,16018],{},[412,66692,1716],{"stretchy":435},[409,66694,688],{"mathvariant":678},[447,66696,66697],{"encoding":449},"\\lim_{\\alpha\\to1}D_\\alpha(P\\|Q)=D_{KL}(P\\|Q),\n\\qquad\n\\lim_{\\alpha\\to-1}D_\\alpha(P\\|Q)=D_{KL}(Q\\|P).",[224,66699,66701,66830,67031],{"className":66700,"ariaHidden":455},[454],[224,66702,66704,66708,66763,66766,66806,66809,66812,66815,66818,66821,66824,66827],{"className":66703},[459],[224,66705],{"className":66706,"style":66707},[463],"height:1.4671em;vertical-align:-0.7171em;",[224,66709,66711],{"className":66710},[21531,33665],[224,66712,66714,66754],{"className":66713},[754,755],[224,66715,66717,66751],{"className":66716},[759],[224,66718,66720,66741],{"className":66719,"style":7642},[763],[224,66721,66723,66726],{"style":66722},"top:-2.3829em;margin-left:0em;",[224,66724],{"className":66725,"style":2108},[771],[224,66727,66729],{"className":66728},[776,777,778,779],[224,66730,66732,66735,66738],{"className":66731},[468,779],[224,66733,64666],{"className":66734,"style":64696},[468,469,779],[224,66736,1144],{"className":66737},[500,779],[224,66739,1438],{"className":66740},[468,779],[224,66742,66743,66746],{"style":2120},[224,66744],{"className":66745,"style":2108},[771],[224,66747,66748],{},[224,66749,66584],{"className":66750},[21531],[224,66752,787],{"className":66753},[786],[224,66755,66757],{"className":66756},[759],[224,66758,66761],{"className":66759,"style":66760},[763],"height:0.7171em;",[224,66762],{},[224,66764],{"className":66765,"style":724},[473],[224,66767,66769,66772],{"className":66768},[468],[224,66770,20303],{"className":66771,"style":2473},[468,469],[224,66773,66775],{"className":66774},[750],[224,66776,66778,66798],{"className":66777},[754,755],[224,66779,66781,66795],{"className":66780},[759],[224,66782,66784],{"className":66783,"style":64841},[763],[224,66785,66786,66789],{"style":20390},[224,66787],{"className":66788,"style":772},[771],[224,66790,66792],{"className":66791},[776,777,778,779],[224,66793,64666],{"className":66794,"style":64696},[468,469,779],[224,66796,787],{"className":66797},[786],[224,66799,66801],{"className":66800},[759],[224,66802,66804],{"className":66803,"style":794},[763],[224,66805],{},[224,66807,239],{"className":66808},[542],[224,66810,16018],{"className":66811,"style":653},[468,469],[224,66813,20312],{"className":66814},[468],[224,66816,20315],{"className":66817},[468,469],[224,66819,1716],{"className":66820},[549],[224,66822],{"className":66823,"style":496},[473],[224,66825,278],{"className":66826},[500],[224,66828],{"className":66829,"style":496},[473],[224,66831,66833,66837,66883,66886,66889,66892,66895,66898,66901,66904,66907,66964,66967,67007,67010,67013,67016,67019,67022,67025,67028],{"className":66832},[459],[224,66834],{"className":66835,"style":66836},[463],"height:1.5254em;vertical-align:-0.7754em;",[224,66838,66840,66843],{"className":66839},[468],[224,66841,20303],{"className":66842,"style":2473},[468,469],[224,66844,66846],{"className":66845},[750],[224,66847,66849,66875],{"className":66848},[754,755],[224,66850,66852,66872],{"className":66851},[759],[224,66853,66855],{"cl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对称地给出反向 KL。所以 α-散度把 KL（两个方向）与 Hellinger 连成一条曲线，",[224,68984,68986,69000],{"className":68985,"translate":388},[392],[224,68987,68989],{"className":68988},[396],[398,68990,68991],{"xmlns":400},[403,68992,68993,68997],{},[406,68994,68995],{},[409,68996,64666],{},[447,68998,68999],{"encoding":449},"\\alpha",[224,69001,69003],{"className":69002,"ariaHidden":455},[454],[224,69004,69006,69009],{"className":69005},[459],[224,69007],{"className":69008,"style":839},[463],[224,69010,64666],{"className":69011,"style":64696},[468,469]," 是「在对称与不对称之间连续插值」的旋钮。",[11,69014,69015,69017,69018,69046,69047,69081,69082,69110,69111,69161,69162,69165,69166,69194,69195,69223,69224,69252,69253,69281],{},[378,69016,44131],{}," α-散度的价值在于它把「方向」与「强度」一并参数化：",[224,69019,69021,69034],{"className":69020,"translate":388},[392],[224,69022,69024],{"className":69023},[396],[398,69025,69026],{"xmlns":400},[403,69027,69028,69032],{},[406,69029,69030],{},[409,69031,64666],{},[447,69033,68999],{"encoding":449},[224,69035,69037],{"className":69036,"ariaHidden":455},[454],[224,69038,69040,69043],{"className":69039},[459],[224,69041],{"className":69042,"style":839},[463],[224,69044,64666],{"className":69045,"style":64696},[468,469]," 从 ",[224,69048,69050,69066],{"className":69049,"translate":388},[392],[224,69051,69053],{"className":69052},[396],[398,69054,69055],{"xmlns":400},[403,69056,69057,69063],{},[406,69058,69059,69061],{},[412,69060,5268],{},[419,69062,1438],{},[447,69064,69065],{"encoding":449},"-1",[224,69067,69069],{"className":69068,"ariaHidden":455},[454],[224,69070,69072,69075,69078],{"className":69071},[459],[224,69073],{"className":69074,"style":7047},[463],[224,69076,5268],{"className":69077},[468],[224,69079,1438],{"className":69080},[468]," 扫到 ",[224,69083,69085,69098],{"className":69084,"translate":388},[392],[224,69086,69088],{"className":69087},[396],[398,69089,69090],{"xmlns":400},[403,69091,69092,69096],{},[406,69093,69094],{},[419,69095,1438],{},[447,69097,1438],{"encoding":449},[224,69099,69101],{"className":69100,"ariaHidden":455},[454],[224,69102,69104,69107],{"className":69103},[459],[224,69105],{"className":69106,"style":510},[463],[224,69108,1438],{"className":69109},[468]," 时，散度由反向 KL（mode-seeking）连续过渡到正向 KL（mass-covering），中间在 ",[224,69112,69114,69131],{"className":69113,"translate":388},[392],[224,69115,69117],{"className":69116},[396],[398,69118,69119],{"xmlns":400},[403,69120,69121,69129],{},[406,69122,69123,69125,69127],{},[409,69124,64666],{},[412,69126,278],{},[419,69128,284],{},[447,69130,66399],{"encoding":449},[224,69132,69134,69152],{"className":69133,"ariaHidden":455},[454],[224,69135,69137,69140,69143,69146,69149],{"className":69136},[459],[224,69138],{"className":69139,"style":839},[463],[224,69141,64666],{"className":69142,"style":64696},[468,469],[224,69144],{"className":69145,"style":496},[473],[224,69147,278],{"className":69148},[500],[224,69150],{"className":69151,"style":496},[473],[224,69153,69155,69158],{"className":69154},[459],[224,69156],{"className":69157,"style":510},[463],[224,69159,284],{"className":69160},[468]," 处恰好经过 Hellinger 这个对称点。在 Amari 的",[378,69163,69164],{},"信息几何","里，α-族是唯一同时拥有 ",[224,69167,69169,69182],{"className":69168,"translate":388},[392],[224,69170,69172],{"className":69171},[396],[398,69173,69174],{"xmlns":400},[403,69175,69176,69180],{},[406,69177,69178],{},[409,69179,64666],{},[447,69181,68999],{"encoding":449},[224,69183,69185],{"className":69184,"ariaHidden":455},[454],[224,69186,69188,69191],{"className":69187},[459],[224,69189],{"className":69190,"style":839},[463],[224,69192,64666],{"className":69193,"style":64696},[468,469],"-对偶平坦结构的单参数散度族：它诱导的度量是 Fisher 度量，诱导的仿射联络是 ",[224,69196,69198,69211],{"className":69197,"translate":388},[392],[224,69199,69201],{"className":69200},[396],[398,69202,69203],{"xmlns":400},[403,69204,69205,69209],{},[406,69206,69207],{},[409,69208,64666],{},[447,69210,68999],{"encoding":449},[224,69212,69214],{"className":69213,"ariaHidden":455},[454],[224,69215,69217,69220],{"className":69216},[459],[224,69218],{"className":69219,"style":839},[463],[224,69221,64666],{"className":69222,"style":64696},[468,469],"-联络，这正是它被称为「散度范本」的原因。在机器学习中，",[224,69225,69227,69240],{"className":69226,"translate":388},[392],[224,69228,69230],{"className":69229},[396],[398,69231,69232],{"xmlns":400},[403,69233,69234,69238],{},[406,69235,69236],{},[409,69237,64666],{},[447,69239,68999],{"encoding":449},[224,69241,69243],{"className":69242,"ariaHidden":455},[454],[224,69244,69246,69249],{"className":69245},[459],[224,69247],{"className":69248,"style":839},[463],[224,69250,64666],{"className":69251,"style":64696},[468,469],"-GAN 与 ",[224,69254,69256,69269],{"className":69255,"translate":388},[392],[224,69257,69259],{"className":69258},[396],[398,69260,69261],{"xmlns":400},[403,69262,69263,69267],{},[406,69264,69265],{},[409,69266,64666],{},[447,69268,68999],{"encoding":449},[224,69270,69272],{"className":69271,"ariaHidden":455},[454],[224,69273,69275,69278],{"className":69274},[459],[224,69276],{"className":69277,"style":839},[463],[224,69279,64666],{"className":69280,"style":64696},[468,469],"-VI 用一个参数调节「对异常值的鲁棒性」与「模式覆盖」之间的取舍；在信号处理中它用于独立成分分析（ICA）与鲁棒估计。",[613,69283,69285],{"id":69284},"_37-与-rényi-散度的关系","3.7 与 Rényi 散度的关系",[11,69287,69288],{},"Rényi 散度定义为",[224,69290,69292],{"className":69291,"translate":388},[387],[224,69293,69295,69422],{"className":69294,"translate":388},[392],[224,69296,69298],{"className":69297},[396],[398,69299,69300],{"xmlns":400,"display":401},[403,69301,69302,69419],{},[5209,69303,69304,69384],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,69305,69306,69330],{},[5218,69307,69308],{},[5221,69309,69310],{"scriptlevel":284,"displaystyle":455},[406,69311,69312,69320,69322,69324,69326,69328],{},[27527,69313,69314,69316,69318],{},[409,69315,20303],{},[409,69317,64666],{},[409,69319,21092],{},[412,69321,239],{"stretchy":435},[409,69323,16018],{},[409,69325,20312],{"mathvariant":678},[409,69327,20315],{},[412,69329,1716],{"stretchy":435},[5218,69331,69332],{},[5221,69333,69334],{"scriptlevel":284,"displaystyle":455},[406,69335,69336,69338,69340,69352,69354,69356,69358,69364,69376,69378,69380,69382],{},[406,69337],{},[412,69339,278],{},[20338,69341,69342,69344],{},[419,69343,1438],{},[406,69345,69346,69348,69350],{},[409,69347,64666],{},[412,69349,5268],{},[419,69351,1438],{},[409,69353,24131],{},[412,69355,24134],{},[412,69357,21338],{},[3248,69359,69360,69362],{},[409,69361,11],{},[409,69363,64666],{},[3248,69365,69366,69368],{},[409,69367,2023],{},[406,69369,69370,69372,69374],{},[419,69371,1438],{},[412,69373,5268],{},[409,69375,64666],{},[423,69377,682],{},[409,69379,13182],{},[409,69381,371],{},[412,69383,1877],{"separator":455},[5215,69385,69386,69392],{},[5218,69387,69388],{},[5221,69389,69390],{"scriptlevel":284,"displaystyle":455},[406,69391],{},[5218,69393,69394],{},[5221,69395,69396],{"scriptlevel":284,"displaystyle":455},[406,69397,69398,69400,69402,69404,69406,69408,69410,69412,69414,69416],{},[406,69399],{},[473,69401],{"width":4247},[409,69403,64666],{},[412,69405,325],{},[419,69407,284],{},[412,69409,1877],{"separator":455},[423,69411,15054],{},[409,69413,64666],{},[412,69415,2048],{"mathvariant":678},[419,69417,69418],{},"1.",[447,69420,69421],{"encoding":449},"\\begin{aligned}\nD^{R}_{\\alpha}(P\\|Q)\n&=\\frac{1}{\\alpha-1}\n\\log\\int p^{\\alpha}q^{1-\\alpha}\\,dx,\\\\[2pt]\n&\\qquad \\alpha>0,\\ \\alpha\\ne1 .\n\\end{aligned}",[224,69423,69425],{"className":69424,"ariaHidden":455},[454],[224,69426,69428,69432],{"className":69427},[459],[224,69429],{"className":69430,"style":69431},[463],"height:3.7223em;vertical-align:-1.6111em;",[224,69433,69435],{"className":69434},[468],[224,69436,69438,69555],{"className":69437},[5209],[224,69439,69441],{"className":69440},[5331],[224,69442,69444,69546],{"className":69443},[754,755],[224,69445,69447,69543],{"className":69446},[759],[224,69448,69451,69534],{"className":69449,"style":69450},[763],"height:2.1111em;",[224,69452,69454,69457],{"style":69453},"top:-4.1111em;",[224,69455],{"className":69456,"style":26276},[771],[224,69458,69460,69519,69522,69525,69528,69531],{"className":69459},[468],[224,69461,69463,69466],{"className":69462},[468],[224,69464,20303],{"className":69465,"style":2473},[468,469],[224,69467,69469],{"className":69468},[750],[224,69470,69472,69511],{"className":69471},[754,755],[224,69473,69475,69508],{"className":69474},[759],[224,69476,69478,69493],{"className":69477,"style":58433},[763],[224,69479,69481,69484],{"style":69480},"top:-2.453em;margin-left:-0.0278em;margin-right:0.05em;",[224,69482],{"className":69483,"style":772},[771],[224,69485,69487],{"className":69486},[776,777,778,779],[224,69488,69490],{"className":69489},[468,779],[224,69491,64666],{"className":69492,"style":64696},[468,469,779],[224,69494,69495,69498],{"style":5074},[224,69496],{"className":69497,"style":772},[771],[224,69499,69501],{"className":69500},[776,777,778,779],[224,69502,69504],{"className":69503},[468,779],[224,69505,21092],{"className":69506,"style":69507},[468,469,779],"margin-right:0.0077em;",[224,69509,787],{"className":69510},[786],[224,69512,69514],{"className":69513},[759],[224,69515,69517],{"className":69516,"style":28265},[763],[224,69518],{},[224,69520,239],{"className":69521},[542],[224,69523,16018],{"className":69524,"style":653},[468,469],[224,69526,20312],{"className":69527},[468],[224,69529,20315],{"className":69530},[468,469],[224,69532,1716],{"className":69533},[549],[224,69535,69537,69540],{"style":69536},"top:-2.1089em;",[224,69538],{"className":69539,"style":26276},[771],[224,69541],{"className":69542},[468],[224,69544,787],{"className":69545},[786],[224,69547,69549],{"className":69548},[759],[224,69550,69553],{"className":69551,"style":69552},[763],"height:1.6111em;",[224,69554],{},[224,69556,69558],{"className":69557},[5399],[224,69559,69561,69849],{"className":69560},[754,755],[224,69562,69564,69846],{"className":69563},[759],[224,69565,69567,69760],{"className":69566,"style":69450},[763],[224,69568,69569,69572],{"style":69453},[224,69570],{"className":69571,"style":26276},[771],[224,69573,69575,69578,69581,69584,69587,69661,69664,69669,69672,69675,69678,69710,69748,69751,69754,69757],{"className":69574},[468],[224,69576],{"className":69577},[468],[224,69579],{"className":69580,"style":496},[473],[224,69582,278],{"className":69583},[500],[224,69585],{"className":69586,"style":496},[473],[224,69588,69590,69593,69658],{"className":69589},[468],[224,69591],{"className":69592},[542,8479],[224,69594,69596],{"className":69595},[20338],[224,69597,69599,69650],{"className":69598},[754,755],[224,69600,69602,69647],{"className":69601},[759],[224,69603,69605,69628,69636],{"className":69604,"style":42827},[763],[224,69606,69607,69610],{"style":20531},[224,69608],{"className":69609,"style":2108},[771],[224,69611,69613,69616,69619,69622,69625],{"className":69612},[468],[224,69614,64666],{"className":69615,"style":64696},[468,469],[224,69617],{"className":69618,"style":474},[473],[224,69620,5268],{"className":69621},[478],[224,69623],{"className":69624,"style":474},[473],[224,69626,1438],{"className":69627},[468],[224,69629,69630,69633],{"style":20546},[224,69631],{"className":69632,"style":2108},[771],[224,69634],{"className":69635,"style":20554},[20553],[224,69637,69638,69641],{"style":20557},[224,69639],{"className":69640,"style":2108},[771],[224,69642,69644],{"className":69643},[468],[224,69645,1438],{"className":69646},[468],[224,69648,787],{"className":69649},[786],[224,69651,69653],{"className":69652},[759],[224,69654,69656],{"className":69655,"style":36958},[763],[224,69657],{},[224,69659],{"className":69660},[549,8479],[224,69662],{"className":69663,"style":724},[473],[224,69665,24185,69667],{"className":69666},[21531],[224,69668,24189],{"style":24188},[224,69670],{"className":69671,"style":724},[473],[224,69673,21338],{"className":69674,"style":21534},[21531,21532,21533],[224,69676],{"className":69677,"style":724},[473],[224,69679,69681,69684],{"className":69680},[468],[224,69682,11],{"className":69683},[468,469],[224,69685,69687],{"className":69686},[750],[224,69688,69690],{"className":69689},[754],[224,69691,69693],{"className":69692},[759],[224,69694,69696],{"className":69695,"style":65822},[763],[224,69697,69698,69701],{"style":5074},[224,69699],{"className":69700,"style":772},[771],[224,69702,69704],{"className":69703},[776,777,778,779],[224,69705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",[224,69861,69863,69878],{"className":69862,"translate":388},[392],[224,69864,69866],{"className":69865},[396],[398,69867,69868],{"xmlns":400},[403,69869,69870,69876],{},[406,69871,69872,69874],{},[409,69873,24131],{},[412,69875,24134],{},[447,69877,41521],{"encoding":449},[224,69879,69881],{"className":69880,"ariaHidden":455},[454],[224,69882,69884,69887],{"className":69883},[459],[224,69885],{"className":69886,"style":2565},[463],[224,69888,24185,69890],{"className":69889},[21531],[224,69891,24189],{"style":24188}," 包在积分外面（这正是它满足可乘可加性、但一般不满足数据处理不等式的根源：",[224,69894,69896,69914],{"className":69895,"translate":388},[392],[224,69897,69899],{"className":69898},[396],[398,69900,69901],{"xmlns":400},[403,69902,69903,69911],{},[406,69904,69905,69907,69909],{},[409,69906,64666],{},[412,69908,325],{},[419,69910,1438],{},[447,69912,69913],{"encoding":449},"\\alpha>1",[224,69915,69917,69935],{"className":69916,"ariaHidden":455},[454],[224,69918,69920,69923,69926,69929,69932],{"className":69919},[459],[224,69921],{"className":69922,"style":2810},[463],[224,69924,64666],{"className":69925,"style":64696},[468,469],[224,69927],{"className":69928,"style":496},[473],[224,69930,325],{"className":69931},[500],[224,69933],{"className":69934,"style":496},[473],[224,69936,69938,69941],{"className":69937},[459],[224,69939],{"className":69940,"style":510},[463],[224,69942,1438],{"className":69943},[468]," 的 Rényi 散度可以在随机变换下增大）。它与 α-散度是\"同一积分的两种包装\"：沿用 3.6 节的指数记号 ",[224,69946,69948,69987],{"className":69947,"translate":388},[392],[224,69949,69951],{"className":69950},[396],[398,69952,69953],{"xmlns":400},[403,69954,69955,69985],{},[406,69956,69957,69959,69961,69973,69975,69977,69979,69981,69983],{},[409,69958,15],{},[412,69960,278],{},[20338,69962,69963,69971],{},[406,69964,69965,69967,69969],{},[419,69966,1438],{},[412,69968,319],{},[409,69970,64666],{},[419,69972,1447],{},[412,69974,12248],{},[412,69976,239],{"stretchy":435},[419,69978,284],{},[412,69980,1877],{"separator":455},[419,69982,1438],{},[412,69984,1716],{"stretchy":435},[447,69986,65120],{"encoding":449},[224,69988,69990,70008,70097],{"className":69989,"ariaHidden":455},[454],[224,69991,69993,69996,69999,70002,70005],{"className":69992},[459],[224,69994],{"className":69995,"style":839},[463],[224,69997,15],{"className":69998},[468,469],[224,70000],{"className":70001,"style":496},[473],[224,70003,278],{"className":70004},[500],[224,70006],{"className":70007,"style":496},[473],[224,70009,70011,70014,70088,70091,70094],{"className":70010},[459],[224,70012],{"className":70013,"style":22544},[463],[224,70015,70017,70020,70085],{"className":70016},[468],[224,70018],{"className":70019},[542,8479],[224,70021,70023],{"className":70022},[20338],[224,70024,70026,70077],{"className":70025},[754,755],[224,70027,70029,70074],{"className":70028},[759],[224,70030,70032,70046,70054],{"className":70031,"style":22563},[763],[224,70033,70034,70037],{"style":22566},[224,70035],{"className":70036,"style":2108},[771],[224,70038,70040],{"className":70039},[776,777,778,779],[224,70041,70043],{"className":70042},[468,779],[224,70044,1447],{"className":70045},[468,779],[224,70047,70048,70051],{"style":20546},[224,70049],{"className":70050,"style":2108},[771],[224,70052],{"className":70053,"style":20554},[20553],[224,70055,70056,70059],{"style":22589},[224,70057],{"className":70058,"style":2108},[771],[224,70060,70062],{"className":70061},[776,777,778,779],[224,70063,70065,70068,70071],{"className":70064},[468,779],[224,70066,1438],{"className":70067},[468,779],[224,70069,319],{"className":70070},[478,779],[224,70072,64666],{"className":70073,"style":64696},[468,469,779],[224,70075,787],{"className":70076},[786],[224,70078,70080],{"className":70079},[759],[224,70081,70083],{"className":70082,"style":22611},[763],[224,70084],{},[224,70086],{"className":70087},[549,8479],[224,70089],{"className":70090,"style":496},[473],[224,70092,12248],{"className":70093},[500],[224,70095],{"className":70096,"style":496},[473],[224,70098,70100,70103,70106,70109,70112,70115,70118],{"className":70099},[459],[224,70101],{"className":70102,"style":535},[463],[224,70104,239],{"className":70105},[542],[224,70107,284],{"className":70108},[468],[224,70110,1877],{"className":70111},[1917],[224,70113],{"className":70114,"style":724},[473],[224,70116,1438],{"className":70117},[468],[224,70119,1716],{"className":70120},[549],"， ",[224,70123,70125,70171],{"className":70124,"translate":388},[392],[224,70126,70128],{"className":70127},[396],[398,70129,70130],{"xmlns":400},[403,70131,70132,70168],{},[406,70133,70134,70140,70142,70144,70150,70162,70164,70166],{},[695,70135,70136,70138],{},[409,70137,12252],{},[409,70139,15],{},[412,70141,278],{},[412,70143,21338],{},[3248,70145,70146,70148],{},[409,70147,11],{},[409,70149,15],{},[3248,70151,70152,70154],{},[409,70153,2023],{},[406,70155,70156,70158,70160],{},[419,70157,1438],{},[412,70159,5268],{},[409,70161,15],{},[423,70163,682],{},[409,70165,13182],{},[409,70167,371],{},[447,70169,70170],{"encoding":449},"Z_a=\\int p^{a}q^{1-a}\\,dx",[224,70172,70174,70230],{"className":70173,"ariaHidden":455},[454],[224,70175,70177,70180,70221,70224,70227],{"className":70176},[459],[224,70178],{"className":70179,"style":35905},[463],[224,70181,70183,70186],{"className":70182},[468],[224,70184,12252],{"className":70185,"style":30740},[468,469],[224,70187,70189],{"className":70188},[750],[224,70190,70192,70213],{"className":70191},[754,755],[224,70193,70195,70210],{"className":70194},[759],[224,70196,70198],{"className":70197,"style":64841},[763],[224,70199,70201,70204],{"style":70200},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[224,70202],{"className":70203,"style":772},[771],[224,70205,70207],{"className":70206},[776,777,778,779],[224,70208,15],{"className":70209},[468,469,779],[224,70211,787],{"className":70212},[786],[224,70214,70216],{"className":70215},[759],[224,70217,70219],{"className":70218,"style":794},[763],[224,70220],{},[224,70222],{"className":70223,"style":496},[473],[224,70225,278],{"className":70226},[500],[224,70228],{"className":70229,"style":496},[473],[224,70231,70233,70237,70240,70243,70275,70313,70316,70319],{"className":70232},[459],[224,70234],{"className":70235,"style":70236},[463],"height:1.1202em;vertical-align:-0.3061em;",[224,70238,21338],{"className":70239,"style":25626},[21531,21532,25625],[224,70241],{"className":70242,"style":724},[473],[224,70244,70246,70249],{"className":70245},[468],[224,70247,11],{"className":70248},[468,469],[224,70250,70252],{"className":70251},[750],[224,70253,70255],{"className":70254},[754],[224,70256,70258],{"className":70257},[759],[224,70259,70261],{"className":70260,"style":3267},[763],[224,70262,70263,70266],{"style":3288},[224,70264],{"className":70265,"style":772},[771],[224,70267,70269],{"className":70268},[776,777,778,779],[224,70270,70272],{"className":70271},[468,779],[224,70273,15],{"className":70274},[468,469,779],[224,70276,70278,70281],{"className":70277},[468],[224,70279,2023],{"className":70280,"style":492},[468,469],[224,70282,70284],{"className":70283},[750],[224,70285,70287],{"className":70286},[754],[224,70288,70290],{"className":70289},[759],[224,70291,70293],{"className":70292,"style":5671},[763],[224,70294,70295,70298],{"style":3288},[224,70296],{"className":70297,"style":772},[771],[224,70299,70301],{"className":70300},[776,777,778,779],[224,70302,70304,70307,70310],{"className":70303},[468,779],[224,70305,1438],{"className":70306},[468,779],[224,70308,5268],{"className":70309},[478,779],[224,70311,15],{"className":70312},[468,469,779],[224,70314],{"className":70315,"style":724},[473],[224,70317,13182],{"className":70318},[468,469],[224,70320,371],{"className":70321},[468,469],"，则 3.6 节给出",[224,70324,70326],{"className":70325,"translate":388},[387],[224,70327,70329,70375],{"className":70328,"translate":388},[392],[224,70330,70332],{"className":70331},[396],[398,70333,70334],{"xmlns":400,"display":401},[403,70335,70336,70372],{},[406,70337,70338,70344,70346,70348,70350,70352,70354,70356,70358,70360,70362,70364,70370],{},[695,70339,70340,70342],{},[409,70341,12252],{},[409,70343,15],{},[412,70345,278],{},[419,70347,1438],{},[412,70349,5268],{},[409,70351,15],{},[412,70353,239],{"stretchy":435},[419,70355,1438],{},[412,70357,5268],{},[409,70359,15],{},[412,70361,1716],{"stretchy":435},[423,70363,682],{},[695,70365,70366,70368],{},[409,70367,20303],{},[409,70369,64666],{},[412,70371,1877],{"separator":455},[447,70373,70374],{"encoding":449},"Z_a=1-a(1-a)\\,D_\\alpha ,",[224,70376,70378,70433,70451,70475],{"className":70377,"ariaHidden":455},[454],[224,70379,70381,70384,70424,70427,70430],{"className":70380},[459],[224,70382],{"className":70383,"style":35905},[463],[224,70385,70387,70390],{"className":70386},[468],[224,70388,12252],{"className":70389,"style":30740},[468,469],[224,70391,70393],{"className":70392},[750],[224,70394,70396,70416],{"className":70395},[754,755],[224,70397,70399,70413],{"className":70398},[759],[224,70400,70402],{"className":70401,"style":64841},[763],[224,70403,70404,70407],{"style":70200},[224,70405],{"className":70406,"style":772},[771],[224,70408,70410],{"className":70409},[776,777,778,779],[224,70411,15],{"className":70412},[468,469,779],[224,70414,787],{"className":70415},[786],[224,70417,70419],{"className":70418},[759],[224,70420,70422],{"className":70421,"style":794},[763],[224,70423],{},[224,70425],{"className":70426,"style":496},[473],[224,70428,278],{"className":70429},[500],[224,70431],{"className":70432,"style":496},[473],[224,70434,70436,70439,70442,70445,70448],{"className":70435},[459],[224,70437],{"className":70438,"style":7047},[463],[224,70440,1438],{"className":70441},[468],[224,70443],{"className":70444,"style":474},[473],[224,70446,5268],{"className":70447},[478],[224,70449],{"className":70450,"style":474},[473],[224,70452,70454,70457,70460,70463,70466,70469,70472],{"className":70453},[459],[224,70455],{"className":70456,"style":535},[463],[224,70458,15],{"className":70459},[468,469],[224,70461,239],{"className":70462},[542],[224,70464,1438],{"className":70465},[468],[224,70467],{"className":70468,"style":474},[473],[224,70470,5268],{"className":70471},[478],[224,70473],{"className":70474,"style":474},[473],[224,70476,70478,70481,70484,70487,70490,70530],{"className":70477},[459],[224,70479],{"className":70480,"style":535},[463],[224,70482,15],{"className":70483},[468,469],[224,70485,1716],{"className":70486},[549],[224,70488],{"className":70489,"style":724},[473],[224,70491,70493,70496],{"className":70492},[468],[224,70494,20303],{"className":70495,"style":2473},[468,469],[224,70497,70499],{"className":70498},[750],[224,70500,70502,70522],{"className":70501},[754,755],[224,70503,70505,70519],{"className":70504},[759],[224,70506,70508],{"className":70507,"style":64841},[763],[224,70509,70510,70513],{"style":20390},[224,70511],{"className":70512,"style":772},[771],[224,70514,70516],{"className":70515},[776,777,778,779],[224,70517,64666],{"className":70518,"style":64696},[468,469,779],[224,70520,787],{"className":70521},[786],[224,70523,70525],{"className":70524},[759],[224,70526,70528],{"className":70527,"style":794},[763],[224,70529],{},[224,70531,1877],{"className":70532},[1917],[11,70534,70535,70536,70539,70540,70568,70569,70631,70632,70660],{},"而把 Rényi 的",[378,70537,70538],{},"阶数","取为 ",[224,70541,70543,70556],{"className":70542,"translate":388},[392],[224,70544,70546],{"className":70545},[396],[398,70547,70548],{"xmlns":400},[403,70549,70550,70554],{},[406,70551,70552],{},[409,70553,15],{},[447,70555,15],{"encoding":449},[224,70557,70559],{"className":70558,"ariaHidden":455},[454],[224,70560,70562,70565],{"className":70561},[459],[224,70563],{"className":70564,"style":839},[463],[224,70566,15],{"className":70567},[468,469],"（注意 order ",[224,70570,70572,70586],{"className":70571,"translate":388},[392],[224,70573,70575],{"className":70574},[396],[398,70576,70577],{"xmlns":400},[403,70578,70579,70583],{},[406,70580,70581],{},[412,70582,2048],{"mathvariant":678},[447,70584,70585],{"encoding":449},"\\neq",[224,70587,70589],{"className":70588,"ariaHidden":455},[454],[224,70590,70592,70595],{"className":70591},[459],[224,70593],{"className":70594,"style":2565},[463],[224,70596,70598,70625,70628],{"className":70597},[500],[224,70599,70601],{"className":70600},[500],[224,70602,70604],{"className":70603},[468,2553],[224,70605,70607],{"className":70606},[2557],[224,70608,70610,70613,70622],{"className":70609},[2561],[224,70611],{"className":70612,"style":2565},[463],[224,70614,70616],{"className":70615},[2569],[224,70617,70619],{"className":70618},[468],[224,70620,2576],{"className":70621},[500],[224,70623],{"className":70624},[2580],[224,70626],{"className":70627},[473,2584],[224,70629,278],{"className":70630},[500]," 3.6 的参数 ",[224,70633,70635,70648],{"className":70634,"translate":388},[392],[224,70636,70638],{"className":70637},[396],[398,70639,70640],{"xmlns":400},[403,70641,70642,70646],{},[406,70643,70644],{},[409,70645,64666],{},[447,70647,68999],{"encoding":449},[224,70649,70651],{"className":70650,"ariaHidden":455},[454],[224,70652,70654,70657],{"className":70653},[459],[224,70655],{"className":70656,"style":839},[463],[224,70658,64666],{"className":70659,"style":64696},[468,469],"，只是经过线性换算）：",[224,70662,70664],{"className":70663,"translate":388},[387],[224,70665,70667,70771],{"className":70666,"translate":388},[392],[224,70668,70670],{"className":70669},[396],[398,70671,70672],{"xmlns":400,"display":401},[403,70673,70674,70768],{},[406,70675,70676,70684,70686,70688,70690,70692,70694,70696,70708,70710,70712,70718,70720,70722,70734,70736,70738,70740,70742,70744,70746,70748,70750,70752,70754,70756,70758,70764,70766],{},[27527,70677,70678,70680,70682],{},[409,70679,20303],{},[409,70681,15],{},[409,70683,21092],{},[412,70685,239],{"stretchy":435},[409,70687,16018],{},[409,70689,20312],{"mathvariant":678},[409,70691,20315],{},[412,70693,1716],{"stretchy":435},[412,70695,278],{},[20338,70697,70698,70700],{},[419,70699,1438],{},[406,70701,70702,70704,70706],{},[409,70703,15],{},[412,70705,5268],{},[419,70707,1438],{},[409,70709,24131],{},[412,70711,24134],{},[695,70713,70714,70716],{},[409,70715,12252],{},[409,70717,15],{},[412,70719,278],{},[412,70721,5268],{},[20338,70723,70724,70726],{},[419,70725,1438],{},[406,70727,70728,70730,70732],{},[419,70729,1438],{},[412,70731,5268],{},[409,70733,15],{},[423,70735,682],{},[409,70737,24131],{},[412,70739,24134],{},[412,70741,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[419,70743,1438],{},[412,70745,5268],{},[409,70747,15],{},[412,70749,239],{"stretchy":435},[419,70751,1438],{},[412,70753,5268],{},[409,70755,15],{},[412,70757,1716],{"stretchy":435},[695,70759,70760,70762],{},[409,70761,20303],{},[409,70763,64666],{},[412,70765,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,70767,688],{"mathvariant":678},[447,70769,70770],{"encoding":449},"D^{R}_{a}(P\\|Q)=\\frac{1}{a-1}\\log Z_a\n=-\\frac{1}{1-a}\\,\\log\\big(1-a(1-a)D_\\alpha\\big).",[224,70772,70774,70862,71002,71117,71141],{"className":70773,"ariaHidden":455},[454],[224,70775,70777,70781,70838,70841,70844,70847,70850,70853,70856,70859],{"className":70776},[459],[224,70778],{"className":70779,"style":70780},[463],"height:1.1413em;vertical-align:-0.25em;",[224,70782,70784,70787],{"className":70783},[468],[224,70785,20303],{"className":70786,"style":2473},[468,469],[224,70788,70790],{"className":70789},[750],[224,70791,70793,70830],{"className":70792},[754,755],[224,70794,70796,70827],{"className":70795},[759],[224,70797,70799,70813],{"className":70798,"style":58433},[763],[224,70800,70801,70804],{"style":69480},[224,70802],{"className":70803,"style":772},[771],[224,70805,70807],{"className":70806},[776,777,778,779],[224,70808,70810],{"className":70809},[468,779],[224,70811,15],{"className":70812},[468,469,779],[224,70814,70815,70818],{"style":5074},[224,70816],{"className":70817,"style":772},[771],[224,70819,70821],{"className":70820},[776,777,778,779],[224,70822,70824],{"className":70823},[468,779],[224,70825,21092],{"className":70826,"style":69507},[468,469,779],[224,70828,787],{"className":70829},[786],[224,70831,70833],{"className":70832},[759],[224,70834,70836],{"className":70835,"style":28265},[763],[224,70837],{},[224,70839,239],{"className":70840},[542],[224,70842,16018],{"className":70843,"style":653},[468,469],[224,70845,20312],{"className":70846},[468],[224,70848,20315],{"className":70849},[468,469],[224,70851,1716],{"className":70852},[549],[224,70854],{"className":70855,"style":496},[473],[224,70857,278],{"className":70858},[500],[224,70860],{"className":70861,"style":496},[473],[224,70863,70865,70868,70942,70945,70950,70953,70993,70996,70999],{"className":70864},[459],[224,70866],{"className":70867,"style":46546},[463],[224,70869,70871,70874,70939],{"className":70870},[468],[224,70872],{"className":70873},[542,8479],[224,70875,70877],{"className":70876},[20338],[224,70878,70880,70931],{"className":70879},[754,755],[224,70881,70883,70928],{"className":70882},[759],[224,70884,70886,70909,70917],{"className":70885,"style":42827},[763],[224,70887,70888,70891],{"style":20531},[224,70889],{"className":70890,"style":2108},[771],[224,70892,70894,70897,70900,70903,70906],{"className":70893},[468],[224,70895,15],{"className":70896},[468,469],[224,70898],{"className":70899,"style":474},[473],[224,70901,5268],{"className":70902},[478],[224,70904],{"className":70905,"style":474},[473],[224,70907,1438],{"className":70908},[468],[224,70910,70911,70914],{"style":20546},[224,70912],{"className":70913,"style":2108},[771],[224,70915],{"className":70916,"style":20554},[20553],[224,70918,70919,70922],{"style":20557},[224,70920],{"className":70921,"style":2108},[771],[224,70923,70925],{"className":70924},[468],[224,70926,1438],{"className":70927},[468],[224,70929,787],{"className":70930},[786],[224,70932,70934],{"className":70933},[759],[224,70935,70937],{"className":70936,"style":36958},[763],[224,70938],{},[224,70940],{"className":70941},[549,8479],[224,70943],{"className":70944,"style":724},[473],[224,70946,24185,70948],{"className":70947},[21531],[224,70949,24189],{"style":24188},[224,70951],{"className":70952,"style":724},[473],[224,70954,70956,70959],{"className":70955},[468],[224,70957,12252],{"className":70958,"style":30740},[468,469],[224,70960,70962],{"className":70961},[750],[224,70963,70965,70985],{"className":70964},[754,755],[224,70966,70968,70982],{"className":70967},[759],[224,70969,70971],{"className":70970,"style":64841},[763],[224,70972,70973,70976],{"style":70200},[224,70974],{"className":70975,"style":772},[771],[224,70977,70979],{"className":70978},[776,777,778,779],[224,70980,15],{"className":70981},[468,469,779],[224,70983,787],{"className":70984},[786],[224,70986,70988],{"className":70987},[759],[224,70989,70991],{"className":70990,"style":794},[763],[224,70992],{},[224,70994],{"className":70995,"style":496},[473],[224,70997,278],{"className":70998},[500],[224,71000],{"className":71001,"style":496},[473],[224,71003,71005,71008,71011,71085,71088,71091,71096,71099,71105,71108,71111,71114],{"className":71004},[459],[224,71006],{"className":71007,"style":46546},[463],[224,71009,5268],{"className":71010},[468],[224,71012,71014,71017,71082],{"className":71013},[468],[224,71015],{"className":71016},[542,8479],[224,71018,71020],{"className":71019},[20338],[224,71021,71023,71074],{"className":71022},[754,755],[224,71024,71026,71071],{"className":71025},[759],[224,71027,71029,71052,71060],{"className":71028,"style":42827},[763],[224,71030,71031,71034],{"style":20531},[224,71032],{"className":71033,"style":2108},[771],[224,71035,71037,71040,71043,71046,71049],{"className":71036},[468],[224,71038,1438],{"className":71039},[468],[224,71041],{"className":71042,"style":474},[473],[224,71044,5268],{"className":71045},[478],[224,71047],{"className":71048,"style":474},[473],[224,71050,15],{"className":71051},[468,469],[224,71053,71054,71057],{"style":20546},[224,71055],{"className":71056,"style":2108},[771],[224,71058],{"className":71059,"style":20554},[20553],[224,71061,71062,71065],{"style":20557},[224,71063],{"className":71064,"style":2108},[771],[224,71066,71068],{"className":71067},[468],[224,71069,1438],{"className":71070},[468],[224,71072,787],{"className":71073},[786],[224,71075,71077],{"className":71076},[759],[224,71078,71080],{"className":71079,"style":36958},[763],[224,71081],{},[224,71083],{"className":71084},[549,8479],[224,71086],{"className":71087,"style":724},[473],[224,71089],{"className":71090,"style":724},[473],[224,71092,24185,71094],{"className":71093},[21531],[224,71095,24189],{"style":24188},[224,71097],{"className":71098,"style":724},[473],[224,71100,71102],{"className":71101},[468],[224,71103,239],{"className":71104},[2363,2364],[224,71106,1438],{"className":71107},[468],[224,71109],{"className":71110,"style":474},[473],[224,71112,5268],{"className":71113},[478],[224,71115],{"className":71116,"style":474},[473],[224,71118,71120,71123,71126,71129,71132,71135,71138],{"className":71119},[459],[224,71121],{"className":71122,"style":535},[463],[224,71124,15],{"className":71125},[468,469],[224,71127,239],{"className":71128},[542],[224,71130,1438],{"className":71131},[468],[224,71133],{"className":71134,"style":474},[473],[224,71136,5268],{"className":71137},[478],[224,71139],{"className":71140,"style":474},[473],[224,71142,71144,71147,71150,71153,71193,71199],{"className":71143},[459],[224,71145],{"className":71146,"style":2663},[463],[224,71148,15],{"className":71149},[468,469],[224,71151,1716],{"className":71152},[549],[224,71154,71156,71159],{"className":71155},[468],[224,71157,20303],{"className":71158,"style":2473},[468,469],[224,71160,71162],{"className":71161},[750],[224,71163,71165,71185],{"className":71164},[754,755],[224,71166,71168,71182],{"className":71167},[759],[224,71169,71171],{"className":71170,"style":64841},[763],[224,71172,71173,71176],{"style":20390},[224,71174],{"className":71175,"style":772},[771],[224,71177,71179],{"className":71178},[776,777,778,779],[224,71180,64666],{"className":71181,"style":64696},[468,469,779],[224,71183,787],{"className":71184},[786],[224,71186,71188],{"className":71187},[759],[224,71189,71191],{"className":71190,"style":794},[763],[224,71192],{},[224,71194,71196],{"className":71195},[468],[224,71197,1716],{"className":71198},[2363,2364],[224,71200,688],{"className":71201},[468],[11,71203,71204,71205,71352,71353,71381,71382,71414],{},"这说明两者是一条\"幂—对数\"的两级包装链：f-散度是对似然比幂 ",[224,71206,71208,71234],{"className":71207,"translate":388},[392],[224,71209,71211],{"className":71210},[396],[398,71212,71213],{"xmlns":400},[403,71214,71215,71231],{},[406,71216,71217,71219,71225],{},[412,71218,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[20338,71220,71221,71223],{},[409,71222,11],{},[409,71224,2023],{},[3248,71226,71227,71229],{},[412,71228,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,71230,15],{},[447,71232,71233],{"encoding":449},"\\big(\\frac pq\\big)^a",[224,71235,71237],{"className":71236,"ariaHidden":455},[454],[224,71238,71240,71244,71250,71318],{"className":71239},[459],[224,71241],{"className":71242,"style":71243},[463],"height:1.3854em;vertical-align:-0.4811em;",[224,71245,71247],{"className":71246},[468],[224,71248,239],{"className":71249},[2363,2364],[224,71251,71253,71256,71315],{"className":71252},[468],[224,71254],{"className":71255},[542,8479],[224,71257,71259],{"className":71258},[20338],[224,71260,71262,71307],{"className":71261},[754,755],[224,71263,71265,71304],{"className":71264},[759],[224,71266,71268,71282,71290],{"className":71267,"style":23136},[763],[224,71269,71270,71273],{"style":22566},[224,71271],{"className":71272,"style":2108},[771],[224,71274,71276],{"className":71275},[776,777,778,779],[224,71277,71279],{"className":71278},[468,779],[224,71280,2023],{"className":71281,"style":492},[468,469,779],[224,71283,71284,71287],{"style":20546},[224,71285],{"className":71286,"style":2108},[771],[224,71288],{"className":71289,"style":20554},[20553],[224,71291,71292,71295],{"style":22889},[224,71293],{"className":71294,"style":2108},[771],[224,71296,71298],{"className":71297},[776,777,778,779],[224,71299,71301],{"className":71300},[468,779],[224,71302,11],{"className":71303},[468,469,779],[224,71305,787],{"className":71306},[786],[224,71308,71310],{"className":71309},[759],[224,71311,71313],{"className":71312,"style":22776},[763],[224,71314],{},[224,71316],{"className":71317},[549,8479],[224,71319,71321,71327],{"className":71320},[468],[224,71322,71324],{"className":71323},[468],[224,71325,1716],{"className":71326},[2363,2364],[224,71328,71330],{"className":71329},[750],[224,71331,71333],{"className":71332},[754],[224,71334,71336],{"className":71335},[759],[224,71337,71340],{"className":71338,"style":71339},[763],"height:0.9043em;",[224,71341,71343,71346],{"style":71342},"top:-3.3029em;margin-right:0.05em;",[224,71344],{"className":71345,"style":772},[771],[224,71347,71349],{"className":71348},[776,777,778,779],[224,71350,15],{"className":71351},[468,469,779]," 做 ",[224,71354,71356,71369],{"className":71355,"translate":388},[392],[224,71357,71359],{"className":71358},[396],[398,71360,71361],{"xmlns":400},[403,71362,71363,71367],{},[406,71364,71365],{},[409,71366,20315],{},[447,71368,20315],{"encoding":449},[224,71370,71372],{"className":71371,"ariaHidden":455},[454],[224,71373,71375,71378],{"className":71374},[459],[224,71376],{"className":71377,"style":20637},[463],[224,71379,20315],{"className":71380},[468,469]," 下的线性平均，Rényi 在外面再套一层对数。所以 f-散度族与 Rényi 族\"同源\"：后者是前者经 ",[224,71383,71385,71400],{"className":71384,"translate":388},[392],[224,71386,71388],{"className":71387},[396],[398,71389,71390],{"xmlns":400},[403,71391,71392,71398],{},[406,71393,71394,71396],{},[409,71395,24131],{},[412,71397,24134],{},[447,71399,41521],{"encoding":449},[224,71401,71403],{"className":71402,"ariaHidden":455},[454],[224,71404,71406,71409],{"className":71405},[459],[224,71407],{"className":71408,"style":2565},[463],[224,71410,24185,71412],{"className":71411},[21531],[224,71413,24189],{"style":24188}," 变换后的\"信息量\"版本。",[11,71416,71417,71419,71420,71423],{},[378,71418,44131],{}," Rényi 散度在",[378,71421,71422],{},"假设检验","中直接就是误差指数：Chernoff 信息",[224,71425,71427],{"className":71426,"translate":388},[387],[224,71428,71430,71553],{"className":71429,"translate":388},[392],[224,71431,71433],{"className":71432},[396],[398,71434,71435],{"xmlns":400,"display":401},[403,71436,71437,71550],{},[406,71438,71439,71441,71443,71445,71447,71449,71451,71453,71473,71475,71477,71479,71481,71483,71491,71493,71495,71497,71499,71501,71503,71505,71507,71509,71530,71532,71538],{},[409,71440,16021],{},[412,71442,239],{"stretchy":435},[409,71444,16018],{},[412,71446,1877],{"separator":455},[409,71448,20315],{},[412,71450,1716],{"stretchy":435},[412,71452,278],{},[1965,71454,71455,71461],{},[406,71456,71457,71459],{},[409,71458,55354],{},[412,71460,24134],{},[406,71462,71463,71465,71467,71469,71471],{},[419,71464,284],{},[412,71466,1577],{},[409,71468,31295],{},[412,71470,1577],{},[419,71472,1438],{},[412,71474,239],{"stretchy":435},[419,71476,1438],{},[412,71478,5268],{},[409,71480,31295],{},[412,71482,1716],{"stretchy":435},[27527,71484,71485,71487,71489],{},[409,71486,20303],{},[409,71488,31295],{},[409,71490,21092],{},[412,71492,239],{"stretchy":435},[409,71494,16018],{},[409,71496,20312],{"mathvariant":678},[409,71498,20315],{},[412,71500,1716],{"stretchy":435},[412,71502,278],{},[412,71504,5268],{},[409,71506,24131],{},[412,71508,24134],{},[1965,71510,71511,71518],{},[406,71512,71513,71516],{},[409,71514,71515],{},"min",[412,71517,24134],{},[406,71519,71520,71522,71524,71526,71528],{},[419,71521,284],{},[412,71523,2070],{},[409,71525,31295],{},[412,71527,2070],{},[419,71529,1438],{},[412,71531,21338],{},[3248,71533,71534,71536],{},[409,71535,11],{},[409,71537,31295],{},[3248,71539,71540,71542],{},[409,71541,2023],{},[406,71543,71544,71546,71548],{},[419,71545,1438],{},[412,71547,5268],{},[409,71549,31295],{},[447,71551,71552],{"encoding":449},"C(P,Q)=\\sup_{0\u003C\\lambda\u003C1}\n(1-\\lambda)D^{R}_{\\lambda}(P\\|Q)\n=-\\log\\min_{0\\le\\lambda\\le1}\n\\int p^\\lambda 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引理给出的单侧指数则是 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时退化为 Shannon 熵、在 ",[224,72219,72221,72239],{"className":72220,"translate":388},[392],[224,72222,72224],{"className":72223},[396],[398,72225,72226],{"xmlns":400},[403,72227,72228,72236],{},[406,72229,72230,72232,72234],{},[409,72231,64666],{},[412,72233,1144],{},[409,72235,21085],{"mathvariant":678},[447,72237,72238],{"encoding":449},"\\alpha\\to\\infty",[224,72240,72242,72260],{"className":72241,"ariaHidden":455},[454],[224,72243,72245,72248,72251,72254,72257],{"className":72244},[459],[224,72246],{"className":72247,"style":839},[463],[224,72249,64666],{"className":72250,"style":64696},[468,469],[224,72252],{"className":72253,"style":496},[473],[224,72255,1144],{"className":72256},[500],[224,72258],{"className":72259,"style":496},[473],[224,72261,72263,72266],{"className":72262},[459],[224,72264],{"className":72265,"style":839},[463],[224,72267,21085],{"className":72268},[468]," 时退化为 min-entropy 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或任一控制测度），",[224,75046,75048,75073],{"className":75047,"translate":388},[392],[224,75049,75051],{"className":75050},[396],[398,75052,75053],{"xmlns":400},[403,75054,75055,75071],{},[406,75056,75057,75059,75061,75063,75065,75067,75069],{},[409,75058,1435],{},[412,75060,278],{},[409,75062,13182],{},[409,75064,16018],{},[409,75066,12413],{"mathvariant":678},[409,75068,13182],{},[409,75070,20315],{},[447,75072,25343],{"encoding":449},[224,75074,75076,75094],{"className":75075,"ariaHidden":455},[454],[224,75077,75079,75082,75085,75088,75091],{"className":75078},[459],[224,75080],{"className":75081,"style":25353},[463],[224,75083,1435],{"className":75084},[468,469],[224,75086],{"className":75087,"style":496},[473],[224,75089,278],{"className":75090},[500],[224,75092],{"className":75093,"style":496},[473],[224,75095,75097,75100,75103,75106,75109,75112],{"className":75096},[459],[224,75098],{"className":75099,"style":535},[463],[224,75101,13182],{"className":75102},[468,469],[224,75104,16018],{"className":75105,"style":653},[468,469],[224,75107,12413],{"className":75108},[468],[224,75110,13182],{"className":75111},[468,469],[224,75113,20315],{"className":75114},[468,469],[613,75116,75118],{"id":75117},"_41-非负性","4.1 非负性",[21055,75120,75122,75166,75291],{"title":75121},"命题 4.1（非负性）",[11,75123,75124,75125,7098],{},"对一切概率分布 ",[224,75126,75128,75145],{"className":75127,"translate":388},[392],[224,75129,75131],{"className":75130},[396],[398,75132,75133],{"xmlns":400},[403,75134,75135,75143],{},[406,75136,75137,75139,75141],{},[409,75138,16018],{},[412,75140,1877],{"separator":455},[409,75142,20315],{},[447,75144,20627],{"encoding":449},[224,75146,75148],{"className":75147,"ariaHidden":455},[454],[224,75149,75151,75154,75157,75160,75163],{"className":75150},[459],[224,75152],{"className":75153,"style":20637},[463],[224,75155,16018],{"className":75156,"style":653},[468,469],[224,75158,1877],{"className":75159},[1917],[224,75161],{"className":75162,"style":724},[473],[224,75164,20315],{"className":75165},[468,469],[224,75167,75169],{"className":75168,"translate":388},[387],[224,75170,75172,75206],{"className":75171,"translate":388},[392],[224,75173,75175],{"className":75174},[396],[398,75176,75177],{"xmlns":400,"display":401},[403,75178,75179,75203],{},[406,75180,75181,75187,75189,75191,75193,75195,75197,75199,75201],{},[695,75182,75183,75185],{},[409,75184,20303],{},[409,75186,1705],{},[412,75188,239],{"stretchy":435},[409,75190,16018],{},[409,75192,20312],{"mathvariant":678},[409,75194,20315],{},[412,75196,1716],{"stretchy":435},[412,75198,9766],{},[419,75200,284],{},[412,75202,1877],{"separator":455},[447,75204,75205],{"encoding":449},"D_f(P\\|Q)\\ge 0,",[224,75207,75209,75279],{"className":75208,"ariaHidden":455},[454],[224,75210,75212,75215,75255,75258,75261,75264,75267,75270,75273,75276],{"className":75211},[459],[224,75213],{"className":75214,"style":20368},[463],[224,75216,75218,75221],{"className":75217},[468],[224,75219,20303],{"className":75220,"style":2473},[468,469],[224,75222,75224],{"className":75223},[750],[224,75225,75227,75247],{"className":75226},[754,755],[224,75228,75230,75244],{"className":75229},[759],[224,75231,75233],{"className":75232,"style":20387},[763],[224,75234,75235,75238],{"style":20390},[224,75236],{"className":75237,"style":772},[771],[224,75239,75241],{"className":75240},[776,777,778,779],[224,75242,1705],{"className":75243,"style":1740},[468,469,779],[224,75245,787],{"className":75246},[786],[224,75248,75250],{"className":75249},[759],[224,75251,75253],{"className":75252,"style":20409},[763],[224,75254],{},[224,75256,239],{"className":75257},[542],[224,75259,16018],{"className":75260,"style":653},[468,469],[224,75262,20312],{"className":75263},[468],[224,75265,20315],{"className":75266},[468,469],[224,75268,1716],{"className":75269},[549],[224,75271],{"className":75272,"style":496},[473],[224,75274,9766],{"className":75275},[500],[224,75277],{"className":75278,"style":496},[473],[224,75280,75282,75285,75288],{"className":75281},[459],[224,75283],{"className":75284,"style":8688},[463],[224,75286,284],{"className":75287},[468],[224,75289,1877],{"className":75290},[1917],[11,75292,75293,75294,75322,75323,3117],{},"且当 ",[224,75295,75297,75310],{"className":75296,"translate":388},[392],[224,75298,75300],{"className":75299},[396],[398,75301,75302],{"xmlns":400},[403,75303,75304,75308],{},[406,75305,75306],{},[409,75307,1705],{},[447,75309,1705],{"encoding":449},[224,75311,75313],{"className":75312,"ariaHidden":455},[454],[224,75314,75316,75319],{"className":75315},[459],[224,75317],{"className":75318,"style":2565},[463],[224,75320,1705],{"className":75321,"style":1740},[468,469]," 严格凸时，等号成立当且仅当 ",[224,75324,75326,75343],{"className":75325,"translate":388},[392],[224,75327,75329],{"className":75328},[396],[398,75330,75331],{"xmlns":400},[403,75332,75333,75341],{},[406,75334,75335,75337,75339],{},[409,75336,16018],{},[412,75338,278],{},[409,75340,20315],{},[447,75342,30229],{"encoding":449},[224,75344,75346,75364],{"className":75345,"ariaHidden":455},[454],[224,75347,75349,75352,75355,75358,75361],{"className":75348},[459],[224,75350],{"className":75351,"style":649},[463],[224,75353,16018],{"className":75354,"style":653},[468,469],[224,75356],{"className":75357,"style":496},[473],[224,75359,278],{"className":75360},[500],[224,75362],{"className":75363,"style":496},[473],[224,75365,75367,75370],{"className":75366},[459],[224,75368],{"className":75369,"style":20637},[463],[224,75371,20315],{"className":75372},[468,469],[11,75374,75375,75378,75379,75408,75409,75437,75438,23884,75567,75595,75596,75646,75647,75675,75676,75844,75845,75937,75938,75988,75989,76039],{},[378,75376,75377],{},"证明。"," Jensen 已给出 ",[224,75380,75382,75396],{"className":75381,"translate":388},[392],[224,75383,75385],{"className":75384},[396],[398,75386,75387],{"xmlns":400},[403,75388,75389,75393],{},[406,75390,75391],{},[412,75392,9766],{},[447,75394,75395],{"encoding":449},"\\ge",[224,75397,75399],{"className":75398,"ariaHidden":455},[454],[224,75400,75402,75405],{"className":75401},[459],[224,75403],{"className":75404,"style":3204},[463],[224,75406,9766],{"className":75407},[500],"（见 2.1）。等号条件：若 ",[224,75410,75412,75425],{"className":75411,"translate":388},[392],[224,75413,75415],{"className":75414},[396],[398,75416,75417],{"xmlns":400},[403,75418,75419,75423],{},[406,75420,75421],{},[409,75422,1705],{},[447,75424,1705],{"encoding":449},[224,75426,75428],{"className":75427,"ariaHidden":455},[454],[224,75429,75431,75434],{"className":75430},[459],[224,75432],{"className":75433,"style":2565},[463],[224,75435,1705],{"className":75436,"style":1740},[468,469]," 严格凸，Jensen 取等号当且仅当 ",[224,75439,75441,75473],{"className":75440,"translate":388},[392],[224,75442,75444],{"className":75443},[396],[398,75445,75446],{"xmlns":400},[403,75447,75448,75470],{},[406,75449,75450,75452,75454,75460,75462,75464,75466,75468],{},[409,75451,1435],{},[412,75453,278],{},[695,75455,75456,75458],{},[409,75457,20324],{"mathvariant":12251},[409,75459,20315],{},[412,75461,268],{"stretchy":435},[409,75463,1435],{},[412,75465,440],{"stretchy":435},[412,75467,278],{},[419,75469,1438],{},[447,75471,75472],{"encoding":449},"t=\\mathbb E_Q[t]=1",[224,75474,75476,75494,75558],{"className":75475,"ariaHidden":455},[454],[224,75477,75479,75482,75485,75488,75491],{"className":75478},[459],[224,75480],{"className":75481,"style":25353},[463],[224,75483,1435],{"className":75484},[468,469],[224,75486],{"className":75487,"style":496},[473],[224,75489,278],{"className":75490},[500],[224,75492],{"className":75493,"style":496},[473],[224,75495,75497,75500,75540,75543,75546,75549,75552,75555],{"className":75496},[459],[224,75498],{"className":75499,"style":20368},[463],[224,75501,75503,75506],{"className":75502},[468],[224,75504,20324],{"className":75505},[468,12389],[224,75507,75509],{"className":75508},[750],[224,75510,75512,75532],{"className":75511},[754,755],[224,75513,75515,75529],{"className":75514},[759],[224,75516,75518],{"className":75517,"style":764},[763],[224,75519,75520,75523],{"style":767},[224,75521],{"className":75522,"style":772},[771],[224,75524,75526],{"className":75525},[776,777,778,779],[224,75527,20315],{"className":75528},[468,469,779],[224,75530,787],{"className":75531},[786],[224,75533,75535],{"className":75534},[759],[224,75536,75538],{"className":75537,"style":20409},[763],[224,75539],{},[224,75541,268],{"className":75542},[542],[224,75544,1435],{"className":75545},[468,469],[224,75547,440],{"className":75548},[549],[224,75550],{"className":75551,"style":496},[473],[224,75553,278],{"className":75554},[500],[224,75556],{"className":75557,"style":496},[473],[224,75559,75561,75564],{"className":75560},[459],[224,75562],{"className":75563,"style":510},[463],[224,75565,1438],{"className":75566},[468],[224,75568,75570,75583],{"className":75569,"translate":388},[392],[224,75571,75573],{"className":75572},[396],[398,75574,75575],{"xmlns":400},[403,75576,75577,75581],{},[406,75578,75579],{},[409,75580,20315],{},[447,75582,20315],{"encoding":449},[224,75584,75586],{"className":75585,"ariaHidden":455},[454],[224,75587,75589,75592],{"className":75588},[459],[224,75590],{"className":75591,"style":20637},[463],[224,75593,20315],{"className":75594},[468,469]," 下几乎必然成立，即 ",[224,75597,75599,75616],{"className":75598,"translate":388},[392],[224,75600,75602],{"className":75601},[396],[398,75603,75604],{"xmlns":400},[403,75605,75606,75614],{},[406,75607,75608,75610,75612],{},[409,75609,16018],{},[412,75611,278],{},[409,75613,20315],{},[447,75615,30229],{"encoding":449},[224,75617,75619,75637],{"className":75618,"ariaHidden":455},[454],[224,75620,75622,75625,75628,75631,75634],{"className":75621},[459],[224,75623],{"className":75624,"style":649},[463],[224,75626,16018],{"className":75627,"style":653},[468,469],[224,75629],{"className":75630,"style":496},[473],[224,75632,278],{"className":75633},[500],[224,75635],{"className":75636,"style":496},[473],[224,75638,75640,75643],{"className":75639},[459],[224,75641],{"className":75642,"style":20637},[463],[224,75644,20315],{"className":75645},[468,469],"。若 ",[224,75648,75650,75663],{"className":75649,"translate":388},[392],[224,75651,75653],{"className":75652},[396],[398,75654,75655],{"xmlns":400},[403,75656,75657,75661],{},[406,75658,75659],{},[409,75660,1705],{},[447,75662,1705],{"encoding":449},[224,75664,75666],{"className":75665,"ariaHidden":455},[454],[224,75667,75669,75672],{"className":75668},[459],[224,75670],{"className":75671,"style":2565},[463],[224,75673,1705],{"className":75674,"style":1740},[468,469]," 只是凸而非严格凸（如 ",[224,75677,75679,75716],{"className":75678,"translate":388},[392],[224,75680,75682],{"className":75681},[396],[398,75683,75684],{"xmlns":400},[403,75685,75686,75714],{},[406,75687,75688,75690,75692,75694,75696,75698,75704,75706,75708,75710,75712],{},[409,75689,1705],{},[412,75691,239],{"stretchy":435},[409,75693,1435],{},[412,75695,1716],{"stretchy":435},[412,75697,278],{},[20338,75699,75700,75702],{},[419,75701,1438],{},[419,75703,1447],{},[409,75705,1083],{"mathvariant":678},[409,75707,1435],{},[412,75709,5268],{},[419,75711,1438],{},[409,75713,1083],{"mathvariant":678},[447,75715,54479],{"encoding":449},[224,75717,75719,75746,75835],{"className":75718,"ariaHidden":455},[454],[224,75720,75722,75725,75728,75731,75734,75737,75740,75743],{"className":75721},[459],[224,75723],{"className":75724,"style":535},[463],[224,75726,1705],{"className":75727,"style":1740},[468,469],[224,75729,239],{"className":75730},[542],[224,75732,1435],{"className":75733},[468,469],[224,75735,1716],{"className":75736},[549],[224,75738],{"className":75739,"style":496},[473],[224,75741,278],{"className":75742},[500],[224,75744],{"className":75745,"style":496},[473],[224,75747,75749,75752,75820,75823,75826,75829,75832],{"className":75748},[459],[224,75750],{"className":75751,"style":22544},[463],[224,75753,75755,75758,75817],{"className":75754},[468],[224,75756],{"className":75757},[542,8479],[224,75759,75761],{"className":75760},[20338],[224,75762,75764,75809],{"className":75763},[754,755],[224,75765,75767,75806],{"className":75766},[759],[224,75768,75770,75784,75792],{"className":75769,"style":22563},[763],[224,75771,75772,75775],{"style":22566},[224,75773],{"className":75774,"style":2108},[771],[224,75776,75778],{"className":75777},[776,777,778,779],[224,75779,75781],{"className":75780},[468,779],[224,75782,1447],{"className":75783},[468,779],[224,75785,75786,75789],{"style":20546},[224,75787],{"className":75788,"style":2108},[771],[224,75790],{"className":75791,"style":20554},[20553],[224,75793,75794,75797],{"style":22589},[224,75795],{"className":75796,"style":2108},[771],[224,75798,75800],{"className":75799},[776,777,778,779],[224,75801,75803],{"className":75802},[468,779],[224,75804,1438],{"className":75805},[468,779],[224,75807,787],{"className":75808},[786],[224,75810,75812],{"className":75811},[759],[224,75813,75815],{"className":75814,"style":22611},[763],[224,75816],{},[224,75818],{"className":75819},[549,8479],[224,75821,1083],{"className":75822},[468],[224,75824,1435],{"className":75825},[468,469],[224,75827],{"className":75828,"style":474},[473],[224,75830,5268],{"className":75831},[478],[224,75833],{"className":75834,"style":474},[473],[224,75836,75838,75841],{"className":75837},[459],[224,75839],{"className":75840,"style":535},[463],[224,75842,54602],{"className":75843},[468],"），则只能保证 ",[224,75846,75848,75870],{"className":75847,"translate":388},[392],[224,75849,75851],{"className":75850},[396],[398,75852,75853],{"xmlns":400},[403,75854,75855,75867],{},[406,75856,75857,75863,75865],{},[695,75858,75859,75861],{},[409,75860,20303],{},[409,75862,1705],{},[412,75864,278],{},[419,75866,284],{},[447,75868,75869],{"encoding":449},"D_f=0",[224,75871,75873,75928],{"className":75872,"ariaHidden":455},[454],[224,75874,75876,75879,75919,75922,75925],{"className":75875},[459],[224,75877],{"className":75878,"style":24063},[463],[224,75880,75882,75885],{"className":75881},[468],[224,75883,20303],{"className":75884,"style":2473},[468,469],[224,75886,75888],{"className":75887},[750],[224,75889,75891,75911],{"className":75890},[754,755],[224,75892,75894,75908],{"className":75893},[759],[224,75895,75897],{"className":75896,"style":20387},[763],[224,75898,75899,75902],{"style":20390},[224,75900],{"className":75901,"style":772},[771],[224,75903,75905],{"className":75904},[776,777,778,779],[224,75906,1705],{"className":75907,"style":1740},[468,469,779],[224,75909,787],{"className":75910},[786],[224,75912,75914],{"className":75913},[759],[224,75915,75917],{"className":75916,"style":20409},[763],[224,75918],{},[224,75920],{"className":75921,"style":496},[473],[224,75923,278],{"className":75924},[500],[224,75926],{"className":75927,"style":496},[473],[224,75929,75931,75934],{"className":75930},[459],[224,75932],{"className":75933,"style":510},[463],[224,75935,284],{"className":75936},[468]," 时不必然 ",[224,75939,75941,75958],{"className":75940,"translate":388},[392],[224,75942,75944],{"className":75943},[396],[398,75945,75946],{"xmlns":400},[403,75947,75948,75956],{},[406,75949,75950,75952,75954],{},[409,75951,16018],{},[412,75953,278],{},[409,75955,20315],{},[447,75957,30229],{"encoding":449},[224,75959,75961,75979],{"className":75960,"ariaHidden":455},[454],[224,75962,75964,75967,75970,75973,75976],{"className":75963},[459],[224,75965],{"className":75966,"style":649},[463],[224,75968,16018],{"className":75969,"style":653},[468,469],[224,75971],{"className":75972,"style":496},[473],[224,75974,278],{"className":75975},[500],[224,75977],{"className":75978,"style":496},[473],[224,75980,75982,75985],{"className":75981},[459],[224,75983],{"className":75984,"style":20637},[463],[224,75986,20315],{"className":75987},[468,469],"：TV 是反例，",[224,75990,75992,76009],{"className":75991,"translate":388},[392],[224,75993,75995],{"className":75994},[396],[398,75996,75997],{"xmlns":400},[403,75998,75999,76007],{},[406,76000,76001,76003,76005],{},[409,76002,16018],{},[412,76004,278],{},[409,76006,20315],{},[447,76008,30229],{"encoding":449},[224,76010,76012,76030],{"className":76011,"ariaHidden":455},[454],[224,76013,76015,76018,76021,76024,76027],{"className":76014},[459],[224,76016],{"className":76017,"style":649},[463],[224,76019,16018],{"className":76020,"style":653},[468,469],[224,76022],{"className":76023,"style":496},[473],[224,76025,278],{"className":76026},[500],[224,76028],{"className":76029,"style":496},[473],[224,76031,76033,76036],{"className":76032},[459],[224,76034],{"className":76035,"style":20637},[463],[224,76037,20315],{"className":76038},[468,469]," 时才是唯一使 TV 为 0 的一对（TV 也对）。",[613,76041,76043],{"id":76042},"_42-联合凸性","4.2 联合凸性",[21055,76045,76047,76516],{"title":76046},"命题 4.2（联合凸性）",[11,76048,76049,76050,7098,76246,7098,76442,61551],{},"若 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这是 ",[224,77007,77009,77022],{"className":77008,"translate":388},[392],[224,77010,77012],{"className":77011},[396],[398,77013,77014],{"xmlns":400},[403,77015,77016,77020],{},[406,77017,77018],{},[409,77019,1705],{},[447,77021,1705],{"encoding":449},[224,77023,77025],{"className":77024,"ariaHidden":455},[454],[224,77026,77028,77031],{"className":77027},[459],[224,77029],{"className":77030,"style":2565},[463],[224,77032,1705],{"className":77033,"style":1740},[468,469]," 凸性加上「",[224,77036,77038,77054],{"className":77037,"translate":388},[392],[224,77039,77041],{"className":77040},[396],[398,77042,77043],{"xmlns":400},[403,77044,77045,77051],{},[406,77046,77047,77049],{},[409,77048,55354],{},[412,77050,24134],{},[447,77052,77053],{"encoding":449},"\\sup",[224,77055,77057],{"className":77056,"ariaHidden":455},[454],[224,77058,77060,77063],{"className":77059},[459],[224,77061],{"className":77062,"style":806},[463],[224,77064,55354],{"className":77065},[21531]," 与线性运算」的标准结果，也可用 f-散度的变分表示（4.4 节）直接看：",[224,77068,77070,77150],{"className":77069,"translate":388},[392],[224,77071,77073],{"className":77072},[396],[398,77074,77075],{"xmlns":400},[403,77076,77077,77147],{},[406,77078,77079,77085,77087,77089,77091,77093,77095,77097,77107,77109,77115,77117,77119,77121,77123,77129,77131,77137,77139,77141,77143,77145],{},[695,77080,77081,77083],{},[409,77082,20303],{},[409,77084,1705],{},[412,77086,239],{"stretchy":435},[409,77088,16018],{},[409,77090,20312],{"mathvariant":678},[409,77092,20315],{},[412,77094,1716],{"stretchy":435},[412,77096,278],{},[695,77098,77099,77105],{},[406,77100,77101,77103],{},[409,77102,55354],{},[412,77104,24134],{},[409,77106,24189],{},[412,77108,8110],{"stretchy":435},[695,77110,77111,77113],{},[409,77112,20324],{"mathvariant":12251},[409,77114,16018],{},[412,77116,268],{"stretchy":435},[409,77118,24189],{},[412,77120,440],{"stretchy":435},[412,77122,5268],{},[695,77124,77125,77127],{},[409,77126,20324],{"mathvariant":12251},[409,77128,20315],{},[412,77130,268],{"stretchy":435},[3248,77132,77133,77135],{},[409,77134,1705],{},[412,77136,27543],{},[412,77138,239],{"stretchy":435},[409,77140,24189],{},[412,77142,1716],{"stretchy":435},[412,77144,440],{"stretchy":435},[412,77146,15077],{"stretchy":435},[447,77148,77149],{"encoding":449},"D_f(P\\|Q)=\\sup_g\\{\\mathbb E_P[g]-\\mathbb E_Q[f^*(g)]\\}",[224,77151,77153,77223,77332],{"className":77152,"ariaHidden":455},[454],[224,77154,77156,77159,77199,77202,77205,77208,77211,77214,77217,77220],{"className":77155},[459],[224,77157],{"className":77158,"style":20368},[463],[224,77160,77162,77165],{"className":77161},[468],[224,77163,20303],{"className":77164,"style":2473},[468,469],[224,77166,77168],{"className":77167},[750],[224,77169,77171,77191],{"className":77170},[754,755],[224,77172,77174,77188],{"className":77173},[759],[224,77175,77177],{"className":77176,"style":20387},[763],[224,77178,77179,77182],{"style":20390},[224,77180],{"className":77181,"style":772},[771],[224,77183,77185],{"className":77184},[776,777,778,779],[224,77186,1705],{"className":77187,"style":1740},[468,469,779],[224,77189,787],{"className":77190},[786],[224,77192,77194],{"className":77193},[759],[224,77195,77197],{"className":77196,"style":20409},[763],[224,77198],{},[224,77200,239],{"className":77201},[542],[224,77203,16018],{"className":77204,"style":653},[468,469],[224,77206,20312],{"className":77207},[468],[224,77209,20315],{"className":77210},[468,469],[224,77212,1716],{"className":77213},[549],[224,77215],{"className":77216,"style":496},[473],[224,77218,278],{"className":77219},[500],[224,77221],{"className":77222,"style":496},[473],[224,77224,77226,77230,77271,77274,77314,77317,77320,77323,77326,77329],{"className":77225},[459],[224,77227],{"className":77228,"style":77229},[463],"height:1.1302em;vertical-align:-0.3802em;",[224,77231,77233,77236],{"className":77232},[21531],[224,77234,55354],{"className":77235},[21531],[224,77237,77239],{"className":77238},[750],[224,77240,77242,77262],{"className":77241},[754,755],[224,77243,77245,77259],{"className":77244},[759],[224,77246,77248],{"className":77247,"style":56921},[763],[224,77249,77250,77253],{"style":55500},[224,77251],{"className":77252,"style":772},[771],[224,77254,77256],{"className":77255},[776,777,778,779],[224,77257,24189],{"className":77258,"style":492},[468,469,779],[224,77260,787],{"className":77261},[786],[224,77263,77265],{"className":77264},[759],[224,77266,77269],{"className":77267,"style":77268},[763],"height:0.3802em;",[224,77270],{},[224,77272,8110],{"className":77273},[542],[224,77275,77277,77280],{"className":77276},[468],[224,77278,20324],{"className":77279},[468,12389],[224,77281,77283],{"className":77282},[750],[224,77284,77286,77306],{"className":77285},[754,755],[224,77287,77289,77303],{"className":77288},[759],[224,77290,77292],{"className":77291,"style":764},[763],[224,77293,77294,77297],{"style":767},[224,77295],{"className":77296,"style":772},[771],[224,77298,77300],{"className":77299},[776,777,778,779],[224,77301,16018],{"className":77302,"style":653},[468,469,779],[224,77304,787],{"className":77305},[786],[224,77307,77309],{"className":77308},[759],[224,77310,77312],{"className":77311,"style":794},[763],[224,77313],{},[224,77315,268],{"className":77316},[542],[224,77318,24189],{"className":77319,"style":492},[468,469],[224,77321,440],{"className":77322},[549],[224,77324],{"className":77325,"style":474},[473],[224,77327,5268],{"className":77328},[478],[224,77330],{"className":77331,"style":474},[473],[224,77333,77335,77338,77378,77381,77410,77413,77416],{"className":77334},[459],[224,77336],{"className":77337,"style":20368},[463],[224,77339,77341,77344],{"className":77340},[468],[224,77342,20324],{"className":77343},[468,12389],[224,77345,77347],{"className":77346},[750],[224,77348,77350,77370],{"className":77349},[754,755],[224,77351,77353,77367],{"className":77352},[759],[224,77354,77356],{"className":77355,"style":764},[763],[224,77357,77358,77361],{"style":767},[224,77359],{"className":77360,"style":772},[771],[224,77362,77364],{"className":77363},[776,777,778,779],[224,77365,20315],{"className":77366},[468,469,779],[224,77368,787],{"className":77369},[786],[224,77371,77373],{"className":77372},[759],[224,77374,77376],{"className":77375,"style":20409},[763],[224,77377],{},[224,77379,268],{"className":77380},[542],[224,77382,77384,77387],{"className":77383},[468],[224,77385,1705],{"className":77386,"style":1740},[468,469],[224,77388,77390],{"className":77389},[750],[224,77391,77393],{"className":77392},[754],[224,77394,77396],{"className":77395},[759],[224,77397,77399],{"className":77398,"style":28465},[763],[224,77400,77401,77404],{"style":3288},[224,77402],{"className":77403,"style":772},[771],[224,77405,77407],{"className":77406},[776,777,778,779],[224,77408,27543],{"className":77409},[478,779],[224,77411,239],{"className":77412},[542],[224,77414,24189],{"className":77415,"style":492},[468,469],[224,77417,77419],{"className":77418},[549],")]}","，右端关于 ",[224,77422,77424,77445],{"className":77423,"translate":388},[392],[224,77425,77427],{"className":77426},[396],[398,77428,77429],{"xmlns":400},[403,77430,77431,77443],{},[406,77432,77433,77435,77437,77439,77441],{},[412,77434,239],{"stretchy":435},[409,77436,16018],{},[412,77438,1877],{"separator":455},[409,77440,20315],{},[412,77442,1716],{"stretchy":435},[447,77444,30988],{"encoding":449},[224,77446,77448],{"className":77447,"ariaHidden":455},[454],[224,77449,77451,77454,77457,77460,77463,77466,77469],{"className":77450},[459],[224,77452],{"className":77453,"style":535},[463],[224,77455,239],{"className":77456},[542],[224,77458,16018],{"className":77459,"style":653},[468,469],[224,77461,1877],{"className":77462},[1917],[224,77464],{"className":77465,"style":724},[473],[224,77467,20315],{"className":77468},[468,469],[224,77470,1716],{"className":77471},[549]," 是凸的（",[224,77474,77476,77500],{"className":77475,"translate":388},[392],[224,77477,77479],{"className":77478},[396],[398,77480,77481],{"xmlns":400},[403,77482,77483,77497],{},[406,77484,77485,77491,77493,77495],{},[695,77486,77487,77489],{},[409,77488,20324],{"mathvariant":12251},[409,77490,16018],{},[412,77492,268],{"stretchy":435},[409,77494,24189],{},[412,77496,440],{"stretchy":435},[447,77498,77499],{"encoding":449},"\\mathbb E_P[g]",[224,77501,77503],{"className":77502,"ariaHidden":455},[454],[224,77504,77506,77509,77549,77552,77555],{"className":77505},[459],[224,77507],{"className":77508,"style":535},[463],[224,77510,77512,77515],{"className":77511},[468],[224,77513,20324],{"className":77514},[468,12389],[224,77516,77518],{"className":77517},[750],[224,77519,77521,77541],{"className":77520},[754,755],[224,77522,77524,77538],{"className":77523},[759],[224,77525,77527],{"className":77526,"style":764},[763],[224,77528,77529,77532],{"style":767},[224,77530],{"className":77531,"style":772},[771],[224,77533,77535],{"className":77534},[776,777,778,779],[224,77536,16018],{"className":77537,"style":653},[468,469,779],[224,77539,787],{"className":77540},[786],[224,77542,77544],{"className":77543},[759],[224,77545,77547],{"className":77546,"style":794},[763],[224,77548],{},[224,77550,268],{"className":77551},[542],[224,77553,24189],{"className":77554,"style":492},[468,469],[224,77556,440],{"className":77557},[549]," 线性、",[224,77560,77562,77598],{"className":77561,"translate":388},[392],[224,77563,77565],{"className":77564},[396],[398,77566,77567],{"xmlns":400},[403,77568,77569,77595],{},[406,77570,77571,77573,77579,77581,77587,77589,77591,77593],{},[412,77572,5268],{},[695,77574,77575,77577],{},[409,77576,20324],{"mathvariant":12251},[409,77578,20315],{},[412,77580,268],{"stretchy":435},[3248,77582,77583,77585],{},[409,77584,1705],{},[412,77586,27543],{},[412,77588,239],{"stretchy":435},[409,77590,24189],{},[412,77592,1716],{"stretchy":435},[412,77594,440],{"stretchy":435},[447,77596,77597],{"encoding":449},"-\\mathbb E_Q[f^*(g)]",[224,77599,77601],{"className":77600,"ariaHidden":455},[454],[224,77602,77604,77607,77610,77650,77653,77682,77685,77688],{"className":77603},[459],[224,77605],{"className":77606,"style":20368},[463],[224,77608,5268],{"className":77609},[468],[224,77611,77613,77616],{"className":77612},[468],[224,77614,20324],{"className":77615},[468,12389],[224,77617,77619],{"className":77618},[750],[224,77620,77622,77642],{"className":77621},[754,755],[224,77623,77625,77639],{"className":77624},[759],[224,77626,77628],{"className":77627,"style":764},[763],[224,77629,77630,77633],{"style":767},[224,77631],{"className":77632,"style":772},[771],[224,77634,77636],{"className":77635},[776,777,778,779],[224,77637,20315],{"className":77638},[468,469,779],[224,77640,787],{"className":77641},[786],[224,77643,77645],{"className":77644},[759],[224,77646,77648],{"className":77647,"style":20409},[763],[224,77649],{},[224,77651,268],{"className":77652},[542],[224,77654,77656,77659],{"className":77655},[468],[224,77657,1705],{"className":77658,"style":1740},[468,469],[224,77660,77662],{"className":77661},[750],[224,77663,77665],{"className":77664},[754],[224,77666,77668],{"className":77667},[759],[224,77669,77671],{"className":77670,"style":28465},[763],[224,77672,77673,77676],{"style":3288},[224,77674],{"className":77675,"style":772},[771],[224,77677,77679],{"className":77678},[776,777,778,779],[224,77680,27543],{"className":77681},[478,779],[224,77683,239],{"className":77684},[542],[224,77686,24189],{"className":77687,"style":492},[468,469],[224,77689,40329],{"className":77690},[549]," 线性），取上确界后仍凸。",[11,77693,77694,77695,7098,77723,77841,77842,7098,77870,30923,77987,12570,77990,78106,78107,78135,78136,78164,78165,78193],{},"等价地，固定 ",[224,77696,77698,77711],{"className":77697,"translate":388},[392],[224,77699,77701],{"className":77700},[396],[398,77702,77703],{"xmlns":400},[403,77704,77705,77709],{},[406,77706,77707],{},[409,77708,20315],{},[447,77710,20315],{"encoding":449},[224,77712,77714],{"className":77713,"ariaHidden":455},[454],[224,77715,77717,77720],{"className":77716},[459],[224,77718],{"className":77719,"style":20637},[463],[224,77721,20315],{"className":77722},[468,469],[224,77724,77726,77758],{"className":77725,"translate":388},[392],[224,77727,77729],{"className":77728},[396],[398,77730,77731],{"xmlns":400},[403,77732,77733,77755],{},[406,77734,77735,77737,77739,77745,77747,77749,77751,77753],{},[409,77736,16018],{},[412,77738,28649],{},[695,77740,77741,77743],{},[409,77742,20303],{},[409,77744,1705],{},[412,77746,239],{"stretchy":435},[409,77748,16018],{},[409,77750,20312],{"mathvariant":678},[409,77752,20315],{},[412,77754,1716],{"stretchy":435},[447,77756,77757],{"encoding":449},"P\\mapsto D_f(P\\|Q)",[224,77759,77761,77780],{"className":77760,"ariaHidden":455},[454],[224,77762,77764,77768,77771,77774,77777],{"className":77763},[459],[224,77765],{"className":77766,"style":77767},[463],"height:0.6943em;vertical-align:-0.011em;",[224,77769,16018],{"className":77770,"style":653},[468,469],[224,77772],{"className":77773,"style":496},[473],[224,77775,28649],{"className":77776},[500],[224,77778],{"className":77779,"style":496},[473],[224,77781,77783,77786,77826,77829,77832,77835,77838],{"className":77782},[459],[224,77784],{"className":77785,"style":20368},[463],[224,77787,77789,77792],{"className":77788},[468],[224,77790,20303],{"className":77791,"style":2473},[468,469],[224,77793,77795],{"className":77794},[750],[224,77796,77798,77818],{"className":77797},[754,755],[224,77799,77801,77815],{"className":77800},[759],[224,77802,77804],{"className":77803,"style":20387},[763],[224,77805,77806,77809],{"style":20390},[224,77807],{"className":77808,"style":772},[771],[224,77810,77812],{"className":77811},[776,777,778,779],[224,77813,1705],{"className":77814,"style":1740},[468,469,779],[224,77816,787],{"className":77817},[786],[224,77819,77821],{"className":77820},[759],[224,77822,77824],{"className":77823,"style":20409},[763],[224,77825],{},[224,77827,239],{"className":77828},[542],[224,77830,16018],{"className":77831,"style":653},[468,469],[224,77833,20312],{"className":77834},[468],[224,77836,20315],{"className":77837},[468,469],[224,77839,1716],{"className":77840},[549]," 凸；固定 ",[224,77843,77845,77858],{"className":77844,"translate":388},[392],[224,77846,77848],{"className":77847},[396],[398,77849,77850],{"xmlns":400},[403,77851,77852,77856],{},[406,77853,77854],{},[409,77855,16018],{},[447,77857,16018],{"encoding":449},[224,77859,77861],{"className":77860,"ariaHidden":455},[454],[224,77862,77864,77867],{"className":77863},[459],[224,77865],{"className":77866,"style":649},[463],[224,77868,16018],{"className":77869,"style":653},[468,469],[224,77871,77873,77905],{"className":77872,"translate":388},[392],[224,77874,77876],{"className":77875},[396],[398,77877,77878],{"xmlns":400},[403,77879,77880,77902],{},[406,77881,77882,77884,77886,77892,77894,77896,77898,77900],{},[409,77883,20315],{},[412,77885,28649],{},[695,77887,77888,77890],{},[409,77889,20303],{},[409,77891,1705],{},[412,77893,239],{"stretchy":435},[409,77895,16018],{},[409,77897,20312],{"mathvariant":678},[409,77899,20315],{},[412,77901,1716],{"stretchy":435},[447,77903,77904],{"encoding":449},"Q\\mapsto D_f(P\\|Q)",[224,77906,77908,77926],{"className":77907,"ariaHidden":455},[454],[224,77909,77911,77914,77917,77920,77923],{"className":77910},[459],[224,77912],{"className":77913,"style":20637},[463],[224,77915,20315],{"className":77916},[468,469],[224,77918],{"className":77919,"style":496},[473],[224,77921,28649],{"className":77922},[500],[224,77924],{"className":77925,"style":496},[473],[224,77927,77929,77932,77972,77975,77978,77981,77984],{"className":77928},[459],[224,77930],{"className":77931,"style":20368},[463],[224,77933,77935,77938],{"className":77934},[468],[224,77936,20303],{"className":77937,"style":2473},[468,469],[224,77939,77941],{"className":77940},[750],[224,77942,77944,77964],{"className":77943},[754,755],[224,77945,77947,77961],{"className":77946},[759],[224,77948,77950],{"className":77949,"style":20387},[763],[224,77951,77952,77955],{"style":20390},[224,77953],{"className":77954,"style":772},[771],[224,77956,77958],{"className":77957},[776,777,778,779],[224,77959,1705],{"className":77960,"style":1740},[468,469,779],[224,77962,787],{"className":77963},[786],[224,77965,77967],{"className":77966},[759],[224,77968,77970],{"className":77969,"style":20409},[763],[224,77971],{},[224,77973,239],{"className":77974},[542],[224,77976,16018],{"className":77977,"style":653},[468,469],[224,77979,20312],{"className":77980},[468],[224,77982,20315],{"className":77983},[468,469],[224,77985,1716],{"className":77986},[549],[378,77988,77989],{},"注意",[224,77991,77993,78024],{"className":77992,"translate":388},[392],[224,77994,77996],{"className":77995},[396],[398,77997,77998],{"xmlns":400},[403,77999,78000,78022],{},[406,78001,78002,78004,78006,78012,78014,78016,78018,78020],{},[409,78003,20315],{},[412,78005,28649],{},[695,78007,78008,78010],{},[409,78009,20303],{},[409,78011,1705],{},[412,78013,239],{"stretchy":435},[409,78015,16018],{},[409,78017,20312],{"mathvariant":678},[409,78019,20315],{},[412,78021,1716],{"stretchy":435},[447,78023,77904],{"encoding":449},[224,78025,78027,78045],{"className":78026,"ariaHidden":455},[454],[224,78028,78030,78033,78036,78039,78042],{"className":78029},[459],[224,78031],{"className":78032,"style":20637},[463],[224,78034,20315],{"className":78035},[468,469],[224,78037],{"className":78038,"style":496},[473],[224,78040,28649],{"className":78041},[500],[224,78043],{"className":78044,"style":496},[473],[224,78046,78048,78051,78091,78094,78097,78100,78103],{"className":78047},[459],[224,78049],{"className":78050,"style":20368},[463],[224,78052,78054,78057],{"className":78053},[468],[224,78055,20303],{"className":78056,"style":2473},[468,469],[224,78058,78060],{"className":78059},[750],[224,78061,78063,78083],{"className":78062},[754,755],[224,78064,78066,78080],{"className":78065},[759],[224,78067,78069],{"className":78068,"style":20387},[763],[224,78070,78071,78074],{"style":20390},[224,78072],{"className":78073,"style":772},[771],[224,78075,78077],{"className":78076},[776,777,778,779],[224,78078,1705],{"className":78079,"style":1740},[468,469,779],[224,78081,787],{"className":78082},[786],[224,78084,78086],{"className":78085},[759],[224,78087,78089],{"className":78088,"style":20409},[763],[224,78090],{},[224,78092,239],{"className":78093},[542],[224,78095,16018],{"className":78096,"style":653},[468,469],[224,78098,20312],{"className":78099},[468],[224,78101,20315],{"className":78102},[468,469],[224,78104,1716],{"className":78105},[549]," 凸是「以 ",[224,78108,78110,78123],{"className":78109,"translate":388},[392],[224,78111,78113],{"className":78112},[396],[398,78114,78115],{"xmlns":400},[403,78116,78117,78121],{},[406,78118,78119],{},[409,78120,20315],{},[447,78122,20315],{"encoding":449},[224,78124,78126],{"className":78125,"ariaHidden":455},[454],[224,78127,78129,78132],{"className":78128},[459],[224,78130],{"className":78131,"style":20637},[463],[224,78133,20315],{"className":78134},[468,469]," 为参照」的凸性，它与「",[224,78137,78139,78152],{"className":78138,"translate":388},[392],[224,78140,78142],{"className":78141},[396],[398,78143,78144],{"xmlns":400},[403,78145,78146,78150],{},[406,78147,78148],{},[409,78149,20315],{},[447,78151,20315],{"encoding":449},[224,78153,78155],{"className":78154,"ariaHidden":455},[454],[224,78156,78158,78161],{"className":78157},[459],[224,78159],{"className":78160,"style":20637},[463],[224,78162,20315],{"className":78163},[468,469]," 出现在分母上」的直觉相反，这个凸性正是变分推断里「对 ",[224,78166,78168,78181],{"className":78167,"translate":388},[392],[224,78169,78171],{"className":78170},[396],[398,78172,78173],{"xmlns":400},[403,78174,78175,78179],{},[406,78176,78177],{},[409,78178,2023],{},[447,78180,2023],{"encoding":449},[224,78182,78184],{"className":78183,"ariaHidden":455},[454],[224,78185,78187,78190],{"className":78186},[459],[224,78188],{"className":78189,"style":806},[463],[224,78191,2023],{"className":78192,"style":492},[468,469]," 做平均场优化收敛到全局最优」的根据之一。",[613,78195,78197],{"id":78196},"_43-斜对称关系","4.3 斜对称关系",[21055,78199,78201,78299,78517],{"title":78200},"定义 4.1（斜共轭）",[11,78202,34210,78203,78298],{},[224,78204,78206,78235],{"className":78205,"translate":388},[392],[224,78207,78209],{"className":78208},[396],[398,78210,78211],{"xmlns":400},[403,78212,78213,78233],{},[406,78214,78215,78217,78219,78221,78223,78225,78227,78229,78231],{},[409,78216,1705],{},[412,78218,1071],{},[412,78220,268],{"stretchy":435},[419,78222,284],{},[412,78224,1877],{"separator":455},[409,78226,21085],{"mathvariant":678},[412,78228,1716],{"stretchy":435},[412,78230,1144],{},[409,78232,21092],{"mathvariant":12251},[447,78234,21095],{"encoding":449},[224,78236,78238,78256,78289],{"className":78237,"ariaHidden":455},[454],[224,78239,78241,78244,78247,78250,78253],{"className":78240},[459],[224,78242],{"className":78243,"style":2565},[463],[224,78245,1705],{"className":78246,"style":1740},[468,469],[224,78248],{"className":78249,"style":496},[473],[224,78251,1071],{"className":78252},[500],[224,78254],{"className":78255,"style":496},[473],[224,78257,78259,78262,78265,78268,78271,78274,78277,78280,78283,78286],{"className":78258},[459],[224,78260],{"className":78261,"style":535},[463],[224,78263,268],{"className":78264},[542],[224,78266,284],{"className":78267},[468],[224,78269,1877],{"className":78270},[1917],[224,78272],{"className":78273,"style":724},[473],[224,78275,21085],{"className":78276},[468],[224,78278,1716],{"className":78279},[549],[224,78281],{"className":78282,"style":496},[473],[224,78284,1144],{"className":78285},[500],[224,78287],{"className":78288,"style":496},[473],[224,78290,78292,78295],{"className":78291},[459],[224,78293],{"className":78294,"style":12385},[463],[224,78296,21092],{"className":78297},[468,12389],"，称",[224,78300,78302],{"className":78301,"translate":388},[387],[224,78303,78305,78367],{"className":78304,"translate":388},[392],[224,78306,78308],{"className":78307},[396],[398,78309,78310],{"xmlns":400,"display":401},[403,78311,78312,78364],{},[406,78313,78314,78330,78332,78334,78336,78338,78340,78342,78344,78346,78348,78350,78352,78354,78356,78358,78360,78362],{},[27527,78315,78316,78318,78328],{},[409,78317,1705],{},[406,78319,78320,78322,78324,78326],{},[409,78321,1858],{},[409,78323,7630],{},[409,78325,1861],{},[409,78327,13639],{},[412,78329,27543],{"lspace":5213,"rspace":5213},[412,78331,239],{"stretchy":435},[409,78333,1435],{},[412,78335,1716],{"stretchy":435},[412,78337,278],{},[409,78339,1435],{},[423,78341,682],{},[409,78343,1705],{},[412,78345,239],{"stretchy":435},[419,78347,1438],{},[409,78349,12413],{"mathvariant":678},[409,78351,1435],{},[412,78353,1716],{"stretchy":435},[412,78355,1877],{"separator":455},[473,78357],{"width":4247},[409,78359,1435],{},[412,78361,325],{},[419,78363,284],{},[447,78365,78366],{"encoding":449},"f^{*}_{skew}(t)=t\\,f(1\u002Ft),\\qquad t>0",[224,78368,78370,78460,78508],{"className":78369,"ariaHidden":455},[454],[224,78371,78373,78376,78442,78445,78448,78451,78454,78457],{"className":78372},[459],[224,78374],{"className":78375,"style":535},[463],[224,78377,78379,78382],{"className":78378},[468],[224,78380,1705],{"className":78381,"style":1740},[468,469],[224,78383,78385],{"className":78384},[750],[224,78386,78388,78434],{"className":78387},[754,755],[224,78389,78391,78431],{"className":78390},[759],[224,78392,78394,78417],{"className":78393,"style":28217},[763],[224,78395,78396,78399],{"style":28220},[224,78397],{"className":78398,"style":772},[771],[224,78400,78402],{"className":78401},[776,777,778,779],[224,78403,78405,78408,78411,78414],{"className":78404},[468,779],[224,78406,1858],{"className":78407},[468,469,779],[224,78409,7630],{"className":78410,"style":7646},[468,469,779],[224,78412,1861],{"className":78413},[468,469,779],[224,78415,13639],{"className":78416,"style":13754},[468,469,779],[224,78418,78419,78422],{"style":5074},[224,78420],{"className":78421,"style":772},[771],[224,78423,78425],{"className":78424},[776,777,778,779],[224,78426,78428],{"className":78427},[468,779],[224,78429,27543],{"className":78430},[468,779],[224,78432,787],{"className":78433},[786],[224,78435,78437],{"className":78436},[759],[224,78438,78440],{"className":78439,"style":28265},[763],[224,78441],{},[224,78443,239],{"className":78444},[542],[224,78446,1435],{"className":78447},[468,469],[224,78449,1716],{"className":78450},[549],[224,78452],{"className":78453,"style":496},[473],[224,78455,278],{"className":78456},[500],[224,78458],{"className":78459,"style":496},[473],[224,78461,78463,78466,78469,78472,78475,78478,78481,78484,78487,78490,78493,78496,78499,78502,78505],{"className":78462},[459],[224,78464],{"className":78465,"style":535},[463],[224,78467,1435],{"className":78468},[468,469],[224,78470],{"className":78471,"style":724},[473],[224,78473,1705],{"className":78474,"style":1740},[468,469],[224,78476,239],{"className":78477},[542],[224,78479,28555],{"className":78480},[468],[224,78482,1435],{"className":78483},[468,469],[224,78485,1716],{"className":78486},[549],[224,78488,1877],{"className":78489},[1917],[224,78491],{"className":78492,"style":4416},[473],[224,78494],{"className":78495,"style":724},[473],[224,78497,1435],{"className":78498},[468,469],[224,78500],{"className":78501,"style":496},[473],[224,78503,325],{"className":78504},[500],[224,78506],{"className":78507,"style":496},[473],[224,78509,78511,78514],{"className":78510},[459],[224,78512],{"className":78513,"style":510},[463],[224,78515,284],{"className":78516},[468],[11,78518,78519,78520,53748,78548,3117],{},"为 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f(t)\\,dQ\n=\\int f\\!\\left(\\frac1s\\right) s\\,dP\\\\\n&=\\int 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对称（",[224,79788,79790,79836],{"className":79789,"translate":388},[392],[224,79791,79793],{"className":79792},[396],[398,79794,79795],{"xmlns":400},[403,79796,79797,79833],{},[406,79798,79799,79805,79807,79809,79811,79813,79815,79817,79823,79825,79827,79829,79831],{},[695,79800,79801,79803],{},[409,79802,20303],{},[409,79804,1705],{},[412,79806,239],{"stretchy":435},[409,79808,16018],{},[409,79810,20312],{"mathvariant":678},[409,79812,20315],{},[412,79814,1716],{"stretchy":435},[412,79816,278],{},[695,79818,79819,79821],{},[409,79820,20303],{},[409,79822,1705],{},[412,79824,239],{"stretchy":435},[409,79826,20315],{},[409,79828,20312],{"mathvariant":678},[409,79830,16018],{},[412,79832,1716],{"stretchy":435},[447,79834,79835],{"encoding":449},"D_f(P\\|Q)=D_f(Q\\|P)",[224,79837,79839,79909],{"className":79838,"ariaHidden":455},[454],[224,79840,79842,79845,79885,79888,79891,79894,79897,79900,79903,79906],{"className":79841},[459],[224,79843],{"className":79844,"style":20368},[463],[224,79846,79848,79851],{"className":79847},[468],[224,79849,20303],{"className":79850,"style":2473},[468,469],[224,79852,79854],{"className":79853},[750],[224,79855,79857,79877],{"className":79856},[754,755],[224,79858,79860,79874],{"className":79859},[759],[224,79861,79863],{"className":79862,"style":20387},[763],[224,79864,79865,79868],{"style":20390},[224,79866],{"className":79867,"style":772},[771],[224,79869,79871],{"className":79870},[776,777,778,779],[224,79872,1705],{"className":79873,"style":1740},[468,469,779],[224,79875,787],{"className":79876},[786],[224,79878,79880],{"className":79879},[759],[224,79881,79883],{"className":79882,"style":20409},[763],[224,79884],{},[224,79886,239],{"className":79887},[542],[224,79889,16018],{"className":79890,"style":653},[468,469],[224,79892,20312],{"className":79893},[468],[224,79895,20315],{"className":79896},[468,469],[224,79898,1716],{"className":79899},[549],[224,79901],{"className":79902,"style":496},[473],[224,79904,278],{"className":79905},[500],[224,79907],{"className":79908,"style":496},[473],[224,79910,79912,79915,79955,79958,79961,79964,79967],{"className":79911},[459],[224,79913],{"className":79914,"style":20368},[463],[224,79916,79918,79921],{"className":79917},[468],[224,79919,20303],{"className":79920,"style":2473},[468,469],[224,79922,79924],{"className":79923},[750],[224,79925,79927,79947],{"className":79926},[754,755],[224,79928,79930,79944],{"className":79929},[759],[224,79931,79933],{"className":79932,"style":20387},[763],[224,79934,79935,79938],{"style":20390},[224,79936],{"className":79937,"style":772},[771],[224,79939,79941],{"className":79940},[776,777,778,779],[224,79942,1705],{"className":79943,"style":1740},[468,469,779],[224,79945,787],{"className":79946},[786],[224,79948,79950],{"className":79949},[759],[224,79951,79953],{"className":79952,"style":20409},[763],[224,79954],{},[224,79956,239],{"className":79957},[542],[224,79959,20315],{"className":79960},[468,469],[224,79962,20312],{"className":79963},[468],[224,79965,16018],{"className":79966,"style":653},[468,469],[224,79968,1716],{"className":79969},[549]," 对所有 ",[224,79972,79974,79991],{"className":79973,"translate":388},[392],[224,79975,79977],{"className":79976},[396],[398,79978,79979],{"xmlns":400},[403,79980,79981,79989],{},[406,79982,79983,79985,79987],{},[409,79984,16018],{},[412,79986,1877],{"separator":455},[409,79988,20315],{},[447,79990,20627],{"encoding":449},[224,79992,79994],{"className":79993,"ariaHidden":455},[454],[224,79995,79997,80000,80003,80006,80009],{"className":79996},[459],[224,79998],{"className":79999,"style":20637},[463],[224,80001,16018],{"className":80002,"style":653},[468,469],[224,80004,1877],{"className":80005},[1917],[224,80007],{"className":80008,"style":724},[473],[224,80010,20315],{"className":80011},[468,469],"）当且仅当 ",[224,80014,80016,80047],{"className":80015,"translate":388},[392],[224,80017,80019],{"className":80018},[396],[398,80020,80021],{"xmlns":400},[403,80022,80023,80045],{},[406,80024,80025,80041,80043],{},[27527,80026,80027,80029,80039],{},[409,80028,1705],{},[406,80030,80031,80033,80035,80037],{},[409,80032,1858],{},[409,80034,7630],{},[409,80036,1861],{},[409,80038,13639],{},[412,80040,27543],{"lspace":5213,"rspace":5213},[412,80042,278],{},[409,80044,1705],{},[447,80046,64065],{"encoding":449},[224,80048,80050,80131],{"className":80049,"ariaHidden":455},[454],[224,80051,80053,80056,80122,80125,80128],{"className":80052},[459],[224,80054],{"className":80055,"style":64075},[463],[224,80057,80059,80062],{"className":80058},[468],[224,80060,1705],{"className":80061,"style":1740},[468,469],[224,80063,80065],{"className":80064},[750],[224,80066,80068,80114],{"className":80067},[754,755],[224,80069,80071,80111],{"className":80070},[759],[224,80072,80074,80097],{"className":80073,"style":28465},[763],[224,80075,80076,80079],{"style":28468},[224,80077],{"className":80078,"style":772},[771],[224,80080,80082],{"className":80081},[776,777,778,779],[224,80083,80085,80088,80091,80094],{"className":80084},[468,779],[224,80086,1858],{"className":80087},[468,469,779],[224,80089,7630],{"className":80090,"style":7646},[468,469,779],[224,80092,1861],{"className":80093},[468,469,779],[224,80095,13639],{"className":80096,"style":13754},[468,469,779],[224,80098,80099,80102],{"style":3288},[224,80100],{"className":80101,"style":772},[771],[224,80103,80105],{"className":80104},[776,777,778,779],[224,80106,80108],{"className":80107},[468,779],[224,80109,27543],{"className":80110},[468,779],[224,80112,787],{"className":80113},[786],[224,80115,80117],{"className":80116},[759],[224,80118,80120],{"className":80119,"style":28513},[763],[224,80121],{},[224,80123],{"className":80124,"style":496},[473],[224,80126,278],{"className":80127},[500],[224,80129],{"className":80130,"style":496},[473],[224,80132,80134,80137],{"className":80133},[459],[224,80135],{"className":80136,"style":2565},[463],[224,80138,1705],{"className":80139,"style":1740},[468,469],"（模去 ",[224,80142,80144,80167],{"className":80143,"translate":388},[392],[224,80145,80147],{"className":80146},[396],[398,80148,80149],{"xmlns":400},[403,80150,80151,80165],{},[406,80152,80153,80155,80157,80159,80161,80163],{},[409,80154,219],{},[412,80156,239],{"stretchy":435},[409,80158,1435],{},[412,80160,5268],{},[419,80162,1438],{},[412,80164,1716],{"stretchy":435},[447,80166,26604],{"encoding":449},[224,80168,80170,80194],{"className":80169,"ariaHidden":455},[454],[224,80171,80173,80176,80179,80182,80185,80188,80191],{"className":80172},[459],[224,80174],{"className":80175,"style":535},[463],[224,80177,219],{"className":80178},[468,469],[224,80180,239],{"className":80181},[542],[224,80183,1435],{"className":80184},[468,469],[224,80186],{"className":80187,"style":474},[473],[224,80189,5268],{"className":80190},[478],[224,80192],{"className":80193,"style":474},[473],[224,80195,80197,80200,80203],{"className":80196},[459],[224,80198],{"className":80199,"style":535},[463],[224,80201,1438],{"className":80202},[468],[224,80204,1716],{"className":80205},[549],"）。验证 TV：",[224,80208,80210,80264],{"className":80209,"translate":388},[392],[224,80211,80213],{"className":80212},[396],[398,80214,80215],{"xmlns":400},[403,80216,80217,80261],{},[406,80218,80219,80221,80223,80229,80231,80233,80235,80237,80239,80241,80243,80245,80251,80253,80255,80257,80259],{},[409,80220,1435],{},[412,80222,23331],{},[20338,80224,80225,80227],{},[419,80226,1438],{},[419,80228,1447],{},[409,80230,1083],{"mathvariant":678},[419,80232,1438],{},[409,80234,12413],{"mathvariant":678},[409,80236,1435],{},[412,80238,5268],{},[419,80240,1438],{},[409,80242,1083],{"mathvariant":678},[412,80244,278],{},[20338,80246,80247,80249],{},[419,80248,1438],{},[419,80250,1447],{},[409,80252,1083],{"mathvariant":678},[409,80254,1435],{},[412,80256,5268],{},[419,80258,1438],{},[409,80260,1083],{"mathvariant":678},[447,80262,80263],{"encoding":449},"t\\cdot\\frac12|1\u002Ft-1|=\\frac12|t-1|",[224,80265,80267,80285,80375,80393,80482],{"className":80266,"ariaHidden":455},[454],[224,80268,80270,80273,80276,80279,80282],{"className":80269},[459],[224,80271],{"className":80272,"style":25353},[463],[224,80274,1435],{"className":80275},[468,469],[224,80277],{"className":80278,"style":474},[473],[224,80280,23331],{"className":80281},[478],[224,80283],{"className":80284,"style":474},[473],[224,80286,80288,80291,80359,80363,80366,80369,80372],{"className":80287},[459],[224,80289],{"className":80290,"style":22544},[463],[224,80292,80294,80297,80356],{"className":80293},[468],[224,80295],{"className":80296},[542,8479],[224,80298,80300],{"className":80299},[20338],[224,80301,80303,80348],{"className":80302},[754,755],[224,80304,80306,80345],{"className":80305},[759],[224,80307,80309,80323,80331],{"className":80308,"style":22563},[763],[224,80310,80311,80314],{"style":22566},[224,80312],{"className":80313,"style":2108},[771],[224,80315,80317],{"className":80316},[776,777,778,779],[224,80318,80320],{"className":80319},[468,779],[224,80321,1447],{"className":80322},[468,779],[224,80324,80325,80328],{"style":20546},[224,80326],{"className":80327,"style":2108},[771],[224,80329],{"className":80330,"style":20554},[20553],[224,80332,80333,80336],{"style":22589},[224,80334],{"className":80335,"style":2108},[771],[224,80337,80339],{"className":80338},[776,777,778,779],[224,80340,80342],{"className":80341},[468,779],[224,80343,1438],{"className":80344},[468,779],[224,80346,787],{"className":80347},[786],[224,80349,80351],{"className":80350},[759],[224,80352,80354],{"className":80353,"style":22611},[763],[224,80355],{},[224,80357],{"className":80358},[549,8479],[224,80360,80362],{"className":80361},[468],"∣1\u002F",[224,80364,1435],{"className":80365},[468,469],[224,80367],{"className":80368,"style":474},[473],[224,80370,5268],{"className":80371},[478],[224,80373],{"className":80374,"style":474},[473],[224,80376,80378,80381,80384,80387,80390],{"className":80377},[459],[224,80379],{"className":80380,"style":535},[463],[224,80382,54602],{"className":80383},[468],[224,80385],{"className":80386,"style":496},[473],[224,80388,278],{"className":80389},[500],[224,80391],{"className":80392,"style":496},[473],[224,80394,80396,80399,80467,80470,80473,80476,80479],{"className":80395},[459],[224,80397],{"className":80398,"style":22544},[463],[224,80400,80402,80405,80464],{"className":80401},[468],[224,80403],{"className":80404},[542,8479],[224,80406,80408],{"className":80407},[20338],[224,80409,80411,80456],{"className":80410},[754,755],[224,80412,80414,80453],{"className":80413},[759],[224,80415,80417,80431,80439],{"className":80416,"style":22563},[763],[224,80418,80419,80422],{"style":22566},[224,80420],{"className":80421,"style":2108},[771],[224,80423,80425],{"className":80424},[776,777,778,779],[224,80426,80428],{"className":80427},[468,779],[224,80429,1447],{"className":80430},[468,779],[224,80432,80433,80436],{"style":20546},[224,80434],{"className":80435,"style":2108},[771],[224,80437],{"className":80438,"style":20554},[20553],[224,80440,80441,80444],{"style":22589},[224,80442],{"className":80443,"style":2108},[771],[224,80445,80447],{"className":80446},[776,777,778,779],[224,80448,80450],{"className":80449},[468,779],[224,80451,1438],{"className":80452},[468,779],[224,80454,787],{"className":80455},[786],[224,80457,80459],{"className":80458},[759],[224,80460,80462],{"className":80461,"style":22611},[763],[224,80463],{},[224,80465],{"className":80466},[549,8479],[224,80468,1083],{"className":80469},[468],[224,80471,1435],{"className":80472},[468,469],[224,80474],{"className":80475,"style":474},[473],[224,80477,5268],{"className":80478},[478],[224,80480],{"className":80481,"style":474},[473],[224,80483,80485,80488],{"className":80484},[459],[224,80486],{"className":80487,"style":535},[463],[224,80489,54602],{"className":80490},[468],"；JS 的 ",[224,80493,80495,80530],{"className":80494,"translate":388},[392],[224,80496,80498],{"className":80497},[396],[398,80499,80500],{"xmlns":400},[403,80501,80502,80528],{},[406,80503,80504,80506,80508,80510,80512,80514,80516,80518,80520,80522,80524,80526],{},[409,80505,1435],{},[409,80507,1705],{},[412,80509,239],{"stretchy":435},[419,80511,1438],{},[409,80513,12413],{"mathvariant":678},[409,80515,1435],{},[412,80517,1716],{"stretchy":435},[412,80519,278],{},[409,80521,1705],{},[412,80523,239],{"stretchy":435},[409,80525,1435],{},[412,80527,1716],{"stretchy":435},[447,80529,43507],{"encoding":449},[224,80531,80533,80566],{"className":80532,"ariaHidden":455},[454],[224,80534,80536,80539,80542,80545,80548,80551,80554,80557,80560,80563],{"className":80535},[459],[224,80537],{"className":80538,"style":535},[463],[224,80540,1435],{"className":80541},[468,469],[224,80543,1705],{"className":80544,"style":1740},[468,469],[224,80546,239],{"className":80547},[542],[224,80549,28555],{"className":80550},[468],[224,80552,1435],{"className":80553},[468,469],[224,80555,1716],{"className":80556},[549],[224,80558],{"className":80559,"style":496},[473],[224,80561,278],{"className":80562},[500],[224,80564],{"className":80565,"style":496},[473],[224,80567,80569,80572,80575,80578,80581],{"className":80568},[459],[224,80570],{"className":80571,"style":535},[463],[224,80573,1705],{"className":80574,"style":1740},[468,469],[224,80576,239],{"className":80577},[542],[224,80579,1435],{"className":80580},[468,469],[224,80582,1716],{"className":80583},[549]," 也已验算；KL 显然不满足，故不对称。",[613,80586,80588],{"id":80587},"_44-变分表示","4.4 变分表示",[21055,80590,80592,80683,80909,80912,81288],{"title":80591},"定理 4.1（变分表示 \u002F Fenchel 对偶）",[11,80593,20606,80594,80653,80654,80682],{},[224,80595,80597,80615],{"className":80596,"translate":388},[392],[224,80598,80600],{"className":80599},[396],[398,80601,80602],{"xmlns":400},[403,80603,80604,80612],{},[406,80605,80606],{},[3248,80607,80608,80610],{},[409,80609,1705],{},[412,80611,27543],{},[447,80613,80614],{"encoding":449},"f^*",[224,80616,80618],{"className":80617,"ariaHidden":455},[454],[224,80619,80621,80624],{"className":80620},[459],[224,80622],{"className":80623,"style":2565},[463],[224,80625,80627,80630],{"className":80626},[468],[224,80628,1705],{"className":80629,"style":1740},[468,469],[224,80631,80633],{"className":80632},[750],[224,80634,80636],{"className":80635},[754],[224,80637,80639],{"className":80638},[759],[224,80640,80642],{"className":80641,"style":28465},[763],[224,80643,80644,80647],{"style":3288},[224,80645],{"className":80646,"style":772},[771],[224,80648,80650],{"className":80649},[776,777,778,779],[224,80651,27543],{"className":80652},[478,779]," 为 ",[224,80655,80657,80670],{"className":80656,"translate":388},[392],[224,80658,80660],{"className":80659},[396],[398,80661,80662],{"xmlns":400},[403,80663,80664,80668],{},[406,80665,80666],{},[409,80667,1705],{},[447,80669,1705],{"encoding":449},[224,80671,80673],{"className":80672,"ariaHidden":455},[454],[224,80674,80676,80679],{"className":80675},[459],[224,80677],{"className":80678,"style":2565},[463],[224,80680,1705],{"className":80681,"style":1740},[468,469]," 的凸共轭（Fenchel 共轭）",[224,80684,80686],{"className":80685,"translate":388},[387],[224,80687,80689,80751],{"className":80688,"translate":388},[392],[224,80690,80692],{"className":80691},[396],[398,80693,80694],{"xmlns":400,"display":401},[403,80695,80696,80748],{},[406,80697,80698,80704,80706,80708,80710,80712,80728,80730,80732,80734,80736,80738,80740,80742,80744,80746],{},[3248,80699,80700,80702],{},[409,80701,1705],{},[412,80703,27543],{},[412,80705,239],{"stretchy":435},[409,80707,375],{},[412,80709,1716],{"stretchy":435},[412,80711,278],{},[1965,80713,80714,80720],{},[406,80715,80716,80718],{},[409,80717,55354],{},[412,80719,24134],{},[406,80721,80722,80724,80726],{},[409,80723,1435],{},[412,80725,9766],{},[419,80727,284],{},[412,80729,8110],{"stretchy":435},[409,80731,375],{},[409,80733,1435],{},[412,80735,5268],{},[409,80737,1705],{},[412,80739,239],{"stretchy":435},[409,80741,1435],{},[412,80743,1716],{"stretchy":435},[412,80745,15077],{"stretchy":435},[412,80747,1877],{"separator":455},[447,80749,80750],{"encoding":449},"f^*(y)=\\sup_{t\\ge0}\\{yt-f(t)\\},",[224,80752,80754,80807,80887],{"className":80753,"ariaHidden":455},[454],[224,80755,80757,80760,80789,80792,80795,80798,80801,80804],{"className":80756},[459],[224,80758],{"className":80759,"style":535},[463],[224,80761,80763,80766],{"className":80762},[468],[224,80764,1705],{"className":80765,"style":1740},[468,469],[224,80767,80769],{"className":80768},[750],[224,80770,80772],{"className":80771},[754],[224,80773,80775],{"className":80774},[759],[224,80776,80778],{"className":80777,"style":28217},[763],[224,80779,80780,80783],{"style":5074},[224,80781],{"className":80782,"style":772},[771],[224,80784,80786],{"className":80785},[776,777,778,779],[224,80787,27543],{"className":80788},[478,779],[224,80790,239],{"className":80791},[542],[224,80793,375],{"className":80794,"style":492},[468,469],[224,80796,1716],{"className":80797},[549],[224,80799],{"className":80800,"style":496},[473],[224,80802,278],{"className":80803},[500],[224,80805],{"className":80806,"style":496},[473],[224,80808,80810,80814,80869,80872,80875,80878,80881,80884],{"className":80809},[459],[224,80811],{"className":80812,"style":80813},[463],"height:1.7567em;vertical-align:-1.0067em;",[224,80815,80817],{"className":80816},[21531,33665],[224,80818,80820,80860],{"className":80819},[754,755],[224,80821,80823,80857],{"className":80822},[759],[224,80824,80826,80847],{"className":80825,"style":839},[763],[224,80827,80829,80832],{"style":80828},"top:-2.1885em;margin-left:0em;",[224,80830],{"className":80831,"style":2108},[771],[224,80833,80835],{"className":80834},[776,777,778,779],[224,80836,80838,80841,80844],{"className":80837},[468,779],[224,80839,1435],{"className":80840},[468,469,779],[224,80842,9766],{"className":80843},[500,779],[224,80845,284],{"className":80846},[468,779],[224,80848,80849,80852],{"style":2120},[224,80850],{"className":80851,"style":2108},[771],[224,80853,80854],{},[224,80855,55354],{"className":80856},[21531],[224,80858,787],{"className":80859},[786],[224,80861,80863],{"className":80862},[759],[224,80864,80867],{"className":80865,"style":80866},[763],"height:1.0067em;",[224,80868],{},[224,80870,8110],{"className":80871},[542],[224,80873,375],{"className":80874,"style":492},[468,469],[224,80876,1435],{"className":80877},[468,469],[224,80879],{"className":80880,"style":474},[473],[224,80882,5268],{"className":80883},[478],[224,80885],{"className":80886,"style":474},[473],[224,80888,80890,80893,80896,80899,80902,80906],{"className":80889},[459],[224,80891],{"className":80892,"style":535},[463],[224,80894,1705],{"className":80895,"style":1740},[468,469],[224,80897,239],{"className":80898},[542],[224,80900,1435],{"className":80901},[468,469],[224,80903,80905],{"className":80904},[549],")}",[224,80907,1877],{"className":80908},[1917],[11,80910,80911],{},"则",[224,80913,80915],{"className":80914,"translate":388},[387],[224,80916,80918,81002],{"className":80917,"translate":388},[392],[224,80919,80921],{"className":80920},[396],[398,80922,80923],{"xmlns":400,"display":401},[403,80924,80925,80999],{},[406,80926,80927,80933,80935,80937,80939,80941,80943,80945,80955,80997],{},[695,80928,80929,80931],{},[409,80930,20303],{},[409,80932,1705],{},[412,80934,239],{"stretchy":435},[409,80936,16018],{},[409,80938,20312],{"mathvariant":678},[409,80940,20315],{},[412,80942,1716],{"stretchy":435},[412,80944,278],{},[1965,80946,80947,80953],{},[406,80948,80949,80951],{},[409,80950,55354],{},[412,80952,24134],{},[409,80954,24189],{},[406,80956,80957,80959,80965,80967,80969,80971,80973,80979,80981,80987,80989,80991,80993,80995],{},[412,80958,8110],{"fence":455},[695,80960,80961,80963],{},[409,80962,20324],{"mathvariant":12251},[409,80964,16018],{},[412,80966,268],{"stretchy":435},[409,80968,24189],{},[412,80970,440],{"stretchy":435},[412,80972,5268],{},[695,80974,80975,80977],{},[409,80976,20324],{"mathvariant":12251},[409,80978,20315],{},[412,80980,268],{"stretchy":435},[3248,80982,80983,80985],{},[409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E_P[g]\n-\\mathbb 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代入并对 ",[224,81832,81834,81847],{"className":81833,"translate":388},[392],[224,81835,81837],{"className":81836},[396],[398,81838,81839],{"xmlns":400},[403,81840,81841,81845],{},[406,81842,81843],{},[409,81844,20315],{},[447,81846,20315],{"encoding":449},[224,81848,81850],{"className":81849,"ariaHidden":455},[454],[224,81851,81853,81856],{"className":81852},[459],[224,81854],{"className":81855,"style":20637},[463],[224,81857,20315],{"className":81858},[468,469]," 积分：",[224,81861,81863],{"className":81862,"translate":388},[387],[224,81864,81866,82108],{"className":81865,"translate":388},[392],[224,81867,81869],{"className":81868},[396],[398,81870,81871],{"xmlns":400,"display":401},[403,81872,81873,82105],{},[5209,81874,81875,81961,82035],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,81876,81877,81899],{},[5218,81878,81879],{},[5221,81880,81881],{"scriptlevel":284,"displaystyle":455},[406,81882,81883,81889,81891,81893,81895,81897],{},[695,81884,81885,81887],{},[409,81886,20303],{},[409,81888,1705],{},[412,81890,239],{"stretchy":435},[409,81892,16018],{},[409,81894,20312],{"mathvariant":678},[409,81896,20315],{},[412,81898,1716],{"stretchy":435},[5218,81900,81901],{},[5221,81902,81903],{"scriptlevel":284,"displaystyle":455},[406,81904,81905,81907,81909,81911,81921,81923,81925,81927,81941,81943,81949,81951,81953,81955,81957,81959],{},[406,81906],{},[412,81908,278],{},[412,81910,21338],{},[1965,81912,81913,81919],{},[406,81914,81915,81917],{},[409,81916,55354],{},[412,81918,24134],{},[409,81920,375],{},[412,81922,8110],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[409,81924,375],{},[423,81926,682],{},[20338,81928,81929,81935],{},[406,81930,81931,81933],{},[409,81932,13182],{},[409,81934,16018],{},[406,81936,81937,81939],{},[409,81938,13182],{},[409,81940,20315],{},[412,81942,5268],{},[3248,81944,81945,81947],{},[409,81946,1705],{},[412,81948,27543],{},[412,81950,239],{"stretchy":435},[409,81952,375],{},[412,81954,1716],{"stretchy":435},[412,81956,15077],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[409,81958,13182],{},[409,81960,20315],{},[5215,81962,81963,81969],{},[5218,81964,81965],{},[5221,81966,81967],{"scriptlevel":284,"displaystyle":455},[406,81968],{},[5218,81970,81971],{},[5221,81972,81973],{"scriptlevel":284,"displaystyle":455},[406,81974,81975,81977,81979,81981,81983,81993,81995,81997,81999,82001,82015,82017,82023,82025,82027,82029,82031,82033],{},[406,81976],{},[423,81978,15054],{},[412,81980,9766],{},[423,81982,15054],{},[1965,81984,81985,81991],{},[406,81986,81987,81989],{},[409,81988,55354],{},[412,81990,24134],{},[409,81992,375],{},[412,81994,21338],{},[412,81996,8110],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[409,81998,375],{},[423,82000,682],{},[20338,82002,82003,82009],{},[406,82004,82005,82007],{},[409,82006,13182],{},[409,82008,16018],{},[406,82010,82011,82013],{},[409,82012,13182],{},[409,82014,20315],{},[412,82016,5268],{},[3248,82018,82019,82021],{},[409,82020,1705],{},[412,82022,27543],{},[412,82024,239],{"stretchy":435},[409,82026,375],{},[412,82028,1716],{"stretchy":435},[412,82030,15077],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[409,82032,13182],{},[409,82034,20315],{},[5215,82036,82037,82043],{},[5218,82038,82039],{},[5221,82040,82041],{"scriptlevel":284,"displaystyle":455},[406,82042],{},[5218,82044,82045],{},[5221,82046,82047],{"scriptlevel":284,"displaystyle":455},[406,82048,82049,82051,82053,82063,82065,82071,82073,82075,82077,82079,82085,82087,82093,82095,82097,82099,82101,82103],{},[406,82050],{},[412,82052,278],{},[1965,82054,82055,82061],{},[406,82056,82057,82059],{},[409,82058,55354],{},[412,82060,24134],{},[409,82062,375],{},[412,82064,8110],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[695,82066,82067,82069],{},[409,82068,20324],{"mathvariant":12251},[409,82070,16018],{},[412,82072,268],{"stretchy":435},[409,82074,375],{},[412,82076,440],{"stretchy":435},[412,82078,5268],{},[695,82080,82081,82083],{},[409,82082,20324],{"mathvariant":12251},[409,82084,20315],{},[412,82086,268],{"stretchy":435},[3248,82088,82089,82091],{},[409,82090,1705],{},[412,82092,27543],{},[412,82094,239],{"stretchy":435},[409,82096,375],{},[412,82098,1716],{"stretchy":435},[412,82100,440],{"stretchy":435},[412,82102,15077],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,82104,688],{"mathvariant":678},[447,82106,82107],{"encoding":449},"\\begin{aligned}\nD_f(P\\|Q)\n&=\\int \\sup_y\\Big\\{y\\,\\frac{dP}{dQ}\n-f^*(y)\\Big\\}dQ\\\\\n&\\ \\ge\\ \\sup_y\\int\\Big\\{y\\,\\frac{dP}{dQ}\n-f^*(y)\\Big\\}dQ\\\\\n&=\\sup_y\\big\\{\\mathbb E_P[y]\n-\\mathbb 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",[224,82910,82912,82925],{"className":82911,"translate":388},[392],[224,82913,82915],{"className":82914},[396],[398,82916,82917],{"xmlns":400},[403,82918,82919,82923],{},[406,82920,82921],{},[412,82922,9766],{},[447,82924,75395],{"encoding":449},[224,82926,82928],{"className":82927,"ariaHidden":455},[454],[224,82929,82931,82934],{"className":82930},[459],[224,82932],{"className":82933,"style":3204},[463],[224,82935,9766],{"className":82936},[500],"，因为「积分的上确界 ≥ 上确界的积分」。反向不等式需要一个可测选择\u002F极大值论证：由于 ",[224,82939,82941,82954],{"className":82940,"translate":388},[392],[224,82942,82944],{"className":82943},[396],[398,82945,82946],{"xmlns":400},[403,82947,82948,82952],{},[406,82949,82950],{},[409,82951,1705],{},[447,82953,1705],{"encoding":449},[224,82955,82957],{"className":82956,"ariaHidden":455},[454],[224,82958,82960,82963],{"className":82959},[459],[224,82961],{"className":82962,"style":2565},[463],[224,82964,1705],{"className":82965,"style":1740},[468,469]," 凸，",[224,82968,82970,83024],{"className":82969,"translate":388},[392],[224,82971,82973],{"className":82972},[396],[398,82974,82975],{"xmlns":400},[403,82976,82977,83021],{},[406,82978,82979,82981,82983,82985,82987,82989,82999,83001,83003,83005,83007,83013,83015,83017,83019],{},[409,82980,1705],{},[412,82982,239],{"stretchy":435},[409,82984,1435],{},[412,82986,1716],{"stretchy":435},[412,82988,278],{},[695,82990,82991,82997],{},[406,82992,82993,82995],{},[409,82994,55354],{},[412,82996,24134],{},[409,82998,375],{},[412,83000,8110],{"stretchy":435},[409,83002,375],{},[409,83004,1435],{},[412,83006,5268],{},[3248,83008,83009,83011],{},[409,83010,1705],{},[412,83012,27543],{},[412,83014,239],{"stretchy":435},[409,83016,375],{},[412,83018,1716],{"stretchy":435},[412,83020,15077],{"stretchy":435},[447,83022,83023],{"encoding":449},"f(t)=\\sup_y\\{yt-f^*(y)\\}",[224,83025,83027,83054,83118],{"className":83026,"ariaHidden":455},[454],[224,83028,83030,83033,83036,83039,83042,83045,83048,83051],{"className":83029},[459],[224,83031],{"className":83032,"style":535},[463],[224,83034,1705],{"className":83035,"style":1740},[468,469],[224,83037,239],{"className":83038},[542],[224,83040,1435],{"className":83041},[468,469],[224,83043,1716],{"className":83044},[549],[224,83046],{"className":83047,"style":496},[473],[224,83049,278],{"className":83050},[500],[224,83052],{"className":83053,"style":496},[473],[224,83055,83057,83060,83100,83103,83106,83109,83112,83115],{"className":83056},[459],[224,83058],{"className":83059,"style":77229},[463],[224,83061,83063,83066],{"className":83062},[21531],[224,83064,55354],{"className":83065},[21531],[224,83067,83069],{"className":83068},[750],[224,83070,83072,83092],{"className":83071},[754,755],[224,83073,83075,83089],{"className":83074},[759],[224,83076,83078],{"className":83077,"style":56921},[763],[224,83079,83080,83083],{"style":55500},[224,83081],{"className":83082,"style":772},[771],[224,83084,83086],{"className":83085},[776,777,778,779],[224,83087,375],{"className":83088,"style":492},[468,469,779],[224,83090,787],{"className":83091},[786],[224,83093,83095],{"className":83094},[759],[224,83096,83098],{"className":83097,"style":77268},[763],[224,83099],{},[224,83101,8110],{"className":83102},[542],[224,83104,375],{"className":83105,"style":492},[468,469],[224,83107,1435],{"className":83108},[468,469],[224,83110],{"className":83111,"style":474},[473],[224,83113,5268],{"className":83114},[478],[224,83116],{"className":83117,"style":474},[473],[224,83119,83121,83124,83153,83156,83159],{"className":83120},[459],[224,83122],{"className":83123,"style":535},[463],[224,83125,83127,83130],{"className":83126},[468],[224,83128,1705],{"className":83129,"style":1740},[468,469],[224,83131,83133],{"className":83132},[750],[224,83134,83136],{"className":83135},[754],[224,83137,83139],{"className":83138},[759],[224,83140,83142],{"className":83141,"style":28465},[763],[224,83143,83144,83147],{"style":3288},[224,83145],{"className":83146,"style":772},[771],[224,83148,83150],{"className":83149},[776,777,778,779],[224,83151,27543],{"className":83152},[478,779],[224,83154,239],{"className":83155},[542],[224,83157,375],{"className":83158,"style":492},[468,469],[224,83160,80905],{"className":83161},[549]," 中的 ",[224,83164,83166,83179],{"className":83165,"translate":388},[392],[224,83167,83169],{"className":83168},[396],[398,83170,83171],{"xmlns":400},[403,83172,83173,83177],{},[406,83174,83175],{},[409,83176,375],{},[447,83178,375],{"encoding":449},[224,83180,83182],{"className":83181,"ariaHidden":455},[454],[224,83183,83185,83188],{"className":83184},[459],[224,83186],{"className":83187,"style":806},[463],[224,83189,375],{"className":83190,"style":492},[468,469]," 可取为 ",[224,83193,83195,83219],{"className":83194,"translate":388},[392],[224,83196,83198],{"className":83197},[396],[398,83199,83200],{"xmlns":400},[403,83201,83202,83216],{},[406,83203,83204,83210,83212,83214],{},[3248,83205,83206,83208],{},[409,83207,1705],{},[412,83209,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,83211,239],{"stretchy":435},[409,83213,1435],{},[412,83215,1716],{"stretchy":435},[447,83217,83218],{"encoding":449},"f'(t)",[224,83220,83222],{"className":83221,"ariaHidden":455},[454],[224,83223,83225,83228,83260,83263,83266],{"className":83224},[459],[224,83226],{"className":83227,"style":6966},[463],[224,83229,83231,83234],{"className":83230},[468],[224,83232,1705],{"className":83233,"style":1740},[468,469],[224,83235,83237],{"className":83236},[750],[224,83238,83240],{"className":83239},[754],[224,83241,83243],{"className":83242},[759],[224,83244,83246],{"className":83245,"style":6992},[763],[224,83247,83248,83251],{"style":3288},[224,83249],{"className":83250,"style":772},[771],[224,83252,83254],{"className":83253},[776,777,778,779],[224,83255,83257],{"className":83256},[468,779],[224,83258,6933],{"className":83259},[468,779],[224,83261,239],{"className":83262},[542],[224,83264,1435],{"className":83265},[468,469],[224,83267,1716],{"className":83268},[549],"（在可微点），而 ",[224,83271,83273,83301],{"className":83272,"translate":388},[392],[224,83274,83276],{"className":83275},[396],[398,83277,83278],{"xmlns":400},[403,83279,83280,83298],{},[406,83281,83282,83284,83286,83292,83294,83296],{},[409,83283,1435],{},[412,83285,28649],{},[3248,83287,83288,83290],{},[409,83289,1705],{},[412,83291,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,83293,239],{"stretchy":435},[409,83295,1435],{},[412,83297,1716],{"stretchy":435},[447,83299,83300],{"encoding":449},"t\\mapsto f'(t)",[224,83302,83304,83322],{"className":83303,"ariaHidden":455},[454],[224,83305,83307,83310,83313,83316,83319],{"className":83306},[459],[224,83308],{"className":83309,"style":28676},[463],[224,83311,1435],{"className":83312},[468,469],[224,83314],{"className":83315,"style":496},[473],[224,83317,28649],{"className":83318},[500],[224,83320],{"className":83321,"style":496},[473],[224,83323,83325,83328,83360,83363,83366],{"className":83324},[459],[224,83326],{"className":83327,"style":6966},[463],[224,83329,83331,83334],{"className":83330},[468],[224,83332,1705],{"className":83333,"style":1740},[468,469],[224,83335,83337],{"className":83336},[750],[224,83338,83340],{"className":83339},[754],[224,83341,83343],{"className":83342},[759],[224,83344,83346],{"className":83345,"style":6992},[763],[224,83347,83348,83351],{"style":3288},[224,83349],{"className":83350,"style":772},[771],[224,83352,83354],{"className":83353},[776,777,778,779],[224,83355,83357],{"className":83356},[468,779],[224,83358,6933],{"className":83359},[468,779],[224,83361,239],{"className":83362},[542],[224,83364,1435],{"className":83365},[468,469],[224,83367,1716],{"className":83368},[549]," 单调，于是 ",[224,83371,83373,83419],{"className":83372,"translate":388},[392],[224,83374,83376],{"className":83375},[396],[398,83377,83378],{"xmlns":400},[403,83379,83380,83416],{},[406,83381,83382,83388,83390,83392,83394,83396,83398,83404,83406,83408,83410,83412,83414],{},[3248,83383,83384,83386],{},[409,83385,24189],{},[412,83387,27543],{},[412,83389,239],{"stretchy":435},[409,83391,371],{},[412,83393,1716],{"stretchy":435},[412,83395,1071],{},[412,83397,278],{},[3248,83399,83400,83402],{},[409,83401,1705],{},[412,83403,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,83405,239],{"stretchy":435},[409,83407,1435],{},[412,83409,239],{"stretchy":435},[409,83411,371],{},[412,83413,1716],{"stretchy":435},[412,83415,1716],{"stretchy":435},[447,83417,83418],{"encoding":449},"g^*(x):=f'(t(x))",[224,83420,83422,83476],{"className":83421,"ariaHidden":455},[454],[224,83423,83425,83428,83457,83460,83463,83466,83469,83473],{"className":83424},[459],[224,83426],{"className":83427,"style":535},[463],[224,83429,83431,83434],{"className":83430},[468],[224,83432,24189],{"className":83433,"style":492},[468,469],[224,83435,83437],{"className":83436},[750],[224,83438,83440],{"className":83439},[754],[224,83441,83443],{"className":83442},[759],[224,83444,83446],{"className":83445,"style":28465},[763],[224,83447,83448,83451],{"style":3288},[224,83449],{"className":83450,"style":772},[771],[224,83452,83454],{"className":83453},[776,777,778,779],[224,83455,27543],{"className":83456},[478,779],[224,83458,239],{"className":83459},[542],[224,83461,371],{"className":83462},[468,469],[224,83464,1716],{"className":83465},[549],[224,83467],{"className":83468,"style":496},[473],[224,83470,83472],{"className":83471},[500],":=",[224,83474],{"className":83475,"style":496},[473],[224,83477,83479,83482,83514,83517,83520,83523,83526],{"className":83478},[459],[224,83480],{"className":83481,"style":6966},[463],[224,83483,83485,83488],{"className":83484},[468],[224,83486,1705],{"className":83487,"style":1740},[468,469],[224,83489,83491],{"className":83490},[750],[224,83492,83494],{"className":83493},[754],[224,83495,83497],{"className":83496},[759],[224,83498,83500],{"className":83499,"style":6992},[763],[224,83501,83502,83505],{"style":3288},[224,83503],{"className":83504,"style":772},[771],[224,83506,83508],{"className":83507},[776,777,778,779],[224,83509,83511],{"className":83510},[468,779],[224,83512,6933],{"className":83513},[468,779],[224,83515,239],{"className":83516},[542],[224,83518,1435],{"className":83519},[468,469],[224,83521,239],{"className":83522},[542],[224,83524,371],{"className":83525},[468,469],[224,83527,38989],{"className":83528},[549]," 是可测的，并且对每个 ",[224,83531,83533,83546],{"className":83532,"translate":388},[392],[224,83534,83536],{"className":83535},[396],[398,83537,83538],{"xmlns":400},[403,83539,83540,83544],{},[406,83541,83542],{},[409,83543,371],{},[447,83545,371],{"encoding":449},[224,83547,83549],{"className":83548,"ariaHidden":455},[454],[224,83550,83552,83555],{"className":83551},[459],[224,83553],{"className":83554,"style":839},[463],[224,83556,371],{"className":83557},[468,469]," 达到上确界。于是不等式取等号。",[11,83560,83561,83564],{},[378,83562,83563],{},"最优解。"," 达到上确界的函数为",[224,83566,83568],{"className":83567,"translate":388},[387],[224,83569,83571,83633],{"className":83570,"translate":388},[392],[224,83572,83574],{"className":83573},[396],[398,83575,83576],{"xmlns":400,"display":401},[403,83577,83578,83630],{},[406,83579,83580,83586,83588,83590,83592,83594,83600,83602,83628],{},[3248,83581,83582,83584],{},[409,83583,24189],{},[412,83585,27543],{},[412,83587,239],{"stretchy":435},[409,83589,371],{},[412,83591,1716],{"stretchy":435},[412,83593,278],{},[3248,83595,83596,83598],{},[409,83597,1705],{},[412,83599,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[423,83601,21353],{},[406,83603,83604,83606,83620,83622,83624,83626],{},[412,83605,239],{"fence":455},[20338,83607,83608,83614],{},[406,83609,83610,83612],{},[409,83611,13182],{},[409,83613,16018],{},[406,83615,83616,83618],{},[409,83617,13182],{},[409,83619,20315],{},[412,83621,239],{"stretchy":435},[409,83623,371],{},[412,83625,1716],{"stretchy":435},[412,83627,1716],{"fence":455},[412,83629,1877],{"separator":455},[447,83631,83632],{"encoding":449},"g^*(x)=f'\\!\\left(\\frac{dP}{dQ}(x)\\right),",[224,83634,83636,83689],{"className":83635,"ariaHidden":455},[454],[224,83637,83639,83642,83671,83674,83677,83680,83683,83686],{"className":83638},[459],[224,83640],{"className":83641,"style":535},[463],[224,83643,83645,83648],{"className":83644},[468],[224,83646,24189],{"className":83647,"style":492},[468,469],[224,83649,83651],{"className":83650},[750],[224,83652,83654],{"className":83653},[754],[224,83655,83657],{"className":83656},[759],[224,83658,83660],{"className":83659,"style":28217},[763],[224,83661,83662,83665],{"style":5074},[224,83663],{"className":83664,"style":772},[771],[224,83666,83668],{"className":83667},[776,777,778,779],[224,83669,27543],{"className":83670},[478,779],[224,83672,239],{"className":83673},[542],[224,83675,371],{"className":83676},[468,469],[224,83678,1716],{"className":83679},[549],[224,83681],{"className":83682,"style":496},[473],[224,83684,278],{"className":83685},[500],[224,83687],{"className":83688,"style":496},[473],[224,83690,83692,83695,83727,83730,83733,83825,83828],{"className":83691},[459],[224,83693],{"className":83694,"style":20442},[463],[224,83696,83698,83701],{"className":83697},[468],[224,83699,1705],{"className":83700,"style":1740},[468,469],[224,83702,83704],{"className":83703},[750],[224,83705,83707],{"className":83706},[754],[224,83708,83710],{"className":83709},[759],[224,83711,83713],{"className":83712,"style":48901},[763],[224,83714,83715,83718],{"style":5074},[224,83716],{"className":83717,"style":772},[771],[224,83719,83721],{"className":83720},[776,777,778,779],[224,83722,83724],{"className":83723},[468,779],[224,83725,6933],{"className":83726},[468,779],[224,83728],{"className":83729,"style":21559},[473],[224,83731],{"className":83732,"style":724},[473],[224,83734,83736,83742,83810,83813,83816,83819],{"className":83735},[2091],[224,83737,83739],{"className":83738,"style":8281},[542,8280],[224,83740,239],{"className":83741},[2363,778],[224,83743,83745,83748,83807],{"className":83744},[468],[224,83746],{"className":83747},[542,8479],[224,83749,83751],{"className":83750},[20338],[224,83752,83754,83799],{"className":83753},[754,755],[224,83755,83757,83796],{"className":83756},[759],[224,83758,83760,83774,83782],{"className":83759,"style":20528},[763],[224,83761,83762,83765],{"style":20531},[224,83763],{"className":83764,"style":2108},[771],[224,83766,83768,83771],{"className":83767},[468],[224,83769,13182],{"className":83770},[468,469],[224,83772,20315],{"className":83773},[468,469],[224,83775,83776,83779],{"style":20546},[224,83777],{"className":83778,"style":2108},[771],[224,83780],{"className":83781,"style":20554},[20553],[224,83783,83784,83787],{"style":20557},[224,83785],{"className":83786,"style":2108},[771],[224,83788,83790,83793],{"className":83789},[468],[224,83791,13182],{"className":83792},[468,469],[224,83794,16018],{"className":83795,"style":653},[468,469],[224,83797,787],{"className":83798},[786],[224,83800,83802],{"className":83801},[759],[224,83803,83805],{"className":83804,"style":20579},[763],[224,83806],{},[224,83808],{"className":83809},[549,8479],[224,83811,239],{"className":83812},[542],[224,83814,371],{"className":83815},[468,469],[224,83817,1716],{"className":83818},[549],[224,83820,83822],{"className":83821,"style":8281},[549,8280],[224,83823,1716],{"className":83824},[2363,778],[224,83826],{"className":83827,"style":724},[473],[224,83829,1877],{"className":83830},[1917],[11,83832,83833,83834,23884,83862,83890,83891,83951,83952,84024,84025,65519,84144,84328],{},"条件是 ",[224,83835,83837,83850],{"className":83836,"translate":388},[392],[224,83838,83840],{"className":83839},[396],[398,83841,83842],{"xmlns":400},[403,83843,83844,83848],{},[406,83845,83846],{},[409,83847,1705],{},[447,83849,1705],{"encoding":449},[224,83851,83853],{"className":83852,"ariaHidden":455},[454],[224,83854,83856,83859],{"className":83855},[459],[224,83857],{"className":83858,"style":2565},[463],[224,83860,1705],{"className":83861,"style":1740},[468,469],[224,83863,83865,83878],{"className":83864,"translate":388},[392],[224,83866,83868],{"className":83867},[396],[398,83869,83870],{"xmlns":400},[403,83871,83872,83876],{},[406,83873,83874],{},[409,83875,1435],{},[447,83877,1435],{"encoding":449},[224,83879,83881],{"className":83880,"ariaHidden":455},[454],[224,83882,83884,83887],{"className":83883},[459],[224,83885],{"className":83886,"style":25353},[463],[224,83888,1435],{"className":83889},[468,469]," 处可微、且 ",[224,83892,83894,83912],{"className":83893,"translate":388},[392],[224,83895,83897],{"className":83896},[396],[398,83898,83899],{"xmlns":400},[403,83900,83901,83909],{},[406,83902,83903],{},[3248,83904,83905,83907],{},[409,83906,24189],{},[412,83908,27543],{},[447,83910,83911],{"encoding":449},"g^*",[224,83913,83915],{"className":83914,"ariaHidden":455},[454],[224,83916,83918,83922],{"className":83917},[459],[224,83919],{"className":83920,"style":83921},[463],"height:0.8831em;vertical-align:-0.1944em;",[224,83923,83925,83928],{"className":83924},[468],[224,83926,24189],{"className":83927,"style":492},[468,469],[224,83929,83931],{"className":83930},[750],[224,83932,83934],{"className":83933},[754],[224,83935,83937],{"className":83936},[759],[224,83938,83940],{"className":83939,"style":28465},[763],[224,83941,83942,83945],{"style":3288},[224,83943],{"className":83944,"style":772},[771],[224,83946,83948],{"className":83947},[776,777,778,779],[224,83949,27543],{"className":83950},[478,779]," 落在 ",[224,83953,83955,83977],{"className":83954,"translate":388},[392],[224,83956,83958],{"className":83957},[396],[398,83959,83960],{"xmlns":400},[403,83961,83962,83974],{},[406,83963,83964,83966,83968],{},[409,83965,81313],{"mathvariant":678},[412,83967,24134],{},[3248,83969,83970,83972],{},[409,83971,1705],{},[412,83973,27543],{},[447,83975,83976],{"encoding":449},"\\operatorname{dom} f^*",[224,83978,83980],{"className":83979,"ariaHidden":455},[454],[224,83981,83983,83986,83992,83995],{"className":83982},[459],[224,83984],{"className":83985,"style":2565},[463],[224,83987,83989],{"className":83988},[21531],[224,83990,81313],{"className":83991},[468,1906],[224,83993],{"className":83994,"style":724},[473],[224,83996,83998,84001],{"className":83997},[468],[224,83999,1705],{"className":84000,"style":1740},[468,469],[224,84002,84004],{"className":84003},[750],[224,84005,84007],{"className":84006},[754],[224,84008,84010],{"className":84009},[759],[224,84011,84013],{"className":84012,"style":28465},[763],[224,84014,84015,84018],{"style":3288},[224,84016],{"className":84017,"style":772},[771],[224,84019,84021],{"className":84020},[776,777,778,779],[224,84022,27543],{"className":84023},[478,779]," 内。例如 KL（",[224,84026,84028,84058],{"className":84027,"translate":388},[392],[224,84029,84031],{"className":84030},[396],[398,84032,84033],{"xmlns":400},[403,84034,84035,84055],{},[406,84036,84037,84043,84045,84047,84049,84051,84053],{},[3248,84038,84039,84041],{},[409,84040,1705],{},[412,84042,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,84044,278],{},[409,84046,24131],{},[412,84048,24134],{},[409,84050,1435],{},[412,84052,319],{},[419,84054,1438],{},[447,84056,84057],{"encoding":449},"f'=\\log t+1",[224,84059,84061,84109,84135],{"className":84060,"ariaHidden":455},[454],[224,84062,84064,84068,84100,84103,84106],{"className":84063},[459],[224,84065],{"className":84066,"style":84067},[463],"height:0.9463em;vertical-align:-0.1944em;",[224,84069,84071,84074],{"className":84070},[468],[224,84072,1705],{"className":84073,"style":1740},[468,469],[224,84075,84077],{"className":84076},[750],[224,84078,84080],{"className":84079},[754],[224,84081,84083],{"className":84082},[759],[224,84084,84086],{"className":84085,"style":6992},[763],[224,84087,84088,84091],{"style":3288},[224,84089],{"className":84090,"style":772},[771],[224,84092,84094],{"className":84093},[776,777,778,779],[224,84095,84097],{"className":84096},[468,779],[224,84098,6933],{"className":84099},[468,779],[224,84101],{"className":84102,"style":496},[473],[224,84104,278],{"className":84105},[500],[224,84107],{"className":84108,"style":496},[473],[224,84110,84112,84115,84120,84123,84126,84129,84132],{"className":84111},[459],[224,84113],{"className":84114,"style":2565},[463],[224,84116,24185,84118],{"className":84117},[21531],[224,84119,24189],{"style":24188},[224,84121],{"className":84122,"style":724},[473],[224,84124,1435],{"className":84125},[468,469],[224,84127],{"className":84128,"style":474},[473],[224,84130,319],{"className":84131},[478],[224,84133],{"className":84134,"style":474},[473],[224,84136,84138,84141],{"className":84137},[459],[224,84139],{"className":84140,"style":510},[463],[224,84142,1438],{"className":84143},[468],[224,84145,84147,84181],{"className":84146,"translate":388},[392],[224,84148,84150],{"className":84149},[396],[398,84151,84152],{"xmlns":400},[403,84153,84154,84178],{},[406,84155,84156,84162,84164,84166,84168,84174,84176],{},[3248,84157,84158,84160],{},[409,84159,24189],{},[412,84161,27543],{},[412,84163,278],{},[409,84165,24131],{},[412,84167,24134],{},[20338,84169,84170,84172],{},[409,84171,11],{},[409,84173,2023],{},[412,84175,319],{},[419,84177,1438],{},[447,84179,84180],{"encoding":449},"g^*=\\log\\frac{p}{q}+1",[224,84182,84184,84228,84319],{"className":84183,"ariaHidden":455},[454],[224,84185,84187,84190,84219,84222,84225],{"className":84186},[459],[224,84188],{"className":84189,"style":83921},[463],[224,84191,84193,84196],{"className":84192},[468],[224,84194,24189],{"className":84195,"style":492},[468,469],[224,84197,84199],{"className":84198},[750],[224,84200,84202],{"className":84201},[754],[224,84203,84205],{"className":84204},[759],[224,84206,84208],{"className":84207,"style":28465},[763],[224,84209,84210,84213],{"style":3288},[224,84211],{"className":84212,"style":772},[771],[224,84214,84216],{"className":84215},[776,777,778,779],[224,84217,27543],{"className":84218},[478,779],[224,84220],{"className":84221,"style":496},[473],[224,84223,278],{"className":84224},[500],[224,84226],{"className":84227,"style":496},[473],[224,84229,84231,84234,84239,84242,84310,84313,84316],{"className":84230},[459],[224,84232],{"className":84233,"style":23117},[463],[224,84235,24185,84237],{"className":84236},[21531],[224,84238,24189],{"style":24188},[224,84240],{"className":84241,"style":724},[473],[224,84243,84245,84248,84307],{"className":84244},[468],[224,84246],{"className":84247},[542,8479],[224,84249,84251],{"className":84250},[20338],[224,84252,84254,84299],{"className":84253},[754,755],[224,84255,84257,84296],{"className":84256},[759],[224,84258,84260,84274,84282],{"className":84259,"style":23136},[763],[224,84261,84262,84265],{"style":22566},[224,84263],{"className":84264,"style":2108},[771],[224,84266,84268],{"className":84267},[776,777,778,779],[224,84269,84271],{"className":84270},[468,779],[224,84272,2023],{"className":84273,"style":492},[468,469,779],[224,84275,84276,84279],{"style":20546},[224,84277],{"className":84278,"style":2108},[771],[224,84280],{"className":84281,"style":20554},[20553],[224,84283,84284,84287],{"style":22889},[224,84285],{"className":84286,"style":2108},[771],[224,84288,84290],{"className":84289},[776,777,778,779],[224,84291,84293],{"className":84292},[468,779],[224,84294,11],{"className":84295},[468,469,779],[224,84297,787],{"className":84298},[786],[224,84300,84302],{"className":84301},[759],[224,84303,84305],{"className":84304,"style":22776},[763],[224,84306],{},[224,84308],{"className":84309},[549,8479],[224,84311],{"className":84312,"style":474},[473],[224,84314,319],{"className":84315},[478],[224,84317],{"className":84318,"style":474},[473],[224,84320,84322,84325],{"className":84321},[459],[224,84323],{"className":84324,"style":510},[463],[224,84326,1438],{"className":84327},[468],"；这解释了为什么「对判别器做 log 密度比」是 f-GAN 的最佳策略。",[11,84330,84331,84393,84394,84493,84494,65519],{},[378,84332,84333,84334,84392],{},"常用 ",[224,84335,84337,84354],{"className":84336,"translate":388},[392],[224,84338,84340],{"className":84339},[396],[398,84341,84342],{"xmlns":400},[403,84343,84344,84352],{},[406,84345,84346],{},[3248,84347,84348,84350],{},[409,84349,1705],{},[412,84351,27543],{},[447,84353,80614],{"encoding":449},[224,84355,84357],{"className":84356,"ariaHidden":455},[454],[224,84358,84360,84363],{"className":84359},[459],[224,84361],{"className":84362,"style":2565},[463],[224,84364,84366,84369],{"className":84365},[468],[224,84367,1705],{"className":84368,"style":1740},[468,469],[224,84370,84372],{"className":84371},[750],[224,84373,84375],{"className":84374},[754],[224,84376,84378],{"className":84377},[759],[224,84379,84381],{"className":84380,"style":28465},[763],[224,84382,84383,84386],{"style":3288},[224,84384],{"className":84385,"style":772},[771],[224,84387,84389],{"className":84388},[776,777,778,779],[224,84390,27543],{"className":84391},[478,779]," 表","（结合 ",[224,84395,84397,84427],{"className":84396,"translate":388},[392],[224,84398,84400],{"className":84399},[396],[398,84401,84402],{"xmlns":400},[403,84403,84404,84424],{},[406,84405,84406,84408,84410,84412,84414,84416,84418,84420,84422],{},[412,84407,21338],{},[409,84409,13182],{},[409,84411,16018],{},[412,84413,278],{},[412,84415,21338],{},[409,84417,13182],{},[409,84419,20315],{},[412,84421,278],{},[419,84423,1438],{},[447,84425,84426],{"encoding":449},"\\int dP=\\int dQ=1",[224,84428,84430,84457,84484],{"className":84429,"ariaHidden":455},[454],[224,84431,84433,84436,84439,84442,84445,84448,84451,84454],{"className":84432},[459],[224,84434],{"className":84435,"style":25621},[463],[224,84437,21338],{"className":84438,"style":25626},[21531,21532,25625],[224,84440],{"className":84441,"style":724},[473],[224,84443,13182],{"className":84444},[468,469],[224,84446,16018],{"className":84447,"style":653},[468,469],[224,84449],{"className":84450,"style":496},[473],[224,84452,278],{"className":84453},[500],[224,84455],{"className":84456,"style":496},[473],[224,84458,84460,84463,84466,84469,84472,84475,84478,84481],{"className":84459},[459],[224,84461],{"className":84462,"style":25621},[463],[224,84464,21338],{"className":84465,"style":25626},[21531,21532,25625],[224,84467],{"className":84468,"style":724},[473],[224,84470,13182],{"className":84471},[468,469],[224,84473,20315],{"className":84474},[468,469],[224,84476],{"className":84477,"style":496},[473],[224,84479,278],{"className":84480},[500],[224,84482],{"className":84483,"style":496},[473],[224,84485,84487,84490],{"className":84486},[459],[224,84488],{"className":84489,"style":510},[463],[224,84491,1438],{"className":84492},[468]," 可加任意常数 ",[224,84495,84497,84510],{"className":84496,"translate":388},[392],[224,84498,84500],{"className":84499},[396],[398,84501,84502],{"xmlns":400},[403,84503,84504,84508],{},[406,84505,84506],{},[409,84507,219],{},[447,84509,219],{"encoding":449},[224,84511,84513],{"className":84512,"ariaHidden":455},[454],[224,84514,84516,84519],{"className":84515},[459],[224,84517],{"className":84518,"style":839},[463],[224,84520,219],{"className":84521},[468,469],[72399,84523,84524,84725],{},[72402,84525,84526],{},[72405,84527,84528,84573,84649,84722],{},[72408,84529,84530],{},[224,84531,84533,84552],{"className":84532,"translate":388},[392],[224,84534,84536],{"className":84535},[396],[398,84537,84538],{"xmlns":400},[403,84539,84540,84550],{},[406,84541,84542,84544,84546,84548],{},[409,84543,1705],{},[412,84545,239],{"stretchy":435},[409,84547,1435],{},[412,84549,1716],{"stretchy":435},[447,84551,72435],{"encoding":449},[224,84553,84555],{"className":84554,"ariaHidden":455},[454],[224,84556,84558,84561,84564,84567,84570],{"className":84557},[459],[224,84559],{"className":84560,"style":535},[463],[224,84562,1705],{"className":84563,"style":1740},[468,469],[224,84565,239],{"className":84566},[542],[224,84568,1435],{"className":84569},[468,469],[224,84571,1716],{"className":84572},[549],[72408,84574,84575],{},[224,84576,84578,84602],{"className":84577,"translate":388},[392],[224,84579,84581],{"className":84580},[396],[398,84582,84583],{"xmlns":400},[403,84584,84585,84599],{},[406,84586,84587,84593,84595,84597],{},[3248,84588,84589,84591],{},[409,84590,1705],{},[412,84592,27543],{},[412,84594,239],{"stretchy":435},[409,84596,375],{},[412,84598,1716],{"stretchy":435},[447,84600,84601],{"encoding":449},"f^*(y)",[224,84603,84605],{"className":84604,"ariaHidden":455},[454],[224,84606,84608,84611,84640,84643,84646],{"className":84607},[459],[224,84609],{"className":84610,"style":535},[463],[224,84612,84614,84617],{"className":84613},[468],[224,84615,1705],{"className":84616,"style":1740},[468,469],[224,84618,84620],{"className":84619},[750],[224,84621,84623],{"className":84622},[754],[224,84624,84626],{"className":84625},[759],[224,84627,84629],{"className":84628,"style":28465},[763],[224,84630,84631,84634],{"style":3288},[224,84632],{"className":84633,"style":772},[771],[224,84635,84637],{"className":84636},[776,777,778,779],[224,84638,27543],{"className":84639},[478,779],[224,84641,239],{"className":84642},[542],[224,84644,375],{"className":84645,"style":492},[468,469],[224,84647,1716],{"className":84648},[549],[72408,84650,84651],{},[224,84652,84654,84675],{"className":84653,"translate":388},[392],[224,84655,84657],{"className":84656},[396],[398,84658,84659],{"xmlns":400},[403,84660,84661,84673],{},[406,84662,84663,84665,84667],{},[409,84664,81313],{"mathvariant":678},[412,84666,24134],{},[3248,84668,84669,84671],{},[409,84670,1705],{},[412,84672,27543],{},[447,84674,83976],{"encoding":449},[224,84676,84678],{"className":84677,"ariaHidden":455},[454],[224,84679,84681,84684,84690,84693],{"className":84680},[459],[224,84682],{"className":84683,"style":2565},[463],[224,84685,84687],{"className":84686},[21531],[224,84688,81313],{"className":84689},[468,1906],[224,84691],{"className":84692,"style":724},[473],[224,84694,84696,84699],{"className":84695},[468],[224,84697,1705],{"className":84698,"style":1740},[468,469],[224,84700,84702],{"className":84701},[750],[224,84703,84705],{"className":84704},[754],[224,84706,84708],{"className":84707},[759],[224,84709,84711],{"className":84710,"style":28465},[763],[224,84712,84713,84716],{"style":3288},[224,84714],{"className":84715,"style":772},[771],[224,84717,84719],{"className":84718},[776,777,778,779],[224,84720,27543],{"className":84721},[478,779],[72408,84723,84724],{},"对应 GAN 损失",[72497,84726,84727,84890,85131,85764,86068,86407],{},[72405,84728,84729,84780,84856,84887],{},[72502,84730,84731,84779],{},[224,84732,84734,84753],{"className":84733,"translate":388},[392],[224,84735,84737],{"className":84736},[396],[398,84738,84739],{"xmlns":400},[403,84740,84741,84751],{},[406,84742,84743,84745,84747,84749],{},[409,84744,1435],{},[409,84746,24131],{},[412,84748,24134],{},[409,84750,1435],{},[447,84752,72633],{"encoding":449},[224,84754,84756],{"className":84755,"ariaHidden":455},[454],[224,84757,84759,84762,84765,84768,84773,84776],{"className":84758},[459],[224,84760],{"className":84761,"style":2565},[463],[224,84763,1435],{"className":84764},[468,469],[224,84766],{"className":84767,"style":724},[473],[224,84769,24185,84771],{"className":84770},[21531],[224,84772,24189],{"style":24188},[224,84774],{"className":84775,"style":724},[473],[224,84777,1435],{"className":84778},[468,469],"（KL）",[72502,84781,84782],{},[224,84783,84785,84809],{"className":84784,"translate":388},[392],[224,84786,84788],{"className":84787},[396],[398,84789,84790],{"xmlns":400},[403,84791,84792,84806],{},[406,84793,84794],{},[3248,84795,84796,84798],{},[409,84797,1861],{},[406,84799,84800,84802,84804],{},[409,84801,375],{},[412,84803,5268],{},[419,84805,1438],{},[447,84807,84808],{"encoding":449},"e^{y-1}",[224,84810,84812],{"className":84811,"ariaHidden":455},[454],[224,84813,84815,84818],{"className":84814},[459],[224,84816],{"className":84817,"style":5671},[463],[224,84819,84821,84824],{"className":84820},[468],[224,84822,1861],{"className":84823},[468,469],[224,84825,84827],{"className":84826},[750],[224,84828,84830],{"className":84829},[754],[224,84831,84833],{"className":84832},[759],[224,84834,84836],{"className":84835,"style":5671},[763],[224,84837,84838,84841],{"style":3288},[224,84839],{"className":84840,"style":772},[771],[224,84842,84844],{"className":84843},[776,777,778,779],[224,84845,84847,84850,84853],{"className":84846},[468,779],[224,84848,375],{"className":84849,"style":492},[468,469,779],[224,84851,5268],{"className":84852},[478,779],[224,84854,1438],{"className":84855},[468,779],[72502,84857,84858],{},[224,84859,84861,84875],{"className":84860,"translate":388},[392],[224,84862,84864],{"className":84863},[396],[398,84865,84866],{"xmlns":400},[403,84867,84868,84872],{},[406,84869,84870],{},[409,84871,21092],{"mathvariant":12251},[447,84873,84874],{"encoding":449},"\\mathbb 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,{},[409,86562,375],{},[406,86564,86565,86567,86569],{},[419,86566,1438],{},[412,86568,5268],{},[409,86570,375],{},[447,86572,86573],{"encoding":449},"\\frac{y}{1-y}",[224,86575,86577],{"className":86576,"ariaHidden":455},[454],[224,86578,86580,86583],{"className":86579},[459],[224,86581],{"className":86582,"style":23117},[463],[224,86584,86586,86589,86654],{"className":86585},[468],[224,86587],{"className":86588},[542,8479],[224,86590,86592],{"className":86591},[20338],[224,86593,86595,86646],{"className":86594},[754,755],[224,86596,86598,86643],{"className":86597},[759],[224,86599,86601,86621,86629],{"className":86600,"style":23136},[763],[224,86602,86603,86606],{"style":22566},[224,86604],{"className":86605,"style":2108},[771],[224,86607,86609],{"className":86608},[776,777,778,779],[224,86610,86612,86615,86618],{"className":86611},[468,779],[224,86613,1438],{"className":86614},[468,779],[224,86616,5268],{"className":86617},[478,779],[224,86619,375],{"className":86620,"style":492},[468,469,779],[224,86622,86623,86626],{"style":20546},[224,86624],{"className":86625,"style":2108},[771],[224,86627],{"className":86628,"style":20554},[20553],[224,86630,86631,86634],{"style":22889},[224,86632],{"className":86633,"style":2108},[771],[224,86635,86637],{"className":86636},[776,777,778,779],[224,86638,86640],{"className":86639},[468,779],[224,86641,375],{"className":86642,"style":492},[468,469,779],[224,86644,787],{"className":86645},[786],[224,86647,86649],{"className":86648},[759],[224,86650,86652],{"className":86651,"style":22776},[763],[224,86653],{},[224,86655],{"className":86656},[549,8479],[72502,86658,86659],{},[224,86660,86662,86680],{"className":86661,"translate":388},[392],[224,86663,86665],{"className":86664},[396],[398,86666,86667],{"xmlns":400},[403,86668,86669,86677],{},[406,86670,86671,86673,86675],{},[409,86672,375],{},[412,86674,1577],{},[419,86676,1438],{},[447,86678,86679],{"encoding":449},"y\u003C1",[224,86681,86683,86701],{"className":86682,"ariaHidden":455},[454],[224,86684,86686,86689,86692,86695,86698],{"className":86685},[459],[224,86687],{"className":86688,"style":488},[463],[224,86690,375],{"className":86691,"style":492},[468,469],[224,86693],{"className":86694,"style":496},[473],[224,86696,1577],{"className":86697},[500],[224,86699],{"className":86700,"style":496},[473],[224,86702,86704,86707],{"className":86703},[459],[224,86705],{"className":86706,"style":510},[463],[224,86708,1438],{"className":86709},[468],[72502,86711,86067],{},[11,86713,86714,86715,86773,86774,87020,87021,87182,87183,87448,87449,40894,87609,87725],{},"以 ",[224,86716,86718,86735],{"className":86717,"translate":388},[392],[224,86719,86721],{"className":86720},[396],[398,86722,86723],{"xmlns":400},[403,86724,86725,86733],{},[406,86726,86727],{},[3248,86728,86729,86731],{},[409,86730,57370],{},[419,86732,1447],{},[447,86734,60308],{"encoding":449},[224,86736,86738],{"className":86737,"ariaHidden":455},[454],[224,86739,86741,86744],{"className":86740},[459],[224,86742],{"className":86743,"style":60318},[463],[224,86745,86747,86750],{"className":86746},[468],[224,86748,57370],{"className":86749},[468,469],[224,86751,86753],{"className":86752},[750],[224,86754,86756],{"className":86755},[754],[224,86757,86759],{"className":86758},[759],[224,86760,86762],{"className":86761,"style":5671},[763],[224,86763,86764,86767],{"style":3288},[224,86765],{"className":86766,"style":772},[771],[224,86768,86770],{"className":86769},[776,777,778,779],[224,86771,1447],{"className":86772},[468,779]," 为例验证：",[224,86775,86777,86837],{"className":86776,"translate":388},[392],[224,86778,86780],{"className":86779},[396],[398,86781,86782],{"xmlns":400},[403,86783,86784,86834],{},[406,86785,86786,86792,86794,86796,86798,86800,86810,86812,86814,86816,86818,86820,86822,86824,86826,86832],{},[3248,86787,86788,86790],{},[409,86789,1705],{},[412,86791,27543],{},[412,86793,239],{"stretchy":435},[409,86795,375],{},[412,86797,1716],{"stretchy":435},[412,86799,278],{},[695,86801,86802,86808],{},[406,86803,86804,86806],{},[409,86805,55354],{},[412,86807,24134],{},[409,86809,1435],{},[412,86811,8110],{"stretchy":435},[409,86813,1435],{},[409,86815,375],{},[412,86817,5268],{},[412,86819,239],{"stretchy":435},[409,86821,1435],{},[412,86823,5268],{},[419,86825,1438],{},[3248,86827,86828,86830],{},[412,86829,1716],{"stretchy":435},[419,86831,1447],{},[412,86833,15077],{"stretchy":435},[447,86835,86836],{"encoding":449},"f^*(y)=\\sup_t\\{ty-(t-1)^2\\}",[224,86838,86840,86893,86958,86979],{"className":86839,"ariaHidden":455},[454],[224,86841,86843,86846,86875,86878,86881,86884,86887,86890],{"className":86842},[459],[224,86844],{"className":86845,"style":535},[463],[224,86847,86849,86852],{"className":86848},[468],[224,86850,1705],{"className":86851,"style":1740},[468,469],[224,86853,86855],{"className":86854},[750],[224,86856,86858],{"className":86857},[754],[224,86859,86861],{"className":86860},[759],[224,86862,86864],{"className":86863,"style":28465},[763],[224,86865,86866,86869],{"style":3288},[224,86867],{"className":86868,"style":772},[771],[224,86870,86872],{"className":86871},[776,777,778,779],[224,86873,27543],{"className":86874},[478,779],[224,86876,239],{"className":86877},[542],[224,86879,375],{"className":86880,"style":492},[468,469],[224,86882,1716],{"className":86883},[549],[224,86885],{"className":86886,"style":496},[473],[224,86888,278],{"className":86889},[500],[224,86891],{"className":86892,"style":496},[473],[224,86894,86896,86899,86940,86943,86946,86949,86952,86955],{"className":86895},[459],[224,86897],{"className":86898,"style":535},[463],[224,86900,86902,86905],{"className":86901},[21531],[224,86903,55354],{"className":86904},[21531],[224,86906,86908],{"className":86907},[750],[224,86909,86911,86932],{"className":86910},[754,755],[224,86912,86914,86929],{"className":86913},[759],[224,86915,86918],{"className":86916,"style":86917},[763],"height:0.1864em;",[224,86919,86920,86923],{"style":55500},[224,86921],{"className":86922,"style":772},[771],[224,86924,86926],{"className":86925},[776,777,778,779],[224,86927,1435],{"className":86928},[468,469,779],[224,86930,787],{"className":86931},[786],[224,86933,86935],{"className":86934},[759],[224,86936,86938],{"className":86937,"style":56942},[763],[224,86939],{},[224,86941,8110],{"className":86942},[542],[224,86944,1435],{"className":86945},[468,469],[224,86947,375],{"className":86948,"style":492},[468,469],[224,86950],{"className":86951,"style":474},[473],[224,86953,5268],{"className":86954},[478],[224,86956],{"className":86957,"style":474},[473],[224,86959,86961,86964,86967,86970,86973,86976],{"className":86960},[459],[224,86962],{"className":86963,"style":535},[463],[224,86965,239],{"className":86966},[542],[224,86968,1435],{"className":86969},[468,469],[224,86971],{"className":86972,"style":474},[473],[224,86974,5268],{"className":86975},[478],[224,86977],{"className":86978,"style":474},[473],[224,86980,86982,86985,86988,87017],{"className":86981},[459],[224,86983],{"className":86984,"style":39943},[463],[224,86986,1438],{"className":86987},[468],[224,86989,86991,86994],{"className":86990},[549],[224,86992,1716],{"className":86993},[549],[224,86995,86997],{"className":86996},[750],[224,86998,87000],{"className":86999},[754],[224,87001,87003],{"className":87002},[759],[224,87004,87006],{"className":87005,"style":5671},[763],[224,87007,87008,87011],{"style":3288},[224,87009],{"className":87010,"style":772},[771],[224,87012,87014],{"className":87013},[776,777,778,779],[224,87015,1447],{"className":87016},[468,779],[224,87018,15077],{"className":87019},[549],"，驻点 ",[224,87022,87024,87068],{"className":87023,"translate":388},[392],[224,87025,87027],{"className":87026},[396],[398,87028,87029],{"xmlns":400},[403,87030,87031,87065],{},[406,87032,87033,87035,87037,87039,87041,87043,87045,87047,87049,87051,87053,87055,87057,87059,87061,87063],{},[409,87034,375],{},[412,87036,278],{},[419,87038,1447],{},[412,87040,239],{"stretchy":435},[409,87042,1435],{},[412,87044,5268],{},[419,87046,1438],{},[412,87048,1716],{"stretchy":435},[412,87050,9886],{},[409,87052,1435],{},[412,87054,278],{},[419,87056,1438],{},[412,87058,319],{},[409,87060,375],{},[409,87062,12413],{"mathvariant":678},[419,87064,1447],{},[447,87066,87067],{"encoding":449},"y=2(t-1)\\Rightarrow t=1+y\u002F2",[224,87069,87071,87089,87113,87134,87152,87170],{"className":87070,"ariaHidden":455},[454],[224,87072,87074,87077,87080,87083,87086],{"className":87073},[459],[224,87075],{"className":87076,"style":806},[463],[224,87078,375],{"className":87079,"style":492},[468,469],[224,87081],{"className":87082,"style":496},[473],[224,87084,278],{"className":87085},[500],[224,87087],{"className":87088,"style":496},[473],[224,87090,87092,87095,87098,87101,87104,87107,87110],{"className":87091},[459],[224,87093],{"className":87094,"style":535},[463],[224,87096,1447],{"className":87097},[468],[224,87099,239],{"className":87100},[542],[224,87102,1435],{"className":87103},[468,469],[224,87105],{"className":87106,"style":474},[473],[224,87108,5268],{"className":87109},[478],[224,87111],{"className":87112,"style":474},[473],[224,87114,87116,87119,87122,87125,87128,87131],{"className":87115},[459],[224,87117],{"className":87118,"style":535},[463],[224,87120,1438],{"className":87121},[468],[224,87123,1716],{"className":87124},[549],[224,87126],{"className":87127,"style":496},[473],[224,87129,9886],{"className":87130},[500],[224,87132],{"className":87133,"style":496},[473],[224,87135,87137,87140,87143,87146,87149],{"className":87136},[459],[224,87138],{"className":87139,"style":25353},[463],[224,87141,1435],{"className":87142},[468,469],[224,87144],{"className":87145,"style":496},[473],[224,87147,278],{"className":87148},[500],[224,87150],{"className":87151,"style":496},[473],[224,87153,87155,87158,87161,87164,87167],{"className":87154},[459],[224,87156],{"className":87157,"style":7047},[463],[224,87159,1438],{"className":87160},[468],[224,87162],{"className":87163,"style":474},[473],[224,87165,319],{"className":87166},[478],[224,87168],{"className":87169,"style":474},[473],[224,87171,87173,87176,87179],{"className":87172},[459],[224,87174],{"className":87175,"style":535},[463],[224,87177,375],{"className":87178,"style":492},[468,469],[224,87180,65062],{"className":87181},[468],"，得 ",[224,87184,87186,87250],{"className":87185,"translate":388},[392],[224,87187,87189],{"className":87188},[396],[398,87190,87191],{"xmlns":400},[403,87192,87193,87247],{},[406,87194,87195,87201,87203,87205,87207,87209,87211,87213,87215,87217,87219,87221,87227,87229,87231,87233,87235,87237,87243,87245],{},[3248,87196,87197,87199],{},[409,87198,1705],{},[412,87200,27543],{},[412,87202,278],{},[409,87204,375],{},[412,87206,239],{"stretchy":435},[419,87208,1438],{},[412,87210,319],{},[409,87212,375],{},[409,87214,12413],{"mathvariant":678},[419,87216,1447],{},[412,87218,1716],{"stretchy":435},[412,87220,5268],{},[3248,87222,87223,87225],{},[409,87224,375],{},[419,87226,1447],{},[409,87228,12413],{"mathvariant":678},[419,87230,9409],{},[412,87232,278],{},[409,87234,375],{},[412,87236,319],{},[3248,87238,87239,87241],{},[409,87240,375],{},[419,87242,1447],{},[409,87244,12413],{"mathvariant":678},[419,87246,9409],{},[447,87248,87249],{"encoding":449},"f^*=y(1+y\u002F2)-y^2\u002F4=y+y^2\u002F4",[224,87251,87253,87297,87321,87345,87392,87410],{"className":87252,"ariaHidden":455},[454],[224,87254,87256,87259,87288,87291,87294],{"className":87255},[459],[224,87257],{"className":87258,"style":2565},[463],[224,87260,87262,87265],{"className":87261},[468],[224,87263,1705],{"className":87264,"style":1740},[468,469],[224,87266,87268],{"className":87267},[750],[224,87269,87271],{"className":87270},[754],[224,87272,87274],{"className":87273},[759],[224,87275,87277],{"className":87276,"style":28465},[763],[224,87278,87279,87282],{"style":3288},[224,87280],{"className":87281,"style":772},[771],[224,87283,87285],{"className":87284},[776,777,778,779],[224,87286,27543],{"className":87287},[478,779],[224,87289],{"className":87290,"style":496},[473],[224,87292,278],{"className":87293},[500],[224,87295],{"className":87296,"style":496},[473],[224,87298,87300,87303,87306,87309,87312,87315,87318],{"className":87299},[459],[224,87301],{"className":87302,"style":535},[463],[224,87304,375],{"className":87305,"style":492},[468,469],[224,87307,239],{"className":87308},[542],[224,87310,1438],{"className":87311},[468],[224,87313],{"className":87314,"style":474},[473],[224,87316,319],{"className":87317},[478],[224,87319],{"className":87320,"style":474},[473],[224,87322,87324,87327,87330,87333,87336,87339,87342],{"className":87323},[459],[224,87325],{"className":87326,"style":535},[463],[224,87328,375],{"className":87329,"style":492},[468,469],[224,87331,65062],{"className":87332},[468],[224,87334,1716],{"className":87335},[549],[224,87337],{"className":87338,"style":474},[473],[224,87340,5268],{"className":87341},[478],[224,87343],{"className":87344,"style":474},[473],[224,87346,87348,87351,87380,87383,87386,87389],{"className":87347},[459],[224,87349],{"className":87350,"style":39943},[463],[224,87352,87354,87357],{"className":87353},[468],[224,87355,375],{"className":87356,"style":492},[468,469],[224,87358,87360],{"className":87359},[750],[224,87361,87363],{"className":87362},[754],[224,87364,87366],{"className":87365},[759],[224,87367,87369],{"className":87368,"style":5671},[763],[224,87370,87371,87374],{"style":3288},[224,87372],{"className":87373,"style":772},[771],[224,87375,87377],{"className":87376},[776,777,778,779],[224,87378,1447],{"className":87379},[468,779],[224,87381,60286],{"className":87382},[468],[224,87384],{"className":87385,"style":496},[473],[224,87387,278],{"className":87388},[500],[224,87390],{"className":87391,"style":496},[473],[224,87393,87395,87398,87401,87404,87407],{"className":87394},[459],[224,87396],{"className":87397,"style":4846},[463],[224,87399,375],{"className":87400,"style":492},[468,469],[224,87402],{"className":87403,"style":474},[473],[224,87405,319],{"className":87406},[478],[224,87408],{"className":87409,"style":474},[473],[224,87411,87413,87416,87445],{"className":87412},[459],[224,87414],{"className":87415,"style":39943},[463],[224,87417,87419,87422],{"className":87418},[468],[224,87420,375],{"className":87421,"style":492},[468,469],[224,87423,87425],{"className":87424},[750],[224,87426,87428],{"className":87427},[754],[224,87429,87431],{"className":87430},[759],[224,87432,87434],{"className":87433,"style":5671},[763],[224,87435,87436,87439],{"style":3288},[224,87437],{"className":87438,"style":772},[771],[224,87440,87442],{"className":87441},[776,777,778,779],[224,87443,1447],{"className":87444},[468,779],[224,87446,60286],{"className":87447},[468],"。以 Hellinger² 为例：",[224,87450,87452,87488],{"className":87451,"translate":388},[392],[224,87453,87455],{"className":87454},[396],[398,87456,87457],{"xmlns":400},[403,87458,87459,87485],{},[406,87460,87461,87463,87465,87467,87469,87471,87473,87475,87477,87481,87483],{},[409,87462,1705],{},[412,87464,239],{"stretchy":435},[409,87466,1435],{},[412,87468,1716],{"stretchy":435},[412,87470,278],{},[409,87472,1435],{},[412,87474,5268],{},[419,87476,1447],{},[54262,87478,87479],{},[409,87480,1435],{},[412,87482,319],{},[419,87484,1438],{},[447,87486,87487],{"encoding":449},"f(t)=t-2\\sqrt t+1",[224,87489,87491,87518,87537,87600],{"className":87490,"ariaHidden":455},[454],[224,87492,87494,87497,87500,87503,87506,87509,87512,87515],{"className":87493},[459],[224,87495],{"className":87496,"style":535},[463],[224,87498,1705],{"className":87499,"style":1740},[468,469],[224,87501,239],{"className":87502},[542],[224,87504,1435],{"className":87505},[468,469],[224,87507,1716],{"className":87508},[549],[224,87510],{"className":87511,"style":496},[473],[224,87513,278],{"className":87514},[500],[224,87516],{"className":87517,"style":496},[473],[224,87519,87521,87525,87528,87531,87534],{"className":87520},[459],[224,87522],{"className":87523,"style":87524},[463],"height:0.6984em;vertical-align:-0.0833em;",[224,87526,1435],{"className":87527},[468,469],[224,87529],{"className":87530,"style":474},[473],[224,87532,5268],{"className":87533},[478],[224,87535],{"className":87536,"style":474},[473],[224,87538,87540,87544,87547,87591,87594,87597],{"className":87539},[459],[224,87541],{"className":87542,"style":87543},[463],"height:1.04em;vertical-align:-0.1475em;",[224,87545,1447],{"className":87546},[468],[224,87548,87550],{"className":87549},[468,54290],[224,87551,87553,87583],{"className":87552},[754,755],[224,87554,87556,87580],{"className":87555},[759],[224,87557,87559,87568],{"className":87558,"style":62596},[763],[224,87560,87562,87565],{"className":87561,"style":2120},[2139],[224,87563],{"className":87564,"style":2108},[771],[224,87566,1435],{"className":87567,"style":54310},[468,469],[224,87569,87570,87573],{"style":62608},[224,87571],{"className":87572,"style":2108},[771],[224,87574,87576],{"className":87575,"style":54367},[54366],[2155,87577,87578],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,87579],{"d":54374},[224,87581,787],{"className":87582},[786],[224,87584,87586],{"className":87585},[759],[224,87587,87589],{"className":87588,"style":62628},[763],[224,87590],{},[224,87592],{"className":87593,"style":474},[473],[224,87595,319],{"className":87596},[478],[224,87598],{"className":87599,"style":474},[473],[224,87601,87603,87606],{"className":87602},[459],[224,87604],{"className":87605,"style":510},[463],[224,87607,1438],{"className":87608},[468],[224,87610,87612,87636],{"className":87611,"translate":388},[392],[224,87613,87615],{"className":87614},[396],[398,87616,87617],{"xmlns":400},[403,87618,87619,87633],{},[406,87620,87621,87623,87625,87629,87631],{},[409,87622,4130],{},[412,87624,278],{},[54262,87626,87627],{},[409,87628,1435],{},[412,87630,9766],{},[419,87632,284],{},[447,87634,87635],{"encoding":449},"u=\\sqrt t\\ge0",[224,87637,87639,87657,87716],{"className":87638,"ariaHidden":455},[454],[224,87640,87642,87645,87648,87651,87654],{"className":87641},[459],[224,87643],{"className":87644,"style":839},[463],[224,87646,4130],{"className":87647},[468,469],[224,87649],{"className":87650,"style":496},[473],[224,87652,278],{"className":87653},[500],[224,87655],{"className":87656,"style":496},[473],[224,87658,87660,87663,87707,87710,87713],{"className":87659},[459],[224,87661],{"className":87662,"style":87543},[463],[224,87664,87666],{"className":87665},[468,54290],[224,87667,87669,87699],{"className":87668},[754,755],[224,87670,87672,87696],{"className":87671},[759],[224,87673,87675,87684],{"className":87674,"style":62596},[763],[224,87676,87678,87681],{"className":87677,"style":2120},[2139],[224,87679],{"className":87680,"style":2108},[771],[224,87682,1435],{"className":87683,"style":54310},[468,469],[224,87685,87686,87689],{"style":62608},[224,87687],{"className":87688,"style":2108},[771],[224,87690,87692],{"className":87691,"style":54367},[54366],[2155,87693,87694],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,87695],{"d":54374},[224,87697,787],{"className":87698},[786],[224,87700,87702],{"className":87701},[759],[224,87703,87705],{"className":87704,"style":62628},[763],[224,87706],{},[224,87708],{"className":87709,"style":496},[473],[224,87711,9766],{"className":87712},[500],[224,87714],{"className":87715,"style":496},[473],[224,87717,87719,87722],{"className":87718},[459],[224,87720],{"className":87721,"style":510},[463],[224,87723,284],{"className":87724},[468],"，把共轭写成有限上确界",[224,87727,87729],{"className":87728,"translate":388},[387],[224,87730,87732,87806],{"className":87731,"translate":388},[392],[224,87733,87735],{"className":87734},[396],[398,87736,87737],{"xmlns":400,"display":401},[403,87738,87739,87803],{},[406,87740,87741,87747,87749,87751,87753,87755,87771,87773,87775,87777,87779,87781,87783,87789,87791,87793,87795,87797,87799,87801],{},[3248,87742,87743,87745],{},[409,87744,1705],{},[412,87746,27543],{},[412,87748,239],{"stretchy":435},[409,87750,375],{},[412,87752,1716],{"stretchy":435},[412,87754,278],{},[1965,87756,87757,87763],{},[406,87758,87759,87761],{},[409,87760,55354],{},[412,87762,24134],{},[406,87764,87765,87767,87769],{},[409,87766,4130],{},[412,87768,9766],{},[419,87770,284],{},[412,87772,8110],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[412,87774,239],{"stretchy":435},[409,87776,375],{},[412,87778,5268],{},[419,87780,1438],{},[412,87782,1716],{"stretchy":435},[3248,87784,87785,87787],{},[409,87786,4130],{},[419,87788,1447],{},[412,87790,319],{},[419,87792,1447],{},[409,87794,4130],{},[412,87796,5268],{},[419,87798,1438],{},[412,87800,15077],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[412,87802,1877],{"separator":455},[447,87804,87805],{"encoding":449},"f^*(y)=\\sup_{u\\ge0}\\Big\\{(y-1)u^2+2u-1\\Big\\},",[224,87807,87809,87862,87946,87996,88017],{"className":87808,"ariaHidden":455},[454],[224,87810,87812,87815,87844,87847,87850,87853,87856,87859],{"className":87811},[459],[224,87813],{"className":87814,"style":535},[463],[224,87816,87818,87821],{"className":87817},[468],[224,87819,1705],{"className":87820,"style":1740},[468,469],[224,87822,87824],{"className":87823},[750],[224,87825,87827],{"className":87826},[754],[224,87828,87830],{"className":87829},[759],[224,87831,87833],{"className":87832,"style":28217},[763],[224,87834,87835,87838],{"style":5074},[224,87836],{"className":87837,"style":772},[771],[224,87839,87841],{"className":87840},[776,777,778,779],[224,87842,27543],{"className":87843},[478,779],[224,87845,239],{"className":87846},[542],[224,87848,375],{"className":87849,"style":492},[468,469],[224,87851,1716],{"className":87852},[549],[224,87854],{"className":87855,"style":496},[473],[224,87857,278],{"className":87858},[500],[224,87860],{"className":87861,"style":496},[473],[224,87863,87865,87869,87922,87925,87931,87934,87937,87940,87943],{"className":87864},[459],[224,87866],{"className":87867,"style":87868},[463],"height:2.1567em;vertical-align:-1.0067em;",[224,87870,87872],{"className":87871},[21531,33665],[224,87873,87875,87914],{"className":87874},[754,755],[224,87876,87878,87911],{"className":87877},[759],[224,87879,87881,87901],{"className":87880,"style":839},[763],[224,87882,87883,87886],{"style":80828},[224,87884],{"className":87885,"style":2108},[771],[224,87887,87889],{"className":87888},[776,777,778,779],[224,87890,87892,87895,87898],{"className":87891},[468,779],[224,87893,4130],{"className":87894},[468,469,779],[224,87896,9766],{"className":87897},[500,779],[224,87899,284],{"className":87900},[468,779],[224,87902,87903,87906],{"style":2120},[224,87904],{"className":87905,"style":2108},[771],[224,87907,87908],{},[224,87909,55354],{"className":87910},[21531],[224,87912,787],{"className":87913},[786],[224,87915,87917],{"className":87916},[759],[224,87918,87920],{"className":87919,"style":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",[224,88040,88042,88060],{"className":88041,"translate":388},[392],[224,88043,88045],{"className":88044},[396],[398,88046,88047],{"xmlns":400},[403,88048,88049,88057],{},[406,88050,88051,88053,88055],{},[409,88052,375],{},[412,88054,9766],{},[419,88056,1438],{},[447,88058,88059],{"encoding":449},"y\\ge1",[224,88061,88063,88081],{"className":88062,"ariaHidden":455},[454],[224,88064,88066,88069,88072,88075,88078],{"className":88065},[459],[224,88067],{"className":88068,"style":2644},[463],[224,88070,375],{"className":88071,"style":492},[468,469],[224,88073],{"className":88074,"style":496},[473],[224,88076,9766],{"className":88077},[500],[224,88079],{"className":88080,"style":496},[473],[224,88082,88084,88087],{"className":88083},[459],[224,88085],{"className":88086,"style":510},[463],[224,88088,1438],{"className":88089},[468]," 时上确界为 ",[224,88092,88094,88109],{"className":88093,"translate":388},[392],[224,88095,88097],{"className":88096},[396],[398,88098,88099],{"xmlns":400},[403,88100,88101,88107],{},[406,88102,88103,88105],{},[412,88104,319],{},[409,88106,21085],{"mathvariant":678},[447,88108,24272],{"encoding":449},[224,88110,88112],{"className":88111,"ariaHidden":455},[454],[224,88113,88115,88118,88121],{"className":88114},[459],[224,88116],{"className":88117,"style":464},[463],[224,88119,319],{"className":88120},[468],[224,88122,21085],{"className":88123},[468],"；当 ",[224,88126,88128,88145],{"className":88127,"translate":388},[392],[224,88129,88131],{"className":88130},[396],[398,88132,88133],{"xmlns":400},[403,88134,88135,88143],{},[406,88136,88137,88139,88141],{},[409,88138,375],{},[412,88140,1577],{},[419,88142,1438],{},[447,88144,86679],{"encoding":449},[224,88146,88148,88166],{"className":88147,"ariaHidden":455},[454],[224,88149,88151,88154,88157,88160,88163],{"className":88150},[459],[224,88152],{"className":88153,"style":488},[463],[224,88155,375],{"className":88156,"style":492},[468,469],[224,88158],{"className":88159,"style":496},[473],[224,88161,1577],{"className":88162},[500],[224,88164],{"className":88165,"style":496},[473],[224,88167,88169,88172],{"className":88168},[459],[224,88170],{"className":88171,"style":510},[463],[224,88173,1438],{"className":88174},[468]," 时上式是开口向下的二次函数，最大点 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",[224,89516,89518,89552],{"className":89517,"translate":388},[392],[224,89519,89521],{"className":89520},[396],[398,89522,89523],{"xmlns":400},[403,89524,89525,89549],{},[406,89526,89527,89529,89531,89543,89545,89547],{},[409,89528,375],{},[412,89530,278],{},[27527,89532,89533,89535,89541],{},[409,89534,1705],{},[406,89536,89537,89539],{},[409,89538,43681],{},[409,89540,6924],{},[412,89542,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,89544,239],{"stretchy":435},[409,89546,1435],{},[412,89548,1716],{"stretchy":435},[447,89550,89551],{"encoding":449},"y=f_{JS}'(t)",[224,89553,89555,89573],{"className":89554,"ariaHidden":455},[454],[224,89556,89558,89561,89564,89567,89570],{"className":89557},[459],[224,89559],{"className":89560,"style":806},[463],[224,89562,375],{"className":89563,"style":492},[468,469],[224,89565],{"className":89566,"style":496},[473],[224,89568,278],{"className":89569},[500],[224,89571],{"className":89572,"style":496},[473],[224,89574,89576,89579,89639,89642,89645],{"className":89575},[459],[224,89577],{"className":89578,"style":89272},[463],[224,89580,89582,89585],{"className":89581},[468],[224,89583,1705],{"className":89584,"style":1740},[468,469],[224,89586,89588],{"className":89587},[750],[224,89589,89591,89631],{"className":89590},[754,755],[224,89592,89594,89628],{"className":89593},[759],[224,89595,89597,89614],{"className":89596,"style":6992},[763],[224,89598,89599,89602],{"style":89293},[224,89600],{"className":89601,"style":772},[771],[224,89603,89605],{"className":89604},[776,777,778,779],[224,89606,89608,89611],{"className":89607},[468,779],[224,89609,43681],{"className":89610,"style":44091},[468,469,779],[224,89612,6924],{"className":89613,"style":6970},[468,469,779],[224,89615,89616,89619],{"style":3288},[224,89617],{"className":89618,"style":772},[771],[224,89620,89622],{"className":89621},[776,777,778,779],[224,89623,89625],{"className":89624},[468,779],[224,89626,6933],{"className":89627},[468,779],[224,89629,787],{"className":89630},[786],[224,89632,89634],{"className":89633},[759],[224,89635,89637],{"className":89636,"style":58311},[763],[224,89638],{},[224,89640,239],{"className":89641},[542],[224,89643,1435],{"className":89644},[468,469],[224,89646,1716],{"className":89647},[549],[224,89649,89651,89691],{"className":89650,"translate":388},[392],[224,89652,89654],{"className":89653},[396],[398,89655,89656],{"xmlns":400},[403,89657,89658,89688],{},[406,89659,89660,89670,89672],{},[3248,89661,89662,89664],{},[409,89663,1861],{},[406,89665,89666,89668],{},[419,89667,1447],{},[409,89669,375],{},[412,89671,278],{},[20338,89673,89674,89680],{},[406,89675,89676,89678],{},[419,89677,1447],{},[409,89679,1435],{},[406,89681,89682,89684,89686],{},[409,89683,1435],{},[412,89685,319],{},[419,89687,1438],{},[447,89689,89690],{"encoding":449},"e^{2y}=\\frac{2t}{t+1}",[224,89692,89694,89744],{"className":89693,"ariaHidden":455},[454],[224,89695,89697,89700,89735,89738,89741],{"className":89696},[459],[224,89698],{"className":89699,"style":5671},[463],[224,89701,89703,89706],{"className":89702},[468],[224,89704,1861],{"className":89705},[468,469],[224,89707,89709],{"className":89708},[750],[224,89710,89712],{"className":89711},[754],[224,89713,89715],{"className":89714},[759],[224,89716,89718],{"className":89717,"style":5671},[763],[224,89719,89720,89723],{"style":3288},[224,89721],{"className":89722,"style":772},[771],[224,89724,89726],{"className":89725},[776,777,778,779],[224,89727,89729,89732],{"className":89728},[468,779],[224,89730,1447],{"className":89731},[468,779],[224,89733,375],{"className":89734,"style":492},[468,469,779],[224,89736],{"className":89737,"style":496},[473],[224,89739,278],{"className":89740},[500],[224,89742],{"className":89743,"style":496},[473],[224,89745,89747,89750],{"className":89746},[459],[224,89748],{"className":89749,"style":65316},[463],[224,89751,89753,89756,89824],{"className":89752},[468],[224,89754],{"className":89755},[542,8479],[224,89757,89759],{"className":89758},[20338],[224,89760,89762,89816],{"className":89761},[754,755],[224,89763,89765,89813],{"className":89764},[759],[224,89766,89768,89788,89796],{"className":89767,"style":22563},[763],[224,89769,89770,89773],{"style":22566},[224,89771],{"className":89772,"style":2108},[771],[224,89774,89776],{"className":89775},[776,777,778,779],[224,89777,89779,89782,89785],{"className":89778},[468,779],[224,89780,1435],{"className":89781},[468,469,779],[224,89783,319],{"className":89784},[478,779],[224,89786,1438],{"className":89787},[468,779],[224,89789,89790,89793],{"style":20546},[224,89791],{"className":89792,"style":2108},[771],[224,89794],{"className":89795,"style":20554},[20553],[224,89797,89798,89801],{"style":22589},[224,89799],{"className":89800,"style":2108},[771],[224,89802,89804],{"className":89803},[776,777,778,779],[224,89805,89807,89810],{"className":89806},[468,779],[224,89808,1447],{"className":89809},[468,779],[224,89811,1435],{"className":89812},[468,469,779],[224,89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",[224,89829,89831,89877],{"className":89830,"translate":388},[392],[224,89832,89834],{"className":89833},[396],[398,89835,89836],{"xmlns":400},[403,89837,89838,89874],{},[406,89839,89840,89842,89844],{},[409,89841,1435],{},[412,89843,278],{},[5221,89845,89846],{"scriptlevel":284,"displaystyle":455},[20338,89847,89848,89858],{},[3248,89849,89850,89852],{},[409,89851,1861],{},[406,89853,89854,89856],{},[419,89855,1447],{},[409,89857,375],{},[406,89859,89860,89862,89864],{},[419,89861,1447],{},[412,89863,5268],{},[3248,89865,89866,89868],{},[409,89867,1861],{},[406,89869,89870,89872],{},[419,89871,1447],{},[409,89873,375],{},[447,89875,89876],{"encoding":449},"t=\\dfrac{e^{2y}}{2-e^{2y}}",[224,89878,89880,89898],{"className":89879,"ariaHidden":455},[454],[224,89881,89883,89886,89889,89892,89895],{"className":89882},[459],[224,89884],{"className":89885,"style":25353},[463],[224,89887,1435],{"className":89888},[468,469],[224,89890],{"className":89891,"style":496},[473],[224,89893,278],{"className":89894},[500],[224,89896],{"className":89897,"style":496},[473],[224,89899,89901,89905],{"className":89900},[459],[224,89902],{"className":89903,"style":89904},[463],"height:2.2604em;vertical-align:-0.7693em;",[224,89906,89908,89911,90040],{"className":89907},[468],[224,89909],{"className":89910},[542,8479],[224,89912,89914],{"className":89913},[20338],[224,89915,89917,90032],{"className":89916},[754,755],[224,89918,89920,90029],{"className":89919},[759],[224,89921,89923,89978,89986],{"className":89922,"style":57644},[763],[224,89924,89925,89928],{"style":20531},[224,89926],{"className":89927,"style":2108},[771],[224,89929,89931,89934,89937,89940,89943],{"className":89930},[468],[224,89932,1447],{"className":89933},[468],[224,89935],{"className":89936,"style":474},[473],[224,89938,5268],{"className":89939},[478],[224,89941],{"className":89942,"style":474},[473],[224,89944,89946,89949],{"className":89945},[468],[224,89947,1861],{"className":89948},[468,469],[224,89950,89952],{"className":89951},[750],[224,89953,89955],{"className":89954},[754],[224,89956,89958],{"className":89957},[759],[224,89959,89961],{"className":89960,"style":64944},[763],[224,89962,89963,89966],{"style":64947},[224,89964],{"className":89965,"style":772},[771],[224,89967,89969],{"className":89968},[776,777,778,779],[224,89970,89972,89975],{"className":89971},[468,779],[224,89973,1447],{"className":89974},[468,779],[224,89976,375],{"className":89977,"style":492},[468,469,779],[224,89979,89980,89983],{"style":20546},[224,89981],{"className":89982,"style":2108},[771],[224,89984],{"className":89985,"style":20554},[20553],[224,89987,89988,89991],{"style":20557},[224,89989],{"className":89990,"style":2108},[771],[224,89992,89994],{"className":89993},[468],[224,89995,89997,90000],{"className":89996},[468],[224,89998,1861],{"className":89999},[468,469],[224,90001,90003],{"className":90002},[750],[224,90004,90006],{"className":90005},[754],[224,90007,90009],{"className":90008},[759],[224,90010,90012],{"className":90011,"style":5671},[763],[224,90013,90014,90017],{"style":3288},[224,90015],{"className":90016,"style":772},[771],[224,90018,90020],{"className":90019},[776,777,778,779],[224,90021,90023,90026],{"className":90022},[468,779],[224,90024,1447],{"className":90025},[468,779],[224,90027,375],{"className":90028,"style":492},[468,469,779],[224,90030,787],{"className":90031},[786],[224,90033,90035],{"className":90034},[759],[224,90036,90038],{"className":90037,"style":36958},[763],[224,90039],{},[224,90041],{"className":90042},[549,8479],"（这要求 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一侧，",[224,91762,91764,91820],{"className":91763,"translate":388},[392],[224,91765,91767],{"className":91766},[396],[398,91768,91769],{"xmlns":400},[403,91770,91771,91817],{},[406,91772,91773,91775,91777,91779,91791,91793,91795,91797,91799,91801,91807,91809,91811],{},[409,91774,375],{},[409,91776,1435],{},[412,91778,5268],{},[20338,91780,91781,91789],{},[406,91782,91783,91785,91787],{},[419,91784,1438],{},[412,91786,5268],{},[409,91788,1435],{},[419,91790,1447],{},[412,91792,278],{},[409,91794,1435],{},[412,91796,239],{"stretchy":435},[409,91798,375],{},[412,91800,319],{},[20338,91802,91803,91805],{},[419,91804,1438],{},[419,91806,1447],{},[412,91808,1716],{"stretchy":435},[412,91810,5268],{},[20338,91812,91813,91815],{},[419,91814,1438],{},[419,91816,1447],{},[447,91818,91819],{"encoding":449},"yt-\\frac{1-t}{2}=t(y+\\frac12)-\\frac12",[224,91821,91823,91844,91933,91957,92043],{"className":91822,"ariaHidden":455},[454],[224,91824,91826,91829,91832,91835,91838,91841],{"className":91825},[459],[224,91827],{"className":91828,"style":46205},[463],[224,91830,375],{"className":91831,"style":492},[468,469],[224,91833,1435],{"className":91834},[468,469],[224,91836],{"className":91837,"style":474},[473],[224,91839,5268],{"className":91840},[478],[224,91842],{"className":91843,"style":474},[473],[224,91845,91847,91850,91924,91927,91930],{"className":91846},[459],[224,91848],{"className":91849,"style":22544},[463],[224,91851,91853,91856,91921],{"className":91852},[468],[224,91854],{"className":91855},[542,8479],[224,91857,91859],{"className":91858},[20338],[224,91860,91862,91913],{"className":91861},[754,755],[224,91863,91865,91910],{"className":91864},[759],[224,91866,91868,91882,91890],{"className":91867,"style":22563},[763],[224,91869,91870,91873],{"style":22566},[224,91871],{"className":91872,"style":2108},[771],[224,91874,91876],{"className":91875},[776,777,778,779],[224,91877,91879],{"className":91878},[468,779],[224,91880,1447],{"className":91881},[468,779],[224,91883,91884,91887],{"style":20546},[224,91885],{"className":91886,"style":2108},[771],[224,91888],{"className":91889,"style":20554},[20553],[224,91891,91892,91895],{"style":22589},[224,91893],{"className":91894,"style":2108},[771],[224,91896,91898],{"className":91897},[776,777,778,779],[224,91899,91901,91904,91907],{"className":91900},[468,779],[224,91902,1438],{"className":91903},[468,779],[224,91905,5268],{"className":91906},[478,779],[224,91908,1435],{"className":91909},[468,469,779],[224,91911,787],{"className":91912},[786],[224,91914,91916],{"className":91915},[759],[224,91917,91919],{"className":91918,"style":22611},[763],[224,91920],{},[224,91922],{"className":91923},[549,8479],[224,91925],{"className":91926,"style":496},[473],[224,91928,278],{"className":91929},[500],[224,91931],{"className":91932,"style":496},[473],[224,91934,91936,91939,91942,91945,91948,91951,91954],{"className":91935},[459],[224,91937],{"className":91938,"style":535},[463],[224,91940,1435],{"className":91941},[468,469],[224,91943,239],{"className":91944},[542],[224,91946,375],{"className":91947,"style":492},[468,469],[224,91949],{"className":91950,"style":474},[473],[224,91952,319],{"className":91953},[478],[224,91955],{"className":91956,"style":474},[473],[224,91958,91960,91963,92031,92034,92037,92040],{"className":91959},[459],[224,91961],{"className":91962,"style":22544},[463],[224,91964,91966,91969,92028],{"className":91965},[468],[224,91967],{"className":91968},[542,8479],[224,91970,91972],{"className":91971},[20338],[224,91973,91975,92020],{"className":91974},[754,755],[224,91976,91978,92017],{"className":91977},[759],[224,91979,91981,91995,92003],{"className":91980,"style":22563},[763],[224,91982,91983,91986],{"style":22566},[224,91984],{"className":91985,"style":2108},[771],[224,91987,91989],{"className":91988},[776,777,778,779],[224,91990,91992],{"className":91991},[468,779],[224,91993,1447],{"className":91994},[468,779],[224,91996,91997,92000],{"style":20546},[224,91998],{"className":91999,"style":2108},[771],[224,92001],{"className":92002,"style":20554},[20553],[224,92004,92005,92008],{"style":22589},[224,92006],{"className":92007,"style":2108},[771],[224,92009,92011],{"className":92010},[776,777,778,779],[224,92012,92014],{"className":92013},[468,779],[224,92015,1438],{"className":92016},[468,779],[224,92018,787],{"className":92019},[786],[224,92021,92023],{"className":92022},[759],[224,92024,92026],{"className":92025,"style":22611},[763],[224,92027],{},[224,92029],{"className":92030},[549,8479],[224,92032,1716],{"className":92033},[549],[224,92035],{"className":92036,"style":474},[473],[224,92038,5268],{"className":92039},[478],[224,92041],{"className":92042,"style":474},[473],[224,92044,92046,92049],{"className":92045},[459],[224,92047],{"className":92048,"style":22544},[463],[224,92050,92052,92055,92114],{"className":92051},[468],[224,92053],{"className":92054},[542,8479],[224,92056,92058],{"className":92057},[20338],[224,92059,92061,92106],{"className":92060},[754,755],[224,92062,92064,92103],{"className":92063},[759],[224,92065,92067,92081,92089],{"className":92066,"style":22563},[763],[224,92068,92069,92072],{"style":22566},[224,92070],{"className":92071,"style":2108},[771],[224,92073,92075],{"className":92074},[776,777,778,779],[224,92076,92078],{"className":92077},[468,779],[224,92079,1447],{"className":92080},[468,779],[224,92082,92083,92086],{"style":20546},[224,92084],{"className":92085,"style":2108},[771],[224,92087],{"className":92088,"style":20554},[20553],[224,92090,92091,92094],{"style":22589},[224,92092],{"className":92093,"style":2108},[771],[224,92095,92097],{"className":92096},[776,777,778,779],[224,92098,92100],{"className":92099},[468,779],[224,92101,1438],{"className":92102},[468,779],[224,92104,787],{"className":92105},[786],[224,92107,92109],{"className":92108},[759],[224,92110,92112],{"className":92111,"style":22611},[763],[224,92113],{},[224,92115],{"className":92116},[549,8479],"；在 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时两段的最大值都在 ",[224,92657,92659,92677],{"className":92658,"translate":388},[392],[224,92660,92662],{"className":92661},[396],[398,92663,92664],{"xmlns":400},[403,92665,92666,92674],{},[406,92667,92668,92670,92672],{},[409,92669,1435],{},[412,92671,278],{},[419,92673,1438],{},[447,92675,92676],{"encoding":449},"t=1",[224,92678,92680,92698],{"className":92679,"ariaHidden":455},[454],[224,92681,92683,92686,92689,92692,92695],{"className":92682},[459],[224,92684],{"className":92685,"style":25353},[463],[224,92687,1435],{"className":92688},[468,469],[224,92690],{"className":92691,"style":496},[473],[224,92693,278],{"className":92694},[500],[224,92696],{"className":92697,"style":496},[473],[224,92699,92701,92704],{"className":92700},[459],[224,92702],{"className":92703,"style":510},[463],[224,92705,1438],{"className":92706},[468]," 取到，为 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Kantorovich 型对偶。",[11,94352,94353],{},"这个对偶表示在机器学习中非常有用，例如 f-GAN 就是基于此构造的（第 5 节详述）。",[613,94355,94357],{"id":94356},"_45-数据处理不等式","4.5 数据处理不等式",[21055,94359,94361,94461],{"title":94360},"命题 4.4（数据处理不等式）",[11,94362,20606,94363,94391,94392,372,94426,94460],{},[224,94364,94366,94379],{"className":94365,"translate":388},[392],[224,94367,94369],{"className":94368},[396],[398,94370,94371],{"xmlns":400},[403,94372,94373,94377],{},[406,94374,94375],{},[409,94376,30652],{},[447,94378,30652],{"encoding":449},[224,94380,94382],{"className":94381,"ariaHidden":455},[454],[224,94383,94385,94388],{"className":94384},[459],[224,94386],{"className":94387,"style":649},[463],[224,94389,30652],{"className":94390,"style":30740},[468,469]," 是一个马尔可夫核（随机变换），记 ",[224,94393,94395,94411],{"className":94394,"translate":388},[392],[224,94396,94398],{"className":94397},[396],[398,94399,94400],{"xmlns":400},[403,94401,94402,94408],{},[406,94403,94404,94406],{},[409,94405,16018],{},[409,94407,30652],{},[447,94409,94410],{"encoding":449},"PK",[224,94412,94414],{"className":94413,"ariaHidden":455},[454],[224,94415,94417,94420,94423],{"className":94416},[459],[224,94418],{"className":94419,"style":649},[463],[224,94421,16018],{"className":94422,"style":653},[468,469],[224,94424,30652],{"className":94425,"style":30740},[468,469],[224,94427,94429,94445],{"className":94428,"translate":388},[392],[224,94430,94432],{"className":94431},[396],[398,94433,94434],{"xmlns":400},[403,94435,94436,94442],{},[406,94437,94438,94440],{},[409,94439,20315],{},[409,94441,30652],{},[447,94443,94444],{"encoding":449},"QK",[224,94446,94448],{"className":94447,"ariaHidden":455},[454],[224,94449,94451,94454,94457],{"className":94450},[459],[224,94452],{"className":94453,"style":20637},[463],[224,94455,20315],{"className":94456},[468,469],[224,94458,30652],{"className":94459,"style":30740},[468,469]," 为输出分布，则",[224,94462,94464],{"className":94463,"translate":388},[387],[224,94465,94467,94519],{"className":94466,"translate":388},[392],[224,94468,94470],{"className":94469},[396],[398,94471,94472],{"xmlns":400,"display":401},[403,94473,94474,94516],{},[406,94475,94476,94482,94484,94486,94488,94490,94492,94494,94496,94498,94504,94506,94508,94510,94512,94514],{},[695,94477,94478,94480],{},[409,94479,20303],{},[409,94481,1705],{},[412,94483,239],{"stretchy":435},[409,94485,16018],{},[409,94487,30652],{},[409,94489,20312],{"mathvariant":678},[409,94491,20315],{},[409,94493,30652],{},[412,94495,1716],{"stretchy":435},[412,94497,2070],{},[695,94499,94500,94502],{},[409,94501,20303],{},[409,94503,1705],{},[412,94505,239],{"stretchy":435},[409,94507,16018],{},[409,94509,20312],{"mathvariant":678},[409,94511,20315],{},[412,94513,1716],{"stretchy":435},[409,94515,688],{"mathvariant":678},[447,94517,94518],{"encoding":449},"D_f(PK\\|QK)\\le D_f(P\\|Q).",[224,94520,94522,94598],{"className":94521,"ariaHidden":455},[454],[224,94523,94525,94528,94568,94571,94574,94577,94580,94583,94586,94589,94592,94595],{"className":94524},[459],[224,94526],{"className":94527,"style":20368},[463],[224,94529,94531,94534],{"className":94530},[468],[224,94532,20303],{"className":94533,"style":2473},[468,469],[224,94535,94537],{"className":94536},[750],[224,94538,94540,94560],{"className":94539},[754,755],[224,94541,94543,94557],{"className":94542},[759],[224,94544,94546],{"className":94545,"style":20387},[763],[224,94547,94548,94551],{"style":20390},[224,94549],{"className":94550,"style":772},[771],[224,94552,94554],{"className":94553},[776,777,778,779],[224,94555,1705],{"className":94556,"style":1740},[468,469,779],[224,94558,787],{"className":94559},[786],[224,94561,94563],{"className":94562},[759],[224,94564,94566],{"className":94565,"style":20409},[763],[224,94567],{},[224,94569,239],{"className":94570},[542],[224,94572,16018],{"className":94573,"style":653},[468,469],[224,94575,30652],{"className":94576,"style":30740},[468,469],[224,94578,20312],{"className":94579},[468],[224,94581,20315],{"className":94582},[468,469],[224,94584,30652],{"className":94585,"style":30740},[468,469],[224,94587,1716],{"className":94588},[549],[224,94590],{"className":94591,"style":496},[473],[224,94593,2070],{"className":94594},[500],[224,94596],{"className":94597,"style":496},[473],[224,94599,94601,94604,94644,94647,94650,94653,94656,94659],{"className":94600},[459],[224,94602],{"className":94603,"style":20368},[463],[224,94605,94607,94610],{"className":94606},[468],[224,94608,20303],{"className":94609,"style":2473},[468,469],[224,94611,94613],{"className":94612},[750],[224,94614,94616,94636],{"className":94615},[754,755],[224,94617,94619,94633],{"className":94618},[759],[224,94620,94622],{"className":94621,"style":20387},[763],[224,94623,94624,94627],{"style":20390},[224,94625],{"className":94626,"style":772},[771],[224,94628,94630],{"className":94629},[776,777,778,779],[224,94631,1705],{"className":94632,"style":1740},[468,469,779],[224,94634,787],{"className":94635},[786],[224,94637,94639],{"className":94638},[759],[224,94640,94642],{"className":94641,"style":20409},[763],[224,94643],{},[224,94645,239],{"className":94646},[542],[224,94648,16018],{"className":94649,"style":653},[468,469],[224,94651,20312],{"className":94652},[468],[224,94654,20315],{"className":94655},[468,469],[224,94657,1716],{"className":94658},[549],[224,94660,688],{"className":94661},[468],[11,94663,94664,31017,94667,7098,94723,7098,94773,75646,94859,94909,94910,94971,94972,12570],{},[378,94665,94666],{},"证明（条件 Jensen）。",[224,94668,94670,94690],{"className":94669,"translate":388},[392],[224,94671,94673],{"className":94672},[396],[398,94674,94675],{"xmlns":400},[403,94676,94677,94687],{},[406,94678,94679,94681,94683,94685],{},[409,94680,25410],{},[412,94682,28834],{},[409,94684,20315],{},[409,94686,30652],{},[447,94688,94689],{"encoding":449},"Y\\sim QK",[224,94691,94693,94711],{"className":94692,"ariaHidden":455},[454],[224,94694,94696,94699,94702,94705,94708],{"className":94695},[459],[224,94697],{"className":94698,"style":649},[463],[224,94700,25410],{"className":94701,"style":474},[468,469],[224,94703],{"className":94704,"style":496},[473],[224,94706,28834],{"className":94707},[500],[224,94709],{"className":94710,"style":496},[473],[224,94712,94714,94717,94720],{"className":94713},[459],[224,94715],{"className":94716,"style":20637},[463],[224,94718,20315],{"className":94719},[468,469],[224,94721,30652],{"className":94722,"style":30740},[468,469],[224,94724,94726,94743],{"className":94725,"translate":388},[392],[224,94727,94729],{"className":94728},[396],[398,94730,94731],{"xmlns":400},[403,94732,94733,94741],{},[406,94734,94735,94737,94739],{},[409,94736,20668],{},[412,94738,28834],{},[409,94740,20315],{},[447,94742,28839],{"encoding":449},[224,94744,94746,94764],{"className":94745,"ariaHidden":455},[454],[224,94747,94749,94752,94755,94758,94761],{"className":94748},[459],[224,94750],{"className":94751,"style":649},[463],[224,94753,20668],{"className":94754,"style":28852},[468,469],[224,94756],{"className":94757,"style":496},[473],[224,94759,28834],{"className":94760},[500],[224,94762],{"className":94763,"style":496},[473],[224,94765,94767,94770],{"className":94766},[459],[224,94768],{"className":94769,"style":20637},[463],[224,94771,20315],{"className":94772},[468,469],[224,94774,94776,94808],{"className":94775,"translate":388},[392],[224,94777,94779],{"className":94778},[396],[398,94780,94781],{"xmlns":400},[403,94782,94783,94805],{},[406,94784,94785,94787,94789,94791,94793,94795,94797,94799,94801,94803],{},[409,94786,25410],{},[409,94788,1083],{"mathvariant":678},[409,94790,20668],{},[412,94792,28834],{},[409,94794,30652],{},[412,94796,239],{"stretchy":435},[412,94798,23331],{},[409,94800,1083],{"mathvariant":678},[409,94802,20668],{},[412,94804,1716],{"stretchy":435},[447,94806,94807],{"encoding":449},"Y|X\\sim K(\\cdot|X)",[224,94809,94811,94835],{"className":94810,"ariaHidden":455},[454],[224,94812,94814,94817,94820,94823,94826,94829,94832],{"className":94813},[459],[224,94815],{"className":94816,"style":535},[463],[224,94818,25410],{"className":94819,"style":474},[468,469],[224,94821,1083],{"className":94822},[468],[224,94824,20668],{"className":94825,"style":28852},[468,469],[224,94827],{"className":94828,"style":496},[473],[224,94830,28834],{"className":94831},[500],[224,94833],{"className":94834,"style":496},[473],[224,94836,94838,94841,94844,94847,94850,94853,94856],{"className":94837},[459],[224,94839],{"className":94840,"style":535},[463],[224,94842,30652],{"className":94843,"style":30740},[468,469],[224,94845,239],{"className":94846},[542],[224,94848,23331],{"className":94849},[468],[224,94851,1083],{"className":94852},[468],[224,94854,20668],{"className":94855,"style":28852},[468,469],[224,94857,1716],{"className":94858},[549],[224,94860,94862,94879],{"className":94861,"translate":388},[392],[224,94863,94865],{"className":94864},[396],[398,94866,94867],{"xmlns":400},[403,94868,94869,94877],{},[406,94870,94871,94873,94875],{},[409,94872,16018],{},[412,94874,12984],{},[409,94876,20315],{},[447,94878,20786],{"encoding":449},[224,94880,94882,94900],{"className":94881,"ariaHidden":455},[454],[224,94883,94885,94888,94891,94894,94897],{"className":94884},[459],[224,94886],{"className":94887,"style":20796},[463],[224,94889,16018],{"className":94890,"style":653},[468,469],[224,94892],{"className":94893,"style":496},[473],[224,94895,12984],{"className":94896},[500],[224,94898],{"className":94899,"style":496},[473],[224,94901,94903,94906],{"className":94902},[459],[224,94904],{"className":94905,"style":20637},[463],[224,94907,20315],{"className":94908},[468,469]," 则 ",[224,94911,94913,94935],{"className":94912,"translate":388},[392],[224,94914,94916],{"className":94915},[396],[398,94917,94918],{"xmlns":400},[403,94919,94920,94932],{},[406,94921,94922,94924,94926,94928,94930],{},[409,94923,16018],{},[409,94925,30652],{},[412,94927,12984],{},[409,94929,20315],{},[409,94931,30652],{},[447,94933,94934],{"encoding":449},"PK\\ll QK",[224,94936,94938,94959],{"className":94937,"ariaHidden":455},[454],[224,94939,94941,94944,94947,94950,94953,94956],{"className":94940},[459],[224,94942],{"className":94943,"style":20796},[463],[224,94945,16018],{"className":94946,"style":653},[468,469],[224,94948,30652],{"className":94949,"style":30740},[468,469],[224,94951],{"className":94952,"style":496},[473],[224,94954,12984],{"className":94955},[500],[224,94957],{"className":94958,"style":496},[473],[224,94960,94962,94965,94968],{"className":94961},[459],[224,94963],{"className":94964,"style":20637},[463],[224,94966,20315],{"className":94967},[468,469],[224,94969,30652],{"className":94970,"style":30740},[468,469],"，且对任意可测集 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若 ",[224,97514,97516,97529],{"className":97515,"translate":388},[392],[224,97517,97519],{"className":97518},[396],[398,97520,97521],{"xmlns":400},[403,97522,97523,97527],{},[406,97524,97525],{},[409,97526,1705],{},[447,97528,1705],{"encoding":449},[224,97530,97532],{"className":97531,"ariaHidden":455},[454],[224,97533,97535,97538],{"className":97534},[459],[224,97536],{"className":97537,"style":2565},[463],[224,97539,1705],{"className":97540,"style":1740},[468,469]," 严格凸，等号成立当且仅当 ",[224,97543,97545,97573],{"className":97544,"translate":388},[392],[224,97546,97548],{"className":97547},[396],[398,97549,97550],{"xmlns":400},[403,97551,97552,97570],{},[406,97553,97554,97560,97562,97564,97566,97568],{},[695,97555,97556,97558],{},[409,97557,20324],{"mathvariant":12251},[409,97559,20315],{},[412,97561,268],{"stretchy":435},[409,97563,1435],{},[409,97565,1083],{"mathvariant":678},[409,97567,25410],{},[412,97569,440],{"stretchy":435},[447,97571,97572],{"encoding":449},"\\mathbb 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本身是 ",[224,97639,97641,97661],{"className":97640,"translate":388},[392],[224,97642,97644],{"className":97643},[396],[398,97645,97646],{"xmlns":400},[403,97647,97648,97658],{},[406,97649,97650,97652,97654,97656],{},[409,97651,685],{},[412,97653,239],{"stretchy":435},[409,97655,25410],{},[412,97657,1716],{"stretchy":435},[447,97659,97660],{"encoding":449},"\\sigma(Y)",[224,97662,97664],{"className":97663,"ariaHidden":455},[454],[224,97665,97667,97670,97673,97676,97679],{"className":97666},[459],[224,97668],{"className":97669,"style":535},[463],[224,97671,685],{"className":97672,"style":492},[468,469],[224,97674,239],{"className":97675},[542],[224,97677,25410],{"className":97678,"style":474},[468,469],[224,97680,1716],{"className":97681},[549],"-可测的（对每个 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可表为 ",[224,97784,97786,97799],{"className":97785,"translate":388},[392],[224,97787,97789],{"className":97788},[396],[398,97790,97791],{"xmlns":400},[403,97792,97793,97797],{},[406,97794,97795],{},[409,97796,25410],{},[447,97798,25410],{"encoding":449},[224,97800,97802],{"className":97801,"ariaHidden":455},[454],[224,97803,97805,97808],{"className":97804},[459],[224,97806],{"className":97807,"style":649},[463],[224,97809,25410],{"className":97810,"style":474},[468,469]," 的函数。此时 ",[224,97813,97815,97828],{"className":97814,"translate":388},[392],[224,97816,97818],{"className":97817},[396],[398,97819,97820],{"xmlns":400},[403,97821,97822,97826],{},[406,97823,97824],{},[409,97825,25410],{},[447,97827,25410],{"encoding":449},[224,97829,97831],{"className":97830,"ariaHidden":455},[454],[224,97832,97834,97837],{"className":97833},[459],[224,97835],{"className":97836,"style":649},[463],[224,97838,25410],{"className":97839,"style":474},[468,469]," 是关于 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h^2\\,dP\u003C\\infty",[224,98080,98082,98141],{"className":98081,"ariaHidden":455},[454],[224,98083,98085,98088,98091,98094,98123,98126,98129,98132,98135,98138],{"className":98084},[459],[224,98086],{"className":98087,"style":70236},[463],[224,98089,21338],{"className":98090,"style":25626},[21531,21532,25625],[224,98092],{"className":98093,"style":724},[473],[224,98095,98097,98100],{"className":98096},[468],[224,98098,58586],{"className":98099},[468,469],[224,98101,98103],{"className":98102},[750],[224,98104,98106],{"className":98105},[754],[224,98107,98109],{"className":98108},[759],[224,98110,98112],{"className":98111,"style":5671},[763],[224,98113,98114,98117],{"style":3288},[224,98115],{"className":98116,"style":772},[771],[224,98118,98120],{"className":98119},[776,777,778,779],[224,98121,1447],{"className":98122},[468,779],[224,98124],{"className":98125,"style":724},[473],[224,98127,13182],{"className":98128},[468,469],[224,98130,16018],{"className":98131,"style":653},[468,469],[224,98133],{"className":98134,"style":496},[473],[224,98136,1577],{"className":98137},[500],[224,98139],{"className":98140,"style":496},[473],[224,98142,98144,98147],{"className":98143},[459],[224,98145],{"className":98146,"style":839},[463],[224,98148,21085],{"className":98149},[468],[224,98151,98153,98193],{"className":98152,"translate":388},[392],[224,98154,98156],{"className":98155},[396],[398,98157,98158],{"xmlns":400},[403,98159,98160,98190],{},[406,98161,98162,98164,98170,98172,98174,98176,98178,98180,98182,98184,98186,98188],{},[409,98163,13182],{},[695,98165,98166,98168],{},[409,98167,20315],{},[409,98169,58583],{},[412,98171,278],{},[412,98173,239],{"stretchy":435},[419,98175,1438],{},[412,98177,319],{},[409,98179,58583],{},[409,98181,58586],{},[412,98183,1716],{"stretchy":435},[423,98185,682],{},[409,98187,13182],{},[409,98189,16018],{},[447,98191,98192],{"encoding":449},"dQ_\\varepsilon=(1+\\varepsilon h)\\,dP",[224,98194,98196,98254,98275],{"className":98195,"ariaHidden":455},[454],[224,98197,98199,98202,98205,98245,98248,98251],{"className":98198},[459],[224,98200],{"className":98201,"style":2565},[463],[224,98203,13182],{"className":98204},[468,469],[224,98206,98208,98211],{"className":98207},[468],[224,98209,20315],{"className":98210},[468,469],[224,98212,98214],{"className":98213},[750],[224,98215,98217,98237],{"className":98216},[754,755],[224,98218,98220,98234],{"className":98219},[759],[224,98221,98223],{"className":98222,"style":64841},[763],[224,98224,98225,98228],{"style":767},[224,98226],{"className":98227,"style":772},[771],[224,98229,98231],{"className":98230},[776,777,778,779],[224,98232,58583],{"className":98233},[468,469,779],[224,98235,787],{"className":98236},[786],[224,98238,98240],{"className":98239},[759],[224,98241,98243],{"className":98242,"style":794},[763],[224,98244],{},[224,98246],{"className":98247,"style":496},[473],[224,98249,278],{"className":98250},[500],[224,98252],{"className":98253,"style":496},[473],[224,98255,98257,98260,98263,98266,98269,98272],{"className":98256},[459],[224,98258],{"className":98259,"style":535},[463],[224,98261,239],{"className":98262},[542],[224,98264,1438],{"className":98265},[468],[224,98267],{"className":98268,"style":474},[473],[224,98270,319],{"className":98271},[478],[224,98273],{"className":98274,"style":474},[473],[224,98276,98278,98281,98284,98287,98290,98293,98296],{"className":98277},[459],[224,98279],{"className":98280,"style":535},[463],[224,98282,58583],{"className":98283},[468,469],[224,98285,58586],{"className":98286},[468,469],[224,98288,1716],{"className":98289},[549],[224,98291],{"className":98292,"style":724},[473],[224,98294,13182],{"className":98295},[468,469],[224,98297,16018],{"className":98298,"style":653},[468,469],"，则当 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记 ",[224,98824,98826,98866],{"className":98825,"translate":388},[392],[224,98827,98829],{"className":98828},[396],[398,98830,98831],{"xmlns":400},[403,98832,98833,98863],{},[406,98834,98835,98837,98839,98841,98847,98849,98851,98853,98855,98857,98859,98861],{},[409,98836,1435],{},[412,98838,278],{},[409,98840,13182],{},[695,98842,98843,98845],{},[409,98844,20315],{},[409,98846,58583],{},[409,98848,12413],{"mathvariant":678},[409,98850,13182],{},[409,98852,16018],{},[412,98854,278],{},[419,98856,1438],{},[412,98858,319],{},[409,98860,58583],{},[409,98862,58586],{},[447,98864,98865],{"encoding":449},"t=dQ_\\varepsilon\u002FdP=1+\\varepsilon h",[224,98867,98869,98887,98954,98972],{"className":98868,"ariaHidden":455},[454],[224,98870,98872,98875,98878,98881,98884],{"className":98871},[459],[224,98873],{"className":98874,"style":25353},[463],[224,98876,1435],{"className":98877},[468,469],[224,98879],{"className":98880,"style":496},[473],[224,98882,278],{"className":98883},[500],[224,98885],{"className":98886,"style":496},[473],[224,98888,98890,98893,98896,98936,98939,98942,98945,98948,98951],{"className":98889},[459],[224,98891],{"className":98892,"style":535},[463],[224,98894,13182],{"className":98895},[468,469],[224,98897,98899,98902],{"className":98898},[468],[224,98900,20315],{"className":98901},[468,469],[224,98903,98905],{"className":98904},[750],[224,98906,98908,98928],{"className":98907},[754,755],[224,98909,98911,98925],{"className":98910},[759],[224,98912,98914],{"className":98913,"style":64841},[763],[224,98915,98916,98919],{"style":767},[224,98917],{"className":98918,"style":772},[771],[224,98920,98922],{"className":98921},[776,777,778,779],[224,98923,58583],{"className":98924},[468,469,779],[224,98926,787],{"className":98927},[786],[224,98929,98931],{"className":98930},[759],[224,98932,98934],{"className":98933,"style":794},[763],[224,98935],{},[224,98937,12413],{"className":98938},[468],[224,98940,13182],{"className":98941},[468,469],[224,98943,16018],{"className":98944,"style":653},[468,469],[224,98946],{"className":98947,"style":496},[473],[224,98949,278],{"className":98950},[500],[224,98952],{"className":98953,"style":496},[473],[224,98955,98957,98960,98963,98966,98969],{"className":98956},[459],[224,98958],{"className":98959,"style":7047},[463],[224,98961,1438],{"className":98962},[468],[224,98964],{"className":98965,"style":474},[473],[224,98967,319],{"className":98968},[478],[224,98970],{"className":98971,"style":474},[473],[224,98973,98975,98978,98981],{"className":98974},[459],[224,98976],{"className":98977,"style":7642},[463],[224,98979,58583],{"className":98980},[468,469],[224,98982,58586],{"className":98983},[468,469],"，由 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做 Taylor 展开：",[224,99125,99127],{"className":99126,"translate":388},[387],[224,99128,99130,99358],{"className":99129,"translate":388},[392],[224,99131,99133],{"className":99132},[396],[398,99134,99135],{"xmlns":400,"display":401},[403,99136,99137,99355],{},[5209,99138,99139,99201,99275],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,99140,99141,99167],{},[5218,99142,99143],{},[5221,99144,99145],{"scriptlevel":284,"displaystyle":455},[406,99146,99147,99153,99155,99161,99163,99165],{},[695,99148,99149,99151],{},[409,99150,20303],{},[409,99152,1705],{},[412,99154,239],{"stretchy":435},[695,99156,99157,99159],{},[409,99158,20315],{},[409,99160,58583],{},[409,99162,20312],{"mathvariant":678},[409,99164,16018],{},[412,99166,1716],{"stretchy":435},[5218,99168,99169],{},[5221,99170,99171],{"scriptlevel":284,"displaystyle":455},[406,99172,99173,99175,99177,99183,99185,99187,99189,99191,99193,99195,99197,99199],{},[406,99174],{},[412,99176,278],{},[695,99178,99179,99181],{},[409,99180,20324],{"mathvariant":12251},[409,99182,16018],{},[412,99184,268],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,99186,1705],{},[412,99188,239],{"stretchy":435},[419,99190,1438],{},[412,99192,319],{},[409,99194,58583],{},[409,99196,58586],{},[412,99198,1716],{"stretchy":435},[412,99200,440],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[5215,99202,99203,99209],{},[5218,99204,99205],{},[5221,99206,99207],{"scriptlevel":284,"displaystyle":455},[406,99208],{},[5218,99210,99211],{},[5221,99212,99213],{"scriptlevel":284,"displaystyle":455},[406,99214,99215,99217,99219,99237,99239,99241,99247,99249,99251,99253],{},[406,99216],{},[412,99218,278],{},[1965,99220,99221,99235],{},[1965,99222,99223,99233],{},[406,99224,99225,99227,99229,99231],{},[409,99226,1705],{},[412,99228,239],{"stretchy":435},[419,99230,1438],{},[412,99232,1716],{"stretchy":435},[412,99234,1987],{"stretchy":455},[419,99236,284],{},[412,99238,319],{},[409,99240,58583],{},[3248,99242,99243,99245],{},[409,99244,1705],{},[412,99246,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,99248,239],{"stretchy":435},[419,99250,1438],{},[412,99252,1716],{"stretchy":435},[1965,99254,99255,99273],{},[1965,99256,99257,99271],{},[406,99258,99259,99265,99267,99269],{},[695,99260,99261,99263],{},[409,99262,20324],{"mathvariant":12251},[409,99264,16018],{},[412,99266,268],{"stretchy":435},[409,99268,58586],{},[412,99270,440],{"stretchy":435},[412,99272,1987],{"stretchy":455},[419,99274,284],{},[5215,99276,99277,99283],{},[5218,99278,99279],{},[5221,99280,99281],{"scriptlevel":284,"displaystyle":455},[406,99282],{},[5218,99284,99285],{},[5221,99286,99287],{"scriptlevel":284,"displaystyle":455},[406,99288,99289,99291,99293,99295,99305,99315,99317,99319,99321,99323,99329,99331,99337,99339,99341,99343,99345,99351,99353],{},[406,99290],{},[473,99292],{"width":4247},[412,99294,319],{},[20338,99296,99297,99303],{},[3248,99298,99299,99301],{},[409,99300,58583],{},[419,99302,1447],{},[419,99304,1447],{},[3248,99306,99307,99309],{},[409,99308,1705],{},[406,99310,99311,99313],{},[412,99312,6933],{"mathvariant":678},[412,99314,6933],{"mathvariant":678},[412,99316,239],{"stretchy":435},[419,99318,1438],{},[412,99320,1716],{"stretchy":435},[423,99322,682],{},[695,99324,99325,99327],{},[409,99326,20324],{"mathvariant":12251},[409,99328,16018],{},[412,99330,268],{"stretchy":435},[3248,99332,99333,99335],{},[409,99334,58586],{},[419,99336,1447],{},[412,99338,440],{"stretchy":435},[412,99340,319],{},[409,99342,39163],{},[412,99344,239],{"stretchy":435},[3248,99346,99347,99349],{},[409,99348,58583],{},[419,99350,3594],{},[412,99352,1716],{"stretchy":435},[409,99354,688],{"mathvariant":678},[447,99356,99357],{"encoding":449},"\\begin{aligned}\nD_f(Q_\\varepsilon\\|P)\n&=\\mathbb E_P\\big[f(1+\\varepsilon h)\\big]\\\\[2pt]\n&=\\underbrace{f(1)}_{0}\n+\\varepsilon f'(1)\n\\underbrace{\\mathbb E_P[h]}_{0}\\\\\n&\\qquad+\\frac{\\varepsilon^2}{2}f''(1)\\,\n\\mathbb 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f-散度都只差一个常数 ",[224,100272,100274,100301],{"className":100273,"translate":388},[392],[224,100275,100277],{"className":100276},[396],[398,100278,100279],{"xmlns":400},[403,100280,100281,100299],{},[406,100282,100283,100293,100295,100297],{},[3248,100284,100285,100287],{},[409,100286,1705],{},[406,100288,100289,100291],{},[412,100290,6933],{"mathvariant":678},[412,100292,6933],{"mathvariant":678},[412,100294,239],{"stretchy":435},[419,100296,1438],{},[412,100298,1716],{"stretchy":435},[447,100300,59450],{"encoding":449},[224,100302,100304],{"className":100303,"ariaHidden":455},[454],[224,100305,100307,100310,100342,100345,100348],{"className":100306},[459],[224,100308],{"className":100309,"style":6966},[463],[224,100311,100313,100316],{"className":100312},[468],[224,100314,1705],{"className":100315,"style":1740},[468,469],[224,100317,100319],{"className":100318},[750],[224,100320,100322],{"className":100321},[754],[224,100323,100325],{"className":100324},[759],[224,100326,100328],{"className":100327,"style":6992},[763],[224,100329,100330,100333],{"style":3288},[224,100331],{"className":100332,"style":772},[771],[224,100334,100336],{"className":100335},[776,777,778,779],[224,100337,100339],{"className":100338},[468,779],[224,100340,41686],{"className":100341},[468,779],[224,100343,239],{"className":100344},[542],[224,100346,1438],{"className":100347},[468],[224,100349,1716],{"className":100350},[549],"，形状完全一致。",[224,100353,100355,100384],{"className":100354,"translate":388},[392],[224,100356,100358],{"className":100357},[396],[398,100359,100360],{"xmlns":400},[403,100361,100362,100382],{},[406,100363,100364,100366,100368,100370,100372,100374,100376,100378,100380],{},[409,100365,1705],{},[412,100367,239],{"stretchy":435},[409,100369,1435],{},[412,100371,1716],{"stretchy":435},[412,100373,278],{},[409,100375,1435],{},[409,100377,24131],{},[412,100379,24134],{},[409,100381,1435],{},[447,100383,24139],{"encoding":449},[224,100385,100387,100414],{"className":100386,"ariaHidden":455},[454],[224,100388,100390,100393,100396,100399,100402,100405,100408,100411],{"className":100389},[459],[224,100391],{"className":100392,"style":535},[463],[224,100394,1705],{"className":100395,"style":1740},[468,469],[224,100397,239],{"className":100398},[542],[224,100400,1435],{"className":100401},[468,469],[224,100403,1716],{"className":100404},[549],[224,100406],{"className":100407,"style":496},[473],[224,100409,278],{"className":100410},[500],[224,100412],{"className":100413,"style":496},[473],[224,100415,100417,100420,100423,100426,100431,100434],{"className":100416},[459],[224,100418],{"className":100419,"style":2565},[463],[224,100421,1435],{"className":100422},[468,469],[224,100424],{"className":100425,"style":724},[473],[224,100427,24185,100429],{"className":100428},[21531],[224,100430,24189],{"style":24188},[224,100432],{"className":100433,"style":724},[473],[224,100435,1435],{"className":100436},[468,469],[224,100438,100440,100472],{"className":100439,"translate":388},[392],[224,100441,100443],{"className":100442},[396],[398,100444,100445],{"xmlns":400},[403,100446,100447,100469],{},[406,100448,100449,100459,100461,100463,100465,100467],{},[3248,100450,100451,100453],{},[409,100452,1705],{},[406,100454,100455,100457],{},[412,100456,6933],{"mathvariant":678},[412,100458,6933],{"mathvariant":678},[412,100460,239],{"stretchy":435},[419,100462,1438],{},[412,100464,1716],{"stretchy":435},[412,100466,278],{},[419,100468,1438],{},[447,100470,100471],{"encoding":449},"f''(1)=1",[224,100473,100475,100531],{"className":100474,"ariaHidden":455},[454],[224,100476,100478,100481,100513,100516,100519,100522,100525,100528],{"className":100477},[459],[224,100479],{"className":100480,"style":6966},[463],[224,100482,100484,100487],{"className":100483},[468],[224,100485,1705],{"className":100486,"style":1740},[468,469],[224,100488,100490],{"className":100489},[750],[224,100491,100493],{"className":100492},[754],[224,100494,100496],{"className":100495},[759],[224,100497,100499],{"className":100498,"style":6992},[763],[224,100500,100501,100504],{"style":3288},[224,100502],{"className":100503,"style":772},[771],[224,100505,100507],{"className":100506},[776,777,778,779],[224,100508,100510],{"className":100509},[468,779],[224,100511,41686],{"className":100512},[468,779],[224,100514,239],{"className":100515},[542],[224,100517,1438],{"className":100518},[468],[224,100520,1716],{"className":100521},[549],[224,100523],{"className":100524,"style":496},[473],[224,100526,278],{"className":100527},[500],[224,100529],{"className":100530,"style":496},[473],[224,100532,100534,100537],{"className":100533},[459],[224,100535],{"className":100536,"style":510},[463],[224,100538,1438],{"className":100539},[468],"）即得 ",[224,100542,100544,100580],{"className":100543,"translate":388},[392],[224,100545,100547],{"className":100546},[396],[398,100548,100549],{"xmlns":400},[403,100550,100551,100577],{},[406,100552,100553,100563,100565,100571],{},[695,100554,100555,100557],{},[409,100556,20303],{},[406,100558,100559,100561],{},[409,100560,30652],{},[409,100562,41831],{},[412,100564,52884],{},[20338,100566,100567,100569],{},[419,100568,1438],{},[419,100570,1447],{},[3248,100572,100573,100575],{},[409,100574,57370],{},[419,100576,1447],{},[447,100578,100579],{"encoding":449},"D_{KL}\\approx\\frac12\\chi^2",[224,100581,100583,100644],{"className":100582,"ariaHidden":455},[454],[224,100584,100586,100589,100635,100638,100641],{"className":100585},[459],[224,100587],{"className":100588,"style":35905},[463],[224,100590,100592,100595],{"className":100591},[468],[224,100593,20303],{"className":100594,"style":2473},[468,469],[224,100596,100598],{"className":100597},[750],[224,100599,100601,100627],{"className":100600},[754,755],[224,100602,100604,100624],{"className":100603},[759],[224,100605,100607],{"className":100606,"style":764},[763],[224,100608,100609,100612],{"style":20390},[224,100610],{"className":100611,"style":772},[771],[224,100613,100615],{"className":100614},[776,777,778,779],[224,100616,100618,100621],{"className":100617},[468,779],[224,100619,30652],{"className":100620,"style":30740},[468,469,779],[224,100622,41831],{"className":100623},[468,469,779],[224,100625,787],{"className":100626},[786],[224,100628,100630],{"className":100629},[759],[224,100631,100633],{"className":100632,"style":794},[763],[224,100634],{},[224,100636],{"className":100637,"style":496},[473],[224,100639,52884],{"className":100640},[500],[224,100642],{"className":100643,"style":496},[473],[224,100645,100647,100650,100718],{"className":100646},[459],[224,100648],{"className":100649,"style":22544},[463],[224,100651,100653,100656,100715],{"className":100652},[468],[224,100654],{"className":100655},[542,8479],[224,100657,100659],{"className":100658},[20338],[224,100660,100662,100707],{"className":100661},[754,755],[224,100663,100665,100704],{"className":100664},[759],[224,100666,100668,100682,100690],{"className":100667,"style":22563},[763],[224,100669,100670,100673],{"style":22566},[224,100671],{"className":100672,"style":2108},[771],[224,100674,100676],{"className":100675},[776,777,778,779],[224,100677,100679],{"className":100678},[468,779],[224,100680,1447],{"className":100681},[468,779],[224,100683,100684,100687],{"style":20546},[224,100685],{"className":100686,"style":2108},[771],[224,100688],{"className":100689,"style":20554},[20553],[224,100691,100692,100695],{"style":22589},[224,100693],{"className":100694,"style":2108},[771],[224,100696,100698],{"className":100697},[776,777,778,779],[224,100699,100701],{"className":100700},[468,779],[224,100702,1438],{"className":100703},[468,779],[224,100705,787],{"className":100706},[786],[224,100708,100710],{"className":100709},[759],[224,100711,100713],{"className":100712,"style":22611},[763],[224,100714],{},[224,100716],{"className":100717},[549,8479],[224,100719,100721,100724],{"className":100720},[468],[224,100722,57370],{"className":100723},[468,469],[224,100725,100727],{"className":100726},[750],[224,100728,100730],{"className":100729},[754],[224,100731,100733],{"className":100732},[759],[224,100734,100736],{"className":100735,"style":5671},[763],[224,100737,100738,100741],{"style":3288},[224,100739],{"className":100740,"style":772},[771],[224,100742,100744],{"className":100743},[776,777,778,779],[224,100745,1447],{"className":100746},[468,779],"；取 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2 或换算成 ",[224,101399,101401,101421],{"className":101400,"translate":388},[392],[224,101402,101404],{"className":101403},[396],[398,101405,101406],{"xmlns":400},[403,101407,101408,101418],{},[406,101409,101410,101412,101414,101416],{},[419,101411,1438],{},[412,101413,5268],{},[409,101415,9889],{},[409,101417,16021],{},[447,101419,101420],{"encoding":449},"1-BC",[224,101422,101424,101442],{"className":101423,"ariaHidden":455},[454],[224,101425,101427,101430,101433,101436,101439],{"className":101426},[459],[224,101428],{"className":101429,"style":7047},[463],[224,101431,1438],{"className":101432},[468],[224,101434],{"className":101435,"style":474},[473],[224,101437,5268],{"className":101438},[478],[224,101440],{"className":101441,"style":474},[473],[224,101443,101445,101448,101451],{"className":101444},[459],[224,101446],{"className":101447,"style":649},[463],[224,101449,9889],{"className":101450,"style":9923},[468,469],[224,101452,16021],{"className":101453,"style":30740},[468,469]," 就是 3.4 节引用的 Bhattacharyya 约定 ",[224,101456,101458,101490],{"className":101457,"translate":388},[392],[224,101459,101461],{"className":101460},[396],[398,101462,101463],{"xmlns":400},[403,101464,101465,101487],{},[406,101466,101467,101473,101475,101481],{},[3248,101468,101469,101471],{},[409,101470,53807],{},[419,101472,1447],{},[412,101474,52884],{},[20338,101476,101477,101479],{},[419,101478,1438],{},[419,101480,5430],{},[3248,101482,101483,101485],{},[409,101484,57370],{},[419,101486,1447],{},[447,101488,101489],{"encoding":449},"H^2\\approx\\frac18\\chi^2",[224,101491,101493,101537],{"className":101492,"ariaHidden":455},[454],[224,101494,101496,101499,101528,101531,101534],{"className":101495},[459],[224,101497],{"className":101498,"style":5671},[463],[224,101500,101502,101505],{"className":101501},[468],[224,101503,53807],{"className":101504,"style":53961},[468,469],[224,101506,101508],{"className":101507},[750],[224,101509,101511],{"className":101510},[754],[224,101512,101514],{"className":101513},[759],[224,101515,101517],{"className":101516,"style":5671},[763],[224,101518,101519,101522],{"style":3288},[224,101520],{"className":101521,"style":772},[771],[224,101523,101525],{"className":101524},[776,777,778,779],[224,101526,1447],{"className":101527},[468,779],[224,101529],{"className":101530,"style":496},[473],[224,101532,52884],{"className":101533},[500],[224,101535],{"className":101536,"style":496},[473],[224,101538,101540,101543,101611],{"className":101539},[459],[224,101541],{"className":101542,"style":22544},[463],[224,101544,101546,101549,101608],{"className":101545},[468],[224,101547],{"className":101548},[542,8479],[224,101550,101552],{"className":101551},[20338],[224,101553,101555,101600],{"className":101554},[754,755],[224,101556,101558,101597],{"className":101557},[759],[224,101559,101561,101575,101583],{"className":101560,"style":22563},[763],[224,101562,101563,101566],{"style":22566},[224,101564],{"className":101565,"style":2108},[771],[224,101567,101569],{"className":101568},[776,777,778,779],[224,101570,101572],{"className":101571},[468,779],[224,101573,5430],{"className":101574},[468,779],[224,101576,101577,101580],{"style":20546},[224,101578],{"className":101579,"style":2108},[771],[224,101581],{"className":101582,"style":20554},[20553],[224,101584,101585,101588],{"style":22589},[224,101586],{"className":101587,"style":2108},[771],[224,101589,101591],{"className":101590},[776,777,778,779],[224,101592,101594],{"className":101593},[468,779],[224,101595,1438],{"className":101596},[468,779],[224,101598,787],{"className":101599},[786],[224,101601,101603],{"className":101602},[759],[224,101604,101606],{"className":101605,"style":22611},[763],[224,101607],{},[224,101609],{"className":101610},[549,8479],[224,101612,101614,101617],{"className":101613},[468],[224,101615,57370],{"className":101616},[468,469],[224,101618,101620],{"className":101619},[750],[224,101621,101623],{"className":101622},[754],[224,101624,101626],{"className":101625},[759],[224,101627,101629],{"className":101628,"style":5671},[763],[224,101630,101631,101634],{"style":3288},[224,101632],{"className":101633,"style":772},[771],[224,101635,101637],{"className":101636},[776,777,778,779],[224,101638,1447],{"className":101639},[468,779],"。共同因子 ",[224,101642,101644,101696],{"className":101643,"translate":388},[392],[224,101645,101647],{"className":101646},[396],[398,101648,101649],{"xmlns":400},[403,101650,101651,101693],{},[406,101652,101653,101659,101661,101667,101669,101671,101673,101679,101681,101687,101689,101691],{},[3248,101654,101655,101657],{},[409,101656,58583],{},[419,101658,1447],{},[412,101660,21338],{},[3248,101662,101663,101665],{},[409,101664,58586],{},[419,101666,1447],{},[409,101668,13182],{},[409,101670,16018],{},[412,101672,278],{},[3248,101674,101675,101677],{},[409,101676,57370],{},[419,101678,1447],{},[412,101680,239],{"stretchy":435},[695,101682,101683,101685],{},[409,101684,20315],{},[409,101686,58583],{},[409,101688,20312],{"mathvariant":678},[409,101690,16018],{},[412,101692,1716],{"stretchy":435},[447,101694,101695],{"encoding":449},"\\varepsilon^2\\int h^2 dP=\\chi^2(Q_\\varepsilon\\|P)",[224,101697,101699,101787],{"className":101698,"ariaHidden":455},[454],[224,101700,101702,101705,101734,101737,101740,101743,101772,101775,101778,101781,101784],{"className":101701},[459],[224,101703],{"className":101704,"style":70236},[463],[224,101706,101708,101711],{"className":101707},[468],[224,101709,58583],{"className":101710},[468,469],[224,101712,101714],{"className":101713},[750],[224,101715,101717],{"className":101716},[754],[224,101718,101720],{"className":101719},[759],[224,101721,101723],{"className":101722,"style":5671},[763],[224,101724,101725,101728],{"style":3288},[224,101726],{"className":101727,"style":772},[771],[224,101729,101731],{"className":101730},[776,777,778,779],[224,101732,1447],{"className":101733},[468,779],[224,101735],{"className":101736,"style":724},[473],[224,101738,21338],{"className":101739,"style":25626},[21531,21532,25625],[224,101741],{"className":101742,"style":724},[473],[224,101744,101746,101749],{"className":101745},[468],[224,101747,58586],{"className":101748},[468,469],[224,101750,101752],{"className":101751},[750],[224,101753,101755],{"className":101754},[754],[224,101756,101758],{"className":101757},[759],[224,101759,101761],{"className":101760,"style":5671},[763],[224,101762,101763,101766],{"style":3288},[224,101764],{"className":101765,"style":772},[771],[224,101767,101769],{"className":101768},[776,777,778,779],[224,101770,1447],{"className":101771},[468,779],[224,101773,13182],{"className":101774},[468,469],[224,101776,16018],{"className":101777,"style":653},[468,469],[224,101779],{"className":101780,"style":496},[473],[224,101782,278],{"className":101783},[500],[224,101785],{"className":101786,"style":496},[473],[224,101788,101790,101793,101822,101825,101865,101868,101871],{"className":101789},[459],[224,101791],{"className":101792,"style":39943},[463],[224,101794,101796,101799],{"className":101795},[468],[224,101797,57370],{"className":101798},[468,469],[224,101800,101802],{"className":101801},[750],[224,101803,101805],{"className":101804},[754],[224,101806,101808],{"className":101807},[759],[224,101809,101811],{"className":101810,"style":5671},[763],[224,101812,101813,101816],{"style":3288},[224,101814],{"className":101815,"style":772},[771],[224,101817,101819],{"className":101818},[776,777,778,779],[224,101820,1447],{"className":101821},[468,779],[224,101823,239],{"className":101824},[542],[224,101826,101828,101831],{"className":101827},[468],[224,101829,20315],{"className":101830},[468,469],[224,101832,101834],{"className":101833},[750],[224,101835,101837,101857],{"className":101836},[754,755],[224,101838,101840,101854],{"className":101839},[759],[224,101841,101843],{"className":101842,"style":64841},[763],[224,101844,101845,101848],{"style":767},[224,101846],{"className":101847,"style":772},[771],[224,101849,101851],{"className":101850},[776,777,778,779],[224,101852,58583],{"className":101853},[468,469,779],[224,101855,787],{"className":101856},[786],[224,101858,101860],{"className":101859},[759],[224,101861,101863],{"className":101862,"style":794},[763],[224,101864],{},[224,101866,20312],{"className":101867},[468],[224,101869,16018],{"className":101870,"style":653},[468,469],[224,101872,1716],{"className":101873},[549]," 就是",[378,101876,101877],{},"Fisher 信息度量","：它给出统计流形上的黎曼度量，而 ",[224,101880,101882,101909],{"className":101881,"translate":388},[392],[224,101883,101885],{"className":101884},[396],[398,101886,101887],{"xmlns":400},[403,101888,101889,101907],{},[406,101890,101891,101901,101903,101905],{},[3248,101892,101893,101895],{},[409,101894,1705],{},[406,101896,101897,101899],{},[412,101898,6933],{"mathvariant":678},[412,101900,6933],{"mathvariant":678},[412,101902,239],{"stretchy":435},[419,101904,1438],{},[412,101906,1716],{"stretchy":435},[447,101908,59450],{"encoding":449},[224,101910,101912],{"className":101911,"ariaHidden":455},[454],[224,101913,101915,101918,101950,101953,101956],{"className":101914},[459],[224,101916],{"className":101917,"style":6966},[463],[224,101919,101921,101924],{"className":101920},[468],[224,101922,1705],{"className":101923,"style":1740},[468,469],[224,101925,101927],{"className":101926},[750],[224,101928,101930],{"className":101929},[754],[224,101931,101933],{"className":101932},[759],[224,101934,101936],{"className":101935,"style":6992},[763],[224,101937,101938,101941],{"style":3288},[224,101939],{"className":101940,"style":772},[771],[224,101942,101944],{"className":101943},[776,777,778,779],[224,101945,101947],{"className":101946},[468,779],[224,101948,41686],{"className":101949},[468,779],[224,101951,239],{"className":101952},[542],[224,101954,1438],{"className":101955},[468],[224,101957,1716],{"className":101958},[549]," 只是「单位选择」。这也解释了为什么 GAN 中「选 JS 还是选 KL」在小偏差区几乎无差别，差别只在远离平衡或支撑不重叠时才显现。",[613,101961,101963],{"id":101962},"_47-不等式网络","4.7 不等式网络",[21055,101965,101967,102161,102666,103126],{"title":101966},"命题 4.6（不等式网络）",[11,101968,101969,101970,102160],{},"记 ",[224,101971,101973,102011],{"className":101972,"translate":388},[392],[224,101974,101976],{"className":101975},[396],[398,101977,101978],{"xmlns":400},[403,101979,101980,102008],{},[406,101981,101982,101988,101990,101998,102000,102002,102004,102006],{},[3248,101983,101984,101986],{},[409,101985,53807],{},[419,101987,1447],{},[412,101989,278],{},[27527,101991,101992,101994,101996],{},[409,101993,53807],{},[423,101995,62050],{},[419,101997,1447],{},[412,101999,278],{},[419,102001,1438],{},[412,102003,5268],{},[409,102005,9889],{},[409,102007,16021],{},[447,102009,102010],{"encoding":449},"H^2=H^2_{\\text{Bhattacharyya}}=1-BC",[224,102012,102014,102058,102130,102148],{"className":102013,"ariaHidden":455},[454],[224,102015,102017,102020,102049,102052,102055],{"className":102016},[459],[224,102018],{"className":102019,"style":5671},[463],[224,102021,102023,102026],{"className":102022},[468],[224,102024,53807],{"className":102025,"style":53961},[468,469],[224,102027,102029],{"className":102028},[750],[224,102030,102032],{"className":102031},[754],[224,102033,102035],{"className":102034},[759],[224,102036,102038],{"className":102037,"style":5671},[763],[224,102039,102040,102043],{"style":3288},[224,102041],{"className":102042,"style":772},[771],[224,102044,102046],{"className":102045},[776,777,778,779],[224,102047,1447],{"className":102048},[468,779],[224,102050],{"className":102051,"style":496},[473],[224,102053,278],{"className":102054},[500],[224,102056],{"className":102057,"style":496},[473],[224,102059,102061,102064,102121,102124,102127],{"className":102060},[459],[224,102062],{"className":102063,"style":62798},[463],[224,102065,102067,102070],{"className":102066},[468],[224,102068,53807],{"className":102069,"style":53961},[468,469],[224,102071,102073],{"className":102072},[750],[224,102074,102076,102113],{"className":102075},[754,755],[224,102077,102079,102110],{"className":102078},[759],[224,102080,102082,102099],{"className":102081,"style":5671},[763],[224,102083,102084,102087],{"style":62822},[224,102085],{"className":102086,"style":772},[771],[224,102088,102090],{"className":102089},[776,777,778,779],[224,102091,102093],{"className":102092},[468,779],[224,102094,102096],{"className":102095},[468,11373,779],[224,102097,62050],{"className":102098},[468,779],[224,102100,102101,102104],{"style":3288},[224,102102],{"className":102103,"style":772},[771],[224,102105,102107],{"className":102106},[776,777,778,779],[224,102108,1447],{"className":102109},[468,779],[224,102111,787],{"className":102112},[786],[224,102114,102116],{"className":102115},[759],[224,102117,102119],{"className":102118,"style":62858},[763],[224,102120],{},[224,102122],{"className":102123,"style":496},[473],[224,102125,278],{"className":102126},[500],[224,102128],{"className":102129,"style":496},[473],[224,102131,102133,102136,102139,102142,102145],{"className":102132},[459],[224,102134],{"className":102135,"style":7047},[463],[224,102137,1438],{"className":102138},[468],[224,102140],{"className":102141,"style":474},[473],[224,102143,5268],{"className":102144},[478],[224,102146],{"className":102147,"style":474},[473],[224,102149,102151,102154,102157],{"className":102150},[459],[224,102152],{"className":102153,"style":649},[463],[224,102155,9889],{"className":102156,"style":9923},[468,469],[224,102158,16021],{"className":102159,"style":30740},[468,469],"，则对概率测度有",[224,102162,102164],{"className":102163,"translate":388},[387],[224,102165,102167,102287],{"className":102166,"translate":388},[392],[224,102168,102170],{"className":102169},[396],[398,102171,102172],{"xmlns":400,"display":401},[403,102173,102174,102284],{},[5209,102175,102176,102210],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,102177,102178,102184],{},[5218,102179,102180],{},[5221,102181,102182],{"scriptlevel":284,"displaystyle":455},[406,102183],{},[5218,102185,102186],{},[5221,102187,102188],{"scriptlevel":284,"displaystyle":455},[406,102189,102190,102192],{},[406,102191],{},[406,102193,102194,102196,102198,102200,102202,102204,102206,102208],{},[409,102195,16018],{"mathvariant":678},[409,102197,1880],{"mathvariant":678},[409,102199,3254],{"mathvariant":678},[409,102201,1858],{"mathvariant":678},[409,102203,7630],{"mathvariant":678},[409,102205,1861],{"mathvariant":678},[409,102207,2061],{"mathvariant":678},[412,102209,1071],{},[5215,102211,102212,102218],{},[5218,102213,102214],{},[5221,102215,102216],{"scriptlevel":284,"displaystyle":455},[406,102217],{},[5218,102219,102220],{},[5221,102221,102222],{"scriptlevel":284,"displaystyle":455},[406,102223,102224,102226,102228,102238,102240,102242,102244,102246,102248,102250,102282],{},[406,102225],{},[473,102227],{"width":4247},[695,102229,102230,102232],{},[409,102231,20303],{},[406,102233,102234,102236],{},[409,102235,637],{},[409,102237,16059],{},[412,102239,239],{"stretchy":435},[409,102241,16018],{},[412,102243,1877],{"separator":455},[409,102245,20315],{},[412,102247,1716],{"stretchy":435},[412,102249,2070],{},[54262,102251,102252],{},[406,102253,102254,102262,102272,102274,102276,102278,102280],{},[5221,102255,102256],{"scriptlevel":284,"displaystyle":435},[20338,102257,102258,102260],{},[419,102259,1438],{},[419,102261,1447],{},[695,102263,102264,102266],{},[409,102265,20303],{},[406,102267,102268,102270],{},[409,102269,30652],{},[409,102271,41831],{},[412,102273,239],{"stretchy":435},[409,102275,16018],{},[409,102277,20312],{"mathvariant":678},[409,102279,20315],{},[412,102281,1716],{"stretchy":435},[412,102283,1877],{"separator":455},[447,102285,102286],{"encoding":449},"\\begin{aligned}\n&\\mathrm{Pinsker:}\\\\\n&\\qquad D_{TV}(P,Q)\\le\n\\sqrt{\\tfrac12 D_{KL}(P\\|Q)},\n\\end{aligned}",[224,102288,102290],{"className":102289,"ariaHidden":455},[454],[224,102291,102293,102297],{"className":102292},[459],[224,102294],{"className":102295,"style":102296},[463],"height:3.34em;vertical-align:-1.42em;",[224,102298,102300],{"className":102299},[468],[224,102301,102303,102347],{"className":102302},[5209],[224,102304,102306],{"className":102305},[5331],[224,102307,102309,102338],{"className":102308},[754,755],[224,102310,102312,102335],{"className":102311},[759],[224,102313,102316,102326],{"className":102314,"style":102315},[763],"height:1.92em;",[224,102317,102319,102323],{"style":102318},"top:-4.3639em;",[224,102320],{"className":102321,"style":102322},[771],"height:3.2839em;",[224,102324],{"className":102325},[468],[224,102327,102329,102332],{"style":102328},"top:-2.42em;",[224,102330],{"className":102331,"style":102322},[771],[224,102333],{"className":102334},[468],[224,102336,787],{"className":102337},[786],[224,102339,102341],{"className":102340},[759],[224,102342,102345],{"className":102343,"style":102344},[763],"height:1.42em;",[224,102346],{},[224,102348,102350],{"className":102349},[5399],[224,102351,102353,102658],{"className":102352},[754,755],[224,102354,102356,102655],{"className":102355},[759],[224,102357,102359,102383],{"className":102358,"style":102315},[763],[224,102360,102361,102364],{"style":102318},[224,102362],{"className":102363,"style":102322},[771],[224,102365,102367,102370],{"className":102366},[468],[224,102368],{"className":102369},[468],[224,102371,102373,102377,102380],{"className":102372},[468],[224,102374,102376],{"className":102375},[468,1906],"Pinsker",[224,102378],{"className":102379,"style":496},[473],[224,102381,1071],{"className":102382},[500],[224,102384,102385,102388],{"style":102328},[224,102386],{"className":102387,"style":102322},[771],[224,102389,102391,102394,102397,102443,102446,102449,102452,102455,102458,102461,102464,102467,102470,102652],{"className":102390},[468],[224,102392],{"className":102393},[468],[224,102395],{"className":102396,"style":4416},[473],[224,102398,102400,102403],{"className":102399},[468],[224,102401,20303],{"className":102402,"style":2473},[468,469],[224,102404,102406],{"className":102405},[750],[224,102407,102409,102435],{"className":102408},[754,755],[224,102410,102412,102432],{"className":102411},[759],[224,102413,102415],{"className":102414,"style":764},[763],[224,102416,102417,102420],{"style":20390},[224,102418],{"className":102419,"style":772},[771],[224,102421,102423],{"className":102422},[776,777,778,779],[224,102424,102426,102429],{"className":102425},[468,779],[224,102427,637],{"className":102428,"style":653},[468,469,779],[224,102430,16059],{"className":102431,"style":474},[468,469,779],[224,102433,787],{"className":102434},[786],[224,102436,102438],{"className":102437},[759],[224,102439,102441],{"className":102440,"style":794},[763],[224,102442],{},[224,102444,239],{"className":102445},[542],[224,102447,16018],{"className":102448,"style":653},[468,469],[224,102450,1877],{"className":102451},[1917],[224,102453],{"className":102454,"style":724},[473],[224,102456,20315],{"className":102457},[468,469],[224,102459,1716],{"className":102460},[549],[224,102462],{"className":102463,"style":496},[473],[224,102465,2070],{"className":102466},[500],[224,102468],{"className":102469,"style":496},[473],[224,102471,102473],{"className":102472},[468,54290],[224,102474,102476,102643],{"className":102475},[754,755],[224,102477,102479,102640],{"className":102478},[759],[224,102480,102483,102623],{"className":102481,"style":102482},[763],"height:1.2839em;",[224,102484,102487,102491],{"className":102485,"style":102486},[2139],"top:-3.8em;",[224,102488],{"className":102489,"style":102490},[771],"height:3.8em;",[224,102492,102494,102562,102608,102611,102614,102617,102620],{"className":102493,"style":61074},[468],[224,102495,102497,102500,102559],{"className":102496},[468],[224,102498],{"className":102499},[542,8479],[224,102501,102503],{"className":102502},[20338],[224,102504,102506,102551],{"className":102505},[754,755],[224,102507,102509,102548],{"className":102508},[759],[224,102510,102512,102526,102534],{"className":102511,"style":22563},[763],[224,102513,102514,102517],{"style":22566},[224,102515],{"className":102516,"style":2108},[771],[224,102518,102520],{"className":102519},[776,777,778,779],[224,102521,102523],{"className":102522},[468,779],[224,102524,1447],{"className":102525},[468,779],[224,102527,102528,102531],{"style":20546},[224,102529],{"className":102530,"style":2108},[771],[224,102532],{"className":102533,"style":20554},[20553],[224,102535,102536,102539],{"style":22589},[224,102537],{"className":102538,"style":2108},[771],[224,102540,102542],{"className":102541},[776,777,778,779],[224,102543,102545],{"className":102544},[468,779],[224,102546,1438],{"className":102547},[468,779],[224,102549,787],{"className":102550},[786],[224,102552,102554],{"className":102553},[759],[224,102555,102557],{"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压成两个点，用 4.5 节的 DPI：",[224,103931,103933],{"className":103932,"translate":388},[387],[224,103934,103936,104061],{"className":103935,"translate":388},[392],[224,103937,103939],{"className":103938},[396],[398,103940,103941],{"xmlns":400,"display":401},[403,103942,103943,104058],{},[406,103944,103945,103955,103957,103959,103961,103963,103965,103967,103969,103971,103981,103983,103993,103995,103997,103999,104001,104003,104005,104015,104017,104020,104022,104024,104026,104028,104030,104032,104034,104036,104038,104040,104042,104044,104046,104048,104050,104052,104054,104056],{},[695,103946,103947,103949],{},[409,103948,20303],{},[406,103950,103951,103953],{},[409,103952,30652],{},[409,103954,41831],{},[412,103956,239],{"stretchy":435},[409,103958,16018],{},[409,103960,20312],{"mathvariant":678},[409,103962,20315],{},[412,103964,1716],{"stretchy":435},[423,103966,15054],{},[412,103968,9766],{},[423,103970,15054],{},[695,103972,103973,103975],{},[409,103974,20303],{},[406,103976,103977,103979],{},[409,103978,30652],{},[409,103980,41831],{},[412,103982,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[406,103984,103985,103987,103989,103991],{},[409,103986,9889],{"mathvariant":678},[409,103988,1861],{"mathvariant":678},[409,103990,2061],{"mathvariant":678},[409,103992,3254],{"mathvariant":678},[412,103994,239],{"stretchy":435},[409,103996,64666],{},[412,103998,1716],{"stretchy":435},[423,104000,682],{},[409,104002,20312],{"mathvariant":678},[423,104004,682],{},[406,104006,104007,104009,104011,104013],{},[409,104008,9889],{"mathvariant":678},[409,104010,1861],{"mathvariant":678},[409,104012,2061],{"mathvariant":678},[409,104014,3254],{"mathvariant":678},[412,104016,239],{"stretchy":435},[409,104018,104019],{},"β",[412,104021,1716],{"stretchy":435},[412,104023,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[412,104025,1877],{"separator":455},[473,104027],{"width":4247},[409,104029,64666],{},[412,104031,278],{},[409,104033,16018],{},[412,104035,239],{"stretchy":435},[409,104037,1097],{},[412,104039,1716],{"stretchy":435},[412,104041,1877],{"separator":455},[423,104043,15054],{},[409,104045,104019],{},[412,104047,278],{},[409,104049,20315],{},[412,104051,239],{"stretchy":435},[409,104053,1097],{},[412,104055,1716],{"stretchy":435},[409,104057,688],{"mathvariant":678},[447,104059,104060],{"encoding":449},"D_{KL}(P\\|Q)\\ \\ge\\ \nD_{KL}\\big(\\mathrm{Bern}(\\alpha)\\,\\|\\,\n\\mathrm{Bern}(\\beta)\\big),\n\\qquad \\alpha=P(A),\\ 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2 步（二元问题的精确最小值）。"," 固定差 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TV 值），对 ",[224,104435,104437,104450],{"className":104436,"translate":388},[392],[224,104438,104440],{"className":104439},[396],[398,104441,104442],{"xmlns":400},[403,104443,104444,104448],{},[406,104445,104446],{},[409,104447,64666],{},[447,104449,68999],{"encoding":449},[224,104451,104453],{"className":104452,"ariaHidden":455},[454],[224,104454,104456,104459],{"className":104455},[459],[224,104457],{"className":104458,"style":839},[463],[224,104460,64666],{"className":104461,"style":64696},[468,469]," 最小化二元 KL。取对称中心 ",[224,104464,104466,104494],{"className":104465,"translate":388},[392],[224,104467,104469],{"className":104468},[396],[398,104470,104471],{"xmlns":400},[403,104472,104473,104491],{},[406,104474,104475,104477,104479],{},[409,104476,64666],{},[412,104478,278],{},[20338,104480,104481,104489],{},[406,104482,104483,104485,104487],{},[419,104484,1438],{},[412,104486,319],{},[409,104488,35657],{},[419,104490,1447],{},[447,104492,104493],{"encoding":449},"\\alpha=\\frac{1+\\delta}{2}",[224,104495,104497,104515],{"className":104496,"ariaHidden":455},[454],[224,104498,104500,104503,104506,104509,104512],{"className":104499},[459],[224,104501],{"className":104502,"style":839},[463],[224,104504,64666],{"className":104505,"style":64696},[468,469],[224,104507],{"className":104508,"style":496},[473],[224,104510,278],{"className":104511},[500],[224,104513],{"className":104514,"style":496},[473],[224,104516,104518,104522],{"className":104517},[459],[224,104519],{"className":104520,"style":104521},[463],"height:1.2251em;vertical-align:-0.345em;",[224,104523,104525,104528,104593],{"className":104524},[468],[224,104526],{"className":104527},[542,8479],[224,104529,104531],{"className":104530},[20338],[224,104532,104534,104585],{"className":104533},[754,755],[224,104535,104537,104582],{"className":104536},[759],[224,104538,104540,104554,104562],{"className":104539,"style":22724},[763],[224,104541,104542,104545],{"style":22566},[224,104543],{"className":104544,"style":2108},[771],[224,104546,104548],{"className":104547},[776,777,778,779],[224,104549,104551],{"className":104550},[468,779],[224,104552,1447],{"className":104553},[468,779],[224,104555,104556,104559],{"style":20546},[224,104557],{"className":104558,"style":2108},[771],[224,104560],{"className":104561,"style":20554},[20553],[224,104563,104564,104567],{"style":22589},[224,104565],{"className":104566,"style":2108},[771],[224,104568,104570],{"className":104569},[776,777,778,779],[224,104571,104573,104576,104579],{"className":104572},[468,779],[224,104574,1438],{"className":104575},[468,779],[224,104577,319],{"className":104578},[478,779],[224,104580,35657],{"className":104581,"style":35706},[468,469,779],[224,104583,787],{"className":104584},[786],[224,104586,104588],{"className":104587},[759],[224,104589,104591],{"className":104590,"style":22611},[763],[224,104592],{},[224,104594],{"className":104595},[549,8479],[224,104597,104599,104627],{"className":104598,"translate":388},[392],[224,104600,104602],{"className":104601},[396],[398,104603,104604],{"xmlns":400},[403,104605,104606,104624],{},[406,104607,104608,104610,104612],{},[409,104609,104019],{},[412,104611,278],{},[20338,104613,104614,104622],{},[406,104615,104616,104618,104620],{},[419,104617,1438],{},[412,104619,5268],{},[409,104621,35657],{},[419,104623,1447],{},[447,104625,104626],{"encoding":449},"\\beta=\\frac{1-\\delta}{2}",[224,104628,104630,104648],{"className":104629,"ariaHidden":455},[454],[224,104631,104633,104636,104639,104642,104645],{"className":104632},[459],[224,104634],{"className":104635,"style":2565},[463],[224,104637,104019],{"className":104638,"style":104241},[468,469],[224,104640],{"className":104641,"style":496},[473],[224,104643,278],{"className":104644},[500],[224,104646],{"className":104647,"style":496},[473],[224,104649,104651,104654],{"className":104650},[459],[224,104652],{"className":104653,"style":104521},[463],[224,104655,104657,104660,104725],{"className":104656},[468],[224,104658],{"className":104659},[542,8479],[224,104661,104663],{"className":104662},[20338],[224,104664,104666,104717],{"className":104665},[754,755],[224,104667,104669,104714],{"className":104668},[759],[224,104670,104672,104686,104694],{"className":104671,"style":22724},[763],[224,104673,104674,104677],{"style":22566},[224,104675],{"className":104676,"style":2108},[771],[224,104678,104680],{"className":104679},[776,777,778,779],[224,104681,104683],{"className":104682},[468,779],[224,104684,1447],{"className":104685},[468,779],[224,104687,104688,104691],{"style":20546},[224,104689],{"className":104690,"style":2108},[771],[224,104692],{"className":104693,"style":20554},[20553],[224,104695,104696,104699],{"style":22589},[224,104697],{"className":104698,"style":2108},[771],[224,104700,104702],{"className":104701},[776,777,778,779],[224,104703,104705,104708,104711],{"className":104704},[468,779],[224,104706,1438],{"className":104707},[468,779],[224,104709,5268],{"className":104710},[478,779],[224,104712,35657],{"className":104713,"style":35706},[468,469,779],[224,104715,787],{"className":104716},[786],[224,104718,104720],{"className":104719},[759],[224,104721,104723],{"className":104722,"style":22611},[763],[224,104724],{},[224,104726],{"className":104727},[549,8479],"，此时 ",[224,104730,104732,104754],{"className":104731,"translate":388},[392],[224,104733,104735],{"className":104734},[396],[398,104736,104737],{"xmlns":400},[403,104738,104739,104751],{},[406,104740,104741,104743,104745,104747,104749],{},[419,104742,1438],{},[412,104744,5268],{},[409,104746,64666],{},[412,104748,278],{},[409,104750,104019],{},[447,104752,104753],{"encoding":449},"1-\\alpha=\\beta",[224,104755,104757,104775,104793],{"className":104756,"ariaHidden":455},[454],[224,104758,104760,104763,104766,104769,104772],{"className":104759},[459],[224,104761],{"className":104762,"style":7047},[463],[224,104764,1438],{"className":104765},[468],[224,104767],{"className":104768,"style":474},[473],[224,104770,5268],{"className":104771},[478],[224,104773],{"className":104774,"style":474},[473],[224,104776,104778,104781,104784,104787,104790],{"className":104777},[459],[224,104779],{"className":104780,"style":839},[463],[224,104782,64666],{"className":104783,"style":64696},[468,469],[224,104785],{"className":104786,"style":496},[473],[224,104788,278],{"className":104789},[500],[224,104791],{"className":104792,"style":496},[473],[224,104794,104796,104799],{"className":104795},[459],[224,104797],{"className":104798,"style":2565},[463],[224,104800,104019],{"className":104801,"style":104241},[468,469],[224,104803,104805,104827],{"className":104804,"translate":388},[392],[224,104806,104808],{"className":104807},[396],[398,104809,104810],{"xmlns":400},[403,104811,104812,104824],{},[406,104813,104814,104816,104818,104820,104822],{},[419,104815,1438],{},[412,104817,5268],{},[409,104819,104019],{},[412,104821,278],{},[409,104823,64666],{},[447,104825,104826],{"encoding":449},"1-\\beta=\\alpha",[224,104828,104830,104848,104866],{"className":104829,"ariaHidden":455},[454],[224,104831,104833,104836,104839,104842,104845],{"className":104832},[459],[224,104834],{"className":104835,"style":7047},[463],[224,104837,1438],{"className":104838},[468],[224,104840],{"className":104841,"style":474},[473],[224,104843,5268],{"className":104844},[478],[224,104846],{"className":104847,"style":474},[473],[224,104849,104851,104854,104857,104860,104863],{"className":104850},[459],[224,104852],{"className":104853,"style":2565},[463],[224,104855,104019],{"className":104856,"style":104241},[468,469],[224,104858],{"className":104859,"style":496},[473],[224,104861,278],{"className":104862},[500],[224,104864],{"className":104865,"style":496},[473],[224,104867,104869,104872],{"className":104868},[459],[224,104870],{"className":104871,"style":839},[463],[224,104873,64666],{"className":104874,"style":64696},[468,469],"，算得",[224,104877,104879],{"className":104878,"translate":388},[387],[224,104880,104882,105138],{"className":104881,"translate":388},[392],[224,104883,104885],{"className":104884},[396],[398,104886,104887],{"xmlns":400,"display":401},[403,104888,104889,105135],{},[5209,104890,104891,104949,104999,105073],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,104892,104893,104899],{},[5218,104894,104895],{},[5221,104896,104897],{"scriptlevel":284,"displaystyle":455},[406,104898],{},[5218,104900,104901],{},[5221,104902,104903],{"scriptlevel":284,"displaystyle":455},[406,104904,104905,104907,104917,104919,104929,104931,104945,104947],{},[406,104906],{},[695,104908,104909,104911],{},[409,104910,20303],{},[406,104912,104913,104915],{},[409,104914,30652],{},[409,104916,41831],{},[412,104918,239],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[406,104920,104921,104923,104925,104927],{},[409,104922,9889],{"mathvariant":678},[409,104924,1861],{"mathvariant":678},[409,104926,2061],{"mathvariant":678},[409,104928,3254],{"mathvariant":678},[412,104930,239],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[5221,104932,104933],{"scriptlevel":284,"displaystyle":435},[20338,104934,104935,104943],{},[406,104936,104937,104939,104941],{},[419,104938,1438],{},[412,104940,319],{},[409,104942,35657],{},[419,104944,1447],{},[412,104946,1716],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[412,104948,1716],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[5215,104950,104951,104957],{},[5218,104952,104953],{},[5221,104954,104955],{"scriptlevel":284,"displaystyle":455},[406,104956],{},[5218,104958,104959],{},[5221,104960,104961],{"scriptlevel":284,"displaystyle":455},[406,104962,104963,104965,104967,104969,104979,104981,104995,104997],{},[406,104964],{},[473,104966],{"width":4247},[412,104968,20312],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[406,104970,104971,104973,104975,104977],{},[409,104972,9889],{"mathvariant":678},[409,104974,1861],{"mathvariant":678},[409,104976,2061],{"mathvariant":678},[409,104978,3254],{"mathvariant":678},[412,104980,239],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[5221,104982,104983],{"scriptlevel":284,"displaystyle":435},[20338,104984,104985,104993],{},[406,104986,104987,104989,104991],{},[419,104988,1438],{},[412,104990,5268],{},[409,10499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时不平衡质量挤出 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时值为 ",[224,106294,106296,106324],{"className":106295,"translate":388},[392],[224,106297,106299],{"className":106298},[396],[398,106300,106301],{"xmlns":400},[403,106302,106303,106321],{},[406,106304,106305,106307,106309,106311,106313,106315,106317,106319],{},[412,106306,5268],{},[409,106308,24131],{},[412,106310,24134],{},[412,106312,239],{"stretchy":435},[419,106314,1438],{},[412,106316,5268],{},[409,106318,35657],{},[412,106320,1716],{"stretchy":435},[447,106322,106323],{"encoding":449},"-\\log(1-\\delta)",[224,106325,106327,106359],{"className":106326,"ariaHidden":455},[454],[224,106328,106330,106333,106336,106339,106344,106347,106350,106353,106356],{"className":106329},[459],[224,106331],{"className":106332,"style":535},[463],[224,106334,5268],{"className":106335},[468],[224,106337],{"className":106338,"style":724},[473],[224,106340,24185,106342],{"className":106341},[21531],[224,106343,24189],{"style":24188},[224,106345,239],{"className":106346},[542],[224,106348,1438],{"className":106349},[468],[224,106351],{"className":106352,"style":474},[473],[224,106354,5268],{"className":106355},[478],[224,106357],{"className":106358,"style":474},[473],[224,106360,106362,106365,106368],{"className":106361},[459],[224,106363],{"className":106364,"style":535},[463],[224,106366,35657],{"className":106367,"style":35706},[468,469],[224,106369,1716],{"className":106370},[549],"，均大于中心值）可得：唯一驻点在 ",[224,106373,106375,106402],{"className":106374,"translate":388},[392],[224,106376,106378],{"className":106377},[396],[398,106379,106380],{"xmlns":400},[403,106381,106382,106400],{},[406,106383,106384,106386,106388],{},[409,106385,64666],{},[412,106387,278],{},[20338,106389,106390,106398],{},[406,106391,106392,106394,106396],{},[419,106393,1438],{},[412,106395,319],{},[409,106397,35657],{},[419,106399,1447],{},[447,106401,104493],{"encoding":449},[224,106403,106405,106423],{"className":106404,"ariaHidden":455},[454],[224,106406,106408,106411,106414,106417,106420],{"className":106407},[459],[224,106409],{"className":106410,"style":839},[463],[224,106412,64666],{"className":106413,"style":64696},[468,469],[224,106415],{"className":106416,"style":496},[473],[224,106418,278],{"className":106419},[500],[224,106421],{"className":106422,"style":496},[473],[224,106424,106426,106429],{"className":106425},[459],[224,106427],{"className":106428,"style":104521},[463],[224,106430,106432,106435,106500],{"className":106431},[468],[224,106433],{"className":106434},[542,8479],[224,106436,106438],{"className":106437},[20338],[224,106439,106441,106492],{"className":106440},[754,755],[224,106442,106444,106489],{"className":106443},[759],[224,106445,106447,106461,106469],{"className":106446,"style":22724},[763],[224,106448,106449,106452],{"style":22566},[224,106450],{"className":106451,"style":2108},[771],[224,106453,106455],{"className":106454},[776,777,778,779],[224,106456,106458],{"className":106457},[468,779],[224,106459,1447],{"className":106460},[468,779],[224,106462,106463,106466],{"style":20546},[224,106464],{"className":106465,"style":2108},[771],[224,106467],{"className":106468,"style":20554},[20553],[224,106470,106471,106474],{"style":22589},[224,106472],{"className":106473,"style":2108},[771],[224,106475,106477],{"className":106476},[776,777,778,779],[224,106478,106480,106483,106486],{"className":106479},[468,779],[224,106481,1438],{"className":106482},[468,779],[224,106484,319],{"className":106485},[478,779],[224,106487,35657],{"className":106488,"style":35706},[468,469,779],[224,106490,787],{"className":106491},[786],[224,106493,106495],{"className":106494},[759],[224,106496,106498],{"className":106497,"style":22611},[763],[224,106499],{},[224,106501],{"className":106502},[549,8479],"，且是",[378,106505,106506],{},"精确最小值","（Vajda 界）：",[224,106509,106511,106567],{"className":106510,"translate":388},[392],[224,106512,106514],{"className":106513},[396],[398,106515,106516],{"xmlns":400},[403,106517,106518,106564],{},[406,106519,106520,106530,106532,106534,106536,106538,106540,106542,106544,106546,106552,106554,106556,106558,106560,106562],{},[695,106521,106522,106524],{},[409,106523,20303],{},[406,106525,106526,106528],{},[409,106527,30652],{},[409,106529,41831],{},[412,106531,239],{"stretchy":435},[409,106533,16018],{},[409,106535,20312],{"mathvariant":678},[409,106537,20315],{},[412,106539,1716],{"stretchy":435},[412,106541,9766],{},[409,106543,58586],{},[412,106545,239],{"stretchy":435},[406,106547,106548,106550],{},[409,106549,637],{"mathvariant":678},[409,106551,16059],{"mathvariant":678},[412,106553,239],{"stretchy":435},[409,106555,16018],{},[412,106557,1877],{"separator":455},[409,106559,20315],{},[412,106561,1716],{"stretchy":435},[412,106563,1716],{"stretchy":435},[447,106565,106566],{"encoding":449},"D_{KL}(P\\|Q)\\ge h(\\mathrm{TV}(P,Q))",[224,106568,106570,106646],{"className":106569,"ariaHidden":455},[454],[224,106571,106573,106576,106622,106625,106628,106631,106634,106637,106640,106643],{"className":106572},[459],[224,106574],{"className":106575,"style":535},[463],[224,106577,106579,106582],{"className":106578},[468],[224,106580,20303],{"className":106581,"style":2473},[468,469],[224,106583,106585],{"className":106584},[750],[224,106586,106588,106614],{"className":106587},[754,755],[224,106589,106591,106611],{"className":106590},[759],[224,106592,106594],{"className":106593,"style":764},[763],[224,106595,106596,106599],{"style":20390},[224,106597],{"className":106598,"style":772},[771],[224,106600,106602],{"className":106601},[776,777,778,779],[224,106603,106605,106608],{"className":106604},[468,779],[224,106606,30652],{"className":106607,"style":30740},[468,469,779],[224,106609,41831],{"className":106610},[468,469,779],[224,106612,787],{"className":106613},[786],[224,106615,106617],{"className":106616},[759],[224,106618,106620],{"className":106619,"style":794},[763],[224,106621],{},[224,106623,239],{"className":106624},[542],[224,106626,16018],{"className":106627,"style":653},[468,469],[224,106629,20312],{"className":106630},[468],[224,106632,20315],{"className":106633},[468,469],[224,106635,1716],{"className":106636},[549],[224,106638],{"className":106639,"style":496},[473],[224,106641,9766],{"className":106642},[500],[224,106644],{"className":106645,"style":496},[473],[224,106647,106649,106652,106655,106658,106665,106668,106671,106674,106677,106680],{"className":106648},[459],[224,106650],{"className":106651,"style":535},[463],[224,106653,58586],{"className":106654},[468,469],[224,106656,239],{"className":106657},[542],[224,106659,106661],{"className":106660},[468],[224,106662,106664],{"className":106663,"style":24188},[468,1906],"TV",[224,106666,239],{"className":106667},[542],[224,106669,16018],{"className":106670,"style":653},[468,469],[224,106672,1877],{"className":106673},[1917],[224,106675],{"className":106676,"style":724},[473],[224,106678,20315],{"className":106679},[468,469],[224,106681,38989],{"className":106682},[549],"，其中 ",[224,106685,106687,106733],{"className":106686,"translate":388},[392],[224,106688,106690],{"className":106689},[396],[398,106691,106692],{"xmlns":400},[403,106693,106694,106730],{},[406,106695,106696,106698,106700,106702,106704,106706,106708,106710,106712],{},[409,106697,58586],{},[412,106699,239],{"stretchy":435},[409,106701,35657],{},[412,106703,1716],{"stretchy":435},[412,106705,278],{},[409,106707,35657],{},[409,106709,24131],{},[412,106711,24134],{},[20338,106713,106714,106722],{},[406,106715,106716,106718,106720],{},[419,106717,1438],{},[412,106719,319],{},[409,106721,35657],{},[406,106723,106724,106726,106728],{},[419,106725,1438],{},[412,106727,5268],{},[409,106729,35657],{},[447,106731,106732],{"encoding":449},"h(\\delta)=\\delta\\log\\frac{1+\\delta}{1-\\delta}",[224,106734,106736,106763],{"className":106735,"ariaHidden":455},[454],[224,106737,106739,106742,106745,106748,106751,106754,106757,106760],{"className":106738},[459],[224,106740],{"className":106741,"style":535},[463],[224,106743,58586],{"className":106744},[468,469],[224,106746,239],{"className":106747},[542],[224,106749,35657],{"className":106750,"style":35706},[468,469],[224,106752,1716],{"className":106753},[549],[224,106755],{"className":106756,"style":496},[473],[224,106758,278],{"className":106759},[500],[224,106761],{"className":106762,"style":496},[473],[224,106764,106766,106769,106772,106775,106780,106783],{"className":106765},[459],[224,106767],{"className":106768,"style":36297},[463],[224,106770,35657],{"className":106771,"style":35706},[468,469],[224,106773],{"className":106774,"style":724},[473],[224,106776,24185,106778],{"className":106777},[21531],[224,106779,24189],{"style":24188},[224,106781],{"className":106782,"style":724},[473],[224,106784,106786,106789,106860],{"className":106785},[468],[224,106787],{"className":106788},[542,8479],[224,106790,106792],{"className":106791},[20338],[224,106793,106795,106852],{"className":106794},[754,755],[224,106796,106798,106849],{"className":106797},[759],[224,106799,106801,106821,106829],{"className":106800,"style":22724},[763],[224,106802,106803,106806],{"style":22566},[224,106804],{"className":106805,"style":2108},[771],[224,106807,106809],{"className":106808},[776,777,778,779],[224,106810,106812,106815,106818],{"className":106811},[468,779],[224,106813,1438],{"className":106814},[468,779],[224,106816,5268],{"className":106817},[478,779],[224,106819,35657],{"className":106820,"style":35706},[468,469,779],[224,106822,106823,106826],{"style":20546},[224,106824],{"className":106825,"style":2108},[771],[224,106827],{"className":106828,"style":20554},[20553],[224,106830,106831,106834],{"style":22589},[224,106832],{"className":106833,"style":2108},[771],[224,106835,106837],{"className":106836},[776,777,778,779],[224,106838,106840,106843,106846],{"className":106839},[468,779],[224,106841,1438],{"className":106842},[468,779],[224,106844,319],{"className":106845},[478,779],[224,106847,35657],{"className":106848,"style":35706},[468,469,779],[224,106850,787],{"className":106851},[786],[224,106853,106855],{"className":106854},[759],[224,106856,106858],{"className":106857,"style":36373},[763],[224,106859],{},[224,106861],{"className":106862},[549,8479],[224,106864,106866,106884],{"className":106865,"translate":388},[392],[224,106867,106869],{"className":106868},[396],[398,106870,106871],{"xmlns":400},[403,106872,106873,106881],{},[406,106874,106875,106877,106879],{},[409,106876,35657],{},[412,106878,1144],{},[419,106880,1438],{},[447,106882,106883],{"encoding":449},"\\delta\\to1",[224,106885,106887,106905],{"className":106886,"ariaHidden":455},[454],[224,106888,106890,106893,106896,106899,106902],{"className":106889},[459],[224,106891],{"className":106892,"style":7642},[463],[224,106894,35657],{"className":106895,"style":35706},[468,469],[224,106897],{"className":106898,"style":496},[473],[224,106900,1144],{"className":106901},[500],[224,106903],{"className":106904,"style":496},[473],[224,106906,106908,106911],{"className":106907},[459],[224,106909],{"className":106910,"style":510},[463],[224,106912,1438],{"className":106913},[468]," 时发散（与 KL 在支撑分离时为 ",[224,106916,106918,106931],{"className":106917,"translate":388},[392],[224,106919,106921],{"className":106920},[396],[398,106922,106923],{"xmlns":400},[403,106924,106925,106929],{},[406,106926,106927],{},[409,106928,21085],{"mathvariant":678},[447,106930,23901],{"encoding":449},[224,106932,106934],{"className":106933,"ariaHidden":455},[454],[224,106935,106937,106940],{"className":106936},[459],[224,106938],{"className":106939,"style":839},[463],[224,106941,21085],{"className":106942},[468]," 一致）。",[11,106945,106946,106949],{},[378,106947,106948],{},"第 3 步（初等放缩）。"," 由",[224,106951,106953],{"className":106952,"translate":388},[387],[224,106954,106956,107041],{"className":106955,"translate":388},[392],[224,106957,106959],{"className":106958},[396],[398,106960,106961],{"xmlns":400,"display":401},[403,106962,106963,107038],{},[406,106964,106965,106967,106969,106987,106989,106991,107030,107032,107034,107036],{},[409,106966,24131],{},[412,106968,24134],{},[20338,106970,106971,106979],{},[406,106972,106973,106975,106977],{},[419,106974,1438],{},[412,106976,319],{},[409,106978,35657],{},[406,106980,106981,106983,106985],{},[419,106982,1438],{},[412,106984,5268],{},[409,106986,35657],{},[412,106988,278],{},[419,106990,1447],{},[406,106992,106993,106995,106997,106999,107009,107011,107021,107023,107026,107028],{},[412,106994,239],{"fence":455},[409,106996,35657],{},[412,106998,319],{},[20338,107000,107001,107007],{},[3248,107002,107003,107005],{},[409,107004,35657],{},[419,107006,3594],{},[419,107008,3594],{},[412,107010,319],{},[20338,107012,107013,107019],{},[3248,107014,107015,107017],{},[409,107016,35657],{},[419,107018,445],{},[419,107020,445],{},[412,107022,319],{},[412,107024,107025],{},"⋯",[423,107027,682],{},[412,107029,1716],{"fence":455},[412,107031,9766],{},[419,107033,1447],{},[409,107035,35657],{},[412,107037,1877],{"separator":455},[447,107039,107040],{"encoding":449},"\\log\\frac{1+\\delta}{1-\\delta}\n=2\\left(\\delta+\\frac{\\delta^3}3\n+\\frac{\\delta^5}5+\\cdots\\right)\n\\ge 2\\delta,",[224,107042,107044,107153,107402],{"className":107043,"ariaHidden":455},[454],[224,107045,107047,107050,107055,107058,107144,107147,107150],{"className":107046},[459],[224,107048],{"className":107049,"style":37994},[463],[224,107051,24185,107053],{"className":107052},[21531],[224,107054,24189],{"style":24188},[224,107056],{"className":107057,"style":724},[473],[224,107059,107061,107064,107141],{"className":107060},[468],[224,107062],{"className":107063},[542,8479],[224,107065,107067],{"className":107066},[20338],[224,107068,107070,107133],{"className":107069},[754,755],[224,107071,107073,107130],{"className":107072},[759],[224,107074,107076,107099,107107],{"className":107075,"style":20528},[763],[224,107077,107078,107081],{"style":20531},[224,107079],{"className":107080,"style":2108},[771],[224,107082,107084,107087,107090,107093,107096],{"className":107083},[468],[224,107085,1438],{"className":107086},[468],[224,107088],{"className":107089,"style":474},[473],[224,107091,5268],{"className":107092},[478],[224,107094],{"className":107095,"style":474},[473],[224,107097,35657],{"className":107098,"style":35706},[468,469],[224,107100,107101,107104],{"style":20546},[224,107102],{"className":107103,"style":2108},[771],[224,107105],{"className":107106,"style":20554},[20553],[224,107108,107109,107112],{"style":20557},[224,107110],{"className":107111,"style":2108},[771],[224,107113,107115,107118,107121,107124,107127],{"className":107114},[468],[224,107116,1438],{"className":107117},[468],[224,107119],{"className":107120,"style":474},[473],[224,107122,319],{"className":107123},[478],[224,107125],{"className":107126,"style":474},[473],[224,107128,35657],{"className":107129,"style":35706},[468,469],[224,107131,787],{"className":107132},[786],[224,107134,107136],{"className":107135},[759],[224,107137,107139],{"className":107138,"style":36958},[763],[224,107140],{},[224,107142],{"className":107143},[549,8479],[224,107145],{"className":107146,"style":496},[473],[224,107148,278],{"className":107149},[500],[224,107151],{"className":107152,"style":496},[473],[224,107154,107156,107160,107163,107166,107393,107396,107399],{"className":107155},[459],[224,107157],{"className":107158,"style":107159},[463],"height:2.4411em;vertical-align:-0.95em;",[224,107161,1447],{"className":107162},[468],[224,107164],{"className":107165,"style":724},[473],[224,107167,107169,107175,107178,107181,107184,107187,107275,107278,107281,107284,107372,107375,107378,107381,107384,107387],{"className":107168},[2091],[224,107170,107172],{"className":107171,"style":8281},[542,8280],[224,107173,239],{"className":107174},[2363,778],[224,107176,35657],{"className":107177,"style":35706},[468,469],[224,107179],{"className":107180,"style":474},[473],[224,107182,319],{"className":107183},[478],[224,107185],{"className":107186,"style":474},[473],[224,107188,107190,107193,107272],{"className":107189},[468],[224,107191],{"className":107192},[542,8479],[224,107194,107196],{"className":107195},[20338],[224,107197,107199,107264],{"className":107198},[754,755],[224,107200,107202,107261],{"className":107201},[759],[224,107203,107205,107216,107224],{"className":107204,"style":57644},[763],[224,107206,107207,107210],{"style":20531},[224,107208],{"className":107209,"style":2108},[771],[224,107211,107213],{"className":107212},[468],[224,107214,3594],{"className":107215},[468],[224,107217,107218,107221],{"style":20546},[224,107219],{"className":107220,"style":2108},[771],[224,107222],{"className":107223,"style":20554},[20553],[224,107225,107226,107229],{"style":20557},[224,107227],{"className":107228,"style":2108},[771],[224,107230,107232],{"className":107231},[468],[224,107233,107235,107238],{"className":107234},[468],[224,107236,35657],{"className":107237,"style":35706},[468,469],[224,107239,107241],{"className":107240},[750],[224,107242,107244],{"className":107243},[754],[224,107245,107247],{"className":107246},[759],[224,107248,107250],{"className":107249,"style":5671},[763],[224,107251,107252,107255],{"style":3288},[224,107253],{"className":107254,"style":772},[771],[224,107256,107258],{"className":107257},[776,777,778,779],[224,107259,3594],{"className":107260},[468,779],[224,107262,787],{"className":107263},[786],[224,107265,107267],{"className":107266},[759],[224,107268,107270],{"className":107269,"style":23809},[763],[224,107271],{},[224,107273],{"className":107274},[549,8479],[224,107276],{"className":107277,"style":474},[473],[224,107279,319],{"className":107280},[478],[224,107282],{"className":107283,"style":474},[473],[224,107285,107287,107290,107369],{"className":107286},[468],[224,107288],{"className":107289},[542,8479],[224,107291,107293],{"className":107292},[20338],[224,107294,107296,107361],{"className":107295},[754,755],[224,107297,107299,107358],{"className":107298},[759],[224,107300,107302,107313,107321],{"className":107301,"style":57644},[763],[224,107303,107304,107307],{"style":20531},[224,107305],{"className":107306,"style":2108},[771],[224,10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\\ge\\ h(\\mathrm{TV})\n\\ \\ge\\ 2\\,\\mathrm{TV}(P,Q)^2 ,\\\\[2pt]\n&\\text{即}\\quad\n\\mathrm{TV}(P,Q)\\ \\le\\\n\\sqrt{\\tfrac12 D_{KL}(P\\|Q)} .\\ \\square\n\\end{aligned}",[224,107603,107605],{"className":107604,"ariaHidden":455},[454],[224,107606,107608,107612],{"className":107607},[459],[224,107609],{"className":107610,"style":107611},[463],"height:3.5641em;vertical-align:-1.5321em;",[224,107613,107615],{"className":107614},[468],[224,107616,107618,107661],{"className":107617},[5209],[224,107619,107621],{"className":107620},[5331],[224,107622,107624,107652],{"className":107623},[754,755],[224,107625,107627,107649],{"className":107626},[759],[224,107628,107631,107640],{"className":107629,"style":107630},[763],"height:2.0321em;",[224,107632,107634,107637],{"style":107633},"top:-4.4519em;",[224,107635],{"className":107636,"style":102322},[771],[224,107638],{"className":107639},[468],[224,107641,107643,107646],{"style":107642},"top:-2.3079em;",[224,107644],{"className":107645,"style":102322},[771],[224,107647],{"className":107648},[468],[224,107650,787],{"className":107651},[786],[224,107653,107655],{"className":107654},[759],[224,107656,107659],{"className":107657,"style":107658},[763],"height:1.5321em;",[224,107660],{},[224,107662,107664],{"className":107663},[5399],[224,107665,107667,108093],{"className":107666},[754,755],[224,107668,107670,108090],{"className":107669},[759],[224,107671,107673,107849],{"className":107672,"style":107630},[763],[224,107674,107675,107678],{"style":107633},[224,107676],{"className":107677,"style":102322},[771],[224,107679,107681,107684,107730,107733,107736,107739,107742,107745,107748,107751,107754,107757,107760,107763,107766,107772,107775,107778,107781,107784,107787,107790,107793,107796,107802,107805,107808,107811,107814,107817,107846],{"className":107680},[468],[224,107682],{"className":107683},[468],[224,107685,107687,107690],{"className":107686},[468],[224,107688,20303],{"className":107689,"style":2473},[468,469],[224,107691,107693],{"className":107692},[750],[224,107694,107696,107722],{"className":107695},[754,755],[224,107697,107699,107719],{"className":107698},[759],[224,107700,107702],{"className":107701,"style":764},[763],[224,107703,107704,107707],{"style":20390},[224,107705],{"className":107706,"style":772},[771],[224,107708,107710],{"className":107709},[776,777,778,779],[224,107711,107713,107716],{"className":107712},[468,779],[224,107714,30652],{"className":107715,"style":30740},[468,469,779],[224,107717,41831],{"className":107718},[468,469,779],[224,107720,787],{"className":107721},[786],[224,107723,107725],{"className":107724},[759],[224,107726,107728],{"className":107727,"style":794},[763],[224,107729],{},[224,107731,239],{"className":107732},[542],[224,107734,16018],{"className":107735,"style":653},[468,469],[224,107737,20312],{"className":107738},[468],[224,107740,20315],{"className":107741},[468,469],[224,107743,1716],{"className":107744},[549],[224,107746,15054],{"className":107747},[473],[224,107749],{"className":107750,"style":496},[473],[224,107752,9766],{"className":107753},[500],[224,107755,15054],{"className":107756},[473],[224,107758],{"className":107759,"style":496},[473],[224,107761,58586],{"className":107762},[468,469],[224,107764,239],{"className":107765},[542],[224,107767,107769],{"className":107768},[468],[224,107770,106664],{"className":107771,"style":24188},[468,1906],[224,107773,1716],{"className":107774},[549],[224,107776,15054],{"className":107777},[473],[224,107779],{"className":107780,"style":496},[473],[224,107782,9766],{"className":107783},[500],[224,107785,15054],{"className":107786},[473],[224,107788],{"className":107789,"style":496},[473],[224,107791,1447],{"className":107792},[468],[224,107794],{"className":107795,"style":724},[473],[224,107797,107799],{"className":107798},[468],[224,107800,106664],{"className":107801,"style":24188},[468,1906],[224,107803,239],{"className":107804},[542],[224,107806,16018],{"className":107807,"style":653},[468,469],[224,107809,1877],{"className":107810},[1917],[224,107812],{"className":107813,"style":724},[473],[224,107815,20315],{"className":107816},[468,469],[224,107818,107820,107823],{"className":107819},[549],[224,107821,1716],{"className":107822},[549],[224,107824,107826],{"className":107825},[750],[224,107827,107829],{"className":107828},[754],[224,107830,107832],{"className":107831},[759],[224,107833,107835],{"className":107834,"style":5071},[763],[224,107836,107837,107840],{"style":5074},[224,107838],{"className":107839,"style":772},[771],[224,107841,107843],{"className":107842},[776,777,778,779],[224,107844,1447],{"className":107845},[468,779],[224,107847,1877],{"className":107848},[1917],[224,107850,107851,107854],{"style":107642},[224,107852],{"className":107853,"style":102322},[771],[224,107855,107857,107860,107866,107869,107875,107878,107881,107884,107887,107890,107893,107896,107899,107902,107905,107908,108081,108084,108087],{"className":107856},[468],[224,107858],{"className":107859},[468],[224,107861,107863],{"className":107862},[468,11373],[224,107864,107536],{"className":107865},[468,16159],[224,107867],{"className":107868,"st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Pinsker 不等式中的 ",[224,108108,108110,108130],{"className":108109,"translate":388},[392],[224,108111,108113],{"className":108112},[396],[398,108114,108115],{"xmlns":400},[403,108116,108117,108127],{},[406,108118,108119],{},[5221,108120,108121],{"scriptlevel":284,"displaystyle":435},[20338,108122,108123,108125],{},[419,108124,1438],{},[419,108126,1447],{},[447,108128,108129],{"encoding":449},"\\tfrac12",[224,108131,108133],{"className":108132,"ariaHidden":455},[454],[224,108134,108136,108139],{"className":108135},[459],[224,108137],{"className":108138,"style":22544},[463],[224,108140,108142,108145,108204],{"className":108141},[468],[224,108143],{"className":108144},[542,8479],[224,108146,108148],{"className":108147},[20338],[224,108149,108151,108196],{"className":108150},[754,755],[224,108152,108154,108193],{"className":108153},[759],[224,108155,108157,108171,108179],{"className":108156,"style":22563},[763],[224,108158,108159,108162],{"style":22566},[224,108160],{"className":108161,"style":2108},[771],[224,108163,108165],{"className":108164},[776,777,778,779],[224,108166,108168],{"className":108167},[468,779],[224,108169,1447],{"className":108170},[468,779],[224,108172,108173,108176],{"style":20546},[224,108174],{"className":108175,"style":2108},[771],[224,108177],{"className":108178,"style":20554},[20553],[224,108180,108181,108184],{"style":22589},[224,108182],{"className":108183,"style":2108},[771],[224,108185,108187],{"className":108186},[776,777,778,779],[224,108188,108190],{"className":108189},[468,779],[224,108191,1438],{"className":108192},[468,779],[224,108194,787],{"className":108195},[786],[224,108197,108199],{"className":108198},[759],[224,108200,108202],{"className":108201,"style":22611},[763],[224,108203],{},[224,108205],{"className":108206},[549,8479],"（即 ",[224,108209,108211,108245],{"className":108210,"translate":388},[392],[224,108212,108214],{"className":108213},[396],[398,108215,108216],{"xmlns":400},[403,108217,108218,108242],{},[406,108219,108220,108226,108228,108230,108232],{},[406,108221,108222,108224],{},[409,108223,30652],{"mathvariant":678},[409,108225,41831],{"mathvariant":678},[412,108227,9766],{},[419,108229,1447],{},[423,108231,682],{},[3248,108233,108234,108240],{},[406,108235,108236,108238],{},[409,108237,637],{"mathvariant":678},[409,108239,16059],{"mathvariant":678},[419,108241,1447],{},[447,108243,108244],{"encoding":449},"\\mathrm{KL}\\ge2\\,\\mathrm{TV}^2",[224,108246,108248,108270],{"className":108247,"ariaHidden":455},[454],[224,108249,108251,108255,108261,108264,108267],{"className":108250},[459],[224,108252],{"className":108253,"style":108254},[463],"height:0.8193em;vertical-align:-0.136em;",[224,108256,108258],{"className":108257},[468],[224,108259,52588],{"className":108260},[468,1906],[224,108262],{"className":108263,"style":496},[473],[224,108265,9766],{"className":108266},[500],[224,108268],{"className":108269,"style":496},[473],[224,108271,108273,108277,108280,108283],{"className":108272},[459],[224,108274],{"className":108275,"style":108276},[463],"height:0.8873em;",[224,108278,1447],{"className":108279},[468],[224,108281],{"className":108282,"style":724},[473],[224,108284,108286,108292],{"className":108285},[468],[224,108287,108289],{"className":108288},[468],[224,108290,106664],{"className":108291,"style":24188},[468,1906],[224,108293,108295],{"className":108294},[750],[224,108296,108298],{"className":108297},[754],[224,108299,108301],{"className":108300},[759],[224,108302,108304],{"className":108303,"style":108276},[763],[224,108305,108307,108310],{"style":108306},"top:-3.1362em;margin-right:0.05em;",[224,108308],{"className":108309,"style":772},[771],[224,108311,108313],{"className":108312},[776,777,778,779],[224,108314,1447],{"className":108315},[468,779]," 的系数 2）是最优的：取两分布仅在极小 ",[224,108318,108320,108334],{"className":108319,"translate":388},[392],[224,108321,108323],{"className":108322},[396],[398,108324,108325],{"xmlns":400},[403,108326,108327,108331],{},[406,108328,108329],{},[409,108330,58583],{},[447,108332,108333],{"encoding":449},"\\varepsilon",[224,108335,108337],{"className":108336,"ariaHidden":455},[454],[224,108338,108340,108343],{"className":108339},[459],[224,108341],{"className":108342,"style":839},[463],[224,108344,58583],{"className":108345},[468,469]," 上偏离最优对称配置，可让 ",[224,108348,108350,108382],{"className":108349,"translate":388},[392],[224,108351,108353],{"className":108352},[396],[398,108354,108355],{"xmlns":400},[403,108356,108357,108379],{},[406,108358,108359,108361,108363,108365,108367,108369,108375,108377],{},[409,108360,58586],{},[412,108362,239],{"stretchy":435},[409,108364,35657],{},[412,108366,1716],{"stretchy":435},[412,108368,12413],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[3248,108370,108371,108373],{},[409,108372,35657],{},[419,108374,1447],{},[412,108376,1144],{},[419,108378,1447],{},[447,108380,108381],{"encoding":449},"h(\\delta)\\big\u002F\\delta^2\\to 2",[224,108383,108385,108447],{"className":108384,"ariaHidden":455},[454],[224,108386,108388,108391,108394,108397,108400,108403,108409,108438,108441,108444],{"className":108387},[459],[224,108389],{"className":108390,"style":2663},[463],[224,108392,58586],{"className":108393},[468,469],[224,108395,239],{"className":108396},[542],[224,108398,35657],{"className":108399,"style":35706},[468,469],[224,108401,1716],{"className":108402},[549],[224,108404,108406],{"className":108405},[468],[224,108407,12413],{"className":108408},[2363,2364],[224,108410,108412,108415],{"className":108411},[468],[224,108413,35657],{"className":108414,"style":35706},[468,469],[224,108416,108418],{"className":108417},[750],[224,108419,108421],{"className":108420},[754],[224,108422,108424],{"className":108423},[759],[224,108425,108427],{"className":108426,"style":5671},[763],[224,108428,108429,108432],{"style":3288},[224,108430],{"className":108431,"style":772},[771],[224,108433,108435],{"className":108434},[776,777,778,779],[224,108436,1447],{"className":108437},[468,779],[224,108439],{"className":108440,"style":496},[473],[224,108442,1144],{"className":108443},[500],[224,108445],{"className":108446,"style":496},[473],[224,108448,108450,108453],{"className":108449},[459],[224,108451],{"className":108452,"style":510},[463],[224,108454,1447],{"className":108455},[468],[224,108457,108459,108477],{"className":108458,"translate":388},[392],[224,108460,108462],{"className":108461},[396],[398,108463,108464],{"xmlns":400},[403,108465,108466,108474],{},[406,108467,108468,108470,108472],{},[409,108469,35657],{},[412,108471,1144],{},[419,108473,284],{},[447,108475,108476],{"encoding":449},"\\delta\\to0",[224,108478,108480,108498],{"className":108479,"ariaHidden":455},[454],[224,108481,108483,108486,108489,108492,108495],{"className":108482},[459],[224,108484],{"className":108485,"style":7642},[463],[224,108487,35657],{"className":108488,"style":35706},[468,469],[224,108490],{"className":108491,"style":496},[473],[224,108493,1144],{"className":108494},[500],[224,108496],{"className":108497,"style":496},[473],[224,108499,108501,108504],{"className":108500},[459],[224,108502],{"className":108503,"style":510},[463],[224,108505,284],{"className":108506},[468],"），故常数不能改进。",[11,108509,108510,108511,108580],{},"顺便指出：对一般 f-散度也有平行的\"固定 TV 时最小化 ",[224,108512,108514,108531],{"className":108513,"translate":388},[392],[224,108515,108517],{"className":108516},[396],[398,108518,108519],{"xmlns":400},[403,108520,108521,108529],{},[406,108522,108523],{},[695,108524,108525,108527],{},[409,108526,20303],{},[409,108528,1705],{},[447,108530,24053],{"encoding":449},[224,108532,108534],{"className":108533,"ariaHidden":455},[454],[224,108535,108537,108540],{"className":108536},[459],[224,108538],{"className":108539,"style":24063},[463],[224,108541,108543,108546],{"className":108542},[468],[224,108544,20303],{"className":108545,"style":2473},[468,469],[224,108547,108549],{"className":108548},[750],[224,108550,108552,108572],{"className":108551},[754,755],[224,108553,108555,108569],{"className":108554},[759],[224,108556,108558],{"className":108557,"style":20387},[763],[224,108559,108560,108563],{"style":20390},[224,108561],{"className":108562,"style":772},[771],[224,108564,108566],{"className":108565},[776,777,778,779],[224,108567,1705],{"className":108568,"style":1740},[468,469,779],[224,108570,787],{"className":108571},[786],[224,108573,108575],{"className":108574},[759],[224,108576,108578],{"className":108577,"style":20409},[763],[224,108579],{},"\"函数（Fedotov–Harremoës 2003 给出完整刻画），即每个 f-散度都自带一个 TV 的下界函数；这是 4.7 末拓扑结论的定量基础。",[11,108582,108583,108586,108587,108615,108616,108619],{},[378,108584,108585],{},"Bretagnolle–Huber 的推导。"," 由 Rényi 散度在阶数 ",[224,108588,108590,108603],{"className":108589,"translate":388},[392],[224,108591,108593],{"className":108592},[396],[398,108594,108595],{"xmlns":400},[403,108596,108597,108601],{},[406,108598,108599],{},[409,108600,64666],{},[447,108602,68999],{"encoding":449},[224,108604,108606],{"className":108605,"ariaHidden":455},[454],[224,108607,108609,108612],{"className":108608},[459],[224,108610],{"className":108611,"style":839},[463],[224,108613,64666],{"className":108614,"style":64696},[468,469]," 上",[378,108617,108618],{},"单调不减","（Hölder 不等式的直接推论），",[224,108621,108623],{"className":108622,"translate":388},[387],[224,108624,108626,108724],{"className":108625,"translate":388},[392],[224,108627,108629],{"className":108628},[396],[398,108630,108631],{"xmlns":400,"display":401},[403,108632,108633,108721],{},[406,108634,108635,108645,108647,108649,108651,108653,108655,108657,108665,108667,108669,108671,108673,108675,108677,108679,108681,108695,108697,108699,108701,108703,108705,108707,108709,108711,108713,108715,108717,108719],{},[695,108636,108637,108639],{},[409,108638,20303],{},[406,108640,108641,108643],{},[409,108642,30652],{},[409,108644,41831],{},[412,108646,239],{"stretchy":435},[409,108648,16018],{},[409,108650,20312],{"mathvariant":678},[409,108652,20315],{},[412,108654,1716],{"stretchy":435},[412,108656,278],{},[27527,108658,108659,108661,108663],{},[409,108660,20303],{},[419,108662,1438],{},[409,108664,21092],{},[412,108666,239],{"stretchy":435},[409,108668,16018],{},[409,108670,20312],{"mathvariant":678},[409,108672,20315],{},[412,108674,1716],{"stretchy":435},[423,108676,15054],{},[412,108678,9766],{},[423,108680,15054],{},[27527,108682,108683,108685,108693],{},[409,108684,20303],{},[406,108686,108687,108689,108691],{},[419,108688,1438],{},[409,108690,12413],{"mathvariant":678},[419,108692,1447],{},[409,108694,21092],{},[412,108696,239],{"stretchy":435},[409,108698,16018],{},[409,108700,20312],{"mathvariant":678},[409,108702,20315],{},[412,108704,1716],{"stretchy":435},[412,108706,278],{},[412,108708,5268],{},[419,108710,1447],{},[409,108712,24131],{},[412,108714,24134],{},[409,108716,9889],{},[409,108718,16021],{},[409,108720,688],{"mathvariant":678},[447,108722,108723],{"encoding":449},"D_{KL}(P\\|Q)=D^{R}_{1}(P\\|Q)\n\\ \\ge\\ D^{R}_{1\u002F2}(P\\|Q)=-2\\log BC .",[224,108725,108727,108803,108896,108986],{"className":108726,"ariaHidden":455},[454],[224,108728,108730,108733,108779,108782,108785,108788,108791,108794,108797,108800],{"className":108729},[459],[224,108731],{"className":108732,"style":535},[463],[224,108734,108736,108739],{"className":108735},[468],[224,108737,20303],{"className":108738,"style":2473},[468,469],[224,108740,108742],{"className":108741},[750],[224,108743,108745,108771],{"className":108744},[754,755],[224,108746,108748,108768],{"className":108747},[759],[224,108749,108751],{"className":108750,"style":764},[763],[224,108752,108753,108756],{"style":20390},[224,108754],{"className":108755,"style":772},[771],[224,108757,108759],{"className":108758},[776,777,778,779],[224,108760,108762,108765],{"className":108761},[468,779],[224,108763,30652],{"className":108764,"style":30740},[468,469,779],[224,108766,41831],{"className":108767},[468,469,779],[224,108769,787],{"className":108770},[786],[224,108772,108774],{"className":108773},[759],[224,108775,108777],{"className":108776,"style":794},[763],[224,108778],{},[224,108780,239],{"className":108781},[542],[224,108783,16018],{"className":108784,"style":653},[468,469],[224,108786,20312],{"className":108787},[468],[224,108789,20315],{"className":108790},[468,469],[224,108792,1716],{"className":108793},[549],[224,108795],{"className":108796,"style":496},[473],[224,108798,278],{"className":108799},[500],[224,108801],{"className":108802,"style":496},[473],[224,108804,108806,108809,108866,108869,108872,108875,108878,108881,108884,108887,108890,108893],{"className":108805},[459],[224,108807],{"className":108808,"style":70780},[463],[224,108810,108812,108815],{"className":108811},[468],[224,108813,20303],{"className":108814,"style":2473},[468,469],[224,108816,108818],{"className":108817},[750],[224,108819,108821,108858],{"className":108820},[754,755],[224,108822,108824,108855],{"className":108823},[759],[224,108825,108827,108841],{"className":108826,"style":58433},[763],[224,108828,108829,108832],{"style":69480},[224,108830],{"className":108831,"style":772},[771],[224,108833,108835],{"className":108834},[776,777,778,779],[224,108836,108838],{"className":108837},[468,779],[224,108839,1438],{"className":108840},[468,779],[224,108842,108843,108846],{"style":5074},[224,108844],{"className":108845,"style":772},[771],[224,108847,108849],{"className":108848},[776,777,778,779],[224,108850,108852],{"className":108851},[468,779],[224,108853,21092],{"className":108854,"style":69507},[468,469,779],[224,108856,787],{"className":108857},[786],[224,108859,108861],{"className":108860},[759],[224,108862,108864],{"className":108863,"style":28265},[763],[224,108865],{},[224,108867,239],{"className":108868},[542],[224,108870,16018],{"className":108871,"style":653},[468,469],[224,108873,20312],{"className":108874},[468],[224,108876,20315],{"className":108877},[468,469],[224,108879,1716],{"className":108880},[549],[224,108882,15054],{"className":108883},[473],[224,108885],{"className":108886,"style":496},[473],[224,108888,9766],{"className":108889},[500],[224,108891,15054],{"className":108892},[473],[224,108894],{"className":108895,"style":496},[473],[224,108897,108899,108903,108962,108965,108968,108971,108974,108977,108980,108983],{"className":108898},[459],[224,108900],{"className":108901,"style":108902},[463],"height:1.3383em;vertical-align:-0.447em;",[224,108904,108906,108909],{"className":108905},[468],[224,108907,20303],{"className":108908,"style":2473},[468,469],[224,108910,108912],{"className":108911},[750],[224,108913,108915,108953],{"className":108914},[754,755],[224,108916,108918,108950],{"className":108917},[759],[224,108919,108921,108936],{"className":108920,"style":58433},[763],[224,108922,108924,108927],{"style":108923},"top:-2.428em;margin-left:-0.0278em;margin-right:0.05em;",[224,108925],{"className":108926,"style":772},[771],[224,108928,108930],{"className":108929},[776,777,778,779],[224,108931,108933],{"className":108932},[468,779],[224,108934,53562],{"className":108935},[468,779],[224,108937,108938,108941],{"style":5074},[224,108939],{"className":108940,"style":772},[771],[224,108942,108944],{"className":108943},[776,777,778,779],[224,108945,108947],{"className":108946},[468,779],[224,108948,21092],{"className":108949,"style":69507},[468,469,779],[224,108951,787],{"className":108952},[786],[224,108954,108956],{"className":108955},[759],[224,108957,108960],{"className":108958,"style":108959},[763],"height:0.447em;",[224,108961],{},[224,108963,239],{"className":108964},[542],[224,108966,16018],{"className":108967,"style":653},[468,469],[224,108969,20312],{"className":108970},[468],[224,108972,20315],{"className":108973},[468,469],[224,108975,1716],{"className":108976},[549],[224,108978],{"className":108979,"style":496},[473],[224,108981,278],{"className":108982},[500],[224,108984],{"className":108985,"style":496},[473],[224,108987,108989,108992,108995,108998,109001,109006,109009,109012,109015],{"className":108988},[459],[224,108990],{"className":108991,"style":2565},[463],[224,108993,5268],{"className":108994},[468],[224,108996,1447],{"className":108997},[468],[224,108999],{"className":109000,"style":724},[473],[224,109002,24185,109004],{"className":109003},[21531],[224,109005,24189],{"style":24188},[224,109007],{"className":109008,"style":724},[473],[224,109010,9889],{"className":109011,"style":9923},[468,469],[224,109013,16021],{"className":109014,"style":30740},[468,469],[224,109016,688],{"className":109017},[468],[11,109019,109020,109021,35740,109114,109165,109166,109346],{},"又由 ",[224,109022,109024,109052],{"className":109023,"translate":388},[392],[224,109025,109027],{"className":109026},[396],[398,109028,109029],{"xmlns":400},[403,109030,109031,109049],{},[406,109032,109033,109035,109037,109039,109041,109043,109045,109047],{},[412,109034,5268],{},[409,109036,24131],{},[412,109038,24134],{},[409,109040,371],{},[412,109042,9766],{},[419,109044,1438],{},[412,109046,5268],{},[409,109048,371],{},[447,109050,109051],{"encoding":449},"-\\log x\\ge 1-x",[224,109053,109055,109087,109105],{"className":109054,"ariaHidden":455},[454],[224,109056,109058,109061,109064,109067,109072,109075,109078,109081,109084],{"className":109057},[459],[224,109059],{"className":109060,"style":2565},[463],[224,109062,5268],{"className":109063},[468],[224,109065],{"className":109066,"style":724},[473],[224,109068,24185,109070],{"className":109069},[21531],[224,109071,24189],{"style":24188},[224,109073],{"className":109074,"style":724},[473],[224,109076,371],{"className":109077},[468,469],[224,109079],{"className":109080,"style":496},[473],[224,109082,9766],{"className":109083},[500],[224,109085],{"className":109086,"style":496},[473],[224,109088,109090,109093,109096,109099,109102],{"className":109089},[459],[224,109091],{"className":109092,"style":7047},[463],[224,109094,1438],{"className":109095},[468],[224,109097],{"className":109098,"style":474},[473],[224,109100,5268],{"className":109101},[478],[224,109103],{"className":109104,"style":474},[473],[224,109106,109108,109111],{"className":109107},[459],[224,109109],{"className":109110,"style":839},[463],[224,109112,371],{"className":109113},[468,469],[224,109115,109117,109135],{"className":109116,"translate":388},[392],[224,109118,109120],{"className":109119},[396],[398,109121,109122],{"xmlns":400},[403,109123,109124,109132],{},[406,109125,109126,109128,109130],{},[409,109127,371],{},[412,109129,325],{},[419,109131,284],{},[447,109133,109134],{"encoding":449},"x>0",[224,109136,109138,109156],{"className":109137,"ariaHidden":455},[454],[224,109139,109141,109144,109147,109150,109153],{"className":109140},[459],[224,109142],{"className":109143,"style":2810},[463],[224,109145,371],{"className":109146},[468,469],[224,109148],{"className":109149,"style":496},[473],[224,109151,325],{"className":109152},[500],[224,109154],{"className":109155,"style":496},[473],[224,109157,109159,109162],{"className":109158},[459],[224,109160],{"className":109161,"style":510},[463],[224,109163,284],{"className":109164},[468],"），",[224,109167,109169,109219],{"className":109168,"translate":388},[392],[224,109170,109172],{"className":109171},[396],[398,109173,109174],{"xmlns":400},[403,109175,109176,109216],{},[406,109177,109178,109180,109182,109184,109186,109188,109190,109192,109194,109196,109198,109200,109202,109204,109206,109208,109210],{},[412,109179,5268],{},[419,109181,1447],{},[409,109183,24131],{},[412,109185,24134],{},[409,109187,9889],{},[409,109189,16021],{},[412,109191,9766],{},[419,109193,1447],{},[412,109195,239],{"stretchy":435},[419,109197,1438],{},[412,109199,5268],{},[409,109201,9889],{},[409,109203,16021],{},[412,109205,1716],{"stretchy":435},[412,109207,278],{},[419,109209,1447],{},[3248,109211,109212,109214],{},[409,109213,53807],{},[419,109215,1447],{},[447,109217,109218],{"encoding":449}," -2\\log BC\\ge 2(1-BC)=2H^2",[224,109220,109222,109260,109284,109308],{"className":109221,"ariaHidden":455},[454],[224,109223,109225,109228,109231,109234,109237,109242,109245,109248,109251,109254,109257],{"className":109224},[459],[224,109226],{"className":109227,"style":2565},[463],[224,109229,5268],{"className":109230},[468],[224,109232,1447],{"className":109233},[468],[224,109235],{"className":109236,"style":724},[473],[224,109238,24185,109240],{"className":109239},[21531],[224,109241,24189],{"style":24188},[224,109243],{"className":109244,"style":724},[473],[224,109246,9889],{"className":109247,"style":9923},[468,469],[224,109249,16021],{"className":109250,"style":30740},[468,469],[224,109252],{"className":109253,"style":496},[473],[224,109255,9766],{"className":109256},[500],[224,109258],{"className":109259,"style":496},[473],[224,109261,109263,109266,109269,109272,109275,109278,109281],{"className":109262},[459],[224,109264],{"className":109265,"style":535},[463],[224,109267,1447],{"className":109268},[468],[224,109270,239],{"className":109271},[542],[224,109273,1438],{"className":109274},[468],[224,109276],{"className":109277,"style":474},[473],[224,109279,5268],{"className":109280},[478],[224,109282],{"className":109283,"style":474},[473],[224,109285,109287,109290,109293,109296,109299,109302,109305],{"className":109286},[459],[224,109288],{"className":109289,"style":535},[463],[224,109291,9889],{"className":109292,"style":9923},[468,469],[224,109294,16021],{"className":109295,"style":30740},[468,469],[224,109297,1716],{"className":109298},[549],[224,109300],{"className":109301,"style":496},[473],[224,109303,278],{"className":109304},[500],[224,109306],{"className":109307,"style":496},[473],[224,109309,109311,109314,109317],{"className":109310},[459],[224,109312],{"className":109313,"style":5671},[463],[224,109315,1447],{"className":109316},[468],[224,109318,109320,109323],{"className":109319},[468],[224,109321,53807],{"className":109322,"style":53961},[468,469],[224,109324,109326],{"className":109325},[750],[224,109327,109329],{"className":109328},[754],[224,109330,109332],{"className":109331},[759],[224,109333,109335],{"className":109334,"style":5671},[763],[224,109336,109337,109340],{"style":3288},[224,109338],{"className":109339,"style":772},[771],[224,109341,109343],{"className":109342},[776,777,778,779],[224,109344,1447],{"className":109345},[468,779],"。合并即得",[224,109348,109350],{"className":109349,"translate":388},[387],[224,109351,109353,109407],{"className":109352,"translate":388},[392],[224,109354,109356],{"className":109355},[396],[398,109357,109358],{"xmlns":400,"display":401},[403,109359,109360,109404],{},[406,109361,109362,109368,109370,109378,109388,109390,109392,109394,109396,109398,109400,109402],{},[3248,109363,109364,109366],{},[409,109365,53807],{},[419,109367,1447],{},[412,109369,2070],{},[5221,109371,109372],{"scriptlevel":284,"displaystyle":435},[20338,109373,109374,109376],{},[419,109375,1438],{},[419,109377,1447],{},[695,109379,109380,109382],{},[409,109381,20303],{},[406,109383,109384,109386],{},[409,109385,30652],{},[409,109387,41831],{},[412,109389,239],{"stretchy":435},[409,109391,16018],{},[409,109393,20312],{"mathvariant":678},[409,109395,20315],{},[412,109397,1716],{"stretchy":435},[409,109399,688],{"mathvariant":678},[473,109401],{"width":4247},[409,109403,79158],{"mathvariant":678},[447,109405,109406],{"encoding":449},"H^2\\le \\tfrac12 D_{KL}(P\\|Q). \\qquad\\square",[224,109408,109410,109455],{"className":109409,"ariaHidden":455},[454],[224,109411,109413,109417,109446,109449,109452],{"className":109412},[459],[224,109414],{"className":109415,"style":109416},[463],"height:1.0001em;vertical-align:-0.136em;",[224,109418,109420,109423],{"className":109419},[468],[224,109421,53807],{"className":109422,"style":53961},[468,469],[224,109424,109426],{"className":109425},[750],[224,109427,109429],{"className":109428},[754],[224,109430,109432],{"className":109431},[759],[224,109433,109435],{"className":109434,"style":5071},[763],[224,109436,109437,109440],{"style":5074},[224,109438],{"className":109439,"style":772},[771],[224,109441,109443],{"className":109442},[776,777,778,779],[224,109444,1447],{"className":109445},[468,779],[224,109447],{"className":109448,"style":496},[473],[224,109450,2070],{"className":109451},[500],[224,109453],{"className":109454,"style":496},[473],[224,109456,109458,109461,109529,109575,109578,109581,109584,109587,109590,109593,109596],{"className":109457},[459],[224,109459],{"className":109460,"style":22544},[463],[224,109462,109464,109467,109526],{"className":109463},[468],[224,109465],{"className":109466},[542,8479],[224,109468,109470],{"className":109469},[20338],[224,109471,109473,109518],{"className":109472},[754,755],[224,109474,109476,109515],{"className":109475},[759],[224,109477,109479,109493,109501],{"className":109478,"style":22563},[763],[224,109480,109481,109484],{"style":22566},[224,109482],{"className":109483,"style":2108},[771],[224,109485,109487],{"className":109486},[776,777,778,779],[224,109488,109490],{"className":109489},[468,779],[224,109491,1447],{"className":109492},[468,779],[224,109494,109495,109498],{"style":20546},[224,109496],{"className":109497,"style":2108},[771],[224,109499],{"className":109500,"style":20554},[20553],[224,109502,109503,109506],{"style":22589},[224,109504],{"className":109505,"style":2108},[771],[224,109507,109509],{"className":109508},[776,777,778,779],[224,109510,109512],{"className":109511},[468,779],[224,109513,1438],{"className":109514},[468,779],[224,109516,787],{"className":109517},[786],[224,109519,109521],{"className":109520},[759],[224,109522,109524],{"className":109523,"style":22611},[763],[224,109525],{},[224,109527],{"className":109528},[549,8479],[224,109530,109532,109535],{"className":109531},[468],[224,109533,20303],{"className":109534,"style":2473},[468,469],[224,109536,109538],{"className":109537},[750],[224,109539,109541,109567],{"className":109540},[754,755],[224,109542,109544,109564],{"className":109543},[759],[224,109545,109547],{"className":109546,"style":764},[763],[224,109548,109549,109552],{"style":20390},[224,109550],{"className":109551,"style":772},[771],[224,109553,109555],{"className":109554},[776,777,778,779],[224,109556,109558,109561],{"className":109557},[468,779],[224,109559,30652],{"className":109560,"style":30740},[468,469,779],[224,109562,41831],{"className":109563},[468,469,779],[224,109565,787],{"className":109566},[786],[224,109568,109570],{"className":109569},[759],[224,109571,109573],{"className":109572,"style":794},[763],[224,109574],{},[224,109576,239],{"className":109577},[542],[224,109579,16018],{"className":109580,"style":653},[468,469],[224,109582,20312],{"className":109583},[468],[224,109585,20315],{"className":109586},[468,469],[224,109588,1716],{"className":109589},[549],[224,109591,688],{"className":109592},[468],[224,109594],{"className":109595,"style":4416},[473],[224,109597,79158],{"className":109598},[468,79700],[11,109600,109601,109602,372,109807,46210,109991,110344],{},"顺便验证与 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D_{KL}\\le\\frac12 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与 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的关系。"," 由 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p-\\sqrt q|\\;|\\sqrt p+\\sqrt q|\\,dx\\\\\n&\\le\\sqrt{\\int(\\sqrt p-\\sqrt q)^2}\n\\cdot\\sqrt{\\int(\\sqrt p+\\sqrt 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D_{KL}",[224,111936,111938,111983],{"className":111937,"ariaHidden":455},[454],[224,111939,111941,111945,111974,111977,111980],{"className":111940},[459],[224,111942],{"className":111943,"style":111944},[463],"height:0.9501em;vertical-align:-0.136em;",[224,111946,111948,111951],{"className":111947},[468],[224,111949,53807],{"className":111950,"style":53961},[468,469],[224,111952,111954],{"className":111953},[750],[224,111955,111957],{"className":111956},[754],[224,111958,111960],{"className":111959},[759],[224,111961,111963],{"className":111962,"style":5671},[763],[224,111964,111965,111968],{"style":3288},[224,111966],{"className":111967,"style":772},[771],[224,111969,111971],{"className":111970},[776,777,778,779],[224,111972,1447],{"className":111973},[468,779],[224,111975],{"className":111976,"style":496},[473],[224,111978,2070],{"className":111979},[500],[224,111981],{"className":111982,"style":496},[473],[224,111984,111986,111989,112057],{"className":111985},[459],[224,111987],{"className":111988,"style":22544},[463],[224,111990,111992,111995,112054],{"className":111991},[468],[224,111993],{"className":111994},[542,8479],[224,111996,111998],{"className":111997},[20338],[224,111999,112001,112046],{"className":112000},[754,755],[224,112002,112004,112043],{"className":112003},[759],[224,112005,112007,112021,112029],{"className":112006,"style":22563},[763],[224,112008,112009,112012],{"style":22566},[224,112010],{"className":112011,"style":2108},[771],[224,112013,112015],{"className":112014},[776,777,778,779],[224,112016,112018],{"className":112017},[468,779],[224,112019,1447],{"className":112020},[468,779],[224,112022,112023,112026],{"style":20546},[224,112024],{"className":112025,"style":2108},[771],[224,112027],{"className":112028,"style":20554},[20553],[224,112030,112031,112034],{"style":22589},[224,112032],{"className":112033,"style":2108},[771],[224,112035,112037],{"className":112036},[776,777,778,779],[224,112038,112040],{"className":112039},[468,779],[224,112041,1438],{"className":112042},[468,779],[224,112044,787],{"className":112045},[786],[224,112047,112049],{"className":112048},[759],[224,112050,112052],{"className":112051,"style":22611},[763],[224,112053],{},[224,112055],{"className":112056},[549,8479],[224,112058,112060,112063],{"className":112059},[468],[224,112061,20303],{"className":112062,"style":2473},[468,469],[224,112064,112066],{"className":112065},[750],[224,112067,112069,112095],{"className":112068},[754,755],[224,112070,112072,112092],{"className":112071},[759],[224,112073,112075],{"className":112074,"style":764},[763],[224,112076,112077,112080],{"style":20390},[224,112078],{"className":112079,"style":772},[771],[224,112081,112083],{"className":112082},[776,777,778,779],[224,112084,112086,112089],{"className":112085},[468,779],[224,112087,30652],{"className":112088,"style":30740},[468,469,779],[224,112090,41831],{"className":112091},[468,469,779],[224,112093,787],{"className":112094},[786],[224,112096,112098],{"className":112097},[759],[224,112099,112101],{"className":112100,"style":794},[763],[224,112102],{}," 串起来就有 ",[224,112105,112107,112139],{"className":112106,"translate":388},[392],[224,112108,112110],{"className":112109},[396],[398,112111,112112],{"xmlns":400},[403,112113,112114,112136],{},[406,112115,112116,112122,112124],{},[406,112117,112118,112120],{},[409,112119,637],{"mathvariant":678},[409,112121,16059],{"mathvariant":678},[412,112123,2070],{},[54262,112125,112126],{},[695,112127,112128,112130],{},[409,112129,20303],{},[406,112131,112132,112134],{},[409,112133,30652],{},[409,112135,41831],{},[447,112137,112138],{"encoding":449},"\\mathrm{TV}\\le\\sqrt{D_{KL}}",[224,112140,112142,112163],{"className":112141,"ariaHidden":455},[454],[224,112143,112145,112148,112154,112157,112160],{"className":112144},[459],[224,112146],{"className":112147,"style":108254},[463],[224,112149,112151],{"className":112150},[468],[224,112152,106664],{"className":112153,"style":24188},[468,1906],[224,112155],{"className":112156,"style":496},[473],[224,112158,2070],{"className":112159},[500],[224,112161],{"className":112162,"style":496},[473],[224,112164,112166,112169],{"className":112165},[459],[224,112167],{"className":112168,"style":54286},[463],[224,112170,112172],{"className":112171},[468,54290],[224,112173,112175,112251],{"className":112174},[754,755],[224,112176,112178,112248],{"className":112177},[759],[224,112179,112181,112236],{"className":112180,"style":54300},[763],[224,112182,112184,112187],{"className":112183,"style":2120},[2139],[224,112185],{"className":112186,"style":2108},[771],[224,112188,112190],{"className":112189,"style":54310},[468],[224,112191,112193,112196],{"className":112192},[468],[224,112194,20303],{"className":112195,"style":2473},[468,469],[224,112197,112199],{"className":112198},[750],[224,112200,112202,112228],{"className":112201},[754,755],[224,112203,112205,112225],{"className":112204},[759],[224,112206,112208],{"className":112207,"style":764},[763],[224,112209,112210,112213],{"style":20390},[224,112211],{"className":112212,"style":772},[771],[224,112214,112216],{"className":112215},[776,777,778,779],[224,112217,112219,112222],{"className":112218},[468,779],[224,112220,30652],{"className":112221,"style":30740},[468,469,779],[224,112223,41831],{"className":112224},[468,469,779],[224,112226,787],{"className":112227},[786],[224,112229,112231],{"className":112230},[759],[224,112232,112234],{"className":112233,"style":794},[763],[224,112235],{},[224,112237,112238,112241],{"style":54359},[224,112239],{"className":112240,"style":2108},[771],[224,112242,112244],{"className":112243,"style":54367},[54366],[2155,112245,112246],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,112247],{"d":54374},[224,112249,787],{"className":112250},[786],[224,112252,112254],{"className":112253},[759],[224,112255,112257],{"className":112256,"style":54384},[763],[224,112258],{},"（比 Pinsker 常数差一个 ",[224,112261,112263,112279],{"className":112262,"translate":388},[392],[224,112264,112266],{"className":112265},[396],[398,112267,112268],{"xmlns":400},[403,112269,112270,112276],{},[406,112271,112272],{},[54262,112273,112274],{},[419,112275,1447],{},[447,112277,112278],{"encoding":449},"\\sqrt2",[224,112280,112282],{"className":112281,"ariaHidden":455},[454],[224,112283,112285,112289],{"className":112284},[459],[224,112286],{"className":112287,"style":112288},[463],"height:1.04em;vertical-align:-0.1328em;",[224,112290,112292],{"className":112291},[468,54290],[224,112293,112295,112327],{"className":112294},[754,755],[224,112296,112298,112324],{"className":112297},[759],[224,112299,112302,112311],{"className":112300,"style":112301},[763],"height:0.9072em;",[224,112303,112305,112308],{"className":112304,"style":2120},[2139],[224,112306],{"className":112307,"style":2108},[771],[224,112309,1447],{"className":112310,"style":54310},[468],[224,112312,112314,112317],{"style":112313},"top:-2.8672em;",[224,112315],{"className":112316,"style":2108},[771],[224,112318,112320],{"className":112319,"style":54367},[54366],[2155,112321,112322],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,112323],{"d":54374},[224,112325,787],{"className":112326},[786],[224,112328,112330],{"className":112329},[759],[224,112331,112334],{"className":112332,"style":112333},[763],"height:0.1328em;",[224,112335],{},[21055,112337,112339,112489,114017,115507,115893,116459],{"title":112338},"定理 4.2（拓扑关系的准确版）",[11,112340,20606,112341,112369,112370,112435,112436,112488],{},[224,112342,112344,112357],{"className":112343,"translate":388},[392],[224,112345,112347],{"className":112346},[396],[398,112348,112349],{"xmlns":400},[403,112350,112351,112355],{},[406,112352,112353],{},[409,112354,1705],{},[447,112356,1705],{"encoding":449},[224,112358,112360],{"className":112359,"ariaHidden":455},[454],[224,112361,112363,112366],{"className":112362},[459],[224,112364],{"className":112365,"style":2565},[463],[224,112367,1705],{"className":112368,"style":1740},[468,469]," 凸、",[224,112371,112373,112396],{"className":112372,"translate":388},[392],[224,112374,112376],{"className":112375},[396],[398,112377,112378],{"xmlns":400},[403,112379,112380,112394],{},[406,112381,112382,112384,112386,112388,112390,112392],{},[409,112383,1705],{},[412,112385,239],{"stretchy":435},[419,112387,1438],{},[412,112389,1716],{"stretchy":435},[412,112391,278],{},[419,112393,284],{},[447,112395,20243],{"encoding":449},[224,112397,112399,112426],{"className":112398,"ariaHidden":455},[454],[224,112400,112402,112405,112408,112411,112414,112417,112420,112423],{"className":112401},[459],[224,112403],{"className":112404,"style":535},[463],[224,112406,1705],{"className":112407,"style":1740},[468,469],[224,112409,239],{"className":112410},[542],[224,112412,1438],{"className":112413},[468],[224,112415,1716],{"className":112416},[549],[224,112418],{"className":112419,"style":496},[473],[224,112421,278],{"className":112422},[500],[224,112424],{"className":112425,"style":496},[473],[224,112427,112429,112432],{"className":112428},[459],[224,112430],{"className":112431,"style":510},[463],[224,112433,284],{"className":112434},[468]," 且在 ",[224,112437,112439,112461],{"className":112438,"translate":388},[392],[224,112440,112442],{"className":112441},[396],[398,112443,112444],{"xmlns":400},[403,112445,112446,112458],{},[406,112447,112448,112450,112452,112454,112456],{},[412,112449,239],{"stretchy":435},[419,112451,284],{},[412,112453,1877],{"separator":455},[409,112455,21085],{"mathvariant":678},[412,112457,1716],{"stretchy":435},[447,112459,112460],{"encoding":449},"(0,\\infty)",[224,112462,112464],{"className":112463,"ariaHidden":455},[454],[224,112465,112467,112470,112473,112476,112479,112482,112485],{"className":112466},[459],[224,112468],{"className":112469,"style":535},[463],[224,112471,239],{"className":112472},[542],[224,112474,284],{"className":112475},[468],[224,112477,1877],{"className":112478},[1917],[224,112480],{"className":112481,"style":724},[473],[224,112483,21085],{"className":112484},[468],[224,112486,1716],{"className":112487},[549]," 上连续。",[11,112490,112491,26478,112494,112827,112828,112899,112900,372,113007,113156,113157,113312,113313,113463,113464,113522,113523,113651,113652,113709,113710,114016],{},[378,112492,112493],{},"(a)（\"下来\"方向，一般成立）",[224,112495,112497,112567],{"className":112496,"translate":388},[392],[224,112498,112500],{"className":112499},[396],[398,112501,112502],{"xmlns":400},[403,112503,112504,112564],{},[406,112505,112506,112512,112514,112520,112522,112524,112526,112528,112530,112532,112534,112536,112546,112548,112554,112556,112558,112560,112562],{},[695,112507,112508,112510],{},[409,112509,20303],{},[409,112511,1705],{},[412,112513,239],{"stretchy":435},[695,112515,112516,112518],{},[409,112517,16018],{},[409,112519,3254],{},[409,112521,20312],{"mathvariant":678},[409,112523,20315],{},[412,112525,1716],{"stretchy":435},[412,112527,1144],{},[419,112529,284],{},[423,112531,15054],{},[412,112533,24315],{},[423,112535,15054],{},[695,112537,112538,112540],{},[409,112539,20303],{},[406,112541,112542,112544],{},[409,112543,637],{},[409,112545,16059],{},[412,112547,239],{"stretchy":435},[695,112549,112550,112552],{},[409,112551,16018],{},[409,112553,3254],{},[412,112555,1877],{"separator":455},[409,112557,20315],{},[412,112559,1716],{"stretchy":435},[412,112561,1144],{},[419,112563,284],{},[447,112565,112566],{"encoding":449},"D_f(P_n\\|Q)\\to0\\ \\Longrightarrow\\ D_{TV}(P_n,Q)\\to0",[224,112568,112570,112677,112702,112818],{"className":112569,"ariaHidden":455},[454],[224,112571,112573,112576,112616,112619,112659,112662,112665,112668,112671,112674],{"className":112572},[459],[224,112574],{"className":112575,"style":20368},[463],[224,112577,112579,112582],{"className":112578},[468],[224,112580,20303],{"className":112581,"style":2473},[468,469],[224,112583,112585],{"className":112584},[750],[224,112586,112588,112608],{"className":112587},[754,755],[224,112589,112591,112605],{"className":112590},[759],[224,112592,112594],{"className":112593,"style":20387},[763],[224,112595,112596,112599],{"style":20390},[224,112597],{"className":112598,"style":772},[771],[224,112600,112602],{"className":112601},[776,777,778,779],[224,112603,1705],{"className":112604,"style":1740},[468,469,779],[224,112606,787],{"className":112607},[786],[224,112609,112611],{"className":112610},[759],[224,112612,112614],{"className":112613,"style":20409},[763],[224,112615],{},[224,112617,239],{"className":112618},[542],[224,112620,112622,112625],{"className":112621},[468],[224,112623,16018],{"className":112624,"style":653},[468,469],[224,112626,112628],{"className":112627},[750],[224,112629,112631,112651],{"className":112630},[754,755],[224,112632,112634,112648],{"className":112633},[759],[224,112635,112637],{"className":112636,"style":64841},[763],[224,112638,112639,112642],{"style":31092},[224,112640],{"className":112641,"style":772},[771],[224,112643,112645],{"className":112644},[776,777,778,779],[224,112646,3254],{"className":112647},[468,469,779],[224,112649,787],{"className":112650},[786],[224,112652,112654],{"className":112653},[759],[224,112655,112657],{"className":112656,"style":794},[763],[224,112658],{},[224,112660,20312],{"className":112661},[468],[224,112663,20315],{"className":112664},[468,469],[224,112666,1716],{"className":112667},[549],[224,112669],{"className":112670,"style":496},[473],[224,112672,1144],{"className":112673},[500],[224,112675],{"className":112676,"style":496},[473],[224,112678,112680,112684,112687,112690,112693,112696,112699],{"className":112679},[459],[224,112681],{"className":112682,"style":112683},[463],"height:0.6684em;vertical-align:-0.024em;",[224,112685,284],{"className":112686},[468],[224,112688,15054],{"className":112689},[473],[224,112691],{"className":112692,"style":496},[473],[224,112694,24315],{"className":112695},[500],[224,112697,15054],{"className":112698},[473],[224,112700],{"className":112701,"style":496},[473],[224,112703,112705,112708,112754,112757,112797,112800,112803,112806,112809,112812,112815],{"className":112704},[459],[224,112706],{"className":112707,"style":535},[463],[224,112709,112711,112714],{"className":112710},[468],[224,112712,20303],{"className":112713,"style":2473},[468,469],[224,112715,112717],{"className":112716},[750],[224,112718,112720,112746],{"className":112719},[754,755],[224,112721,112723,112743],{"className":112722},[759],[224,112724,112726],{"className":112725,"style":764},[763],[224,112727,112728,112731],{"style":20390},[224,112729],{"className":112730,"style":772},[771],[224,112732,112734],{"className":112733},[776,777,778,779],[224,112735,112737,112740],{"className":112736},[468,779],[224,112738,637],{"className":112739,"style":653},[468,469,779],[224,112741,16059],{"className":112742,"style":474},[468,469,779],[224,112744,787],{"className":112745},[786],[224,112747,112749],{"className":112748},[759],[224,112750,112752],{"className":112751,"style":794},[763],[224,112753],{},[224,112755,239],{"className":112756},[542],[224,112758,112760,112763],{"className":112759},[468],[224,112761,16018],{"className":112762,"style":653},[468,469],[224,112764,112766],{"className":112765},[750],[224,112767,112769,112789],{"className":112768},[754,755],[224,112770,112772,112786],{"className":112771},[759],[224,112773,112775],{"className":112774,"style":64841},[763],[224,112776,112777,112780],{"style":31092},[224,112778],{"className":112779,"style":772},[771],[224,112781,112783],{"className":112782},[776,777,778,779],[224,112784,3254],{"className":112785},[468,469,779],[224,112787,787],{"className":112788},[786],[224,112790,112792],{"className":112791},[759],[224,112793,112795],{"className":112794,"style":794},[763],[224,112796],{},[224,112798,1877],{"className":112799},[1917],[224,112801],{"className":112802,"style":724},[473],[224,112804,20315],{"className":112805},[468,469],[224,112807,1716],{"className":112808},[549],[224,112810],{"className":112811,"style":496},[473],[224,112813,1144],{"className":112814},[500],[224,112816],{"className":112817,"style":496},[473],[224,112819,112821,112824],{"className":112820},[459],[224,112822],{"className":112823,"style":510},[463],[224,112825,284],{"className":112826},[468],"。每个适度正则的 f-散度都自带一个连续的 TV 下界函数 ",[224,112829,112831,112849],{"className":112830,"translate":388},[392],[224,112832,112834],{"className":112833},[396],[398,112835,112836],{"xmlns":400},[403,112837,112838,112846],{},[406,112839,112840],{},[695,112841,112842,112844],{},[409,112843,24189],{},[409,112845,1705],{},[447,112847,112848],{"encoding":449},"g_f",[224,112850,112852],{"className":112851,"ariaHidden":455},[454],[224,112853,112855,112859],{"className":112854},[459],[224,112856],{"className":112857,"style":112858},[463],"height:0.7167em;vertical-align:-0.2861em;",[224,112860,112862,112865],{"className":112861},[468],[224,112863,24189],{"className":112864,"style":492},[468,469],[224,112866,112868],{"className":112867},[750],[224,112869,112871,112891],{"className":112870},[754,755],[224,112872,112874,112888],{"className":112873},[759],[224,112875,112877],{"className":112876,"style":20387},[763],[224,112878,112879,112882],{"style":3854},[224,112880],{"className":112881,"style":772},[771],[224,112883,112885],{"className":112884},[776,777,778,779],[224,112886,1705],{"className":112887,"style":1740},[468,469,779],[224,112889,787],{"className":112890},[786],[224,112892,112894],{"className":112893},[759],[224,112895,112897],{"className":112896,"style":20409},[763],[224,112898],{},"（Vajda \u002F Fedotov–Harremoës），",[224,112901,112903,112931],{"className":112902,"translate":388},[392],[224,112904,112906],{"className":112905},[396],[398,112907,112908],{"xmlns":400},[403,112909,112910,112928],{},[406,112911,112912,112918,112920,112922,112924,112926],{},[695,112913,112914,112916],{},[409,112915,24189],{},[409,112917,1705],{},[412,112919,239],{"stretchy":435},[419,112921,284],{},[412,112923,1716],{"stretchy":435},[412,112925,278],{},[419,112927,284],{},[447,112929,112930],{"encoding":449},"g_f(0)=0",[224,112932,112934,112998],{"className":112933,"ariaHidden":455},[454],[224,112935,112937,112940,112980,112983,112986,112989,112992,112995],{"className":112936},[459],[224,112938],{"className":112939,"style":20368},[463],[224,112941,112943,112946],{"className":112942},[468],[224,112944,24189],{"className":112945,"style":492},[468,469],[224,112947,112949],{"className":112948},[750],[224,112950,112952,112972],{"className":112951},[754,755],[224,112953,112955,112969],{"className":112954},[759],[224,112956,112958],{"className":112957,"style":20387},[763],[224,112959,112960,112963],{"style":3854},[224,112961],{"className":112962,"style":772},[771],[224,112964,112966],{"className":112965},[776,777,778,779],[224,112967,1705],{"className":112968,"style":1740},[468,469,779],[224,112970,787],{"className":112971},[786],[224,112973,112975],{"className":112974},[759],[224,112976,112978],{"className":112977,"style":20409},[763],[224,112979],{},[224,112981,239],{"className":112982},[542],[224,112984,284],{"className":112985},[468],[224,112987,1716],{"className":112988},[549],[224,112990],{"className":112991,"style":496},[473],[224,112993,278],{"className":112994},[500],[224,112996],{"className":112997,"style":496},[473],[224,112999,113001,113004],{"className":113000},[459],[224,113002],{"className":113003,"style":510},[463],[224,113005,284],{"className":113006},[468],[224,113008,113010,113050],{"className":113009,"translate":388},[392],[224,113011,113013],{"className":113012},[396],[398,113014,113015],{"xmlns":400},[403,113016,113017,113047],{},[406,113018,113019,113025,113027,113029,113031,113033,113035,113037,113039,113041,113043,113045],{},[695,113020,113021,113023],{},[409,113022,24189],{},[409,113024,1705],{},[412,113026,239],{"stretchy":435},[409,113028,1435],{},[412,113030,1716],{"stretchy":435},[412,113032,325],{},[419,113034,284],{},[423,113036,15054],{},[412,113038,239],{"stretchy":435},[409,113040,1435],{},[412,113042,325],{},[419,113044,284],{},[412,113046,1716],{"stretchy":435},[447,113048,113049],{"encoding":449},"g_f(t)>0\\ (t>0)",[224,113051,113053,113117,113144],{"className":113052,"ariaHidden":455},[454],[224,113054,113056,113059,113099,113102,113105,113108,113111,113114],{"className":113055},[459],[224,113057],{"className":113058,"style":20368},[463],[224,113060,113062,113065],{"className":113061},[468],[224,113063,24189],{"className":113064,"style":492},[468,469],[224,113066,113068],{"className":113067},[750],[224,113069,113071,113091],{"className":113070},[754,755],[224,113072,113074,113088],{"className":113073},[759],[224,113075,113077],{"className":113076,"style":20387},[763],[224,113078,113079,113082],{"style":3854},[224,113080],{"className":113081,"style":772},[771],[224,113083,113085],{"className":113084},[776,777,778,779],[224,113086,1705],{"className":113087,"style":1740},[468,469,779],[224,113089,787],{"className":113090},[786],[224,113092,113094],{"className":113093},[759],[224,113095,113097],{"className":113096,"style":20409},[763],[224,113098],{},[224,113100,239],{"className":113101},[542],[224,113103,1435],{"className":113104},[468,469],[224,113106,1716],{"className":113107},[549],[224,113109],{"className":113110,"style":496},[473],[224,113112,325],{"className":113113},[500],[224,113115],{"className":113116,"style":496},[473],[224,113118,113120,113123,113126,113129,113132,113135,113138,113141],{"className":113119},[459],[224,113121],{"className":113122,"style":535},[463],[224,113124,284],{"className":113125},[468],[224,113127,15054],{"className":113128},[473],[224,113130,239],{"className":113131},[542],[224,113133,1435],{"className":113134},[468,469],[224,113136],{"className":113137,"style":496},[473],[224,113139,325],{"className":113140},[500],[224,113142],{"className":113143,"style":496},[473],[224,113145,113147,113150,113153],{"className":113146},[459],[224,113148],{"className":113149,"style":535},[463],[224,113151,284],{"className":113152},[468],[224,113154,1716],{"className":113155},[549],"，使 ",[224,113158,113160,113196],{"className":113159,"translate":388},[392],[224,113161,113163],{"className":113162},[396],[398,113164,113165],{"xmlns":400},[403,113166,113167,113193],{},[406,113168,113169,113175,113177,113183,113185,113191],{},[695,113170,113171,113173],{},[409,113172,20303],{},[409,113174,1705],{},[412,113176,9766],{},[695,113178,113179,113181],{},[409,113180,24189],{},[409,113182,1705],{},[412,113184,239],{"stretchy":435},[406,113186,113187,113189],{},[409,113188,637],{"mathvariant":678},[409,113190,16059],{"mathvariant":678},[412,113192,1716],{"stretchy":435},[447,113194,113195],{"encoding":449},"D_f\\ge g_f(\\mathrm{TV})",[224,113197,113199,113254],{"className":113198,"ariaHidden":455},[454],[224,113200,113202,113205,113245,113248,113251],{"className":113201},[459],[224,113203],{"className":113204,"style":24063},[463],[224,113206,113208,113211],{"className":113207},[468],[224,113209,20303],{"className":113210,"style":2473},[468,469],[224,113212,113214],{"className":113213},[750],[224,113215,113217,113237],{"className":113216},[754,755],[224,113218,113220,113234],{"className":113219},[759],[224,113221,113223],{"className":113222,"style":20387},[763],[224,113224,113225,113228],{"style":20390},[224,113226],{"className":113227,"style":772},[771],[224,113229,113231],{"className":113230},[776,777,778,779],[224,113232,1705],{"className":113233,"style":1740},[468,469,779],[224,113235,787],{"className":113236},[786],[224,113238,113240],{"className":113239},[759],[224,113241,113243],{"className":113242,"style":20409},[763],[224,113244],{},[224,113246],{"className":113247,"style":496},[473],[224,113249,9766],{"className":113250},[500],[224,113252],{"className":113253,"style":496},[473],[224,113255,113257,113260,113300,113303,113309],{"className":113256},[459],[224,113258],{"className":113259,"style":20368},[463],[224,113261,113263,113266],{"className":113262},[468],[224,113264,24189],{"className":113265,"style":492},[468,469],[224,113267,113269],{"className":113268},[750],[224,113270,113272,113292],{"className":113271},[754,755],[224,113273,113275,113289],{"className":113274},[759],[224,113276,113278],{"className":113277,"style":20387},[763],[224,113279,113280,113283],{"style":3854},[224,113281],{"className":113282,"style":772},[771],[224,113284,113286],{"className":113285},[776,777,778,779],[224,113287,1705],{"className":113288,"style":1740},[468,469,779],[224,113290,787],{"className":113291},[786],[224,113293,113295],{"className":113294},[759],[224,113296,113298],{"className":113297,"style":20409},[763],[224,113299],{},[224,113301,239],{"className":113302},[542],[224,113304,113306],{"className":113305},[468],[224,113307,106664],{"className":113308,"style":24188},[468,1906],[224,113310,1716],{"className":113311},[549],"；常用成员的具体形式：KL 给 ",[224,113314,113316,113354],{"className":113315,"translate":388},[392],[224,113317,113319],{"className":113318},[396],[398,113320,113321],{"xmlns":400},[403,113322,113323,113351],{},[406,113324,113325,113327,113329,113339,113341],{},[419,113326,1447],{},[423,113328,682],{},[3248,113330,113331,113337],{},[406,113332,113333,113335],{},[409,113334,637],{"mathvariant":678},[409,113336,16059],{"mathvariant":678},[419,113338,1447],{},[412,113340,2070],{},[695,113342,113343,113345],{},[409,113344,20303],{},[406,113346,113347,113349],{},[409,113348,30652],{},[409,113350,41831],{},[447,113352,113353],{"encoding":449},"2\\,\\mathrm{TV}^2\\le D_{KL}",[224,113355,113357,113411],{"className":113356,"ariaHidden":455},[454],[224,113358,113360,113364,113367,113370,113402,113405,113408],{"className":113359},[459],[224,113361],{"className":113362,"style":113363},[463],"height:1.0233em;vertical-align:-0.136em;",[224,113365,1447],{"className":113366},[468],[224,113368],{"className":113369,"style":724},[473],[224,113371,113373,113379],{"className":113372},[468],[224,113374,113376],{"className":113375},[468],[224,113377,106664],{"className":113378,"style":24188},[468,1906],[224,113380,113382],{"className":113381},[750],[224,113383,113385],{"className":113384},[754],[224,113386,113388],{"className":113387},[759],[224,113389,113391],{"className":113390,"style":108276},[763],[224,113392,113393,113396],{"style":108306},[224,113394],{"className":113395,"style":772},[771],[224,113397,113399],{"className":113398},[776,777,778,779],[224,113400,1447],{"className":113401},[468,779],[224,113403],{"className":113404,"style":496},[473],[224,113406,2070],{"className":113407},[500],[224,113409],{"className":113410,"style":496},[473],[224,113412,113414,113417],{"className":113413},[459],[224,113415],{"className":113416,"style":35905},[463],[224,113418,113420,113423],{"className":113419},[468],[224,113421,20303],{"className":113422,"style":2473},[468,469],[224,113424,113426],{"className":113425},[750],[224,113427,113429,113455],{"className":113428},[754,755],[224,113430,113432,113452],{"className":113431},[759],[224,113433,113435],{"className":113434,"style":764},[763],[224,113436,113437,113440],{"style":20390},[224,113438],{"className":113439,"style":772},[771],[224,113441,113443],{"className":113442},[776,777,778,779],[224,113444,113446,113449],{"className":113445},[468,779],[224,113447,30652],{"className":113448,"style":30740},[468,469,779],[224,113450,41831],{"className":113451},[468,469,779],[224,113453,787],{"className":113454},[786],[224,113456,113458],{"className":113457},[759],[224,113459,113461],{"className":113460,"style":794},[763],[224,113462],{},"（Pinsker），",[224,113465,113467,113484],{"className":113466,"translate":388},[392],[224,113468,113470],{"className":113469},[396],[398,113471,113472],{"xmlns":400},[403,113473,113474,113482],{},[406,113475,113476],{},[3248,113477,113478,113480],{},[409,113479,57370],{},[419,113481,1447],{},[447,113483,60308],{"encoding":449},[224,113485,113487],{"className":113486,"ariaHidden":455},[454],[224,113488,113490,113493],{"className":113489},[459],[224,113491],{"className":113492,"style":60318},[463],[224,113494,113496,113499],{"className":113495},[468],[224,113497,57370],{"className":113498},[468,469],[224,113500,113502],{"className":113501},[750],[224,113503,113505],{"className":113504},[754],[224,113506,113508],{"className":113507},[759],[224,113509,113511],{"className":113510,"style":5671},[763],[224,113512,113513,113516],{"style":3288},[224,113514],{"className":113515,"style":772},[771],[224,113517,113519],{"className":113518},[776,777,778,779],[224,113520,1447],{"className":113521},[468,779]," 给 ",[224,113524,113526,113560],{"className":113525,"translate":388},[392],[224,113527,113529],{"className":113528},[396],[398,113530,113531],{"xmlns":400},[403,113532,113533,113557],{},[406,113534,113535,113541,113543,113545,113547],{},[3248,113536,113537,113539],{},[409,113538,57370],{},[419,113540,1447],{},[412,113542,9766],{},[419,113544,9409],{},[423,113546,682],{},[3248,113548,113549,113555],{},[406,113550,113551,113553],{},[409,113552,637],{"mathvariant":678},[409,113554,16059],{"mathvariant":678},[419,113556,1447],{},[447,113558,113559],{"encoding":449},"\\chi^2\\ge4\\,\\mathrm{TV}^2",[224,113561,113563,113607],{"className":113562,"ariaHidden":455},[454],[224,113564,113566,113569,113598,113601,113604],{"className":113565},[459],[224,113567],{"className":113568,"style":60318},[463],[224,113570,113572,113575],{"className":113571},[468],[224,113573,57370],{"className":113574},[468,469],[224,113576,113578],{"className":113577},[750],[224,113579,113581],{"className":113580},[754],[224,113582,113584],{"className":113583},[759],[224,113585,113587],{"className":113586,"style":5671},[763],[224,113588,113589,113592],{"style":3288},[224,113590],{"className":113591,"style":772},[771],[224,113593,113595],{"className":113594},[776,777,778,779],[224,113596,1447],{"className":113597},[468,779],[224,113599],{"className":113600,"style":496},[473],[224,113602,9766],{"className":113603},[500],[224,113605],{"className":113606,"style":496},[473],[224,113608,113610,113613,113616,113619],{"className":113609},[459],[224,113611],{"className":113612,"style":108276},[463],[224,113614,9409],{"className":113615},[468],[224,113617],{"className":113618,"style":724},[473],[224,113620,113622,113628],{"className":113621},[468],[224,113623,113625],{"className":113624},[468],[224,113626,106664],{"className":113627,"style":24188},[468,1906],[224,113629,113631],{"className":113630},[750],[224,113632,113634],{"className":113633},[754],[224,113635,113637],{"className":113636},[759],[224,113638,113640],{"className":113639,"style":108276},[763],[224,113641,113642,113645],{"style":108306},[224,113643],{"className":113644,"style":772},[771],[224,113646,113648],{"className":113647},[776,777,778,779],[224,113649,1447],{"className":113650},[468,779],"，Hellinger",[224,113653,113655,113672],{"className":113654,"translate":388},[392],[224,113656,113658],{"className":113657},[396],[398,113659,113660],{"xmlns":400},[403,113661,113662,113670],{},[406,113663,113664],{},[3248,113665,113666,113668],{},[406,113667],{},[419,113669,1447],{},[447,113671,64181],{"encoding":449},[224,113673,113675],{"className":113674,"ariaHidden":455},[454],[224,113676,113678,113681],{"className":113677},[459],[224,113679],{"className":113680,"style":5671},[463],[224,113682,113684,113686],{"className":113683},[468],[224,113685],{},[224,113687,113689],{"className":113688},[750],[224,113690,113692],{"className":113691},[754],[224,113693,113695],{"className":113694},[759],[224,113696,113698],{"className":113697,"style":5671},[763],[224,113699,113700,113703],{"style":3288},[224,113701],{"className":113702,"style":772},[771],[224,113704,113706],{"className":113705},[776,777,778,779],[224,113707,1447],{"className":113708},[468,779]," 经由 ",[224,113711,113713,113771],{"className":113712,"translate":388},[392],[224,113714,113716],{"className":113715},[396],[398,113717,113718],{"xmlns":400},[403,113719,113720,113768],{},[406,113721,113722,113728,113730,113754,113756],{},[406,113723,113724,113726],{},[409,113725,637],{"mathvariant":678},[409,113727,16059],{"mathvariant":678},[412,113729,2070],{},[54262,113731,113732],{},[406,113733,113734,113740,113742,113744,113746,113752],{},[3248,113735,113736,113738],{},[409,113737,53807],{},[419,113739,1447],{},[412,113741,239],{"stretchy":435},[419,113743,1447],{},[412,113745,5268],{},[3248,113747,113748,113750],{},[409,113749,53807],{},[419,113751,1447],{},[412,113753,1716],{"stretchy":435},[412,113755,2070],{},[54262,113757,113758],{},[406,113759,113760,113762],{},[419,113761,1447],{},[3248,113763,113764,113766],{},[409,113765,53807],{},[419,113767,1447],{},[447,113769,113770],{"encoding":449},"\\mathrm{TV}\\le\\sqrt{H^2(2-H^2)}\\le\\sqrt{2H^2}",[224,113772,113774,113795,113930],{"className":113773,"ariaHidden":455},[454],[224,113775,113777,113780,113786,113789,113792],{"className":113776},[459],[224,113778],{"className":113779,"style":108254},[463],[224,113781,113783],{"className":113782},[468],[224,113784,106664],{"className":113785,"style":24188},[468,1906],[224,113787],{"className":113788,"style":496},[473],[224,113790,2070],{"className":113791},[500],[224,113793],{"className":113794,"style":496},[473],[224,113796,113798,113801,113921,113924,113927],{"className":113797},[459],[224,113799],{"className":113800,"style":111771},[463],[224,113802,113804],{"className":113803},[468,54290],[224,113805,113807,113913],{"className":113806},[754,755],[224,113808,113810,113910],{"className":113809},[759],[224,113811,113813,113898],{"className":113812,"style":111784},[763],[224,113814,113816,113819],{"className":113815,"style":103438},[2139],[224,113817],{"className":113818,"style":34794},[771],[224,113820,113822,113851,113854,113857,113860,113863,113866,113895],{"className":113821,"style":61074},[468],[224,113823,113825,113828],{"className":113824},[468],[224,113826,53807],{"className":113827,"style":53961},[468,469],[224,113829,113831],{"className":113830},[750],[224,113832,113834],{"className":113833},[754],[224,113835,113837],{"className":113836},[759],[224,113838,113840],{"className":113839,"style":64944},[763],[224,113841,113842,113845],{"style":64947},[224,113843],{"className":113844,"style":772},[771],[224,113846,113848],{"className":113847},[776,777,778,779],[224,113849,1447],{"className":113850},[468,779],[224,113852,239],{"className":113853},[542],[224,113855,1447],{"className":113856},[468],[224,113858],{"className":113859,"style":474},[473],[224,113861,5268],{"className":113862},[478],[224,113864],{"className":113865,"style":474},[473],[224,113867,113869,113872],{"className":113868},[468],[224,113870,53807],{"className":113871,"style":53961},[468,469],[224,113873,113875],{"className":113874},[750],[224,113876,113878],{"className":113877},[754],[224,113879,113881],{"className":113880},[759],[224,113882,113884],{"className":113883,"style":64944},[763],[224,113885,113886,113889],{"style":64947},[224,113887],{"className":113888,"style":772},[771],[224,113890,113892],{"className":113891},[776,777,778,779],[224,113893,1447],{"className":113894},[468,779],[224,113896,1716],{"className":113897},[549],[224,113899,113900,113903],{"style":111872},[224,113901],{"className":113902,"style":34794},[771],[224,113904,113906],{"className":113905,"style":103530},[54366],[2155,113907,113908],{"xmlns":2157,"width":2158,"height":103533,"viewBox":103534,"preserveAspectRatio":2161},[2163,113909],{"d":103537},[224,113911,787],{"className":113912},[786],[224,113914,113916],{"className":113915},[759],[224,113917,113919],{"className":113918,"style":111892},[763],[224,113920],{},[224,113922],{"className":113923,"style":496},[473],[224,113925,2070],{"className":113926},[500],[224,113928],{"className":113929,"style":496},[473],[224,113931,113933,113937],{"className":113932},[459],[224,113934],{"className":113935,"style":113936},[463],"height:1.04em;vertical-align:-0.0849em;",[224,113938,113940],{"className":113939},[468,54290],[224,113941,113943,114007],{"className":113942},[754,755],[224,113944,113946,114004],{"className":113945},[759],[224,113947,113950,113991],{"className":113948,"style":113949},[763],"height:0.9551em;",[224,113951,113953,113956],{"className":113952,"style":2120},[2139],[224,113954],{"className":113955,"style":2108},[771],[224,113957,113959,113962],{"className":113958,"style":54310},[468],[224,113960,1447],{"className":113961},[468],[224,113963,113965,113968],{"className":113964},[468],[224,113966,53807],{"className":113967,"style":53961},[468,469],[224,113969,113971],{"className":113970},[750],[224,113972,113974],{"className":113973},[754],[224,113975,113977],{"className":113976},[759],[224,113978,113980],{"className":113979,"style":64944},[763],[224,113981,113982,113985],{"style":64947},[224,113983],{"className":113984,"style":772},[771],[224,113986,113988],{"className":113987},[776,777,778,779],[224,113989,1447],{"className":113990},[468,779],[224,113992,113994,113997],{"style":113993},"top:-2.9151em;",[224,113995],{"className":113996,"style":2108},[771],[224,113998,114000],{"className":113999,"style":54367},[54366],[2155,114001,114002],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,114003],{"d":54374},[224,114005,787],{"className":114006},[786],[224,114008,114010],{"className":114009},[759],[224,114011,114014],{"className":114012,"style":114013},[763],"height:0.0849em;",[224,114015],{},"，TV 本身平凡。",[11,114018,114019,26478,114022,114354,114355,114358,114359,114410,114411,7098,114462,114789,114790,372,114972,115104,115105,115295,115296,115506],{},[378,114020,114021],{},"(b)（\"上去\"方向，需要条件）",[224,114023,114025,114095],{"className":114024,"translate":388},[392],[224,114026,114028],{"className":114027},[396],[398,114029,114030],{"xmlns":400},[403,114031,114032,114092],{},[406,114033,114034,114044,114046,114052,114054,114056,114058,114060,114062,114064,114066,114068,114074,114076,114082,114084,114086,114088,114090],{},[695,114035,114036,114038],{},[409,114037,20303],{},[406,114039,114040,114042],{},[409,114041,637],{},[409,114043,16059],{},[412,114045,239],{"stretchy":435},[695,114047,114048,114050],{},[409,114049,16018],{},[409,114051,3254],{},[412,114053,1877],{"separator":455},[409,114055,20315],{},[412,114057,1716],{"stretchy":435},[412,114059,1144],{},[419,114061,284],{},[423,114063,15054],{},[412,114065,24315],{},[423,114067,15054],{},[695,114069,114070,114072],{},[409,114071,20303],{},[409,114073,1705],{},[412,114075,239],{"stretchy":435},[695,114077,114078,114080],{},[409,114079,16018],{},[409,114081,3254],{},[409,114083,20312],{"mathvariant":678},[409,114085,20315],{},[412,114087,1716],{"stretchy":435},[412,114089,1144],{},[419,114091,284],{},[447,114093,114094],{"encoding":449},"D_{TV}(P_n,Q)\\to0\\ \\Longrightarrow\\ D_f(P_n\\|Q)\\to0",[224,114096,114098,114214,114238,114345],{"className":114097,"ariaHidden":455},[454],[224,114099,114101,114104,114150,114153,114193,114196,114199,114202,114205,114208,114211],{"className":114100},[459],[224,114102],{"className":114103,"style":535},[463],[224,114105,114107,114110],{"className":114106},[468],[224,114108,20303],{"className":114109,"style":2473},[468,469],[224,114111,114113],{"className":114112},[750],[224,114114,114116,114142],{"className":114115},[754,755],[224,114117,114119,114139],{"className":114118},[759],[224,114120,114122],{"className":114121,"style":764},[763],[224,114123,114124,114127],{"style":20390},[224,114125],{"className":114126,"style":772},[771],[224,114128,114130],{"className":114129},[776,777,778,779],[224,114131,114133,114136],{"className":114132},[468,779],[224,114134,637],{"className":114135,"style":653},[468,469,779],[224,114137,16059],{"className":114138,"style":474},[468,469,779],[224,114140,787],{"className":114141},[786],[224,114143,114145],{"className":114144},[759],[224,114146,114148],{"className":114147,"style":794},[763],[224,114149],{},[224,114151,239],{"className":114152},[542],[224,114154,114156,114159],{"className":114155},[468],[224,114157,16018],{"className":114158,"style":653},[468,469],[224,114160,114162],{"className":114161},[750],[224,114163,114165,114185],{"className":114164},[754,755],[224,114166,114168,114182],{"className":114167},[759],[224,114169,114171],{"className":114170,"style":64841},[763],[224,114172,114173,114176],{"style":31092},[224,114174],{"className":114175,"style":772},[771],[224,114177,114179],{"className":114178},[776,777,778,779],[224,114180,3254],{"className":114181},[468,469,779],[224,114183,787],{"className":114184},[786],[224,114186,114188],{"className":114187},[759],[224,114189,114191],{"className":114190,"style":794},[763],[224,114192],{},[224,114194,1877],{"className":114195},[1917],[224,114197],{"className":114198,"style":724},[473],[224,114200,20315],{"className":114201},[468,469],[224,114203,1716],{"className":114204},[549],[224,114206],{"className":114207,"style":496},[473],[224,114209,1144],{"className":114210},[500],[224,114212],{"className":114213,"style":496},[473],[224,114215,114217,114220,114223,114226,114229,114232,114235],{"className":114216},[459],[224,114218],{"className":114219,"style":112683},[463],[224,114221,284],{"className":114222},[468],[224,114224,15054],{"className":114225},[473],[224,114227],{"className":114228,"style":496},[473],[224,114230,24315],{"className":114231},[500],[224,114233,15054],{"className":114234},[473],[224,114236],{"className":114237,"style":496},[473],[224,114239,114241,114244,114284,114287,114327,114330,114333,114336,114339,114342],{"className":114240},[459],[224,114242],{"className":114243,"style":20368},[463],[224,114245,114247,114250],{"className":114246},[468],[224,114248,20303],{"className":114249,"style":2473},[468,469],[224,114251,114253],{"className":114252},[750],[224,114254,114256,114276],{"className":114255},[754,755],[224,114257,114259,114273],{"className":114258},[759],[224,114260,114262],{"className":114261,"style":20387},[763],[224,114263,114264,114267],{"style":20390},[224,114265],{"className":114266,"style":772},[771],[224,114268,114270],{"className":114269},[776,777,778,779],[224,114271,1705],{"className":114272,"style":1740},[468,469,779],[224,114274,787],{"className":114275},[786],[224,114277,114279],{"className":114278},[759],[224,114280,114282],{"className":114281,"style":20409},[763],[224,114283],{},[224,114285,239],{"className":114286},[542],[224,114288,114290,114293],{"className":114289},[468],[224,114291,16018],{"className":114292,"style":653},[468,469],[224,114294,114296],{"className":114295},[750],[224,114297,114299,114319],{"className":114298},[754,755],[224,114300,114302,114316],{"className":114301},[759],[224,114303,114305],{"className":114304,"style":64841},[763],[224,114306,114307,114310],{"style":31092},[224,114308],{"className":114309,"style":772},[771],[224,114311,114313],{"className":114312},[776,777,778,779],[224,114314,3254],{"className":114315},[468,469,779],[224,114317,787],{"className":114318},[786],[224,114320,114322],{"className":114321},[759],[224,114323,114325],{"className":114324,"style":794},[763],[224,114326],{},[224,114328,20312],{"className":114329},[468],[224,114331,20315],{"className":114332},[468,469],[224,114334,1716],{"className":114335},[549],[224,114337],{"className":114338,"style":496},[473],[224,114340,1144],{"className":114341},[500],[224,114343],{"className":114344,"style":496},[473],[224,114346,114348,114351],{"className":114347},[459],[224,114349],{"className":114350,"style":510},[463],[224,114352,284],{"className":114353},[468]," 一般",[378,114356,114357],{},"不成立","。反例：在 ",[224,114360,114362,114383],{"className":114361,"translate":388},[392],[224,114363,114365],{"className":114364},[396],[398,114366,114367],{"xmlns":400},[403,114368,114369,114381],{},[406,114370,114371,114373,114375,114377,114379],{},[412,114372,268],{"stretchy":435},[419,114374,284],{},[412,114376,1877],{"separator":455},[419,114378,1438],{},[412,114380,440],{"stretchy":435},[447,114382,56736],{"encoding":449},[224,114384,114386],{"className":114385,"ariaHidden":455},[454],[224,114387,114389,114392,114395,114398,114401,114404,114407],{"className":114388},[459],[224,114390],{"className":114391,"style":535},[463],[224,114393,268],{"className":114394},[542],[224,114396,284],{"className":114397},[468],[224,114399,1877],{"className":114400},[1917],[224,114402],{"className":114403,"style":724},[473],[224,114405,1438],{"className":114406},[468],[224,114408,440],{"className":114409},[549]," 上取 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",[224,114791,114793,114828],{"className":114792,"translate":388},[392],[224,114794,114796],{"className":114795},[396],[398,114797,114798],{"xmlns":400},[403,114799,114800,114825],{},[406,114801,114802,114808,114810],{},[695,114803,114804,114806],{},[409,114805,35657],{},[409,114807,3254],{},[412,114809,278],{},[5221,114811,114812],{"scriptlevel":284,"displaystyle":435},[20338,114813,114814,114816],{},[419,114815,1438],{},[406,114817,114818,114821,114823],{},[409,114819,114820],{},"ln",[412,114822,24134],{},[409,114824,3254],{},[447,114826,114827],{"encoding":449},"\\delta_n=\\tfrac1{\\ln n}",[224,114829,114831,114886],{"className":114830,"ariaHidden":455},[454],[224,114832,114834,114837,114877,114880,114883],{"className":114833},[459],[224,114835],{"className":114836,"style":31429},[463],[224,114838,114840,114843],{"className":114839},[468],[224,114841,35657],{"className":114842,"style":35706},[468,469],[224,114844,114846],{"className":114845},[750],[224,114847,114849,114869],{"className":114848},[754,755],[224,114850,114852,114866],{"className":114851},[759],[224,114853,114855],{"className":114854,"style":64841},[763],[224,114856,114857,114860],{"style":114641},[224,114858],{"className":114859,"style":772},[771],[224,114861,114863],{"className":114862},[776,777,778,779],[224,114864,3254],{"className":114865},[468,469,779],[224,114867,787],{"className":114868},[786],[224,114870,114872],{"className":114871},[759],[224,114873,114875],{"className":114874,"style":794},[763],[224,114876],{},[224,114878],{"className":114879,"style":496},[473],[224,114881,278],{"className":114882},[500],[224,114884],{"className":114885,"style":496},[473],[224,114887,114889,114892],{"className":114888},[459],[224,114890],{"className":114891,"style":22544},[463],[224,114893,114895,114898,114969],{"className":114894},[468],[224,114896],{"className":114897},[542,8479],[224,114899,114901],{"className":114900},[20338],[224,114902,114904,114961],{"className":114903},[754,755],[224,114905,114907,114958],{"className":114906},[759],[224,114908,114910,114936,114944],{"className":114909,"style":22563},[763],[224,114911,114912,114915],{"style":22566},[224,114913],{"className":114914,"style":2108},[771],[224,114916,114918],{"className":114917},[776,777,778,779],[224,114919,114921,114930,114933],{"className":114920},[468,779],[224,114922,114924,114927],{"className":114923},[21531,779],[224,114925,1864],{"className":114926},[779],[224,114928,3254],{"className":114929},[779],[224,114931],{"className":114932,"style":62279},[473,779],[224,114934,3254],{"className":114935},[468,469,779],[224,114937,114938,114941],{"style":20546},[224,114939],{"className":114940,"style":2108},[771],[224,114942],{"className":114943,"style":20554},[20553],[224,114945,114946,114949],{"style":22589},[224,114947],{"className":114948,"style":2108},[771],[224,114950,114952],{"className":114951},[776,777,778,779],[224,114953,114955],{"className":114954},[468,779],[224,114956,1438],{"className":114957},[468,779],[224,114959,787],{"className":114960},[786],[224,114962,114964],{"className":114963},[759],[224,114965,114967],{"className":114966,"style":22611},[763],[224,114968],{},[224,114970],{"className":114971},[549,8479],[224,114973,114975,115005],{"className":114974,"translate":388},[392],[224,114976,114978],{"className":114977},[396],[398,114979,114980],{"xmlns":400},[403,114981,114982,115002],{},[406,114983,114984,114990,114992],{},[695,114985,114986,114988],{},[409,114987,58583],{},[409,114989,3254],{},[412,114991,278],{},[3248,114993,114994,114996],{},[409,114995,1861],{},[406,114997,114998,115000],{},[412,114999,5268],{},[409,115001,3254],{},[447,115003,115004],{"encoding":449},"\\varepsilon_n=e^{-n}",[224,115006,115008,115063],{"className":115007,"ariaHidden":455},[454],[224,115009,115011,115014,115054,115057,115060],{"className":115010},[459],[224,115012],{"className":115013,"style":3625},[463],[224,115015,115017,115020],{"className":115016},[468],[224,115018,58583],{"className":115019},[468,469],[224,115021,115023],{"className":115022},[750],[224,115024,115026,115046],{"className":115025},[754,755],[224,115027,115029,115043],{"className":115028},[759],[224,115030,115032],{"className":115031,"style":64841},[763],[224,115033,115034,115037],{"style":767},[224,115035],{"className":115036,"style":772},[771],[224,115038,115040],{"className":115039},[776,777,778,779],[224,115041,3254],{"className":115042},[468,469,779],[224,115044,787],{"className":115045},[786],[224,115047,115049],{"className":115048},[759],[224,115050,115052],{"className":115051,"style":794},[763],[224,115053],{},[224,115055],{"className":115056,"style":496},[473],[224,115058,278],{"className":115059},[500],[224,115061],{"className":115062,"style":496},[473],[224,115064,115066,115069],{"className":115065},[459],[224,115067],{"className":115068,"style":40947},[463],[224,115070,115072,115075],{"className":115071},[468],[224,115073,1861],{"className":115074},[468,469],[224,115076,115078],{"className":115077},[750],[224,115079,115081],{"className":115080},[754],[224,115082,115084],{"className":115083},[759],[224,115085,115087],{"className":115086,"style":40947},[763],[224,115088,115089,115092],{"style":3288},[224,115090],{"className":115091,"style":772},[771],[224,115093,115095],{"className":115094},[776,777,778,779],[224,115096,115098,115101],{"className":115097},[468,779],[224,115099,5268],{"className":115100},[468,779],[224,115102,3254],{"className":115103},[468,469,779],"。则 ",[224,115106,115108,115152],{"className":115107,"translate":388},[392],[224,115109,115111],{"className":115110},[396],[398,115112,115113],{"xmlns":400},[403,115114,115115,115149],{},[406,115116,115117,115123,115125,115131,115133,115135,115137,115139,115145,115147],{},[406,115118,115119,115121],{},[409,115120,637],{"mathvariant":678},[409,115122,16059],{"mathvariant":678},[412,115124,239],{"stretchy":435},[695,115126,115127,115129],{},[409,115128,2023],{},[409,115130,3254],{},[412,115132,1877],{"separator":455},[409,115134,11],{},[412,115136,1716],{"stretchy":435},[412,115138,52884],{},[695,115140,115141,115143],{},[409,115142,35657],{},[409,115144,3254],{},[412,115146,1144],{},[419,115148,284],{},[447,115150,115151],{"encoding":449},"\\mathrm{TV}(q_n,p)\\approx\\delta_n\\to0",[224,115153,115155,115231,115286],{"className":115154,"ariaHidden":455},[454],[224,115156,115158,115161,115167,115170,115210,115213,115216,115219,115222,115225,115228],{"className":115157},[459],[224,115159],{"className":115160,"style":535},[463],[224,115162,115164],{"className":115163},[468],[224,115165,106664],{"className":115166,"style":24188},[468,1906],[224,115168,239],{"className":115169},[542],[224,115171,115173,115176],{"className":115172},[468],[224,115174,2023],{"className":115175,"style":492},[468,469],[224,115177,115179],{"className":115178},[750],[224,115180,115182,115202],{"className":115181},[754,755],[224,115183,115185,115199],{"className":115184},[759],[224,115186,115188],{"className":115187,"style":64841},[763],[224,115189,115190,115193],{"style":3854},[224,115191],{"className":115192,"style":772},[771],[224,115194,115196],{"className":115195},[776,777,778,779],[224,115197,3254],{"className":115198},[468,469,779],[224,115200,787],{"className":115201},[786],[224,115203,115205],{"className":115204},[759],[224,115206,115208],{"className":115207,"style":794},[763],[224,115209],{},[224,115211,1877],{"className":115212},[1917],[224,115214],{"className":115215,"style":724},[473],[224,115217,11],{"className":115218},[468,469],[224,115220,1716],{"className":115221},[549],[224,115223],{"className":115224,"style":496},[473],[224,115226,52884],{"className":115227},[500],[224,115229],{"className":115230,"style":496},[473],[224,115232,115234,115237,115277,115280,115283],{"className":115233},[459],[224,115235],{"className":115236,"style":31429},[463],[224,115238,115240,115243],{"className":115239},[468],[224,115241,35657],{"className":115242,"style":35706},[468,469],[224,115244,115246],{"className":115245},[750],[224,115247,115249,115269],{"className":115248},[754,755],[224,115250,115252,115266],{"className":115251},[759],[224,115253,115255],{"className":115254,"style":64841},[763],[224,115256,115257,115260],{"style":114641},[224,115258],{"className":115259,"style":772},[771],[224,115261,115263],{"className":115262},[776,777,778,779],[224,115264,3254],{"className":115265},[468,469,779],[224,115267,787],{"className":115268},[786],[224,115270,115272],{"className":115271},[759],[224,115273,115275],{"className":115274,"style":794},[763],[224,115276],{},[224,115278],{"className":115279,"style":496},[473],[224,115281,1144],{"className":115282},[500],[224,115284],{"className":115285,"style":496},[473],[224,115287,115289,115292],{"className":115288},[459],[224,115290],{"className":115291,"style":510},[463],[224,115293,284],{"className":115294},[468],"，但 ",[224,115297,115299,115348],{"className":115298,"translate":388},[392],[224,115300,115302],{"className":115301},[396],[398,115303,115304],{"xmlns":400},[403,115305,115306,115345],{},[406,115307,115308,115314,115316,115322,115324,115326,115328,115331,115337,115339,115341,115343],{},[406,115309,115310,115312],{},[409,115311,30652],{"mathvariant":678},[409,115313,41831],{"mathvariant":678},[412,115315,239],{"stretchy":435},[695,115317,115318,115320],{},[409,115319,2023],{},[409,115321,3254],{},[409,115323,20312],{"mathvariant":678},[409,115325,11],{},[412,115327,1716],{"stretchy":435},[412,115329,115330],{},"≳",[695,115332,115333,115335],{},[409,115334,35657],{},[409,115336,3254],{},[412,115338,23331],{},[409,115340,3254],{},[412,115342,1144],{},[409,115344,21085],{"mathvariant":678},[447,115346,115347],{"encoding":449},"\\mathrm{KL}(q_n\\|p)\\gtrsim \\delta_n\\cdot n\\to\\infty",[224,115349,115351,115424,115479,115497],{"className":115350,"ariaHidden":455},[454],[224,115352,115354,115357,115363,115366,115406,115409,115412,115415,115418,115421],{"className":115353},[459],[224,115355],{"className":115356,"style":535},[463],[224,115358,115360],{"className":115359},[468],[224,115361,52588],{"className":115362},[468,1906],[224,115364,239],{"className":115365},[542],[224,115367,115369,115372],{"className":115368},[468],[224,115370,2023],{"className":115371,"style":492},[468,469],[224,115373,115375],{"className":115374},[750],[224,115376,115378,115398],{"className":115377},[754,755],[224,115379,115381,115395],{"className":115380},[759],[224,115382,115384],{"className":115383,"style":64841},[763],[224,115385,115386,115389],{"style":3854},[224,115387],{"className":115388,"style":772},[771],[224,115390,115392],{"className":115391},[776,777,778,779],[224,115393,3254],{"className":115394},[468,469,779],[224,115396,787],{"className":115397},[786],[224,115399,115401],{"className":115400},[759],[224,115402,115404],{"className":115403,"style":794},[763],[224,115405],{},[224,115407,20312],{"className":115408},[468],[224,115410,11],{"className":115411},[468,469],[224,115413,1716],{"className":115414},[549],[224,115416],{"className":115417,"style":496},[473],[224,115419,115330],{"className":115420},[500,79700],[224,115422],{"className":115423,"style":496},[473],[224,115425,115427,115430,115470,115473,115476],{"className":115426},[459],[224,115428],{"className":115429,"style":31429},[463],[224,115431,115433,115436],{"className":115432},[468],[224,115434,35657],{"className":115435,"style":35706},[468,469],[224,115437,115439],{"className":115438},[750],[224,115440,115442,115462],{"className":115441},[754,755],[224,115443,115445,115459],{"className":115444},[759],[224,115446,115448],{"className":115447,"style":64841},[763],[224,115449,115450,115453],{"style":114641},[224,115451],{"className":115452,"style":772},[771],[224,115454,115456],{"className":115455},[776,777,778,779],[224,115457,3254],{"className":115458},[468,469,779],[224,115460,787],{"className":115461},[786],[224,115463,115465],{"className":115464},[759],[224,115466,115468],{"className":115467,"style":794},[763],[224,115469],{},[224,115471],{"className":115472,"style":474},[473],[224,115474,23331],{"className":115475},[478],[224,115477],{"className":115478,"style":474},[473],[224,115480,115482,115485,115488,115491,115494],{"className":115481},[459],[224,115483],{"className":115484,"style":839},[463],[224,115486,3254],{"className":115487},[468,469],[224,115489],{"className":115490,"style":496},[473],[224,115492,1144],{"className":115493},[500],[224,115495],{"className":115496,"style":496},[473],[224,115498,115500,115503],{"className":115499},[459],[224,115501],{"className":115502,"style":839},[463],[224,115504,21085],{"className":115505},[468],"。反例的机制很典型：TV 只控制\"总质量差\"，控制不了小测度集合上密度的巨大尖峰，而 KL 对小分母极其敏感——这正是 KL\u002Fχ² \"无界\"分歧的拓扑体现。",[11,115508,115509,115512,115513,115541,115542,79970,115608,115673,115674,43562],{},[378,115510,115511],{},"(c)（有限字母表\u002F一致比值界下，全部等价）"," 若似然比一致有界（例如：",[224,115514,115516,115529],{"className":115515,"translate":388},[392],[224,115517,115519],{"className":115518},[396],[398,115520,115521],{"xmlns":400},[403,115522,115523,115527],{},[406,115524,115525],{},[409,115526,20315],{},[447,115528,20315],{"encoding":449},[224,115530,115532],{"className":115531,"ariaHidden":455},[454],[224,115533,115535,115538],{"className":115534},[459],[224,115536],{"className":115537,"style":20637},[463],[224,115539,20315],{"className":115540},[468,469]," 在有限字母表上且 ",[224,115543,115545,115569],{"className":115544,"translate":388},[392],[224,115546,115548],{"className":115547},[396],[398,115549,115550],{"xmlns":400},[403,115551,115552,115566],{},[406,115553,115554,115556,115558,115560,115562,115564],{},[409,115555,2023],{},[412,115557,239],{"stretchy":435},[409,115559,371],{},[412,115561,1716],{"stretchy":435},[412,115563,325],{},[419,115565,284],{},[447,115567,115568],{"encoding":449},"q(x)>0",[224,115570,115572,115599],{"className":115571,"ariaHidden":455},[454],[224,115573,115575,115578,115581,115584,115587,115590,115593,115596],{"className":115574},[459],[224,115576],{"className":115577,"style":535},[463],[224,115579,2023],{"className":115580,"style":492},[468,469],[224,115582,239],{"className":115583},[542],[224,115585,371],{"className":115586},[468,469],[224,115588,1716],{"className":115589},[549],[224,115591],{"className":115592,"style":496},[473],[224,115594,325],{"className":115595},[500],[224,115597],{"className":115598,"style":496},[473],[224,115600,115602,115605],{"className":115601},[459],[224,115603],{"className":115604,"style":510},[463],[224,115606,284],{"className":115607},[468],[224,115609,115611,115634],{"className":115610,"translate":388},[392],[224,115612,115614],{"className":115613},[396],[398,115615,115616],{"xmlns":400},[403,115617,115618,115631],{},[406,115619,115620,115622,115624,115627,115629],{},[409,115621,371],{},[412,115623,12248],{},[409,115625,115626],{"mathvariant":678},"supp",[412,115628,24134],{},[409,115630,16018],{},[447,115632,115633],{"encoding":449},"x\\in\\operatorname{supp}P",[224,115635,115637,115655],{"className":115636,"ariaHidden":455},[454],[224,115638,115640,115643,115646,115649,115652],{"className":115639},[459],[224,115641],{"className":115642,"style":2810},[463],[224,115644,371],{"className":115645},[468,469],[224,115647],{"className":115648,"style":496},[473],[224,115650,12248],{"className":115651},[500],[224,115653],{"className":115654,"style":496},[473],[224,115656,115658,115661,115667,115670],{"className":115657},[459],[224,115659],{"className":115660,"style":20637},[463],[224,115662,115664],{"className":115663},[21531],[224,115665,115626],{"className":115666},[468,1906],[224,115668],{"className":115669,"style":724},[473],[224,115671,16018],{"className":115672,"style":653},[468,469],"——此时 ",[224,115675,115677,115713],{"className":115676,"translate":388},[392],[224,115678,115680],{"className":115679},[396],[398,115681,115682],{"xmlns":400},[403,115683,115684,115710],{},[406,115685,115686,115688,115690,115696,115698,115706,115708],{},[409,115687,1435],{},[412,115689,278],{},[20338,115691,115692,115694],{},[409,115693,11],{},[409,115695,2023],{},[412,115697,2070],{},[695,115699,115700,115702,115704],{},[409,115701,1435],{},[409,115703,72294],{},[412,115705,24134],{},[412,115707,1577],{},[409,115709,21085],{"mathvariant":678},[447,115711,115712],{"encoding":449},"t=\\frac{p}{q}\\le t_{\\max}\u003C\\infty",[224,115714,115716,115734,115817,115884],{"className":115715,"ariaHidden":455},[454],[224,115717,115719,115722,115725,115728,115731],{"className":115718},[459],[224,115720],{"className":115721,"style":25353},[463],[224,115723,1435],{"className":115724},[468,469],[224,115726],{"className":115727,"style":496},[473],[224,115729,278],{"className":115730},[500],[224,115732],{"className":115733,"style":496},[473],[224,115735,115737,115740,115808,115811,115814],{"className":115736},[459],[224,115738],{"className":115739,"style":23117},[463],[224,115741,115743,115746,115805],{"className":115742},[468],[224,115744],{"className":115745},[542,8479],[224,115747,115749],{"className":115748},[20338],[224,115750,115752,115797],{"className":115751},[754,755],[224,115753,115755,115794],{"className":115754},[759],[224,115756,115758,115772,115780],{"className":115757,"style":23136},[763],[224,115759,115760,115763],{"style":22566},[224,115761],{"className":115762,"style":2108},[771],[224,115764,115766],{"className":115765},[776,777,778,779],[224,115767,115769],{"className":115768},[468,779],[224,115770,2023],{"className":115771,"style":492},[468,469,779],[224,115773,115774,115777],{"style":20546},[224,115775],{"className":115776,"style":2108},[771],[224,115778],{"className":115779,"style":20554},[20553],[224,115781,115782,115785],{"style":22889},[224,115783],{"className":115784,"style":2108},[771],[224,115786,115788],{"className":115787},[776,777,778,779],[224,115789,115791],{"className":115790},[468,779],[224,115792,11],{"className":115793},[468,469,779],[224,115795,787],{"className":115796},[786],[224,115798,115800],{"className":115799},[759],[224,115801,115803],{"className":115802,"style":22776},[763],[224,115804],{},[224,115806],{"className":115807},[549,8479],[224,115809],{"className":115810,"style":496},[473],[224,115812,2070],{"className":115813},[500],[224,115815],{"className":115816,"style":496},[473],[224,115818,115820,115823,115875,115878,115881],{"className":115819},[459],[224,115821],{"className":115822,"style":1460},[463],[224,115824,115826,115829],{"className":115825},[468],[224,115827,1435],{"className":115828},[468,469],[224,115830,115832],{"className":115831},[750],[224,115833,115835,115867],{"className":115834},[754,755],[224,115836,115838,115864],{"className":115837},[759],[224,115839,115841],{"className":115840,"style":64841},[763],[224,115842,115843,115846],{"style":767},[224,115844],{"className":115845,"style":772},[771],[224,115847,115849],{"className":115848},[776,777,778,779],[224,115850,115852],{"className":115851},[468,779],[224,115853,115855,115858,115861],{"className":115854},[21531,779],[224,115856,13177],{"className":115857},[779],[224,115859,15],{"className":115860},[779],[224,115862,371],{"className":115863},[779],[224,115865,787],{"className":115866},[786],[224,115868,115870],{"className":115869},[759],[224,115871,115873],{"className":115872,"style":794},[763],[224,115874],{},[224,115876],{"className":115877,"style":496},[473],[224,115879,1577],{"className":115880},[500],[224,115882],{"className":115883,"style":496},[473],[224,115885,115887,115890],{"className":115886},[459],[224,115888],{"className":115889,"style":839},[463],[224,115891,21085],{"className":115892},[468],[224,115894,115896],{"className":115895,"translate":388},[387],[224,115897,115899,116034],{"className":115898,"translate":388},[392],[224,115900,115902],{"className":115901},[396],[398,115903,115904],{"xmlns":400,"display":401},[403,115905,115906,116031],{},[5209,115907,115908,115979],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,115909,115910,115916],{},[5218,115911,115912],{},[5221,115913,115914],{"scriptlevel":284,"displaystyle":455},[406,115915],{},[5218,115917,115918],{},[5221,115919,115920],{"scriptlevel":284,"displaystyle":455},[406,115921,115922,115924,115930,115932,115938,115940,115942,115944,115946,115948,115950,115953,115955,115961,115963,115969,115971,115973,115975,115977],{},[406,115923],{},[695,115925,115926,115928],{},[409,115927,20303],{},[409,115929,1705],{},[412,115931,239],{"stretchy":435},[695,115933,115934,115936],{},[409,115935,16018],{},[409,115937,3254],{},[409,115939,20312],{"mathvariant":678},[409,115941,20315],{},[412,115943,1716],{"stretchy":435},[412,115945,1144],{},[419,115947,284],{},[423,115949,425],{},[412,115951,115952],{},"⟺",[423,115954,425],{},[406,115956,115957,115959],{},[409,115958,637],{"mathvariant":678},[409,115960,16059],{"mathvariant":678},[412,115962,239],{"stretchy":435},[695,115964,115965,115967],{},[409,115966,16018],{},[409,115968,3254],{},[412,115970,1877],{"separator":455},[409,115972,20315],{},[412,115974,1716],{"stretchy":435},[412,115976,1144],{},[419,115978,284],{},[5215,115980,115981,115987],{},[5218,115982,115983],{},[5221,115984,115985],{"scriptlevel":284,"displaystyle":455},[406,115986],{},[5218,115988,115989],{},[5221,115990,115991],{"scriptlevel":284,"displaystyle":455},[406,115992,115993,115995,115997,115999,116001,11600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D_{KL}(P_n\\|Q)\\to0\n\\end{aligned}",[224,116035,116037],{"className":116036,"ariaHidden":455},[454],[224,116038,116040,116044],{"className":116039},[459],[224,116041],{"className":116042,"style":116043},[463],"height:2.7em;vertical-align:-1.1em;",[224,116045,116047],{"className":116046},[468],[224,116048,116050,116093],{"className":116049},[5209],[224,116051,116053],{"className":116052},[5331],[224,116054,116056,116084],{"className":116055},[754,755],[224,116057,116059,116081],{"className":116058},[759],[224,116060,116063,116073],{"className":116061,"style":116062},[763],"height:1.6em;",[224,116064,116066,116070],{"style":116065},"top:-3.6em;",[224,116067],{"className":116068,"style":116069},[771],"height:2.84em;",[224,116071],{"className":116072},[468],[224,116074,116075,116078],{"style":21950},[224,116076],{"className":116077,"style":116069},[771],[224,116079],{"className":116080},[468],[224,116082,787],{"className":116083},[786],[224,116085,116087],{"className":116086},[759],[224,116088,116091],{"className":116089,"style":116090},[763],"height:1.1em;",[224,116092],{},[224,116094,116096],{"className":116095},[5399],[224,116097,116099,116451],{"className":116098},[754,755],[224,116100,116102,116448],{"className":116101},[759],[224,116103,116105,116309],{"className":116104,"style":116062},[763],[224,116106,116108,116111],{"style":116107},"top:-3.76em;",[224,116109],{"className":116110,"style":2108},[771],[224,116112,116114,116117,116157,116160,116200,116203,116206,116209,116212,116215,116218,116221,116224,116227,116230,116233,116236,116242,116245,116285,116288,116291,116294,116297,116300,116303,116306],{"className":116113},[468],[224,116115],{"className":116116},[468],[224,116118,116120,116123],{"className":116119},[468],[224,116121,20303],{"className":116122,"style":2473},[468,469],[224,116124,116126],{"className":116125},[750],[224,116127,116129,116149],{"className":116128},[754,755],[224,116130,116132,116146],{"className":116131},[759],[224,116133,116135],{"className":116134,"style":20387},[763],[224,116136,116137,116140],{"style":20390},[224,116138],{"className":116139,"style":772},[771],[224,116141,116143],{"className":116142},[776,777,778,779],[224,116144,1705],{"className":116145,"style":1740},[468,469,779],[224,116147,787],{"className":116148},[786],[224,116150,116152],{"className":116151},[759],[224,116153,116155],{"className":116154,"style":20409},[763],[224,116156],{},[224,116158,239],{"className":116159},[542],[224,116161,116163,116166],{"className":116162},[468],[224,116164,16018],{"className":116165,"style":653},[468,469],[224,116167,116169],{"className":116168},[750],[224,116170,116172,116192],{"className":116171},[754,755],[224,116173,116175,116189],{"className":116174},[759],[224,116176,116178],{"className":116177,"style":64841},[763],[224,116179,116180,116183],{"style":31092},[224,116181],{"className":116182,"style":772},[771],[224,116184,116186],{"className":116185},[776,777,778,779],[224,116187,3254],{"className":116188},[468,469,779],[224,116190,787],{"className":116191},[786],[224,116193,116195],{"className":116194},[759],[224,116196,116198],{"className":116197,"style":794},[763],[224,116199],{},[224,116201,20312],{"className":116202},[468],[224,116204,20315],{"className":116205},[468,469],[224,116207,1716],{"className":116208},[549],[224,116210],{"className":116211,"style":496},[473],[224,116213,1144],{"className":116214},[500],[224,116216],{"className":116217,"style":496},[473],[224,116219,284],{"className":116220},[468],[224,116222],{"className":116223,"style":496},[473],[224,116225],{"className":116226,"style":496},[473],[224,116228,115952],{"className":116229},[500],[224,116231],{"className":116232,"style":496},[473],[224,116234],{"className":116235,"style":496},[473],[224,116237,116239],{"className":116238},[468],[224,116240,106664],{"className":116241,"style":24188},[468,1906],[224,116243,239],{"className":116244},[542],[224,116246,116248,116251],{"className":116247},[468],[224,116249,16018],{"className":116250,"style":653},[468,469],[224,116252,116254],{"className":116253},[750],[224,116255,116257,116277],{"className":116256},[754,755],[224,116258,116260,116274],{"className":116259},[759],[224,116261,116263],{"className":116262,"style":64841},[763],[224,116264,116265,116268],{"style":31092},[224,116266],{"className":116267,"style":772},[771],[224,116269,116271],{"className":116270},[776,777,778,779],[224,116272,3254],{"className":116273},[468,469,779],[224,116275,787],{"className":116276},[786],[224,116278,116280],{"className":116279},[759],[224,116281,116283],{"className":116282,"style":794},[763],[224,116284],{},[224,116286,1877],{"className":116287},[1917],[224,116289],{"className":116290,"style":724},[473],[224,116292,20315],{"className":116293},[468,469],[224,116295,1716],{"className":116296},[549],[224,116298],{"className":116299,"style":496},[473],[224,116301,1144],{"className":116302},[500],[224,116304],{"className":116305,"style":496},[473],[224,116307,284],{"classNa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",[224,116463,116465,116478],{"className":116464,"translate":388},[392],[224,116466,116468],{"className":116467},[396],[398,116469,116470],{"xmlns":400},[403,116471,116472,116476],{},[406,116473,116474],{},[409,116475,1705],{},[447,116477,1705],{"encoding":449},[224,116479,116481],{"className":116480,"ariaHidden":455},[454],[224,116482,116484,116487],{"className":116483},[459],[224,116485],{"className":116486,"style":2565},[463],[224,116488,1705],{"className":116489,"style":1740},[468,469]," 同时成立（上界方向由 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C|t-1|",[224,116543,116545,116575,116605,116629],{"className":116544,"ariaHidden":455},[454],[224,116546,116548,116551,116554,116557,116560,116563,116566,116569,116572],{"className":116547},[459],[224,116549],{"className":116550,"style":535},[463],[224,116552,1083],{"className":116553},[468],[224,116555,1705],{"className":116556,"style":1740},[468,469],[224,116558,239],{"className":116559},[542],[224,116561,1435],{"className":116562},[468,469],[224,116564,1716],{"className":116565},[549],[224,116567],{"className":116568,"style":474},[473],[224,116570,5268],{"className":116571},[478],[224,116573],{"className":116574,"style":474},[473],[224,116576,116578,116581,116584,116587,116590,116593,116596,116599,116602],{"className":116577},[459],[224,116579],{"className":116580,"style":535},[463],[224,116582,1705],{"className":116583,"style":1740},[468,469],[224,116585,239],{"className":116586},[542],[224,116588,1438],{"className":116589},[468],[224,116591,1716],{"className":116592},[549],[224,116594,1083],{"className":116595},[468],[224,116597],{"className":116598,"style":496},[473],[224,116600,2070],{"className":116601},[500],[224,116603],{"className":116604,"style":496},[473],[224,116606,116608,116611,116614,116617,116620,116623,116626],{"className":116607},[459],[224,116609],{"className":116610,"style":535},[463],[224,116612,16021],{"className":116613,"style":30740},[468,469],[224,116615,1083],{"className":116616},[468],[224,116618,1435],{"className":116619},[468,469],[224,116621],{"className":116622,"style":474},[473],[224,116624,5268],{"className":116625},[478],[224,116627],{"className":116628,"style":474},[473],[224,116630,116632,116635],{"className":116631},[459],[224,116633],{"className":116634,"style":535},[463],[224,116636,54602],{"className":116637},[468]," 于 ",[224,116640,116642,116686],{"className":116641,"translate":388},[392],[224,116643,116645],{"className":116644},[396],[398,116646,116647],{"xmlns":400},[403,116648,116649,116683],{},[406,116650,116651,116653,116655,116657,116659,116661,116669,116671,116673,116681],{},[409,116652,1435],{},[412,116654,12248],{},[412,116656,268],{"stretchy":435},[419,116658,1438],{},[409,116660,12413],{"mathvariant":678},[695,116662,116663,116665,116667],{},[409,116664,1435],{},[409,116666,71515],{},[412,116668,24134],{},[412,116670,1877],{"separator":455},[423,116672,682],{},[695,116674,116675,116677,116679],{},[409,116676,1435],{},[409,116678,72294],{},[412,116680,24134],{},[412,116682,440],{"stretchy":435},[447,116684,116685],{"encoding":449},"t\\in[1\u002Ft_{\\min},\\,t_{\\max}]",[224,116687,116689,116707],{"className":116688,"ariaHidden":455},[454],[224,116690,116692,116695,116698,116701,116704],{"className":116691},[459],[224,116693],{"className":116694,"style":27240},[463],[224,116696,1435],{"className":116697},[468,469],[224,116699],{"className":116700,"style":496},[473],[224,116702,12248],{"className":116703},[500],[224,116705],{"className":116706,"style":496},[473],[224,116708,116710,116713,116716,116719,116772,116775,116778,116781,116833],{"className":116709},[459],[224,116711],{"className":116712,"style":535},[463],[224,116714,268],{"className":116715},[542],[224,116717,28555],{"className":116718},[468],[224,116720,116722,116725],{"className":116721},[468],[224,116723,1435],{"className":116724},[468,469],[224,116726,116728],{"className":116727},[750],[224,116729,116731,116764],{"className":116730},[754,755],[224,116732,116734,116761],{"className":116733},[759],[224,116735,116738],{"className":116736,"style":116737},[763],"height:0.3175em;",[224,116739,116740,116743],{"style":767},[224,116741],{"className":116742,"style":772},[771],[224,116744,116746],{"className":116745},[776,777,778,779],[224,116747,116749],{"className":116748},[468,779],[224,116750,116752,116755,116758],{"className":116751},[21531,779],[224,116753,13177],{"className":116754},[779],[224,116756,1880],{"className":116757},[779],[224,116759,3254],{"className":116760},[779],[224,116762,787],{"className":116763},[786],[224,116765,116767],{"className":116766},[759],[224,116768,116770],{"className":116769,"style":794},[763],[224,116771],{},[224,116773,1877],{"className":116774},[1917],[224,116776],{"className":116777,"style":724},[473],[224,116779],{"className":116780,"style":724},[473],[224,116782,116784,116787],{"className":116783},[468],[224,116785,1435],{"className":116786},[468,469],[224,116788,116790],{"className":116789},[750],[224,116791,116793,116825],{"className":116792},[754,755],[224,116794,116796,116822],{"className":116795},[759],[224,116797,116799],{"className":116798,"style":64841},[763],[224,116800,116801,116804],{"style":767},[224,116802],{"className":116803,"style":772},[771],[224,116805,116807],{"className":116806},[776,777,778,779],[224,116808,116810],{"className":116809},[468,779],[224,116811,116813,116816,116819],{"className":116812},[21531,779],[224,116814,13177],{"className":116815},[779],[224,116817,15],{"className":116818},[779],[224,116820,371],{"className":116821},[779],[224,116823,787],{"className":116824},[786],[224,116826,116828],{"className":116827},[759],[224,116829,116831],{"className":116830,"style":794},[763],[224,116832],{},[224,116834,440],{"className":116835},[549]," 的一致 Lipschitz 性给出）。",[11,116838,116839,116840,116843],{},"也就是说，",[378,116841,116842],{},"\"所有 f-散度诱导同一拓扑\"必须在有限\u002F紧设定或一致比值界下读","：此时它们都等价于全变差拓扑（也叫强拓扑），只是\"度量标尺\"不同；而在一般测度空间上，KL 型拓扑严格强于 TV 拓扑。这一分层对理解第 6 节 f-散度与 Wasserstein 的真正差异很重要：f-散度族的内部差异是\"强度\"差异，而它们与 Wasserstein 的差异是\"性质\"差异。",[613,116845,116847],{"id":116846},"_48-一般不是度量","4.8 一般不是度量",[11,116849,116850],{},"f-散度一般不对称，也不满足三角不等式，因此通常不是距离度量。但某些特殊形式可以诱导出度量：",[2674,116852,116853,118364],{},[2677,116854,116855,116858,116859,117531,117533,117534,117562,117563,117636,117637,117897,117898,118303,118304,118363],{},[378,116856,116857],{},"Hellinger 距离","是度量：",[224,116860,116862],{"className":116861,"translate":388},[387],[224,116863,116865,116969],{"className":116864,"translate":388},[392],[224,116866,116868],{"className":116867},[396],[398,116869,116870],{"xmlns":400,"display":401},[403,116871,116872,116966],{},[406,116873,116874,116880,116882,116884,116886,116888,116890,116892,116930,116932,116944,116946,116964],{},[695,116875,116876,116878],{},[409,116877,64472],{},[409,116879,53807],{},[412,116881,239],{"stretchy":435},[409,116883,16018],{},[412,116885,1877],{"separator":455},[409,116887,20315],{},[412,116889,1716],{"stretchy":435},[412,116891,278],{},[54262,116893,116894],{},[406,116895,116896,116904,116906,116908,116912,116914,116918,116924,116926,116928],{},[5221,116897,116898],{"scriptlevel":284,"displaystyle":435},[20338,116899,116900,116902],{},[419,116901,1438],{},[419,116903,1447],{},[412,116905,21338],{},[412,116907,239],{"stretchy":435},[54262,116909,116910],{},[409,116911,11],{},[412,116913,5268],{},[54262,116915,116916],{},[409,116917,2023],{},[3248,116919,116920,116922],{},[412,116921,1716],{"stretchy":435},[419,116923,1447],{},[423,116925,682],{},[409,116927,13182],{},[409,116929,371],{},[412,116931,278],{},[54262,116933,116934],{},[406,116935,116936,116938,116940,116942],{},[419,116937,1438],{},[412,116939,5268],{},[409,116941,9889],{},[409,116943,16021],{},[412,116945,278],{},[54262,116947,116948],{},[406,116949,116950,116952,116954,116956],{},[419,116951,1447],{},[412,116953,5268],{},[419,116955,1447],{},[27527,116957,116958,116960,116962],{},[409,116959,53807],{},[423,116961,62050],{},[419,116963,1447],{},[409,116965,688],{"mathvariant":678},[447,116967,116968],{"encoding":449},"\\rho_H(P,Q)\n=\\sqrt{\\tfrac12\\int(\\sqrt p-\\sqrt q)^2\\,dx}\n=\\sqrt{1-BC}\n=\\sqrt{2-2H^2_{\\text{Bhattacharyya}}}.",[224,116970,116972,117045,117317,117398],{"className":116971,"ariaHidden":455},[454],[224,116973,116975,116978,117018,117021,117024,117027,117030,117033,117036,117039,117042],{"className":116974},[459],[224,116976],{"className":116977,"style":535},[463],[224,116979,116981,116984],{"className":116980},[468],[224,116982,64472],{"className":116983},[468,469],[224,116985,116987],{"className":116986},[750],[224,116988,116990,117010],{"className":116989},[754,755],[224,116991,116993,117007],{"className":116992},[759],[224,116994,116996],{"className":116995,"style":764},[763],[224,116997,116998,117001],{"style":767},[224,116999],{"className":117000,"style":772},[771],[224,117002,117004],{"className":117003},[776,777,778,779],[224,117005,53807],{"className":117006,"style":53961},[468,469,779],[224,117008,787],{"className":117009},[786],[224,117011,117013],{"className":117012},[759],[224,117014,117016],{"className":117015,"style":794},[763],[224,117017],{},[224,117019,239],{"className":117020},[542],[224,117022,16018],{"className":117023,"style":653},[468,469],[224,117025,1877],{"className":117026},[1917],[224,117028],{"className":117029,"style":724},[473],[224,117031,20315],{"className":117032},[468,469],[224,117034,1716],{"className":117035},[549],[224,117037],{"className":117038,"style":496},[473],[224,117040,278],{"className":117041},[500],[224,117043],{"className":117044,"style":496},[473],[224,117046,117048,117052,117308,117311,117314],{"className":117047},[459],[224,117049],{"className":117050,"style":117051},[463],"height:3.04em;vertical-align:-1.1572em;",[224,117053,117055],{"className":117054},[468,54290],[224,117056,117058,117300],{"className":117057},[754,755],[224,117059,117061,117297],{"className":117060},[759],[224,117062,117064,117285],{"className":117063,"style":111011},[763],[224,117065,117067,117070],{"className":117066,"style":111015},[2139],[224,117068],{"className":117069,"style":111019},[771],[224,117071,117073,117141,117144,117147,117150,117194,117197,117200,117203,117247,117276,117279,117282],{"className":117072,"style":61074},[468],[224,117074,117076,117079,117138],{"className":117075},[468],[224,117077],{"className":117078},[542,8479],[224,117080,117082],{"className":117081},[20338],[224,117083,117085,117130],{"className":117084},[754,755],[224,117086,117088,117127],{"className":117087},[759],[224,117089,117091,117105,117113],{"className":117090,"style":22563},[763],[224,117092,117093,117096],{"style":22566},[224,117094],{"className":117095,"style":2108},[771],[224,117097,117099],{"className":117098},[776,777,778,779],[224,117100,117102],{"className":117101},[468,779],[224,117103,1447],{"className":117104},[468,779],[224,117106,117107,117110],{"style":20546},[224,117108],{"className":117109,"style":2108},[771],[224,117111],{"className":117112,"style":20554},[20553],[224,117114,117115,117118],{"style":22589},[224,117116],{"className":117117,"style":2108},[771],[224,117119,117121],{"className":117120},[776,777,778,779],[224,117122,117124],{"className":117123},[468,779],[224,117125,1438],{"className":117126},[468,779],[224,117128,787],{"className":117129},[786],[224,117131,117133],{"className":117132},[759],[224,117134,117136],{"className":117135,"style":22611},[763],[224,117137],{},[224,117139],{"className":117140},[549,8479],[224,117142],{"className":117143,"style":724},[473],[224,117145,21338],{"className":117146,"style":21534},[21531,21532,21533],[224,117148,239],{"className":117149},[542],[224,117151,117153],{"className":117152},[468,54290],[224,117154,117156,117186],{"className":117155},[754,755],[224,117157,117159,117183],{"className":117158},[759],[224,117160,117162,117171],{"className":117161,"style":6992},[763],[224,117163,117165,117168],{"className":117164,"style":2120},[2139],[224,117166],{"className":117167,"style":2108},[771],[224,117169,11],{"className":117170,"style":54310},[468,469],[224,117172,117173,117176],{"style":61263},[224,117174],{"className":117175,"style":2108},[771],[224,117177,117179],{"className":117178,"style":54367},[54366],[2155,117180,117181],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,117182],{"d":54374},[224,117184,787],{"className":117185},[786],[224,117187,117189],{"className":117188},[759],[224,117190,117192],{"className":117191,"style":61283},[763],[224,117193],{},[224,117195],{"className":117196,"style":474},[473],[224,117198,5268],{"className":117199},[478],[224,117201],{"className":117202,"style":474},[473],[224,117204,117206],{"className":117205},[468,54290],[224,117207,117209,117239],{"className":117208},[754,755],[224,117210,117212,117236],{"className":117211},[759],[224,117213,117215,117224],{"className":117214,"style":6992},[763],[224,117216,117218,117221],{"className":117217,"style":2120},[2139],[224,117219],{"className":117220,"style":2108},[771],[224,117222,2023],{"className":117223,"style":59019},[468,469],[224,117225,117226,1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",[224,117535,117537,117550],{"className":117536,"translate":388},[392],[224,117538,117540],{"className":117539},[396],[398,117541,117542],{"xmlns":400},[403,117543,117544,117548],{},[406,117545,117546],{},[409,117547,16018],{},[447,117549,16018],{"encoding":449},[224,117551,117553],{"className":117552,"ariaHidden":455},[454],[224,117554,117556,117559],{"className":117555},[459],[224,117557],{"className":117558,"style":649},[463],[224,117560,16018],{"className":117561,"style":653},[468,469]," 映到单位球面上的点 ",[224,117564,117566,117582],{"className":117565,"translate":388},[392],[224,117567,117569],{"className":117568},[396],[398,117570,117571],{"xmlns":400},[403,117572,117573,117579],{},[406,117574,117575],{},[54262,117576,117577],{},[409,117578,11],{},[447,117580,117581],{"encoding":449},"\\sqrt p",[224,117583,117585],{"className":117584,"ariaHidden":455},[454],[224,117586,117588,117592],{"className":117587},[459],[224,117589],{"className":117590,"style":117591},[463],"height:1.04em;vertical-align:-0.3369em;",[224,117593,117595],{"className":117594},[468,54290],[224,117596,117598,117628],{"className":117597},[754,755],[224,117599,117601,117625],{"className":117600},[759],[224,117602,117604,117613],{"className":117603,"style":58947},[763],[224,117605,117607,117610],{"className":117606,"style":2120},[2139],[224,117608],{"className":117609,"style":2108},[771],[224,117611,11],{"className":117612,"style":54310},[468,469],[224,117614,117615,117618],{"style":58959},[224,117616],{"className":117617,"style":2108},[771],[224,117619,117621],{"className":117620,"style":54367},[54366],[2155,117622,117623],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,117624],{"d":54374},[224,117626,787],{"className":117627},[786],[224,117629,117631],{"className":117630},[759],[224,117632,117634],{"className":117633,"style":58979},[763],[224,117635],{},"（因 ",[224,117638,117640,117676],{"className":117639,"translate":388},[392],[224,117641,117643],{"className":117642},[396],[398,117644,117645],{"xmlns":400},[403,117646,117647,117673],{},[406,117648,117649,117651,117655,117663,117665,117667,117669,117671],{},[412,117650,20312],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[54262,117652,117653],{},[409,117654,11],{},[27527,117656,117657,117659,117661],{},[412,117658,20312],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[419,117660,1447],{},[419,117662,1447],{},[412,117664,278],{},[412,117666,21338],{},[409,117668,11],{},[412,117670,278],{},[419,117672,1438],{},[447,117674,117675],{"encoding":449},"\\big\\lVert\\sqrt p\\big\\rVert_2^2=\\int p=1",[224,117677,117679,117864,117888],{"className":117678,"ariaHidden":455},[454],[224,117680,117682,117686,117723,117767,117855,117858,117861],{"className":117681},[459],[224,117683],{"className":117684,"style":117685},[463],"height:1.4537em;vertical-align:-0.3997em;",[224,117687,117689],{"className":117688},[468],[224,117690,117692],{"className":117691},[2363,34778],[224,117693,117695,117715],{"className":117694},[754,755],[224,117696,117698,117712],{"className":117697},[759],[224,117699,117701],{"className":117700,"style":2285},[763],[224,117702,117703,117706],{"style":34790},[224,117704],{"className":117705,"style":34794},[771],[224,117707,117708],{"style":34797},[2155,117709,117710],{"xmlns":2157,"width":34800,"height":2004,"viewBox":34801},[2163,117711],{"d":34804},[224,117713,787],{"className":117714},[786],[224,117716,117718],{"className":117717},[759],[224,117719,117721],{"className":117720,"style":34814},[763],[224,117722],{},[224,117724,117726],{"className":117725},[468,54290],[224,117727,117729,117759],{"className":117728},[754,755],[224,117730,117732,117756],{"className":117731},[759],[224,117733,117735,117744],{"className":117734,"style":58947},[763],[224,117736,117738,117741],{"className":117737,"style":2120},[2139],[224,117739],{"className":117740,"style":2108},[771],[224,117742,11],{"className":117743,"style":54310},[468,469],[224,117745,117746,117749],{"style":58959},[224,117747],{"className":117748,"style":2108},[771],[224,117750,117752],{"className":117751,"style":54367},[54366],[2155,117753,117754],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,117755],{"d":54374},[224,117757,787],{"className":117758},[786],[224,117760,117762],{"className":117761},[759],[224,117763,117765],{"className":117764,"style":58979},[763],[224,117766],{},[224,117768,117770,117807],{"className":117769},[468],[224,117771,117773],{"className":117772},[468],[224,117774,117776],{"className":117775},[2363,34778],[224,117777,117779,117799],{"className":117778},[754,755],[224,117780,117782,117796],{"className":117781},[759],[224,117783,117785],{"className":117784,"style":2285},[763],[224,117786,117787,117790],{"style":34790},[224,117788],{"className":117789,"style":34794},[771],[224,117791,117792],{"style":34797},[2155,117793,117794],{"xmlns":2157,"width":34800,"height":2004,"viewBox":34801},[2163,117795],{"d":34804},[224,117797,787],{"className":117798},[786],[224,117800,117802],{"className":117801},[759],[224,117803,117805],{"className":117804,"style":34814},[763],[224,117806],{},[224,117808,117810],{"className":117809},[750],[224,117811,117813,117846],{"className":117812},[754,755],[224,117814,117816,117843],{"className":117815},[759],[224,117817,117820,117832],{"className":117818,"style":117819},[763],"height:1.054em;",[224,117821,117823,117826],{"style":117822},"top:-2.3003em;margin-right:0.05em;",[224,117824],{"className":117825,"style":772},[771],[224,117827,117829],{"className":117828},[776,777,778,779],[224,117830,1447],{"className":117831},[468,779],[224,117833,117834,117837],{"style":71342},[224,117835],{"className":117836,"style":772},[771],[224,117838,117840],{"className":117839},[776,777,778,779],[224,117841,1447],{"className":117842},[468,779],[224,117844,787],{"className":117845},[786],[224,117847,117849],{"className":117848},[759],[224,117850,117853],{"className":117851,"style":117852},[763],"height:0.3997em;",[224,117854],{},[224,117856],{"className":117857,"style":496},[473],[224,117859,278],{"className":117860},[500],[224,117862],{"className":117863,"style":496},[473],[224,117865,117867,117870,117873,117876,117879,117882,117885],{"className":117866},[459],[224,117868],{"className":117869,"style":25621},[463],[224,117871,21338],{"className":117872,"style":25626},[21531,21532,25625],[224,117874],{"className":117875,"style":724},[473],[224,117877,11],{"className":117878},[468,469],[224,117880],{"className":117881,"style":496},[473],[224,117883,278],{"className":117884},[500],[224,117886],{"className":117887,"style":496},[473],[224,117889,117891,117894],{"className":117890},[459],[224,117892],{"className":117893,"style":510},[463],[224,117895,1438],{"className":117896},[468],"），则 ",[224,117899,117901,117959],{"className":117900,"translate":388},[392],[224,117902,117904],{"className":117903},[396],[398,117905,117906],{"xmlns":400},[403,117907,117908,117956],{},[406,117909,117910,117916,117918,117920,117922,117924,117926,117928,117938,117940,117944,117946,117950],{},[695,117911,117912,117914],{},[409,117913,64472],{},[409,117915,53807],{},[412,117917,239],{"stretchy":435},[409,117919,16018],{},[412,117921,1877],{"separator":455},[409,117923,20315],{},[412,117925,1716],{"stretchy":435},[412,117927,278],{},[5221,117929,117930],{"scriptlevel":284,"displaystyle":435},[20338,117931,117932,117934],{},[419,117933,1438],{},[54262,117935,117936],{},[419,117937,1447],{},[412,117939,20312],{"stretchy":435},[54262,117941,117942],{},[409,117943,11],{},[412,117945,5268],{},[54262,117947,117948],{},[409,117949,2023],{},[695,117951,117952,117954],{},[412,117953,20312],{"stretchy":435},[419,117955,1447],{},[447,117957,117958],{"encoding":449},"\\rho_H(P,Q)=\\tfrac{1}{\\sqrt2}\\lVert\\sqrt p-\\sqrt q\\rVert_2",[224,117960,117962,118035,118212],{"className":117961,"ariaHidden":455},[454],[224,117963,117965,117968,118008,118011,118014,118017,118020,118023,118026,118029,118032],{"className":117964},[459],[224,117966],{"className":117967,"style":535},[463],[224,117969,117971,117974],{"className":117970},[468],[224,117972,64472],{"className":117973},[468,469],[224,117975,117977],{"className":117976},[750],[224,117978,117980,118000],{"className":117979},[754,755],[224,117981,117983,117997],{"className":117982},[759],[224,117984,117986],{"className":117985,"style":764},[763],[224,117987,117988,117991],{"style":767},[224,117989],{"className":117990,"style":772},[771],[224,117992,117994],{"className":117993},[776,777,778,779],[224,117995,53807],{"className":117996,"style":53961},[468,469,779],[224,117998,787],{"className":117999},[786],[224,118001,118003],{"className":118002},[759],[224,118004,118006],{"className":118005,"style":794},[763],[224,118007],{},[224,118009,239],{"className":118010},[542],[224,118012,16018],{"className":118013,"style":653},[468,469],[224,118015,1877],{"className":118016},[1917],[224,118018],{"className":118019,"style":724},[473],[224,118021,20315],{"className":118022},[468,469],[224,118024,1716],{"className":118025},[549],[224,118027],{"className":118028,"style":496},[473],[224,118030,278],{"className":118031},[500],[224,118033],{"className":118034,"style":496},[473],[224,118036,118038,118042,118156,118159,118203,118206,118209],{"className":118037},[459],[224,118039],{"className":118040,"style":118041},[463],"height:1.3831em;vertical-align:-0.538em;",[224,118043,118045,118048,118153],{"className":118044},[468],[224,118046],{"className":118047},[542,8479],[224,118049,118051],{"className":118050},[20338],[224,118052,118054,118144],{"className":118053},[754,755],[224,118055,118057,118141],{"className":118056},[759],[224,118058,118060,118119,118127],{"className":118059,"style":22563},[763],[224,118061,118063,118066],{"style":118062},"top:-2.551em;",[224,118064],{"className":118065,"style":2108},[771],[224,118067,118069],{"className":118068},[776,777,778,779],[224,118070,118072],{"className":118071},[468,779],[224,118073,118075],{"className":118074},[468,54290,779],[224,118076,118078,118110],{"className":118077},[754,755],[224,118079,118081,118107],{"className":118080},[759],[224,118082,118085,118094],{"className":118083,"style":118084},[763],"height:0.9128em;",[224,118086,118088,118091],{"className":118087,"style":2120},[2139],[224,118089],{"className":118090,"style":2108},[771],[224,118092,1447],{"className":118093,"style":54310},[468,779],[224,118095,118097,118100],{"style":118096},"top:-2.8728em;",[224,118098],{"className":118099,"style":2108},[771],[224,118101,118103],{"className":118102,"style":54367},[54366,779],[2155,118104,118105],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,118106],{"d":54374},[224,118108,787],{"className":118109},[786],[224,118111,118113],{"className":118112},[759],[224,118114,118117],{"className":118115,"style":118116},[763],"height:0.1272em;",[224,118118],{},[224,118120,118121,118124],{"style":20546},[224,118122],{"className":118123,"style":2108},[771],[224,118125],{"className":118126,"style":20554},[20553],[224,118128,118129,118132],{"style":22589},[224,118130],{"className":118131,"style":2108},[771],[224,118133,118135],{"className":118134},[776,777,778,779],[224,118136,118138],{"className":118137},[468,779],[224,118139,1438],{"className":118140},[468,779],[224,118142,787],{"className":118143},[786],[224,118145,118147],{"className":118146},[759],[224,118148,118151],{"className":118149,"style":118150},[763],"height:0.538em;",[224,118152],{},[224,118154],{"className":118155},[549,8479],[224,118157,20312],{"className":118158},[542],[224,118160,118162],{"className":118161},[468,54290],[224,118163,118165,118195],{"className":118164},[754,755],[224,118166,118168,118192],{"className":118167},[759],[224,118169,118171,118180],{"className":118170,"style":58947},[763],[224,118172,118174,118177],{"className":118173,"style":2120},[2139],[224,118175],{"className":118176,"style":2108},[771],[224,118178,11],{"className":118179,"style":54310},[468,469],[224,118181,118182,118185],{"style":58959},[224,118183],{"className":118184,"style":2108},[771],[224,118186,118188],{"className":118187,"style":54367},[54366],[2155,118189,118190],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,118191],{"d":54374},[224,118193,787],{"className":118194},[786],[224,118196,118198],{"className":118197},[759],[224,118199,118201],{"className":118200,"style":58979},[763],[224,118202],{},[224,118204],{"className":118205,"style":474},[473],[224,118207,5268],{"className":118208},[478],[224,118210],{"className":118211,"style":474},[473],[224,118213,118215,118219,118263],{"className":118214},[459],[224,118216],{"className":118217,"style":118218},[463],"height:1.0869em;vertical-align:-0.3369em;",[224,118220,118222],{"className":118221},[468,54290],[224,118223,118225,118255],{"className":118224},[754,755],[224,118226,118228,118252],{"className":118227},[759],[224,118229,118231,118240],{"className":118230,"style":58947},[763],[224,118232,118234,118237],{"className":118233,"style":2120},[2139],[224,118235],{"className":118236,"style":2108},[771],[224,118238,2023],{"className":118239,"style":59019},[468,469],[224,118241,118242,118245],{"style":58959},[224,118243],{"className":118244,"style":2108},[771],[224,118246,118248],{"className":118247,"style":54367},[54366],[2155,118249,118250],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,118251],{"d":54374},[224,118253,787],{"className":118254},[786],[224,118256,118258],{"className":118257},[759],[224,118259,118261],{"className":118260,"style":58979},[763],[224,118262],{},[224,118264,118266,118269],{"className":118265},[549],[224,118267,20312],{"className":118268},[549],[224,118270,118272],{"className":118271},[750],[224,118273,118275,118295],{"className":118274},[754,755],[224,118276,118278,118292],{"className":118277},[759],[224,118279,118281],{"className":118280,"style":1479},[763],[224,118282,118283,118286],{"style":767},[224,118284],{"className":118285,"style":772},[771],[224,118287,118289],{"className":118288},[776,777,778,779],[224,118290,1447],{"className":118291},[468,779],[224,118293,787],{"className":118294},[786],[224,118296,118298],{"className":118297},[759],[224,118299,118301],{"className":118300,"style":794},[763],[224,118302],{}," 就是单位球面上的欧氏距离——度量公理（对称、三角不等式）由 ",[224,118305,118307,118325],{"className":118306,"translate":388},[392],[224,118308,118310],{"className":118309},[396],[398,118311,118312],{"xmlns":400},[403,118313,118314,118322],{},[406,118315,118316],{},[3248,118317,118318,118320],{},[409,118319,41831],{},[419,118321,1447],{},[447,118323,118324],{"encoding":449},"L^2",[224,118326,118328],{"className":118327,"ariaHidden":455},[454],[224,118329,118331,118334],{"className":118330},[459],[224,118332],{"className":118333,"style":5671},[463],[224,118335,118337,118340],{"className":118336},[468],[224,118338,41831],{"className":118339},[468,469],[224,118341,118343],{"className":118342},[750],[224,118344,118346],{"className":118345},[754],[224,118347,118349],{"className":118348},[759],[224,118350,118352],{"className":118351,"style":5671},[763],[224,118353,118354,118357],{"style":3288},[224,118355],{"className":118356,"style":772},[771],[224,118358,118360],{"className":118359},[776,777,778,779],[224,118361,1447],{"className":118362},[468,779]," 距离直接继承，非退化由\"开平方是单射\"保证；",[2677,118365,118366,118682,118683,118762],{},[378,118367,118368,118506,118507,118681],{},[224,118369,118371,118401],{"className":118370,"translate":388},[392],[224,118372,118374],{"className":118373},[396],[398,118375,118376],{"xmlns":400},[403,118377,118378,118398],{},[406,118379,118380],{},[54262,118381,118382],{},[406,118383,118384,118386,118388],{},[419,118385,1447],{},[423,118387,682],{},[695,118389,118390,118392],{},[409,118391,20303],{},[406,118393,118394,118396],{},[409,118395,43681],{},[409,118397,6924],{},[447,118399,118400],{"encoding":449},"\\sqrt{2\\,D_{JS}}",[224,118402,118404],{"className":118403,"ariaHidden":455},[454],[224,118405,118407,118410],{"className":118406},[459],[224,118408],{"className":118409,"style":54286},[463],[224,118411,118413],{"className":118412},[468,54290],[224,118414,118416,118498],{"className":118415},[754,755],[224,118417,118419,118495],{"className":118418},[759],[224,118420,118422,118483],{"className":118421,"style":54300},[763],[224,118423,118425,118428],{"className":118424,"style":2120},[2139],[224,118426],{"className":118427,"style":2108},[771],[224,118429,118431,118434,118437],{"className":118430,"style":54310},[468],[224,118432,1447],{"className":118433},[468],[224,118435],{"className":118436,"style":724},[473],[224,118438,118440,118443],{"className":118439},[468],[224,118441,20303],{"className":118442,"style":2473},[468,469],[224,118444,118446],{"className":118445},[750],[224,118447,118449,118475],{"className":118448},[754,755],[224,118450,118452,118472],{"className":118451},[759],[224,118453,118455],{"className":118454,"style":764},[763],[224,118456,118457,118460],{"style":20390},[224,118458],{"className":118459,"style":772},[771],[224,118461,118463],{"className":118462},[776,777,778,779],[224,118464,118466,118469],{"className":118465},[468,779],[224,118467,43681],{"className":118468,"style":44091},[468,469,779],[224,118470,6924],{"className":118471,"style":6970},[468,469,779],[224,118473,787],{"className":118474},[786],[224,118476,118478],{"className":118477},[759],[224,118479,118481],{"className":118480,"style":794},[763],[224,118482],{},[224,118484,118485,118488],{"style":54359},[224,118486],{"className":118487,"style":2108},[771],[224,118489,118491],{"className":118490,"style":54367},[54366],[2155,118492,118493],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,118494],{"d":54374},[224,118496,787],{"className":118497},[786],[224,118499,118501],{"className":118500},[759],[224,118502,118504],{"className":118503,"style":54384},[763],[224,118505],{},"（等价地 ",[224,118508,118510,118538],{"className":118509,"translate":388},[392],[224,118511,118513],{"className":118512},[396],[398,118514,118515],{"xmlns":400},[403,118516,118517,118535],{},[406,118518,118519],{},[54262,118520,118521],{},[695,118522,118523,118525],{},[409,118524,20303],{},[695,118526,118527,118529],{},[409,118528,1705],{},[406,118530,118531,118533],{},[409,118532,43681],{},[409,118534,6924],{},[447,118536,118537],{"encoding":449},"\\sqrt{D_{f_{JS}}}",[224,118539,118541],{"className":118540,"ariaHidden":455},[454],[224,118542,118544,118548],{"className":118543},[459],[224,118545],{"className":118546,"style":118547},[463],"height:1.24em;vertical-align:-0.3564em;",[224,118549,118551],{"className":118550},[468,54290],[224,118552,118554,118672],{"className":118553},[754,755],[224,118555,118557,118669],{"className":118556},[759],[224,118558,118561,118656],{"className":118559,"style":118560},[763],"height:0.8836em;",[224,118562,118564,118567],{"className":118563,"style":103438},[2139],[224,118565],{"className":118566,"style":34794},[771],[224,118568,118570],{"className":118569,"style":61074},[468],[224,118571,118573,118576],{"className":118572},[468],[224,118574,20303],{"className":118575,"style":2473},[468,469],[224,118577,118579],{"className":118578},[750],[224,118580,118582,118648],{"className":118581},[754,755],[224,118583,118585,118645],{"className":118584},[759],[224,118586,118588],{"className":118587,"style":20387},[763],[224,118589,118590,118593],{"style":20390},[224,118591],{"className":118592,"style":772},[771],[224,118594,118596],{"className":118595},[776,777,778,779],[224,118597,118599],{"className":118598},[468,779],[224,118600,118602,118605],{"className":118601},[468,779],[224,118603,1705],{"className":118604,"style":1740},[468,469,779],[224,118606,118608],{"className":118607},[750],[224,118609,118611,118637],{"className":118610},[754,755],[224,118612,118614,118634],{"className":118613},[759],[224,118615,118617],{"className":118616,"style":26101},[763],[224,118618,118619,118622],{"style":50132},[224,118620],{"className":118621,"style":50136},[771],[224,118623,118625],{"className":118624},[776,50140,2364,779],[224,118626,118628,118631],{"className":118627},[468,779],[224,118629,43681],{"className":118630,"style":44091},[468,469,779],[224,118632,6924],{"className":118633,"style":6970},[468,469,779],[224,118635,787],{"className":118636},[786],[224,118638,118640],{"className":118639},[759],[224,118641,118643],{"className":118642,"style":50159},[763],[224,118644],{},[224,118646,787],{"className":118647},[786],[224,118649,118651],{"className":118650},[759],[224,118652,118654],{"className":118653,"style":20409},[763],[224,118655],{},[224,118657,118659,118662],{"style":118658},"top:-2.8436em;",[224,118660],{"className":118661,"style":34794},[771],[224,118663,118665],{"className":118664,"style":103530},[54366],[2155,118666,118667],{"xmlns":2157,"width":2158,"height":103533,"viewBox":103534,"preserveAspectRatio":2161},[2163,118668],{"d":103537},[224,118670,787],{"className":118671},[786],[224,118673,118675],{"className":118674},[759],[224,118676,118679],{"className":118677,"style":118678},[763],"height:0.3564em;",[224,118680],{}," 规范化后）是度量","（Endres–Schindelin 的结论），其证明用到了 ",[224,118684,118686,118707],{"className":118685,"translate":388},[392],[224,118687,118689],{"className":118688},[396],[398,118690,118691],{"xmlns":400},[403,118692,118693,118705],{},[406,118694,118695],{},[695,118696,118697,118699],{},[409,118698,1705],{},[406,118700,118701,118703],{},[409,118702,43681],{},[409,118704,6924],{},[447,118706,50685],{"encoding":449},[224,118708,118710],{"className":118709,"ariaHidden":455},[454],[224,118711,118713,118716],{"className":118712},[459],[224,118714],{"className":118715,"style":2565},[463],[224,118717,118719,118722],{"className":118718},[468],[224,118720,1705],{"className":118721,"style":1740},[468,469],[224,118723,118725],{"className":118724},[750],[224,118726,118728,118754],{"className":118727},[754,755],[224,118729,118731,118751],{"className":118730},[759],[224,118732,118734],{"className":118733,"style":764},[763],[224,118735,118736,118739],{"style":47439},[224,118737],{"className":118738,"style":772},[771],[224,118740,118742],{"className":118741},[776,777,778,779],[224,118743,118745,118748],{"className":118744},[468,779],[224,118746,43681],{"className":118747,"style":44091},[468,469,779],[224,118749,6924],{"className":118750,"style":6970},[468,469,779],[224,118752,787],{"className":118753},[786],[224,118755,118757],{"className":118756},[759],[224,118758,118760],{"className":118759,"style":794},[763],[224,118761],{}," 对应生成元的\"负定型\"性质。",[11,118764,118765,118766,118882,118883,118963,118964,118992,118993,119021],{},"反过来，不对称显然排除度量；即使对称，",[224,118767,118769,118789],{"className":118768,"translate":388},[392],[224,118770,118772],{"className":118771},[396],[398,118773,118774],{"xmlns":400},[403,118775,118776,118786],{},[406,118777,118778],{},[54262,118779,118780],{},[695,118781,118782,118784],{},[409,118783,20303],{},[409,118785,1705],{},[447,118787,118788],{"encoding":449},"\\sqrt{D_f}",[224,118790,118792],{"className":118791,"ariaHidden":455},[454],[224,118793,118795,118798],{"className":118794},[459],[224,118796],{"className":118797,"style":118547},[463],[224,118799,118801],{"className":118800},[468,54290],[224,118802,118804,118874],{"className":118803},[754,755],[224,118805,118807,118871],{"className":118806},[759],[224,118808,118810,118859],{"className":118809,"style":118560},[763],[224,118811,118813,118816],{"className":118812,"style":103438},[2139],[224,118814],{"className":118815,"style":34794},[771],[224,118817,118819],{"className":118818,"style":61074},[468],[224,118820,118822,118825],{"className":118821},[468],[224,118823,20303],{"className":118824,"style":2473},[468,469],[224,118826,118828],{"className":118827},[750],[224,118829,118831,118851],{"className":118830},[754,755],[224,118832,118834,118848],{"className":118833},[759],[224,118835,118837],{"className":118836,"style":20387},[763],[224,118838,118839,118842],{"style":20390},[224,118840],{"className":118841,"style":772},[771],[224,118843,118845],{"className":118844},[776,777,778,779],[224,118846,1705],{"className":118847,"style":1740},[468,469,779],[224,118849,787],{"className":118850},[786],[224,118852,118854],{"className":118853},[759],[224,118855,118857],{"className":118856,"style":20409},[763],[224,118858],{},[224,118860,118861,118864],{"style":118658},[224,118862],{"className":118863,"style":34794},[771],[224,118865,118867],{"className":118866,"style":103530},[54366],[2155,118868,118869],{"xmlns":2157,"width":2158,"height":103533,"viewBox":103534,"preserveAspectRatio":2161},[2163,118870],{"d":103537},[224,118872,787],{"className":118873},[786],[224,118875,118877],{"className":118876},[759],[224,118878,118880],{"className":118879,"style":118678},[763],[224,118881],{}," 也不自动是度量。例如全变差 ",[224,118884,118886,118908],{"className":118885,"translate":388},[392],[224,118887,118889],{"className":118888},[396],[398,118890,118891],{"xmlns":400},[403,118892,118893,118905],{},[406,118894,118895],{},[695,118896,118897,118899],{},[409,118898,20303],{},[406,118900,118901,118903],{},[409,118902,637],{},[409,118904,16059],{},[447,118906,118907],{"encoding":449},"D_{TV}",[224,118909,118911],{"className":118910,"ariaHidden":455},[454],[224,118912,118914,118917],{"className":118913},[459],[224,118915],{"className":118916,"style":35905},[463],[224,118918,118920,118923],{"className":118919},[468],[224,118921,20303],{"className":118922,"style":2473},[468,469],[224,118924,118926],{"className":118925},[750],[224,118927,118929,118955],{"className":118928},[754,755],[224,118930,118932,118952],{"className":118931},[759],[224,118933,118935],{"className":118934,"style":764},[763],[224,118936,118937,118940],{"style":20390},[224,118938],{"className":118939,"style":772},[771],[224,118941,118943],{"className":118942},[776,777,778,779],[224,118944,118946,118949],{"className":118945},[468,779],[224,118947,637],{"className":118948,"style":653},[468,469,779],[224,118950,16059],{"className":118951,"style":474},[468,469,779],[224,118953,787],{"className":118954},[786],[224,118956,118958],{"className":118957},[759],[224,118959,118961],{"className":118960,"style":794},[763],[224,118962],{}," 本身就是度量，但若取某凸积 ",[224,118965,118967,118980],{"className":118966,"translate":388},[392],[224,118968,118970],{"className":118969},[396],[398,118971,118972],{"xmlns":400},[403,118973,118974,118978],{},[406,118975,118976],{},[409,118977,1705],{},[447,118979,1705],{"encoding":449},[224,118981,118983],{"className":118982,"ariaHidden":455},[454],[224,118984,118986,118989],{"className":118985},[459],[224,118987],{"className":118988,"style":2565},[463],[224,118990,1705],{"className":118991,"style":1740},[468,469]," 的散度再做开方，三角不等式可能失效——这依赖 ",[224,118994,118996,119009],{"className":118995,"translate":388},[392],[224,118997,118999],{"className":118998},[396],[398,119000,119001],{"xmlns":400},[403,119002,119003,119007],{},[406,119004,119005],{},[409,119006,1705],{},[447,119008,1705],{"encoding":449},[224,119010,119012],{"className":119011,"ariaHidden":455},[454],[224,119013,119015,119018],{"className":119014},[459],[224,119016],{"className":119017,"style":2565},[463],[224,119019,1705],{"className":119020,"style":1740},[468,469]," 是否落在\"负定（negative type）\"函数类中；不是自动成立的。",[11,119023,119024,119027,119028,119030,119031,119081,119082,119110,119111,119190],{},[378,119025,119026],{},"与度量的对照结论："," f-散度的正确身份是",[378,119029,72410],{},"（divergence）：非负、在 ",[224,119032,119034,119051],{"className":119033,"translate":388},[392],[224,119035,119037],{"className":119036},[396],[398,119038,119039],{"xmlns":400},[403,119040,119041,119049],{},[406,119042,119043,119045,119047],{},[409,119044,16018],{},[412,119046,278],{},[409,119048,20315],{},[447,119050,30229],{"encoding":449},[224,119052,119054,119072],{"className":119053,"ariaHidden":455},[454],[224,119055,119057,119060,119063,119066,119069],{"className":119056},[459],[224,119058],{"className":119059,"style":649},[463],[224,119061,16018],{"className":119062,"style":653},[468,469],[224,119064],{"className":119065,"style":496},[473],[224,119067,278],{"className":119068},[500],[224,119070],{"className":119071,"style":496},[473],[224,119073,119075,119078],{"className":119074},[459],[224,119076],{"className":119077,"style":20637},[463],[224,119079,20315],{"className":119080},[468,469]," 处取零，但不要求对称与三角不等式。当我们需要真距离时（例如做收敛分析、中心化、 ",[224,119083,119085,119098],{"className":119084,"translate":388},[392],[224,119086,119088],{"className":119087},[396],[398,119089,119090],{"xmlns":400},[403,119091,119092,119096],{},[406,119093,119094],{},[409,119095,7630],{},[447,119097,7630],{"encoding":449},[224,119099,119101],{"className":119100,"ariaHidden":455},[454],[224,119102,119104,119107],{"className":119103},[459],[224,119105],{"className":119106,"style":7642},[463],[224,119108,7630],{"className":119109,"style":7646},[468,469],"-中心聚类），就从 f-散度族里挑那些 ",[224,119112,119114,119130],{"className":119113,"translate":388},[392],[224,119115,119117],{"className":119116},[396],[398,119118,119119],{"xmlns":400},[403,119120,119121,119127],{},[406,119122,119123],{},[54262,119124,119125],{},[412,119126,23331],{"lspace":5213,"rspace":5213},[447,119128,119129],{"encoding":449},"\\sqrt{\\cdot}",[224,119131,119133],{"className":119132,"ariaHidden":455},[454],[224,119134,119136,119140],{"className":119135},[459],[224,119137],{"className":119138,"style":119139},[463],"height:1.04em;vertical-align:-0.2328em;",[224,119141,119143],{"className":119142},[468,54290],[224,119144,119146,119181],{"className":119145},[754,755],[224,119147,119149,119178],{"className":119148},[759],[224,119150,119153,119165],{"className":119151,"style":119152},[763],"height:0.8072em;",[224,119154,119156,119159],{"className":119155,"style":2120},[2139],[224,119157],{"className":119158,"style":2108},[771],[224,119160,119162],{"className":119161,"style":54310},[468],[224,119163,23331],{"className":119164},[468],[224,119166,119168,119171],{"style":119167},"top:-2.7672em;",[224,119169],{"className":119170,"style":2108},[771],[224,119172,119174],{"className":119173,"style":54367},[54366],[2155,119175,119176],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,119177],{"d":54374},[224,119179,787],{"className":119180},[786],[224,119182,119184],{"className":119183},[759],[224,119185,119188],{"className":119186,"style":119187},[763],"height:0.2328em;",[224,119189],{}," 后成为度量的成员（Hellinger、JS、TV），或干脆换成真正的度量类（第 6 节的 IPM）。",[608,119192,119194],{"id":119193},"_5-变分表示与-f-gan","5. 变分表示与 f-GAN",[613,119196,119198],{"id":119197},"_51-可训练目标","5.1 可训练目标",[11,119200,119201],{},"4.4 节给出精确对偶：",[224,119203,119205],{"className":119204,"translate":388},[387],[224,119206,119208,119302],{"className":119207,"translate":388},[392],[224,119209,119211],{"className":119210},[396],[398,119212,119213],{"xmlns":400,"display":401},[403,119214,119215,119299],{},[406,119216,119217,119223,119225,119227,119229,119231,119233,119235,119245,119247,119253,119255,119257,119259,119261,119263,119265,119267,119273,119275,119281,119283,119285,119287,119289,119291,119293,119295,119297],{},[695,119218,119219,119221],{},[409,119220,20303],{},[409,119222,1705],{},[412,119224,239],{"stretchy":435},[409,119226,16018],{},[409,119228,20312],{"mathvariant":678},[409,119230,20315],{},[412,119232,1716],{"stretchy":435},[412,119234,278],{},[1965,119236,119237,119243],{},[406,119238,119239,119241],{},[409,119240,55354],{},[412,119242,24134],{},[409,119244,24189],{},[412,119246,8110],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[695,119248,119249,119251],{},[409,119250,20324],{"mathvariant":12251},[409,119252,16018],{},[412,119254,268],{"stretchy":435},[409,119256,24189],{},[412,119258,239],{"stretchy":435},[409,119260,20668],{},[412,119262,1716],{"stretchy":435},[412,119264,440],{"stretchy":435},[412,119266,5268],{},[695,119268,119269,119271],{},[409,119270,20324],{"mathvariant":12251},[409,119272,20315],{},[412,119274,268],{"stretchy":435},[3248,119276,119277,119279],{},[409,119278,1705],{},[412,119280,27543],{},[412,119282,239],{"stretchy":435},[409,119284,24189],{},[412,119286,239],{"stretchy":435},[409,119288,20668],{},[412,119290,1716],{"stretchy":435},[412,119292,1716],{"stretchy":435},[412,119294,440],{"stretchy":435},[412,119296,15077],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[409,119298,688],{"mathvariant":678},[447,119300,119301],{"encoding":449},"D_f(P\\|Q)\n=\\sup_{g}\\Big\\{\\mathbb E_P[g(X)]\n-\\mathbb E_Q[f^*(g(X))]\\Big\\}.",[224,119303,119305,119375,119502],{"className":119304,"ariaHidden":455},[454],[224,119306,119308,119311,119351,119354,119357,119360,119363,119366,119369,119372],{"className":119307},[459],[224,119309],{"className":119310,"style":20368},[463],[224,119312,119314,119317],{"className":119313},[468],[224,119315,20303],{"className":119316,"style":2473},[468,469],[224,119318,119320],{"className":119319},[750],[224,119321,119323,119343],{"className":119322},[754,755],[224,119324,119326,119340],{"className":119325},[759],[224,119327,119329],{"className":119328,"style":20387},[763],[224,119330,119331,119334],{"style":20390},[224,119332],{"className":119333,"style":772},[771],[224,119335,119337],{"className":119336},[776,777,778,779],[224,119338,1705],{"className":119339,"style":1740},[468,469,779],[224,119341,787],{"className":119342},[786],[224,119344,119346],{"className":119345},[759],[224,119347,119349],{"className":119348,"style":20409},[763],[224,119350],{},[224,119352,239],{"className":119353},[542],[224,119355,16018],{"className":119356,"style":653},[468,469],[224,119358,20312],{"className":119359},[468],[224,119361,20315],{"className":119362},[468,469],[224,119364,1716],{"className":119365},[549],[224,119367],{"className":119368,"style":496},[473],[224,119370,278],{"className":119371},[500],[224,119373],{"className":119374,"style":496},[473],[224,119376,119378,119382,119429,119432,119438,119478,119481,119484,119487,119490,119493,119496,119499],{"className":119377},[459],[224,119379],{"className":119380,"style":119381},[463],"height:2.1805em;vertical-align:-1.0305em;",[224,119383,119385],{"className":119384},[21531,33665],[224,119386,119388,119421],{"className":119387},[754,755],[224,119389,119391,119418],{"className":119390},[759],[224,119392,119394,119408],{"className":119393,"style":839},[763],[224,119395,119396,119399],{"style":81096},[224,119397],{"className":119398,"style":2108},[771],[224,119400,119402],{"className":119401},[776,777,778,779],[224,119403,119405],{"className":119404},[468,779],[224,119406,24189],{"className":119407,"style":492},[468,469,779],[224,119409,119410,119413],{"style":2120},[224,119411],{"className":119412,"style":2108},[771],[224,119414,119415],{},[224,119416,55354],{"className":119417},[21531],[224,119419,787],{"className":119420},[786],[224,119422,119424],{"className":119423},[759],[224,119425,119427],{"className":119426,"style":81128},[763],[224,119428],{},[224,119430],{"className":119431,"style":724},[473],[224,119433,119435],{"className":119434},[468],[224,119436,8110],{"className":119437},[2363,23751],[224,119439,119441,119444],{"className":119440},[468],[224,119442,20324],{"className":119443},[468,12389],[224,119445,119447],{"className":119446},[750],[224,119448,119450,119470],{"className":119449},[754,755],[224,119451,119453,119467],{"className":119452},[759],[224,119454,119456],{"className":119455,"style":764},[763],[224,119457,119458,119461],{"style":767},[224,119459],{"className":119460,"style":772},[771],[224,119462,119464],{"className":119463},[776,777,778,779],[224,119465,16018],{"className":119466,"style":653},[468,469,779],[224,119468,787],{"className":119469},[786],[224,119471,119473],{"className":119472},[759],[224,119474,119476],{"className":119475,"style":794},[763],[224,119477],{},[224,119479,268],{"className":119480},[542],[224,119482,24189],{"className":119483,"style":492},[468,469],[224,119485,239],{"className":119486},[542],[224,119488,20668],{"className":119489,"style":28852},[468,469],[224,119491,40329],{"className":119492},[549],[224,119494],{"className":119495,"style":474},[473],[224,119497,5268],{"className":119498},[478],[224,119500],{"className":119501,"style":474},[473],[224,119503,119505,119508,119548,119551,119580,119583,119586,119589,119592,119596,119602],{"className":119504},[459],[224,119506],{"className":119507,"style":88023},[463],[224,119509,119511,119514],{"className":119510},[468],[224,119512,20324],{"className":119513},[468,12389],[224,119515,119517],{"className":119516},[750],[224,119518,119520,119540],{"className":119519},[754,755],[224,119521,119523,119537],{"className":119522},[759],[224,119524,119526],{"className":119525,"style":764},[763],[224,119527,119528,119531],{"style":767},[224,119529],{"className":119530,"style":772},[771],[224,119532,119534],{"className":119533},[776,777,778,779],[224,119535,20315],{"className":119536},[468,469,779],[224,119538,787],{"className":119539},[786],[224,119541,119543],{"className":119542},[759],[224,119544,119546],{"className":119545,"style":20409},[763],[224,119547],{},[224,119549,268],{"className":119550},[542],[224,119552,119554,119557],{"className":119553},[468],[224,119555,1705],{"className":119556,"style":1740},[468,469],[224,119558,119560],{"className":119559},[750],[224,119561,119563],{"className":119562},[754],[224,119564,119566],{"className":119565},[759],[224,119567,119569],{"className":119568,"style":28217},[763],[224,119570,119571,119574],{"style":5074},[224,119572],{"className":119573,"style":772},[771],[224,119575,119577],{"className":119576},[776,777,778,779],[224,119578,27543],{"className":119579},[478,779],[224,119581,239],{"className":119582},[542],[224,119584,24189],{"className":119585,"style":492},[468,469],[224,119587,239],{"className":119588},[542],[224,119590,20668],{"className":119591,"style":28852},[468,469],[224,119593,119595],{"className":119594},[549],"))]",[224,119597,119599],{"className":119598},[468],[224,119600,15077],{"className":119601},[2363,23751],[224,119603,688],{"className":119604},[468],[11,119606,88038,119607,119635,119636,119664,119665,119734,119735,119738,119739,119810],{},[224,119608,119610,119623],{"className":119609,"translate":388},[392],[224,119611,119613],{"className":119612},[396],[398,119614,119615],{"xmlns":400},[403,119616,119617,119621],{},[406,119618,119619],{},[409,119620,16018],{},[447,119622,16018],{"encoding":449},[224,119624,119626],{"className":119625,"ariaHidden":455},[454],[224,119627,119629,119632],{"className":119628},[459],[224,119630],{"className":119631,"style":649},[463],[224,119633,16018],{"className":119634,"style":653},[468,469]," 是数据分布、",[224,119637,119639,119652],{"className":119638,"translate":388},[392],[224,119640,119642],{"className":119641},[396],[398,119643,119644],{"xmlns":400},[403,119645,119646,119650],{},[406,119647,119648],{},[409,119649,20315],{},[447,119651,20315],{"encoding":449},[224,119653,119655],{"className":119654,"ariaHidden":455},[454],[224,119656,119658,119661],{"className":119657},[459],[224,119659],{"className":119660,"style":20637},[463],[224,119662,20315],{"className":119663},[468,469]," 是生成分布时，右端可直接用两批样本做蒙特卡洛估计，得到 ",[224,119666,119668,119685],{"className":119667,"translate":388},[392],[224,119669,119671],{"className":119670},[396],[398,119672,119673],{"xmlns":400},[403,119674,119675,119683],{},[406,119676,119677],{},[695,119678,119679,119681],{},[409,119680,20303],{},[409,119682,1705],{},[447,119684,24053],{"encoding":449},[224,119686,119688],{"className":119687,"ariaHidden":455},[454],[224,119689,119691,119694],{"className":119690},[459],[224,119692],{"className":119693,"style":24063},[463],[224,119695,119697,119700],{"className":119696},[468],[224,119698,20303],{"className":119699,"style":2473},[468,469],[224,119701,119703],{"className":119702},[750],[224,119704,119706,119726],{"className":119705},[754,755],[224,119707,119709,119723],{"className":119708},[759],[224,119710,119712],{"className":119711,"style":20387},[763],[224,119713,119714,119717],{"style":20390},[224,119715],{"className":119716,"style":772},[771],[224,119718,119720],{"className":119719},[776,777,778,779],[224,119721,1705],{"className":119722,"style":1740},[468,469,779],[224,119724,787],{"className":119725},[786],[224,119727,119729],{"className":119728},[759],[224,119730,119732],{"className":119731,"style":20409},[763],[224,119733],{}," 的一个",[378,119736,119737],{},"下界","，并用一个网络 ",[224,119740,119742,119761],{"className":119741,"translate":388},[392],[224,119743,119745],{"className":119744},[396],[398,119746,119747],{"xmlns":400},[403,119748,119749,119758],{},[406,119750,119751],{},[695,119752,119753,119755],{},[409,119754,24189],{},[409,119756,119757],{},"ω",[447,119759,119760],{"encoding":449},"g_\\omega",[224,119762,119764],{"className":119763,"ariaHidden":455},[454],[224,119765,119767,119770],{"className":119766},[459],[224,119768],{"className":119769,"style":806},[463],[224,119771,119773,119776],{"className":119772},[468],[224,119774,24189],{"className":119775,"style":492},[468,469],[224,119777,119779],{"className":119778},[750],[224,119780,119782,119802],{"className":119781},[754,755],[224,119783,119785,119799],{"className":119784},[759],[224,119786,119788],{"className":119787,"style":64841},[763],[224,119789,119790,119793],{"style":3854},[224,119791],{"className":119792,"style":772},[771],[224,119794,119796],{"className":119795},[776,777,778,779],[224,119797,119757],{"className":119798,"style":492},[468,469,779],[224,119800,787],{"className":119801},[786],[224,119803,119805],{"className":119804},[759],[224,119806,119808],{"className":119807,"style":794},[763],[224,119809],{}," 逼近最优解。于是",[224,119812,119814],{"className":119813,"translate":388},[387],[224,119815,119817,119946],{"className":119816,"translate":388},[392],[224,119818,119820],{"className":119819},[396],[398,119821,119822],{"xmlns":400,"display":401},[403,119823,119824,119943],{},[406,119825,119826,119837,119839,119841,119843,119845,119847,119853,119855,119857,119863,119877,119883,119885,119891,119893,119895,119901,119915,119921,119923,119929,119931,119937,119939,119941],{},[695,119827,119828,119835],{},[31766,119829,119830,119832],{"accent":455},[409,119831,20303],{},[412,119833,119834],{"stretchy":455},"^",[409,119836,1705],{},[412,119838,239],{"stretchy":435},[409,119840,16018],{},[409,119842,20312],{"mathvariant":678},[409,119844,20315],{},[412,119846,53798],{"separator":455},[695,119848,119849,119851],{},[409,119850,24189],{},[409,119852,119757],{},[412,119854,1716],{"stretchy":435},[412,119856,278],{},[20338,119858,119859,119861],{},[419,119860,1438],{},[409,119862,3254],{},[68229,119864,119865,119867,119875],{},[412,119866,33410],{},[406,119868,119869,119871,119873],{},[409,119870,1880],{},[412,119872,278],{},[419,119874,1438],{},[409,119876,3254],{},[695,119878,119879,119881],{},[409,119880,24189],{},[409,119882,119757],{},[412,119884,239],{"stretchy":435},[695,119886,119887,119889],{},[409,119888,371],{},[409,119890,1880],{},[412,119892,1716],{"stretchy":435},[412,119894,5268],{},[20338,119896,119897,119899],{},[419,119898,1438],{},[409,119900,13177],{},[68229,119902,119903,119905,119913],{},[412,119904,33410],{},[406,119906,119907,119909,119911],{},[409,119908,8101],{},[412,119910,278],{},[419,119912,1438],{},[409,119914,13177],{},[3248,119916,119917,119919],{},[409,119918,1705],{},[412,119920,27543],{},[412,119922,239],{"stretchy":435},[695,119924,119925,119927],{},[409,119926,24189],{},[409,119928,119757],{},[412,119930,239],{"stretchy":435},[695,119932,119933,119935],{},[409,119934,375],{},[409,119936,8101],{},[412,119938,1716],{"stretchy":435},[412,119940,1716],{"stretchy":435},[409,119942,688],{"mathvariant":678},[447,119944,119945],{"encoding":449},"\\widehat D_f(P\\|Q;g_\\omega)\n=\\frac1n\\sum_{i=1}^n g_\\omega(x_i)\n-\\frac1m\\sum_{j=1}^m f^*(g_\\omega(y_j)).",[224,119947,119949,120102,120338],{"className":119948,"ariaHidden":455},[454],[224,119950,119952,119956,120032,120035,120038,120041,120044,120047,120050,120090,120093,120096,120099],{"className":119951},[459],[224,119953],{"className":119954,"style":119955},[463],"height:1.2094em;vertical-align:-0.2861em;",[224,119957,119959,119998],{"className":119958},[468],[224,119960,119962],{"className":119961},[468,31806],[224,119963,119965],{"className":119964},[754],[224,119966,119968],{"className":119967},[759],[224,119969,119972,119980],{"className":119970,"style":119971},[763],"height:0.9233em;",[224,119973,119974,119977],{"style":2120},[224,119975],{"className":119976,"style":2108},[771],[224,119978,20303],{"className":119979,"style":2473},[468,469],[224,119981,119984,119987],{"className":119982,"style":119983},[2139],"width:calc(100% - 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极大化即得估计；把 ",[224,120632,120634,120657],{"className":120633,"translate":388},[392],[224,120635,120637],{"className":120636},[396],[398,120638,120639],{"xmlns":400},[403,120640,120641,120654],{},[406,120642,120643,120645,120647],{},[409,120644,20315],{},[412,120646,278],{},[695,120648,120649,120651],{},[409,120650,20315],{},[409,120652,120653],{},"θ",[447,120655,120656],{"encoding":449},"Q=Q_\\theta",[224,120658,120660,120678],{"className":120659,"ariaHidden":455},[454],[224,120661,120663,120666,120669,120672,120675],{"className":120662},[459],[224,120664],{"className":120665,"style":20637},[463],[224,120667,20315],{"className":120668},[468,469],[224,120670],{"className":120671,"style":496},[473],[224,120673,278],{"className":120674},[500],[224,120676],{"className":120677,"style":496},[473],[224,120679,120681,120684],{"className":120680},[459],[224,120682],{"className":120683,"style":20637},[463],[224,120685,120687,120690],{"className":120686},[468],[224,120688,20315],{"className":120689},[468,469],[224,120691,120693],{"className":120692},[750],[224,120694,120696,120716],{"className":120695},[754,755],[224,120697,120699,120713],{"className":120698},[759],[224,120700,120702],{"className":120701,"style":20387},[763],[224,120703,120704,120707],{"style":767},[224,120705],{"className":120706,"style":772},[771],[224,120708,120710],{"className":120709},[776,777,778,779],[224,120711,120653],{"className":120712,"style":2473},[468,469,779],[224,120714,787],{"className":120715},[786],[224,120717,120719],{"className":120718},[759],[224,120720,120722],{"className":120721,"style":794},[763],[224,120723],{},"（生成器）同时放进去，就得到 f-GAN 的 minimax 目标：",[224,120726,120728],{"className":120727,"translate":388},[387],[224,120729,120731,120833],{"className":120730,"translate":388},[392],[224,120732,120734],{"className":120733},[396],[398,120735,120736],{"xmlns":400,"display":401},[403,120737,120738,120830],{},[406,120739,120740,120750,120752,120762,120764,120766,120772,120774,120780,120782,120784,120786,120788,120790,120800,120802,120808,120810,120816,120818,120820,120822,120824,120826,120828],{},[1965,120741,120742,120748],{},[406,120743,120744,120746],{},[409,120745,71515],{},[412,120747,24134],{},[409,120749,120653],{},[423,120751,15054],{},[1965,120753,120754,120760],{},[406,120755,120756,120758],{},[409,120757,72294],{},[412,120759,24134],{},[409,120761,119757],{},[423,120763,15054],{},[412,120765,8110],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[695,120767,120768,120770],{},[409,120769,20324],{"mathvariant":12251},[409,120771,16018],{},[412,120773,268],{"stretchy":435},[695,120775,120776,120778],{},[409,120777,24189],{},[409,120779,119757],{},[412,120781,239],{"stretchy":435},[409,120783,20668],{},[412,120785,1716],{"stretchy":435},[412,120787,440],{"stretchy":435},[412,120789,5268],{},[695,120791,120792,120794],{},[409,120793,20324],{"mathvariant":12251},[695,120795,120796,120798],{},[409,120797,20315],{},[409,120799,120653],{},[412,120801,268],{"stretchy":435},[3248,120803,120804,120806],{},[409,120805,1705],{},[412,120807,27543],{},[412,120809,239],{"stretchy":435},[695,120811,120812,120814],{},[409,120813,24189],{},[409,120815,119757],{},[412,120817,239],{"stretchy":435},[409,120819,20668],{},[412,120821,1716],{"stretchy":435},[412,120823,1716],{"stretchy":435},[412,120825,440],{"stretchy":435},[412,120827,15077],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[409,120829,688],{"mathvariant":678},[447,120831,120832],{"encoding":449},"\\min_\\theta\\ \\max_\\omega\\\n\\Big\\{\\mathbb E_{P}[g_\\omega(X)]\n-\\mathbb E_{Q_\\theta}[f^*(g_\\omega(X))]\\Big\\}.",[224,120834,120836,121056],{"className":120835,"ariaHidden":455},[454],[224,120837,120839,120843,120888,120891,120894,120940,120943,120946,120952,120995,120998,121038,121041,121044,121047,121050,121053],{"className":120838},[459],[224,120840],{"className":120841,"style":120842},[463],"height:1.9021em;vertical-align:-0.7521em;",[224,120844,120846],{"className":120845},[21531,33665],[224,120847,120849,120879],{"className":120848},[754,755],[224,120850,120852,120876],{"className":120851},[759],[224,120853,120855,120866],{"className":120854,"style":71800},[763],[224,120856,120857,120860],{"style":71803},[224,120858],{"className":120859,"style":2108},[771],[224,120861,120863],{"className":120862},[776,777,778,779],[224,120864,120653],{"className":120865,"style":2473},[468,469,779],[224,120867,120868,120871],{"style":2120},[224,120869],{"className":120870,"style":2108},[771],[224,120872,120873],{},[224,120874,71515],{"className":120875},[21531],[224,120877,787],{"className":120878},[786],[224,120880,120882],{"className":120881},[759],[224,120883,120886],{"className":120884,"style":120885},[763],"height:0.7521em;",[224,120887],{},[224,120889,15054],{"className":120890},[473],[224,120892],{"className":120893,"style":724},[473],[224,120895,120897],{"className":120896},[21531,33665],[224,120898,120900,120931],{"className":120899},[754,755],[224,120901,120903,120928],{"className":120902},[759],[224,120904,120906,120918],{"className":120905,"style":839},[763],[224,120907,120909,120912],{"style":120908},"top:-2.4em;margin-left:0em;",[224,120910],{"className":120911,"style":2108},[771],[224,120913,120915],{"className":120914},[776,777,778,779],[224,120916,119757],{"className":120917,"style":492},[468,469,779],[224,120919,120920,120923],{"style":2120},[224,120921],{"className":120922,"style":2108},[771],[224,120924,120925],{},[224,120926,72294],{"className":120927},[21531],[224,120929,787],{"className":120930},[786],[224,120932,120934],{"className":120933},[759],[224,120935,120938],{"className":120936,"style":120937},[763],"height:0.7em;",[224,120939],{},[224,120941,15054],{"className":120942},[473],[224,120944],{"className":120945,"style":724},[473],[224,120947,120949],{"className":120948},[468],[224,120950,8110],{"className":120951},[2363,23751],[224,120953,120955,120958],{"className":120954},[468],[224,120956,20324],{"className":120957},[468,12389],[224,120959,120961],{"className":120960},[750],[224,120962,120964,120987],{"className":120963},[754,755],[224,120965,120967,120984],{"className":120966},[759],[224,120968,120970],{"className":120969,"style":764},[763],[224,120971,120972,120975],{"style":767},[224,120973],{"className":120974,"style":772},[771],[224,120976,120978],{"className":120977},[776,777,778,779],[224,120979,120981],{"className":120980},[468,779],[224,120982,16018],{"className":120983,"style":653},[468,469,779],[224,120985,787],{"className":120986},[786],[224,120988,120990],{"className":120989},[759],[224,120991,120993],{"className":120992,"style":794},[763],[224,120994],{},[224,120996,268],{"className":120997},[542],[224,120999,121001,121004],{"className":121000},[468],[224,121002,24189],{"className":121003,"style":492},[468,469],[224,121005,121007],{"className":121006},[750],[224,121008,121010,121030],{"className":121009},[754,755],[224,121011,121013,121027],{"className":121012},[759],[224,121014,121016],{"className":121015,"style":64841},[763],[224,121017,121018,121021],{"style":3854},[224,121019],{"className":121020,"style":772},[771],[224,121022,121024],{"className":121023},[776,777,778,779],[224,121025,119757],{"className":121026,"style":492},[468,469,779],[224,121028,787],{"className":121029},[786],[224,121031,121033],{"className":121032},[759],[224,121034,121036],{"className":121035,"style":794},[763],[224,121037],{},[224,121039,239],{"className":121040},[542],[224,121042,20668],{"className":121043,"style":28852},[468,469],[224,121045,40329],{"className":121046},[549],[224,121048],{"className":121049,"style":474},[473],[224,121051,5268],{"className":121052},[478],[224,121054],{"className":121055,"style":474},[473],[224,121057,121059,121062,121144,121147,121176,121179,121219,121222,121225,121228,121234],{"className":121058},[459],[224,121060],{"className":121061,"style":88023},[463],[224,121063,121065,121068],{"className":121064},[468],[224,121066,20324],{"className":121067},[468,12389],[224,121069,121071],{"className":121070},[750],[224,121072,121074,121136],{"className":121073},[754,755],[224,121075,121077,121133],{"className":121076},[759],[224,121078,121080],{"className":121079,"style":764},[763],[224,121081,121082,121085],{"style":767},[224,121083],{"className":121084,"style":772},[771],[224,121086,121088],{"className":121087},[776,777,778,779],[224,121089,121091],{"className":121090},[468,779],[224,121092,121094,121097],{"className":121093},[468,779],[224,121095,20315],{"className":121096},[468,469,779],[224,121098,121100],{"className":121099},[750],[224,121101,121103,121124],{"className":121102},[754,755],[224,121104,121106,121121],{"className":121105},[759],[224,121107,121109],{"className":121108,"style":26101},[763],[224,121110,121112,121115],{"style":121111},"top:-2.3488em;margin-left:0em;margin-right:0.0714em;",[224,121113],{"className":121114,"style":50136},[771],[224,121116,121118],{"className":121117},[776,50140,2364,779],[224,121119,120653],{"className":121120,"style":2473},[468,469,779],[224,121122,787],{"className":121123},[786],[224,121125,121127],{"className":121126},[759],[224,121128,121131],{"className":121129,"style":121130},[763],"height:0.1512em;",[224,121132],{},[224,121134,787],{"className":121135},[786],[224,121137,121139],{"className":121138},[759],[224,121140,121142],{"className":121141,"style":20409},[763],[224,121143],{},[224,121145,268],{"className":121146},[542],[224,121148,121150,121153],{"className":121149},[468],[224,121151,1705],{"className":121152,"style":1740},[468,469],[224,121154,121156],{"className":121155},[750],[224,121157,121159],{"className":121158},[754],[224,121160,121162],{"className":121161},[759],[224,121163,121165],{"className":121164,"style":28217},[763],[224,121166,121167,121170],{"style":5074},[224,121168],{"className":121169,"style":772},[771],[224,121171,121173],{"className":121172},[776,777,778,779],[224,121174,27543],{"className":121175},[478,779],[224,121177,239],{"className":121178},[542],[224,121180,121182,121185],{"className":121181},[468],[224,121183,24189],{"className":121184,"style":492},[468,469],[224,121186,121188],{"className":121187},[750],[224,121189,121191,121211],{"className":121190},[754,755],[224,121192,121194,121208],{"className":121193},[759],[224,121195,121197],{"className":121196,"style":64841},[763],[224,121198,121199,121202],{"style":3854},[224,121200],{"className":121201,"style":772},[771],[224,121203,121205],{"className":121204},[776,777,778,779],[224,121206,119757],{"className":121207,"style":492},[468,469,779],[224,121209,787],{"className":121210},[786],[224,121212,121214],{"className":121213},[759],[224,121215,121217],{"className":121216,"style":794},[763],[224,121218],{},[224,121220,239],{"className":121221},[542],[224,121223,20668],{"className":121224,"style":28852},[468,469],[224,121226,119595],{"className":121227},[549],[224,121229,121231],{"className":121230},[468],[224,121232,15077],{"className":121233},[2363,23751],[224,121235,688],{"className":121236},[468],[613,121238,121240],{"id":121239},"_52-最优判别器","5.2 最优判别器",[21055,121242,121244,121318,121589],{"title":121243},"命题 5.1（最优判别器）",[11,121245,121246,121247,121288,121289,121317],{},"在 5.1 节的 minimax 目标中，固定 ",[224,121248,121250,121267],{"className":121249,"translate":388},[392],[224,121251,121253],{"className":121252},[396],[398,121254,121255],{"xmlns":400},[403,121256,121257,121265],{},[406,121258,121259,121261,121263],{},[409,121260,16018],{},[412,121262,1877],{"separator":455},[409,121264,20315],{},[447,121266,20627],{"encoding":449},[224,121268,121270],{"className":121269,"ariaHidden":455},[454],[224,121271,121273,121276,121279,121282,121285],{"className":121272},[459],[224,121274],{"className":121275,"style":20637},[463],[224,121277,16018],{"className":121278,"style":653},[468,469],[224,121280,1877],{"className":121281},[1917],[224,121283],{"className":121284,"style":724},[473],[224,121286,20315],{"className":121287},[468,469]," 时内层对 ",[224,121290,121292,121305],{"className":121291,"translate":388},[392],[224,121293,121295],{"className":121294},[396],[398,121296,121297],{"xmlns":400},[403,121298,121299,121303],{},[406,121300,121301],{},[409,121302,24189],{},[447,121304,24189],{"encoding":449},[224,121306,121308],{"className":121307,"ariaHidden":455},[454],[224,121309,121311,121314],{"className":121310},[459],[224,121312],{"className":121313,"style":806},[463],[224,121315,24189],{"className":121316,"style":492},[468,469]," 的最优解为",[224,121319,121321],{"className":121320,"translate":388},[387],[224,121322,121324,121388],{"className":121323,"translate":388},[392],[224,121325,121327],{"className":121326},[396],[398,121328,121329],{"xmlns":400,"display":401},[403,121330,121331,121385],{},[406,121332,121333,121339,121341,121343,121345,121347,121353,121355,121383],{},[3248,121334,121335,121337],{},[409,121336,24189],{},[412,121338,27543],{},[412,121340,239],{"stretchy":435},[409,121342,371],{},[412,121344,1716],{"stretchy":435},[412,121346,278],{},[3248,121348,121349,121351],{},[409,121350,1705],{},[412,121352,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[423,121354,21353],{},[406,121356,121357,121359,121381],{},[412,121358,239],{"fence":455},[20338,121360,121361,121371],{},[406,121362,121363,121365,121367,121369],{},[409,121364,11],{},[412,121366,239],{"stretchy":435},[409,121368,371],{},[412,121370,1716],{"stretchy":435},[406,121372,121373,121375,121377,121379],{},[409,121374,2023],{},[412,121376,239],{"stretchy":435},[409,121378,371],{},[412,121380,1716],{"stretchy":435},[412,121382,1716],{"fence":455},[412,121384,1877],{"separator":455},[447,121386,121387],{"encoding":449},"g^*(x)=f'\\!\\left(\\frac{p(x)}{q(x)}\\right),",[224,121389,121391,121444],{"className":121390,"ariaHidden":455},[454],[224,121392,121394,121397,121426,121429,121432,121435,121438,121441],{"className":121393},[459],[224,121395],{"className":121396,"style":535},[463],[224,121398,121400,121403],{"className":121399},[468],[224,121401,24189],{"className":121402,"style":492},[468,469],[224,121404,121406],{"className":121405},[750],[224,121407,121409],{"className":121408},[754],[224,121410,121412],{"className":121411},[759],[224,121413,121415],{"className":121414,"style":28217},[763],[224,121416,121417,121420],{"style":5074},[224,121418],{"className":121419,"style":772},[771],[224,121421,121423],{"className":121422},[776,777,778,779],[224,121424,27543],{"className":121425},[478,779],[224,121427,239],{"className":121428},[542],[224,121430,371],{"className":121431},[468,469],[224,121433,1716],{"className":121434},[549],[224,121436],{"className":121437,"style":496},[473],[224,121439,278],{"className":121440},[500],[224,121442],{"className":121443,"style":496},[473],[224,121445,121447,121450,121482,121485,121488,121583,121586],{"className":121446},[459],[224,121448],{"className":121449,"style":20442},[463],[224,121451,121453,121456],{"className":121452},[468],[224,121454,1705],{"className":121455,"style":1740},[468,469],[224,121457,121459],{"className":121458},[750],[224,121460,121462],{"className":121461},[754],[224,121463,121465],{"className":121464},[759],[224,121466,121468],{"className":121467,"style":48901},[763],[224,121469,121470,121473],{"style":5074},[224,121471],{"className":121472,"style":772},[771],[224,121474,121476],{"className":121475},[776,777,778,779],[224,121477,121479],{"className":121478},[468,779],[224,121480,6933],{"className":121481},[468,779],[224,121483],{"className":121484,"style":21559},[473],[224,121486],{"className":121487,"style":724},[473],[224,121489,121491,121497,121577],{"className":121490},[2091],[224,121492,121494],{"className":121493,"style":8281},[542,8280],[224,121495,239],{"className":121496},[2363,778],[224,121498,121500,121503,121574],{"className":121499},[468],[224,121501],{"className":121502},[542,8479],[224,121504,121506],{"className":121505},[20338],[224,121507,121509,121566],{"className":121508},[754,755],[224,121510,121512,121563],{"className":121511},[759],[224,121513,121515,121535,121543],{"className":121514,"style":21590},[763],[224,121516,121517,121520],{"style":20531},[224,121518],{"className":121519,"style":2108},[771],[224,121521,121523,121526,121529,121532],{"className":121522},[468],[224,121524,2023],{"className":121525,"style":492},[468,469],[224,121527,239],{"className":121528},[542],[224,121530,371],{"className":121531},[468,469],[224,121533,1716],{"className":121534},[549],[224,121536,121537,121540],{"style":20546},[224,121538],{"className":121539,"style":2108},[771],[224,121541],{"className":121542,"style":20554},[20553],[224,121544,121545,121548],{"style":20557},[224,121546],{"className":121547,"style":2108},[771],[224,121549,121551,121554,121557,121560],{"className":121550},[468],[224,121552,11],{"className":121553},[468,469],[224,121555,239],{"className":121556},[542],[224,121558,371],{"className":121559},[468,469],[224,121561,1716],{"className":121562},[549],[224,121564,787],{"className":121565},[786],[224,121567,121569],{"className":121568},[759],[224,121570,121572],{"className":121571,"style":21648},[763],[224,121573],{},[224,121575],{"className":121576},[549,8479],[224,121578,121580],{"className":121579,"style":8281},[549,8280],[224,121581,1716],{"className":121582},[2363,778],[224,121584],{"className":121585,"style":724},[473],[224,121587,1877],{"className":121588},[1917],[11,121590,121591,121592,121654],{},"即判别器学到的量是似然比经 ",[224,121593,121595,121613],{"className":121594,"translate":388},[392],[224,121596,121598],{"className":121597},[396],[398,121599,121600],{"xmlns":400},[403,121601,121602,121610],{},[406,121603,121604],{},[3248,121605,121606,121608],{},[409,121607,1705],{},[412,121609,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[447,121611,121612],{"encoding":449},"f'",[224,121614,121616],{"className":121615,"ariaHidden":455},[454],[224,121617,121619,121622],{"className":121618},[459],[224,121620],{"className":121621,"style":84067},[463],[224,121623,121625,121628],{"className":121624},[468],[224,121626,1705],{"className":121627,"style":1740},[468,469],[224,121629,121631],{"className":121630},[750],[224,121632,121634],{"className":121633},[754],[224,121635,121637],{"className":121636},[759],[224,121638,121640],{"className":121639,"style":6992},[763],[224,121641,121642,121645],{"style":3288},[224,121643],{"className":121644,"style":772},[771],[224,121646,121648],{"className":121647},[776,777,778,779],[224,121649,121651],{"className":121650},[468,779],[224,121652,6933],{"className":121653},[468,779]," 之后的取值。",[11,121656,121657,121659,121660,121704,121705,121868,121869,121912],{},[378,121658,75377],{}," 目标对 ",[224,121661,121663,121683],{"className":121662,"translate":388},[392],[224,121664,121666],{"className":121665},[396],[398,121667,121668],{"xmlns":400},[403,121669,121670,121680],{},[406,121671,121672,121674,121676,121678],{},[409,121673,24189],{},[412,121675,239],{"stretchy":435},[409,121677,371],{},[412,121679,1716],{"stretchy":435},[447,121681,121682],{"encoding":449},"g(x)",[224,121684,121686],{"className":121685,"ariaHidden":455},[454],[224,121687,121689,121692,121695,121698,121701],{"className":121688},[459],[224,121690],{"className":121691,"style":535},[463],[224,121693,24189],{"className":121694,"style":492},[468,469],[224,121696,239],{"className":121697},[542],[224,121699,371],{"className":121700},[468,469],[224,121702,1716],{"className":121703},[549]," 的逐点依赖是 ",[224,121706,121708,121764],{"className":121707,"translate":388},[392],[224,121709,121711],{"className":121710},[396],[398,121712,121713],{"xmlns":400},[403,121714,121715,121761],{},[406,121716,121717,121719,121721,121723,121725,121727,121729,121731,121733,121735,121737,121739,121741,121743,121749,121751,121753,121755,121757,121759],{},[409,121718,11],{},[412,121720,239],{"stretchy":435},[409,121722,371],{},[412,121724,1716],{"stretchy":435},[409,121726,24189],{},[412,121728,239],{"stretchy":435},[409,121730,371],{},[412,121732,1716],{"stretchy":435},[412,121734,5268],{},[409,121736,2023],{},[412,121738,239],{"stretchy":435},[409,121740,371],{},[412,121742,1716],{"stretchy":435},[3248,121744,121745,121747],{},[409,121746,1705],{},[412,121748,27543],{},[412,121750,239],{"stretchy":435},[409,121752,24189],{},[412,121754,239],{"stretchy":435},[409,121756,371],{},[412,121758,1716],{"stretchy":435},[412,121760,1716],{"stretchy":435},[447,121762,121763],{"encoding":449},"p(x)g(x)-q(x)f^*(g(x))",[224,121765,121767,121806],{"className":121766,"ariaHidden":455},[454],[224,121768,121770,121773,121776,121779,121782,121785,121788,121791,121794,121797,121800,121803],{"className":121769},[459],[224,121771],{"className":121772,"style":535},[463],[224,121774,11],{"className":121775},[468,469],[224,121777,239],{"className":121778},[542],[224,121780,371],{"className":121781},[468,469],[224,121783,1716],{"className":121784},[549],[224,121786,24189],{"className":121787,"style":492},[468,469],[224,121789,239],{"className":121790},[542],[224,121792,371],{"className":121793},[468,469],[224,121795,1716],{"className":121796},[549],[224,121798],{"className":121799,"style":474},[473],[224,121801,5268],{"className":121802},[478],[224,121804],{"className":121805,"style":474},[473],[224,121807,121809,121812,121815,121818,121821,121824,121853,121856,121859,121862,121865],{"className":121808},[459],[224,121810],{"className":121811,"style":535},[463],[224,121813,2023],{"className":121814,"style":492},[468,469],[224,121816,239],{"className":121817},[542],[224,121819,371],{"className":121820},[468,469],[224,121822,1716],{"className":121823},[549],[224,121825,121827,121830],{"className":121826},[468],[224,121828,1705],{"className":121829,"style":1740},[468,469],[224,121831,121833],{"className":121832},[750],[224,121834,121836],{"className":121835},[754],[224,121837,121839],{"className":121838},[759],[224,121840,121842],{"className":121841,"style":28465},[763],[224,121843,121844,121847],{"style":3288},[224,121845],{"className":121846,"style":772},[771],[224,121848,121850],{"className":121849},[776,777,778,779],[224,121851,27543],{"className":121852},[478,779],[224,121854,239],{"className":121855},[542],[224,121857,24189],{"className":121858,"style":492},[468,469],[224,121860,239],{"className":121861},[542],[224,121863,371],{"className":121864},[468,469],[224,121866,38989],{"className":121867},[549],"，对 ",[224,121870,121872,121891],{"className":121871,"translate":388},[392],[224,121873,121875],{"className":121874},[396],[398,121876,121877],{"xmlns":400},[403,121878,121879,121889],{},[406,121880,121881,121883,121885,121887],{},[409,121882,24189],{},[412,121884,239],{"stretchy":435},[409,121886,371],{},[412,121888,1716],{"stretchy":435},[447,121890,121682],{"encoding":449},[224,121892,121894],{"className":121893,"ariaHidden":455},[454],[224,121895,121897,121900,121903,121906,121909],{"className":121896},[459],[224,121898],{"className":121899,"style":535},[463],[224,121901,24189],{"className":121902,"style":492},[468,469],[224,121904,239],{"className":121905},[542],[224,121907,371],{"className":121908},[468,469],[224,121910,1716],{"className":121911},[549]," 求导：",[224,121914,121916],{"className":121915,"translate":388},[387],[224,121917,121919,122039],{"className":121918,"translate":388},[392],[224,121920,121922],{"className":121921},[396],[398,121923,121924],{"xmlns":400,"display":401},[403,121925,121926,122036],{},[406,121927,121928,121930,121932,121934,121936,121938,121940,121942,121944,121946,121948,121958,121960,121962,121964,121966,121968,121970,121972,121974,121976,121978,121980,121990,121992,121998,122000,122002,122004,122006,122008,122030,122032,122034],{},[409,121929,11],{},[412,121931,239],{"stretchy":435},[409,121933,371],{},[412,121935,1716],{"stretchy":435},[412,121937,5268],{},[409,121939,2023],{},[412,121941,239],{"stretchy":435},[409,121943,371],{},[412,121945,1716],{"stretchy":435},[423,121947,682],{},[3248,121949,121950,121952],{},[409,121951,1705],{},[406,121953,121954,121956],{},[412,121955,27543],{},[412,121957,6933],{"mathvariant":678},[412,121959,239],{"stretchy":435},[409,121961,24189],{},[412,121963,239],{"stretchy":435},[409,121965,371],{},[412,121967,1716],{"stretchy":435},[412,121969,1716],{"stretchy":435},[412,121971,278],{},[419,121973,284],{},[423,121975,15054],{},[412,121977,24315],{},[423,121979,15054],{},[3248,121981,121982,121984],{},[409,121983,1705],{},[406,121985,121986,121988],{},[412,121987,27543],{},[412,121989,6933],{"mathvariant":678},[412,121991,239],{"stretchy":435},[3248,121993,121994,121996],{},[409,121995,24189],{},[412,121997,27543],{},[412,121999,239],{"stretchy":435},[409,122001,371],{},[412,122003,1716],{"stretchy":435},[412,122005,1716],{"stretchy":435},[412,122007,278],{},[20338,122009,122010,122020],{},[406,122011,122012,122014,122016,122018],{},[409,122013,11],{},[412,122015,239],{"stretchy":435},[409,122017,371],{},[412,122019,1716],{"stretchy":435},[406,122021,122022,122024,122026,122028],{},[409,122023,2023],{},[412,122025,239],{"stretchy":435},[409,122027,371],{},[412,122029,1716],{"stretchy":435},[412,122031,278],{},[409,122033,1435],{},[409,122035,688],{"mathvariant":678},[447,122037,122038],{"encoding":449},"p(x)-q(x)\\,f^{*\\prime}(g(x))=0\n\\ \\Longrightarrow\\\nf^{*\\prime}(g^*(x))=\\frac{p(x)}{q(x)}=t .",[224,122040,122042,122069,122150,122174,122265,122361],{"className":122041,"ariaHidden":455},[454],[224,122043,122045,122048,122051,122054,122057,122060,122063,122066],{"className":122044},[459],[224,122046],{"className":122047,"style":535},[463],[224,122049,11],{"className":122050},[468,469],[224,122052,239],{"className":122053},[542],[224,122055,371],{"className":122056},[468,469],[224,122058,1716],{"className":122059},[549],[224,122061],{"className":122062,"style":474},[473],[224,122064,5268],{"className":122065},[478],[224,122067],{"className":122068,"style":474},[473],[224,122070,122072,122076,122079,122082,122085,122088,122091,122126,122129,122132,122135,122138,122141,122144,122147],{"className":122071},[459],[224,122073],{"className":122074,"style":122075},[463],"height:1.0519em;vertical-align:-0.25em;",[224,122077,2023],{"className":122078,"style":492},[468,469],[224,122080,239],{"className":122081},[542],[224,122083,371],{"className":122084},[468,469],[224,122086,1716],{"className":122087},[549],[224,122089],{"className":122090,"style":724},[473],[224,122092,122094,122097],{"className":122093},[468],[224,122095,1705],{"className":122096,"style":1740},[468,469],[224,122098,122100],{"className":122099},[750],[224,122101,122103],{"className":122102},[754],[224,122104,122106],{"className":122105},[759],[224,122107,122109],{"className":122108,"style":48901},[763],[224,122110,122111,122114],{"style":5074},[224,122112],{"className":122113,"style":772},[771],[224,122115,122117],{"className":122116},[776,777,778,779],[224,122118,122120,122123],{"className":122119},[468,779],[224,122121,27543],{"className":122122},[468,779],[224,122124,6933],{"className":122125},[468,779],[224,122127,239],{"className":122128},[542],[224,122130,24189],{"className":122131,"style":492},[468,469],[224,122133,239],{"className":122134},[542],[224,122136,371],{"className":122137},[468,469],[224,122139,38989],{"className":122140},[549],[224,122142],{"className":122143,"style":496},[473],[224,122145,278],{"className":122146},[500],[224,122148],{"className":122149,"style":496},[473],[224,122151,122153,122156,122159,122162,122165,122168,122171],{"className":122152},[459],[224,122154],{"className":122155,"style":112683},[463],[224,122157,284],{"className":122158},[468],[224,122160,15054],{"className":122161},[473],[224,122163],{"className":122164,"style":496},[473],[224,122166,24315],{"className":122167},[500],[224,122169,15054],{"className":122170},[473],[224,122172],{"className":122173,"style":496},[473],[224,122175,122177,122180,122215,122218,122247,122250,122253,122256,122259,122262],{"className":122176},[459],[224,122178],{"className":122179,"style":122075},[463],[224,122181,122183,122186],{"className":122182},[468],[224,122184,1705],{"className":122185,"style":1740},[468,469],[224,122187,122189],{"className":122188},[750],[224,122190,122192],{"className":122191},[754],[224,122193,122195],{"className":122194},[759],[224,122196,122198],{"className":122197,"style":48901},[763],[224,122199,122200,122203],{"style":5074},[224,122201],{"className":122202,"style":772},[771],[224,122204,122206],{"className":122205},[776,777,778,779],[224,122207,122209,122212],{"className":122208},[468,779],[224,122210,27543],{"className":122211},[468,779],[224,122213,6933],{"className":122214},[468,779],[224,122216,239],{"className":122217},[542],[224,122219,122221,122224],{"className":122220},[468],[224,122222,24189],{"className":122223,"style":492},[468,469],[224,122225,122227],{"className":122226},[750],[224,122228,122230],{"className":122229},[754],[224,122231,122233],{"className":122232},[759],[224,122234,122236],{"className":122235,"style":28217},[763],[224,122237,122238,122241],{"style":5074},[224,122239],{"className":122240,"style":772},[771],[224,122242,122244],{"className":122243},[776,777,778,779],[224,122245,27543],{"className":122246},[478,779],[224,122248,239],{"className":122249},[542],[224,122251,371],{"className":122252},[468,469],[224,122254,38989],{"className":122255},[549],[224,122257],{"className":122258,"style":496},[473],[224,122260,278],{"className":122261},[500],[224,122263],{"className":122264,"style":496},[473],[224,122266,122268,122272,122352,122355,122358],{"className":122267},[459],[224,122269],{"className":122270,"style":122271},[463],"height:2.363em;vertical-align:-0.936em;",[224,122273,122275,122278,122349],{"className":122274},[468],[224,122276],{"className":122277},[542,8479],[224,122279,122281],{"className":122280},[20338],[224,122282,122284,122341],{"className":122283},[754,755],[224,122285,122287,122338],{"className":122286},[759],[224,122288,122290,122310,122318],{"className":122289,"style":21590},[763],[224,122291,122292,122295],{"style":20531},[224,122293],{"className":122294,"style":2108},[771],[224,122296,122298,122301,122304,122307],{"className":122297},[468],[224,122299,2023],{"className":122300,"style":492},[468,469],[224,122302,239],{"className":122303},[542],[224,122305,371],{"className":122306},[468,469],[224,122308,1716],{"className":122309},[549],[224,122311,122312,122315],{"style":20546},[224,122313],{"className":122314,"style":2108},[771],[224,122316],{"className":122317,"style":20554},[20553],[224,122319,122320,122323],{"style":20557},[224,122321],{"className":122322,"style":2108},[771],[224,122324,122326,122329,122332,122335],{"className":122325},[468],[224,122327,11],{"className":122328},[468,469],[224,122330,239],{"className":122331},[542],[224,122333,371],{"className":122334},[468,469],[224,122336,1716],{"className":122337},[549],[224,122339,787],{"className":122340},[786],[224,122342,122344],{"className":122343},[759],[224,122345,122347],{"className":122346,"style":21648},[763],[224,122348],{},[224,122350],{"className":122351},[549,8479],[224,122353],{"className":122354,"style":496},[473],[224,122356,278],{"className":122357},[500],[224,122359],{"className":122360,"style":496},[473],[224,122362,122364,122367,122370],{"className":122363},[459],[224,122365],{"className":122366,"style":25353},[463],[224,122368,1435],{"className":122369},[468,469],[224,122371,688],{"className":122372},[468],[11,122374,54015,122375,122582,122583,122739,122740,122798,122799,122893,122894,122897,122898,122967,122968,123049,123050,123078],{},[224,122376,122378,122424],{"className":122377,"translate":388},[392],[224,122379,122381],{"className":122380},[396],[398,122382,122383],{"xmlns":400},[403,122384,122385,122421],{},[406,122386,122387,122389,122395,122401,122403,122405,122411],{},[412,122388,239],{"stretchy":435},[3248,122390,122391,122393],{},[409,122392,1705],{},[412,122394,27543],{},[3248,122396,122397,122399],{},[412,122398,1716],{"stretchy":435},[412,122400,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,122402,278],{},[412,122404,239],{"stretchy":435},[3248,122406,122407,122409],{},[409,122408,1705],{},[412,122410,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[3248,122412,122413,122415],{},[412,122414,1716],{"stretchy":435},[406,122416,122417,122419],{},[412,122418,5268],{},[419,122420,1438],{},[447,122422,122423],{"encoding":449},"(f^*)'=(f')^{-1}",[224,122425,122427,122506],{"className":122426,"ariaHidden":455},[454],[224,122428,122430,122433,122436,122465,122497,122500,122503],{"className":122429},[459],[224,122431],{"className":122432,"style":6966},[463],[224,122434,239],{"className":122435},[542],[224,122437,122439,122442],{"className":122438},[468],[224,122440,1705],{"className":122441,"style":1740},[468,469],[224,122443,122445],{"className":122444},[750],[224,122446,122448],{"className":122447},[754],[224,122449,122451],{"className":122450},[759],[224,122452,122454],{"className":122453,"style":28465},[763],[224,122455,122456,122459],{"style":3288},[224,122457],{"className":122458,"style":772},[771],[224,122460,122462],{"className":122461},[776,777,778,779],[224,122463,27543],{"className":122464},[478,779],[224,122466,122468,122471],{"className":122467},[549],[224,122469,1716],{"className":122470},[549],[224,122472,122474],{"className":122473},[750],[224,122475,122477],{"className":122476},[754],[224,122478,122480],{"className":122479},[759],[224,122481,122483],{"className":122482,"style":6992},[763],[224,122484,122485,122488],{"style":3288},[224,122486],{"className":122487,"style":772},[771],[224,122489,122491],{"className":122490},[776,777,778,779],[224,122492,122494],{"className":122493},[468,779],[224,122495,6933],{"className":122496},[468,779],[224,122498],{"className":122499,"style":496},[473],[224,122501,278],{"className":122502},[500],[224,122504],{"className":122505,"style":496},[473],[224,122507,122509,122512,122515,122547],{"className":122508},[459],[224,122510],{"className":122511,"style":39943},[463],[224,122513,239],{"className":122514},[542],[224,122516,122518,122521],{"className":122517},[468],[224,122519,1705],{"className":122520,"style":1740},[468,469],[224,122522,122524],{"className":122523},[750],[224,122525,122527],{"className":122526},[754],[224,122528,122530],{"className":122529},[759],[224,122531,122533],{"className":122532,"style":6992},[763],[224,122534,122535,122538],{"style":3288},[224,122536],{"className":122537,"style":772},[771],[224,122539,122541],{"className":122540},[776,777,778,779],[224,122542,122544],{"className":122543},[468,779],[224,122545,6933],{"className":122546},[468,779],[224,122548,122550,122553],{"className":122549},[549],[224,122551,1716],{"className":122552},[549],[224,122554,122556],{"className":122555},[750],[224,122557,122559],{"className":122558},[754],[224,122560,122562],{"className":122561},[759],[224,122563,122565],{"className":122564,"style":5671},[763],[224,122566,122567,122570],{"style":3288},[224,122568],{"className":122569,"style":772},[771],[224,122571,122573],{"className":122572},[776,777,778,779],[224,122574,122576,122579],{"className":122575},[468,779],[224,122577,5268],{"className":122578},[468,779],[224,122580,1438],{"className":122581},[468,779],"（Fenchel 的互逆关系），即得 ",[224,122584,122586,122630],{"className":122585,"translate":388},[392],[224,122587,122589],{"className":122588},[396],[398,122590,122591],{"xmlns":400},[403,122592,122593,122627],{},[406,122594,122595,122601,122603,122605,122607,122609,122615,122617,122619,122621,122623,122625],{},[3248,122596,122597,122599],{},[409,122598,24189],{},[412,122600,27543],{},[412,122602,239],{"stretchy":435},[409,122604,371],{},[412,122606,1716],{"stretchy":435},[412,122608,278],{},[3248,122610,122611,122613],{},[409,122612,1705],{},[412,122614,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,122616,239],{"stretchy":435},[409,122618,1435],{},[412,122620,239],{"stretchy":435},[409,122622,371],{},[412,122624,1716],{"stretchy":435},[412,122626,1716],{"stretchy":435},[447,122628,122629],{"encoding":449},"g^*(x)=f'(t(x))",[224,122631,122633,122686],{"className":122632,"ariaHidden":455},[454],[224,122634,122636,122639,122668,122671,122674,122677,122680,122683],{"className":122635},[459],[224,122637],{"className":122638,"style":535},[463],[224,122640,122642,122645],{"className":122641},[468],[224,122643,24189],{"className":122644,"style":492},[468,469],[224,122646,122648],{"className":122647},[750],[224,122649,122651],{"className":122650},[754],[224,122652,122654],{"className":122653},[759],[224,122655,122657],{"className":122656,"style":28465},[763],[224,122658,122659,122662],{"style":3288},[224,122660],{"className":122661,"style":772},[771],[224,122663,122665],{"className":122664},[776,777,778,779],[224,122666,27543],{"className":122667},[478,779],[224,122669,239],{"className":122670},[542],[224,122672,371],{"className":122673},[468,469],[224,122675,1716],{"className":122676},[549],[224,122678],{"className":122679,"style":496},[473],[224,122681,278],{"className":122682},[500],[224,122684],{"className":122685,"style":496},[473],[224,122687,122689,122692,122724,122727,122730,122733,122736],{"className":122688},[459],[224,122690],{"className":122691,"style":6966},[463],[224,122693,122695,122698],{"className":122694},[468],[224,122696,1705],{"className":122697,"style":1740},[468,469],[224,122699,122701],{"className":122700},[750],[224,122702,122704],{"className":122703},[754],[224,122705,122707],{"className":122706},[759],[224,122708,122710],{"className":122709,"style":6992},[763],[224,122711,122712,122715],{"style":3288},[224,122713],{"className":122714,"style":772},[771],[224,122716,122718],{"className":122717},[776,777,778,779],[224,122719,122721],{"className":122720},[468,779],[224,122722,6933],{"className":122723},[468,779],[224,122725,239],{"className":122726},[542],[224,122728,1435],{"className":122729},[468,469],[224,122731,239],{"className":122732},[542],[224,122734,371],{"className":122735},[468,469],[224,122737,38989],{"className":122738},[549],"。把 ",[224,122741,122743,122760],{"className":122742,"translate":388},[392],[224,122744,122746],{"className":122745},[396],[398,122747,122748],{"xmlns":400},[403,122749,122750,122758],{},[406,122751,122752],{},[3248,122753,122754,122756],{},[409,122755,24189],{},[412,122757,27543],{},[447,122759,83911],{"encoding":449},[224,122761,122763],{"className":122762,"ariaHidden":455},[454],[224,122764,122766,122769],{"className":122765},[459],[224,122767],{"className":122768,"style":83921},[463],[224,122770,122772,122775],{"className":122771},[468],[224,122773,24189],{"className":122774,"style":492},[468,469],[224,122776,122778],{"className":122777},[750],[224,122779,122781],{"className":122780},[754],[224,122782,122784],{"className":122783},[759],[224,122785,122787],{"className":122786,"style":28465},[763],[224,122788,122789,122792],{"style":3288},[224,122790],{"className":122791,"style":772},[771],[224,122793,122795],{"className":122794},[776,777,778,779],[224,122796,27543],{"className":122797},[478,779]," 代回即得 ",[224,122800,122802,122829],{"className":122801,"translate":388},[392],[224,122803,122805],{"className":122804},[396],[398,122806,122807],{"xmlns":400},[403,122808,122809,122827],{},[406,122810,122811,122817,122819,122821,122823,122825],{},[695,122812,122813,122815],{},[409,122814,20303],{},[409,122816,1705],{},[412,122818,239],{"stretchy":435},[409,122820,16018],{},[409,122822,20312],{"mathvariant":678},[409,122824,20315],{},[412,122826,1716],{"stretchy":435},[447,122828,25249],{"encoding":449},[224,122830,122832],{"className":122831,"ariaHidden":455},[454],[224,122833,122835,122838,122878,122881,122884,122887,122890],{"className":122834},[459],[224,122836],{"className":122837,"style":20368},[463],[224,122839,122841,122844],{"className":122840},[468],[224,122842,20303],{"className":122843,"style":2473},[468,469],[224,122845,122847],{"className":122846},[750],[224,122848,122850,122870],{"className":122849},[754,755],[224,122851,122853,122867],{"className":122852},[759],[224,122854,122856],{"className":122855,"style":20387},[763],[224,122857,122858,122861],{"style":20390},[224,122859],{"className":122860,"style":772},[771],[224,122862,122864],{"className":122863},[776,777,778,779],[224,122865,1705],{"className":122866,"style":1740},[468,469,779],[224,122868,787],{"className":122869},[786],[224,122871,122873],{"className":122872},[759],[224,122874,122876],{"className":122875,"style":20409},[763],[224,122877],{},[224,122879,239],{"className":122880},[542],[224,122882,16018],{"className":122883,"style":653},[468,469],[224,122885,20312],{"className":122886},[468],[224,122888,20315],{"className":122889},[468,469],[224,122891,1716],{"className":122892},[549],"（这正是定理 4.1 所给的最优解）。",[378,122895,122896],{},"实践含义："," 判别器网络 ",[224,122899,122901,122918],{"className":122900,"translate":388},[392],[224,122902,122904],{"className":122903},[396],[398,122905,122906],{"xmlns":400},[403,122907,122908,122916],{},[406,122909,122910],{},[695,122911,122912,122914],{},[409,122913,24189],{},[409,122915,119757],{},[447,122917,119760],{"encoding":449},[224,122919,122921],{"className":122920,"ariaHidden":455},[454],[224,122922,122924,122927],{"className":122923},[459],[224,122925],{"className":122926,"style":806},[463],[224,122928,122930,122933],{"className":122929},[468],[224,122931,24189],{"className":122932,"style":492},[468,469],[224,122934,122936],{"className":122935},[750],[224,122937,122939,122959],{"className":122938},[754,755],[224,122940,122942,122956],{"className":122941},[759],[224,122943,122945],{"className":122944,"style":64841},[763],[224,122946,122947,122950],{"style":3854},[224,122948],{"className":122949,"style":772},[771],[224,122951,122953],{"className":122952},[776,777,778,779],[224,122954,119757],{"className":122955,"style":492},[468,469,779],[224,122957,787],{"className":122958},[786],[224,122960,122962],{"className":122961},[759],[224,122963,122965],{"className":122964,"style":794},[763],[224,122966],{}," 学的是 ",[224,122969,122971,122996],{"className":122970,"translate":388},[392],[224,122972,122974],{"className":122973},[396],[398,122975,122976],{"xmlns":400},[403,122977,122978,122993],{},[406,122979,122980,122986,122988,122991],{},[3248,122981,122982,122984],{},[409,122983,1705],{},[412,122985,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,122987,239],{"stretchy":435},[423,122989,122990],{},"似然比",[412,122992,1716],{"stretchy":435},[447,122994,122995],{"encoding":449},"f'(\\text{似然比})",[224,122997,122999],{"className":122998,"ariaHidden":455},[454],[224,123000,123002,123005,123037,123040,123046],{"className":123001},[459],[224,123003],{"className":123004,"style":6966},[463],[224,123006,123008,123011],{"className":123007},[468],[224,123009,1705],{"className":123010,"style":1740},[468,469],[224,123012,123014],{"className":123013},[750],[224,123015,123017],{"className":123016},[754],[224,123018,123020],{"className":123019},[759],[224,123021,123023],{"className":123022,"style":6992},[763],[224,123024,123025,123028],{"style":3288},[224,123026],{"className":123027,"style":772},[771],[224,123029,123031],{"className":123030},[776,777,778,779],[224,123032,123034],{"className":123033},[468,779],[224,123035,6933],{"className":123036},[468,779],[224,123038,239],{"className":123039},[542],[224,123041,123043],{"className":123042},[468,11373],[224,123044,122990],{"className":123045},[468,16159],[224,123047,1716],{"className":123048},[549],"，而不是似然比本身；不同 ",[224,123051,123053,123066],{"className":123052,"translate":388},[392],[224,123054,123056],{"className":123055},[396],[398,123057,123058],{"xmlns":400},[403,123059,123060,123064],{},[406,123061,123062],{},[409,123063,1705],{},[447,123065,1705],{"encoding":449},[224,123067,123069],{"className":123068,"ariaHidden":455},[454],[224,123070,123072,123075],{"className":123071},[459],[224,123073],{"className":123074,"style":2565},[463],[224,123076,1705],{"className":123077,"style":1740},[468,469]," 等价于对似然比做了不同的单调「整形」，因此决定了梯度的尺度与饱和方式。",[613,123080,123082],{"id":123081},"_53-各散度的-gan-目标","5.3 各散度的 GAN 目标",[11,123084,123085,123086,12570],{},"取上表的 ",[224,123087,123089,123106],{"className":123088,"translate":388},[392],[224,123090,123092],{"className":123091},[396],[398,123093,123094],{"xmlns":400},[403,123095,123096,123104],{},[406,123097,123098],{},[3248,123099,123100,123102],{},[409,123101,1705],{},[412,123103,27543],{},[447,123105,80614],{"encoding":449},[224,123107,123109],{"className":123108,"ariaHidden":455},[454],[224,123110,123112,123115],{"className":123111},[459],[224,123113],{"className":123114,"style":2565},[463],[224,123116,123118,123121],{"className":123117},[468],[224,123119,1705],{"className":123120,"style":1740},[468,469],[224,123122,123124],{"className":123123},[750],[224,123125,123127],{"className":123126},[754],[224,123128,123130],{"className":123129},[759],[224,123131,123133],{"className":123132,"style":28465},[763],[224,123134,123135,123138],{"style":3288},[224,123136],{"className":123137,"style":772},[771],[224,123139,123141],{"className":123140},[776,777,778,779],[224,123142,27543],{"className":123143},[478,779],[2674,123145,123146],{},[2677,123147,123148,35740,123150,65519],{},[378,123149,52588],{},[224,123151,123153,123185],{"className":123152,"translate":388},[392],[224,123154,123156],{"className":123155},[396],[398,123157,123158],{"xmlns":400},[403,123159,123160,123182],{},[406,123161,123162,123168,123170],{},[3248,123163,123164,123166],{},[409,123165,1705],{},[412,123167,27543],{},[412,123169,278],{},[3248,123171,123172,123174],{},[409,123173,1861],{},[406,123175,123176,123178,123180],{},[409,123177,375],{},[412,123179,5268],{},[419,123181,1438],{},[447,123183,123184],{"encoding":449},"f^*=e^{y-1}",[224,123186,123188,123232],{"className":123187,"ariaHidden":455},[454],[224,123189,123191,123194,123223,123226,123229],{"className":123190},[459],[224,123192],{"className":123193,"style":2565},[463],[224,123195,123197,123200],{"className":123196},[468],[224,123198,1705],{"className":123199,"style":1740},[468,469],[224,123201,123203],{"className":123202},[750],[224,123204,123206],{"className":123205},[754],[224,123207,123209],{"className":123208},[759],[224,123210,123212],{"className":123211,"style":28465},[763],[224,123213,123214,123217],{"style":3288},[224,123215],{"className":123216,"style":772},[771],[224,123218,123220],{"className":123219},[776,777,778,779],[224,123221,27543],{"className":123222},[478,779],[224,123224],{"className":123225,"style":496},[473],[224,123227,278],{"className":123228},[500],[224,123230],{"className":123231,"style":496},[473],[224,123233,123235,123238],{"className":123234},[459],[224,123236],{"className":123237,"style":5671},[463],[224,123239,123241,123244],{"className":123240},[468],[224,123242,1861],{"className":123243},[468,469],[224,123245,123247],{"className":123246},[750],[224,123248,123250],{"className":123249},[754],[224,123251,123253],{"className":123252},[759],[224,123254,123256],{"className":123255,"style":5671},[763],[224,123257,123258,123261],{"style":3288},[224,123259],{"className":123260,"style":772},[771],[224,123262,123264],{"className":123263},[776,777,778,779],[224,123265,123267,123270,123273],{"className":123266},[468,779],[224,123268,375],{"className":123269,"style":492},[468,469,779],[224,123271,5268],{"className":123272},[478,779],[224,123274,1438],{"className":123275},[468,779],[224,123277,123279],{"className":123278,"translate":388},[387],[224,123280,123282,123368],{"className":123281,"translate":388},[392],[224,123283,123285],{"className":123284},[396],[398,123286,123287],{"xmlns":400,"display":401},[403,123288,123289,123365],{},[406,123290,123291,123301,123303,123305,123307,123309,123311,123313,123323,123325,123331,123333,123335,123337,123339,123345,123347,123359,123361,123363],{},[695,123292,123293,123295],{},[409,123294,20303],{},[406,123296,123297,123299],{},[409,123298,30652],{},[409,123300,41831],{},[412,123302,239],{"stretchy":435},[409,123304,16018],{},[409,123306,20312],{"mathvariant":678},[409,123308,20315],{},[412,123310,1716],{"stretchy":435},[412,123312,278],{},[1965,123314,123315,123321],{},[406,123316,123317,123319],{},[409,123318,55354],{},[412,123320,24134],{},[409,123322,24189],{},[412,123324,8110],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[695,123326,123327,123329],{},[409,123328,20324],{"mathvariant":12251},[409,123330,16018],{},[412,123332,268],{"stretchy":435},[409,123334,24189],{},[412,123336,440],{"stretchy":435},[412,123338,5268],{},[695,123340,123341,123343],{},[409,123342,20324],{"mathvariant":12251},[409,123344,20315],{},[412,123346,268],{"stretchy":435},[3248,123348,123349,123351],{},[409,123350,1861],{},[406,123352,123353,123355,123357],{},[409,123354,24189],{},[412,123356,5268],{},[419,123358,1438],{},[412,123360,440],{"stretchy":435},[412,123362,15077],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[409,123364,688],{"mathvariant":678},[447,123366,123367],{"encoding":449},"D_{KL}(P\\|Q)\n=\\sup_g\\Big\\{\\mathbb E_P[g]\n-\\mathbb E_Q[e^{g-1}]\\Big\\}.",[224,123369,123371,123447,123564],{"className":123370,"ariaHidden":455},[454],[224,123372,123374,123377,123423,123426,123429,123432,123435,123438,123441,123444],{"className":123373},[459],[224,123375],{"className":123376,"style":535},[463],[224,123378,123380,123383],{"className":123379},[468],[224,123381,20303],{"className":123382,"style":2473},[468,469],[224,123384,123386],{"className":123385},[750],[224,123387,123389,123415],{"className":123388},[754,755],[224,123390,123392,123412],{"className":123391},[759],[224,123393,123395],{"className":123394,"style":764},[763],[224,123396,123397,123400],{"style":20390},[224,123398],{"className":123399,"style":772},[771],[224,123401,123403],{"className":123402},[776,777,778,779],[224,123404,123406,123409],{"className":123405},[468,779],[224,123407,30652],{"className":123408,"style":30740},[468,469,779],[224,123410,41831],{"className":123411},[468,469,779],[224,123413,787],{"className":123414},[786],[224,123416,123418],{"className":123417},[759],[224,123419,123421],{"className":123420,"style":794},[763],[224,123422],{},[224,123424,239],{"className":123425},[542],[224,123427,16018],{"className":123428,"style":653},[468,469],[224,123430,20312],{"className":123431},[468],[224,123433,20315],{"className":123434},[468,469],[224,123436,1716],{"className":123437},[549],[224,123439],{"className":123440,"style":496},[473],[224,123442,278],{"className":123443},[500],[224,123445],{"className":123446,"style":496},[473],[224,123448,123450,123453,123497,123500,123506,123546,123549,123552,123555,123558,123561],{"className":123449},[459],[224,123451],{"className":123452,"style":119381},[463],[224,123454,123456],{"className":123455},[21531,33665],[224,123457,123459,123489],{"className":123458},[754,755],[224,123460,123462,123486],{"className":123461},[759],[224,123463,123465,123476],{"className":123464,"style":839},[763],[224,123466,123467,123470],{"style":81096},[224,123468],{"className":123469,"style":2108},[771],[224,123471,123473],{"className":123472},[776,777,778,779],[224,123474,24189],{"className":123475,"style":492},[468,469,779],[224,123477,123478,123481],{"style":2120},[224,123479],{"className":123480,"style":2108},[771],[224,123482,123483],{},[224,123484,55354],{"className":123485},[21531],[224,123487,787],{"className":123488},[786],[224,123490,123492],{"className":123491},[759],[224,123493,123495],{"className":123494,"style":81128},[763],[224,123496],{},[224,123498],{"className":123499,"style":724},[473],[224,123501,123503],{"className":123502},[468],[224,123504,8110],{"className":123505},[2363,23751],[224,123507,123509,123512],{"className":123508},[468],[224,123510,20324],{"className":123511},[468,12389],[224,123513,123515],{"className":123514},[750],[224,123516,123518,123538],{"className":123517},[754,755],[224,123519,123521,123535],{"className":123520},[759],[224,123522,123524],{"className":123523,"style":764},[763],[224,123525,123526,123529],{"style":767},[224,123527],{"className":123528,"style":772},[771],[224,123530,123532],{"className":123531},[776,777,778,779],[224,123533,16018],{"className":123534,"style":653},[468,469,779],[224,123536,787],{"className":123537},[786],[224,123539,123541],{"className":123540},[759],[224,123542,123544],{"className":123543,"style":794},[763],[224,123545],{},[224,123547,268],{"className":123548},[542],[224,123550,24189],{"className":123551,"style":492},[468,469],[224,123553,440],{"className":123554},[549],[224,123556],{"className":123557,"style":474},[473],[224,123559,5268],{"className":123560},[478],[224,123562],{"className":123563,"style":474},[473],[224,123565,123567,123570,123610,123613,123651,123654,123660],{"className":123566},[459],[224,123568],{"className":123569,"style":88023},[463],[224,123571,123573,123576],{"className":123572},[468],[224,123574,20324],{"className":123575},[468,12389],[224,123577,123579],{"className":123578},[750],[224,123580,123582,123602],{"className":123581},[754,755],[224,123583,123585,123599],{"className":123584},[759],[224,123586,123588],{"className":123587,"style":764},[763],[224,123589,123590,123593],{"style":767},[224,123591],{"className":123592,"style":772},[771],[224,123594,123596],{"className":123595},[776,777,778,779],[224,123597,20315],{"className":123598},[468,469,779],[224,123600,787],{"className":123601},[786],[224,123603,123605],{"className":123604},[759],[224,123606,123608],{"className":123607,"style":20409},[763],[224,123609],{},[224,123611,268],{"className":123612},[542],[224,123614,123616,123619],{"className":123615},[468],[224,123617,1861],{"className":123618},[468,469],[224,123620,123622],{"className":123621},[750],[224,123623,123625],{"className":123624},[754],[224,123626,123628],{"className":123627},[759],[224,123629,123631],{"className":123630,"style":5071},[763],[224,123632,123633,123636],{"style":5074},[224,123634],{"className":123635,"style":772},[771],[224,123637,123639],{"className":123638},[776,777,778,779],[224,123640,123642,123645,123648],{"className":123641},[468,779],[224,123643,24189],{"className":123644,"style":492},[468,469,779],[224,123646,5268],{"className":123647},[478,779],[224,123649,1438],{"className":123650},[468,779],[224,123652,440],{"className":123653},[549],[224,123655,123657],{"className":123656},[468],[224,123658,15077],{"className":123659},[2363,23751],[224,123661,688],{"className":123662},[468],[11,123664,123665,123666,123874,123875,123948,123949,123981,123982,124010],{},"（这是「MINE-f」式下界。经典 Donsker–Varadhan 下界是 ",[224,123667,123669,123715],{"className":123668,"translate":388},[392],[224,123670,123672],{"className":123671},[396],[398,123673,123674],{"xmlns":400},[403,123675,123676,123712],{},[406,123677,123678,123684,123686,123688,123690,123692,123694,123696,123702,123704,123710],{},[695,123679,123680,123682],{},[409,123681,20324],{"mathvariant":12251},[409,123683,16018],{},[412,123685,268],{"stretchy":435},[409,123687,24189],{},[412,123689,440],{"stretchy":435},[412,123691,5268],{},[409,123693,24131],{},[412,123695,24134],{},[695,123697,123698,123700],{},[409,123699,20324],{"mathvariant":12251},[409,123701,20315],{},[412,123703,268],{"stretchy":435},[3248,123705,123706,123708],{},[409,123707,1861],{},[409,123709,24189],{},[412,123711,440],{"stretchy":435},[447,123713,123714],{"encoding":449},"\\mathbb E_P[g]-\\log\\mathbb E_Q[e^{g}]",[224,123716,123718,123782],{"className":123717,"ariaHidden":455},[454],[224,123719,123721,123724,123764,123767,123770,123773,123776,123779],{"className":123720},[459],[224,123722],{"className":123723,"style":535},[463],[224,123725,123727,123730],{"className":123726},[468],[224,123728,20324],{"className":123729},[468,12389],[224,123731,123733],{"className":123732},[750],[224,123734,123736,123756],{"className":123735},[754,755],[224,123737,123739,123753],{"className":123738},[759],[224,123740,123742],{"className":123741,"style":764},[763],[224,123743,123744,123747],{"style":767},[224,123745],{"className":123746,"style":772},[771],[224,123748,123750],{"className":123749},[776,777,778,779],[224,123751,16018],{"className":123752,"style":653},[468,469,779],[224,123754,787],{"className":123755},[786],[224,123757,123759],{"className":123758},[759],[224,123760,123762],{"className":123761,"style":794},[763],[224,123763],{},[224,123765,268],{"className":123766},[542],[224,123768,24189],{"className":123769,"style":492},[468,469],[224,123771,440],{"className":123772},[549],[224,123774],{"className":123775,"style":474},[473],[224,123777,5268],{"className":123778},[478],[224,123780],{"className":123781,"style":474},[473],[224,123783,123785,123788,123793,123796,123836,123839,123871],{"className":123784},[459],[224,123786],{"className":123787,"style":20368},[463],[224,123789,24185,123791],{"className":123790},[21531],[224,123792,24189],{"style":24188},[224,123794],{"className":123795,"style":724},[473],[224,123797,123799,123802],{"className":123798},[468],[224,123800,20324],{"className":123801},[468,12389],[224,123803,123805],{"className":123804},[750],[224,123806,123808,123828],{"className":123807},[754,755],[224,123809,123811,123825],{"className":123810},[759],[224,123812,123814],{"className":123813,"style":764},[763],[224,123815,123816,123819],{"style":767},[224,123817],{"className":123818,"style":772},[771],[224,123820,123822],{"className":123821},[776,777,778,779],[224,123823,20315],{"className":123824},[468,469,779],[224,123826,787],{"className":123827},[786],[224,123829,123831],{"className":123830},[759],[224,123832,123834],{"className":123833,"style":20409},[763],[224,123835],{},[224,123837,268],{"className":123838},[542],[224,123840,123842,123845],{"className":123841},[468],[224,123843,1861],{"className":123844},[468,469],[224,123846,123848],{"className":123847},[750],[224,123849,123851],{"className":123850},[754],[224,123852,123854],{"className":123853},[759],[224,123855,123857],{"className":123856,"style":3267},[763],[224,123858,123859,123862],{"style":3288},[224,123860],{"className":123861,"style":772},[771],[224,123863,123865],{"className":123864},[776,777,778,779],[224,123866,123868],{"className":123867},[468,779],[224,123869,24189],{"className":123870,"style":492},[468,469,779],[224,123872,440],{"className":123873},[549],"，二者同源但第二项不同：",[224,123876,123878,123901],{"className":123877,"translate":388},[392],[224,123879,123881],{"className":123880},[396],[398,123882,123883],{"xmlns":400},[403,123884,123885,123899],{},[406,123886,123887],{},[3248,123888,123889,123891],{},[409,123890,1861],{},[406,123892,123893,123895,123897],{},[409,123894,375],{},[412,123896,5268],{},[419,123898,1438],{},[447,123900,84808],{"encoding":449},[224,123902,123904],{"className":123903,"ariaHidden":455},[454],[224,123905,123907,123910],{"className":123906},[459],[224,123908],{"className":123909,"style":5671},[463],[224,123911,123913,123916],{"className":123912},[468],[224,123914,1861],{"className":123915},[468,469],[224,123917,123919],{"className":123918},[750],[224,123920,123922],{"className":123921},[754],[224,123923,123925],{"className":123924},[759],[224,123926,123928],{"className":123927,"style":5671},[763],[224,123929,123930,123933],{"style":3288},[224,123931],{"className":123932,"style":772},[771],[224,123934,123936],{"className":123935},[776,777,778,779],[224,123937,123939,123942,123945],{"className":123938},[468,779],[224,123940,375],{"className":123941,"style":492},[468,469,779],[224,123943,5268],{"className":123944},[478,779],[224,123946,1438],{"className":123947},[468,779]," 是把 ",[224,123950,123952,123967],{"className":123951,"translate":388},[392],[224,123953,123955],{"className":123954},[396],[398,123956,123957],{"xmlns":400},[403,123958,123959,123965],{},[406,123960,123961,123963],{},[409,123962,24131],{},[412,123964,24134],{},[447,123966,41521],{"encoding":449},[224,123968,123970],{"className":123969,"ariaHidden":455},[454],[224,123971,123973,123976],{"className":123972},[459],[224,123974],{"className":123975,"style":2565},[463],[224,123977,24185,123979],{"className":123978},[21531],[224,123980,24189],{"style":24188}," 移到最外层并做 ",[224,123983,123985,123998],{"className":123984,"translate":388},[392],[224,123986,123988],{"className":123987},[396],[398,123989,123990],{"xmlns":400},[403,123991,123992,123996],{},[406,123993,123994],{},[409,123995,1705],{},[447,123997,1705],{"encoding":449},[224,123999,124001],{"className":124000,"ariaHidden":455},[454],[224,124002,124004,124007],{"className":124003},[459],[224,124005],{"className":124006,"style":2565},[463],[224,124008,1705],{"className":124009,"style":1740},[468,469],"-散度对偶后的结果。）",[2674,124012,124013,124239,124632,125063],{},[2677,124014,124015,124018,124019,69081,124052,124080,124081,124136,124137,124187,124188,124238],{},[378,124016,124017],{},"反向 KL \u002F α-GAN","：把 α 从 ",[224,124020,124022,124037],{"className":124021,"translate":388},[392],[224,124023,124025],{"className":124024},[396],[398,124026,124027],{"xmlns":400},[403,124028,124029,124035],{},[406,124030,124031,124033],{},[412,124032,5268],{},[419,124034,1438],{},[447,124036,69065],{"encoding":449},[224,124038,124040],{"className":124039,"ariaHidden":455},[454],[224,124041,124043,124046,124049],{"className":124042},[459],[224,124044],{"className":124045,"style":7047},[463],[224,124047,5268],{"className":124048},[468],[224,124050,1438],{"className":124051},[468],[224,124053,124055,124068],{"className":124054,"translate":388},[392],[224,124056,124058],{"className":124057},[396],[398,124059,124060],{"xmlns":400},[403,124061,124062,124066],{},[406,124063,124064],{},[419,124065,1438],{},[447,124067,1438],{"encoding":449},[224,124069,124071],{"className":124070,"ariaHidden":455},[454],[224,124072,124074,124077],{"className":124073},[459],[224,124075],{"className":124076,"style":510},[463],[224,124078,1438],{"className":124079},[468],"，散度从反向 KL（",[224,124082,124084,124103],{"className":124083,"translate":388},[392],[224,124085,124087],{"className":124086},[396],[398,124088,124089],{"xmlns":400},[403,124090,124091,124101],{},[406,124092,124093,124095,124097,124099],{},[409,124094,64666],{},[412,124096,1144],{},[412,124098,5268],{},[419,124100,1438],{},[447,124102,68948],{"encoding":449},[224,124104,124106,124124],{"className":124105,"ariaHidden":455},[454],[224,124107,124109,124112,124115,124118,124121],{"className":124108},[459],[224,124110],{"className":124111,"style":839},[463],[224,124113,64666],{"className":124114,"style":64696},[468,469],[224,124116],{"className":124117,"style":496},[473],[224,124119,1144],{"className":124120},[500],[224,124122],{"className":124123,"style":496},[473],[224,124125,124127,124130,124133],{"className":124126},[459],[224,124128],{"className":124129,"style":7047},[463],[224,124131,5268],{"className":124132},[468],[224,124134,1438],{"className":124135},[468],"，zero-forcing、模式崩塌倾向）经 Hellinger 型对称散度（",[224,124138,124140,124157],{"className":124139,"translate":388},[392],[224,124141,124143],{"className":124142},[396],[398,124144,124145],{"xmlns":400},[403,124146,124147,124155],{},[406,124148,124149,124151,124153],{},[409,124150,64666],{},[412,124152,278],{},[419,124154,284],{},[447,124156,66399],{"encoding":449},[224,124158,124160,124178],{"className":124159,"ariaHidden":455},[454],[224,124161,124163,124166,124169,124172,124175],{"className":124162},[459],[224,124164],{"className":124165,"style":839},[463],[224,124167,64666],{"className":124168,"style":64696},[468,469],[224,124170],{"className":124171,"style":496},[473],[224,124173,278],{"className":124174},[500],[224,124176],{"className":124177,"style":496},[473],[224,124179,124181,124184],{"className":124180},[459],[224,124182],{"className":124183,"style":510},[463],[224,124185,284],{"className":124186},[468],"）过渡到正向 KL（",[224,124189,124191,124208],{"className":124190,"translate":388},[392],[224,124192,124194],{"className":124193},[396],[398,124195,124196],{"xmlns":400},[403,124197,124198,124206],{},[406,124199,124200,124202,124204],{},[409,124201,64666],{},[412,124203,1144],{},[419,124205,1438],{},[447,124207,67126],{"encoding":449},[224,124209,124211,124229],{"className":124210,"ariaHidden":455},[454],[224,124212,124214,124217,124220,124223,124226],{"className":124213},[459],[224,124215],{"className":124216,"style":839},[463],[224,124218,64666],{"className":124219,"style":64696},[468,469],[224,124221],{"className":124222,"style":496},[473],[224,124224,1144],{"className":124225},[500],[224,124227],{"className":124228,"style":496},[473],[224,124230,124232,124235],{"className":124231},[459],[224,124233],{"className":124234,"style":510},[463],[224,124236,1438],{"className":124237},[468],"，zero-avoiding、模式覆盖倾向），可以在\"崩塌—覆盖\"之间连续调节（见 3.6 节的两端极限）。",[2677,124240,124241,12570,124243,7098,124487,124627,124628,124631],{},[378,124242,73031],{},[224,124244,124246,124296],{"className":124245,"translate":388},[392],[224,124247,124249],{"className":124248},[396],[398,124250,124251],{"xmlns":400},[403,124252,124253,124293],{},[406,124254,124255,124261,124263,124265,124271,124273,124275,124277,124279,124281,124291],{},[3248,124256,124257,124259],{},[409,124258,1705],{},[412,124260,27543],{},[412,124262,278],{},[412,124264,5268],{},[20338,124266,124267,124269],{},[419,124268,1438],{},[419,124270,1447],{},[409,124272,24131],{},[412,124274,24134],{},[412,124276,239],{"stretchy":435},[419,124278,1447],{},[412,124280,5268],{},[3248,124282,124283,124285],{},[409,124284,1861],{},[406,124286,124287,124289],{},[419,124288,1447],{},[409,124290,375],{},[412,124292,1716],{"stretchy":435},[447,124294,124295],{"encoding":449},"f^*=-\\frac12\\log(2-e^{2y})",[224,124297,124299,124343,124443],{"className":124298,"ariaHidden":455},[454],[224,124300,124302,124305,124334,124337,124340],{"className":124301},[459],[224,124303],{"className":124304,"style":2565},[463],[224,124306,124308,124311],{"className":124307},[468],[224,124309,1705],{"className":124310,"style":1740},[468,469],[224,124312,124314],{"className":124313},[750],[224,124315,124317],{"className":124316},[754],[224,124318,124320],{"className":124319},[759],[224,124321,124323],{"className":124322,"style":28465},[763],[224,124324,124325,124328],{"style":3288},[224,124326],{"className":124327,"style":772},[771],[224,124329,124331],{"className":124330},[776,777,778,779],[224,124332,27543],{"className":124333},[478,779],[224,124335],{"className":124336,"style":496},[473],[224,124338,278],{"className":124339},[500],[224,124341],{"className":124342,"style":496},[473],[224,124344,124346,124349,124352,124420,124423,124428,124431,124434,124437,124440],{"className":124345},[459],[224,124347],{"className":124348,"style":22544},[463],[224,124350,5268],{"className":124351},[468],[224,124353,124355,124358,124417],{"className":124354},[468],[224,124356],{"className":124357},[542,8479],[224,124359,124361],{"className":124360},[20338],[224,124362,124364,124409],{"className":124363},[754,755],[224,124365,124367,124406],{"className":124366},[759],[224,124368,124370,124384,124392],{"className":124369,"style":22563},[763],[224,124371,124372,124375],{"style":22566},[224,124373],{"className":124374,"style":2108},[771],[224,124376,124378],{"className":124377},[776,777,778,779],[224,124379,124381],{"className":124380},[468,779],[224,124382,1447],{"className":124383},[468,779],[224,124385,124386,124389],{"style":20546},[224,124387],{"className":124388,"style":2108},[771],[224,124390],{"className":124391,"style":20554},[20553],[224,124393,124394,124397],{"style":22589},[224,124395],{"className":124396,"style":2108},[771],[224,124398,124400],{"className":124399},[776,777,778,779],[224,124401,124403],{"className":124402},[468,779],[224,124404,1438],{"className":124405},[468,779],[224,124407,787],{"className":124408},[786],[224,124410,124412],{"className":124411},[759],[224,124413,124415],{"className":124414,"style":22611},[763],[224,124416],{},[224,124418],{"className":124419},[549,8479],[224,124421],{"className":124422,"style":724},[473],[224,124424,24185,124426],{"className":124425},[21531],[224,124427,24189],{"style":24188},[224,124429,239],{"className":124430},[542],[224,124432,1447],{"className":124433},[468],[224,124435],{"className":124436,"style":474},[473],[224,124438,5268],{"className":124439},[478],[224,124441],{"className":124442,"style":474},[473],[224,124444,124446,124449,124484],{"className":124445},[459],[224,124447],{"className":124448,"style":39943},[463],[224,124450,124452,124455],{"className":124451},[468],[224,124453,1861],{"className":124454},[468,469],[224,124456,124458],{"className":124457},[750],[224,124459,124461],{"className":124460},[754],[224,124462,124464],{"className":124463},[759],[224,124465,124467],{"className":124466,"style":5671},[763],[224,124468,124469,124472],{"style":3288},[224,124470],{"className":124471,"style":772},[771],[224,124473,124475],{"className":124474},[776,777,778,779],[224,124476,124478,124481],{"className":124477},[468,779],[224,124479,1447],{"className":124480},[468,779],[224,124482,375],{"className":124483,"style":492},[468,469,779],[224,124485,1716],{"className":124486},[549],[224,124488,124490,124517],{"className":124489,"translate":388},[392],[224,124491,124493],{"className":124492},[396],[398,124494,124495],{"xmlns":400},[403,124496,124497,124515],{},[406,124498,124499,124501,124503],{},[409,124500,375],{},[412,124502,1577],{},[20338,124504,124505,124513],{},[406,124506,124507,124509,124511],{},[409,124508,24131],{},[412,124510,24134],{},[419,124512,1447],{},[419,124514,1447],{},[447,124516,85649],{"encoding":449},[224,124518,124520,124538],{"className":124519,"ariaHidden":455},[454],[224,124521,124523,124526,124529,124532,124535],{"className":124522},[459],[224,124524],{"className":124525,"style":488},[463],[224,124527,375],{"className":124528,"style":492},[468,469],[224,124530],{"className":124531,"style":496},[473],[224,124533,1577],{"className":124534},[500],[224,124536],{"className":124537,"style":496},[473],[224,124539,124541,124544],{"className":124540},[459],[224,124542],{"className":124543,"style":85677},[463],[224,124545,124547,124550,124624],{"className":124546},[468],[224,124548],{"className":124549},[542,8479],[224,124551,124553],{"className":124552},[20338],[224,124554,124556,124616],{"className":124555},[754,755],[224,124557,124559,124613],{"className":124558},[759],[224,124560,124562,124576,124584],{"className":124561,"style":22861},[763],[224,124563,124564,124567],{"style":22566},[224,124565],{"className":124566,"style":2108},[771],[224,124568,124570],{"className":124569},[776,777,778,779],[224,124571,124573],{"className":124572},[468,779],[224,124574,1447],{"className":124575},[468,779],[224,124577,124578,124581],{"style":20546},[224,124579],{"className":124580,"style":2108},[771],[224,124582],{"className":124583,"style":20554},[20553],[224,124585,124586,124589],{"style":22889},[224,124587],{"className":124588,"style":2108},[771],[224,124590,124592],{"className":124591},[776,777,778,779],[224,124593,124595,124607,124610],{"className":124594},[468,779],[224,124596,124598,124601,124604],{"className":124597},[21531,779],[224,124599,1864],{"className":124600},[779],[224,124602,4127],{"className":124603},[779],[224,124605,24189],{"className":124606,"style":24188},[779],[224,124608],{"className":124609,"style":62279},[473,779],[224,124611,1447],{"className":124612},[468,779],[224,124614,787],{"className":124615},[786],[224,124617,124619],{"className":124618},[759],[224,124620,124622],{"className":124621,"style":22611},[763],[224,124623],{},[224,124625],{"className":124626},[549,8479],"。代回即",[378,124629,124630],{},"原始 GAN 的对抗目标","（见 5.4）。",[2677,124633,124634,12570,124695,124835,124836,125058,125059,125062],{},[378,124635,124636,124694],{},[224,124637,124639,124656],{"className":124638,"translate":388},[392],[224,124640,124642],{"className":124641},[396],[398,124643,124644],{"xmlns":400},[403,124645,124646,124654],{},[406,124647,124648],{},[3248,124649,124650,124652],{},[409,124651,57370],{},[419,124653,1447],{},[447,124655,60308],{"encoding":449},[224,124657,124659],{"className":124658,"ariaHidden":455},[454],[224,124660,124662,124665],{"className":124661},[459],[224,124663],{"className":124664,"style":60318},[463],[224,124666,124668,124671],{"className":124667},[468],[224,124669,57370],{"className":124670},[468,469],[224,124672,124674],{"className":124673},[750],[224,124675,124677],{"className":124676},[754],[224,124678,124680],{"className":124679},[759],[224,124681,124683],{"className":124682,"style":5671},[763],[224,124684,124685,124688],{"style":3288},[224,124686],{"className":124687,"style":772},[771],[224,124689,124691],{"className":124690},[776,777,778,779],[224,124692,1447],{"className":124693},[468,779],"（LSGAN）",[224,124696,124698,124732],{"className":124697,"translate":388},[392],[224,124699,124701],{"className":124700},[396],[398,124702,124703],{"xmlns":400},[403,124704,124705,124729],{},[406,124706,124707,124713,124715,124717,124719,124725,124727],{},[3248,124708,124709,124711],{},[409,124710,1705],{},[412,124712,27543],{},[412,124714,278],{},[409,124716,375],{},[412,124718,319],{},[3248,124720,124721,124723],{},[409,124722,375],{},[419,124724,1447],{},[409,124726,12413],{"mathvariant":678},[419,124728,9409],{},[447,124730,124731],{"encoding":449},"f^*=y+y^2\u002F4",[224,124733,124735,124779,124797],{"className":124734,"ariaHidden":455},[454],[224,124736,124738,124741,124770,124773,124776],{"className":124737},[459],[224,124739],{"className":124740,"style":2565},[463],[224,124742,124744,124747],{"className":124743},[468],[224,124745,1705],{"className":124746,"style":1740},[468,469],[224,124748,124750],{"className":124749},[750],[224,124751,124753],{"className":124752},[754],[224,124754,124756],{"className":124755},[759],[224,124757,124759],{"className":124758,"style":28465},[763],[224,124760,124761,124764],{"style":3288},[224,124762],{"className":124763,"style":772},[771],[224,124765,124767],{"className":124766},[776,777,778,779],[224,124768,27543],{"className":124769},[478,779],[224,124771],{"className":124772,"style":496},[473],[224,124774,278],{"className":124775},[500],[224,124777],{"className":124778,"style":496},[473],[224,124780,124782,124785,124788,124791,124794],{"className":124781},[459],[224,124783],{"className":124784,"style":4846},[463],[224,124786,375],{"className":124787,"style":492},[468,469],[224,124789],{"className":124790,"style":474},[473],[224,124792,319],{"className":124793},[478],[224,124795],{"className":124796,"style":474},[473],[224,124798,124800,124803,124832],{"className":124799},[459],[224,124801],{"className":124802,"style":39943},[463],[224,124804,124806,124809],{"className":124805},[468],[224,124807,375],{"className":124808,"style":492},[468,469],[224,124810,124812],{"className":124811},[750],[224,124813,124815],{"className":124814},[754],[224,124816,124818],{"className":124817},[759],[224,124819,124821],{"className":124820,"style":5671},[763],[224,124822,124823,124826],{"style":3288},[224,124824],{"className":124825,"style":772},[771],[224,124827,124829],{"className":124828},[776,777,778,779],[224,124830,1447],{"className":124831},[468,779],[224,124833,60286],{"className":124834},[468],"，目标变为 ",[224,124837,124839,124889],{"className":124838,"translate":388},[392],[224,124840,124842],{"className":124841},[396],[398,124843,124844],{"xmlns":400},[403,124845,124846,124886],{},[406,124847,124848,124854,124856,124858,124860,124862,124868,124870,124872,124874,124880,124882,124884],{},[695,124849,124850,124852],{},[409,124851,20324],{"mathvariant":12251},[409,124853,16018],{},[412,124855,268],{"stretchy":435},[409,124857,24189],{},[412,124859,440],{"stretchy":435},[412,124861,5268],{},[695,124863,124864,124866],{},[409,124865,20324],{"mathvariant":12251},[409,124867,20315],{},[412,124869,268],{"stretchy":435},[409,124871,24189],{},[412,124873,319],{},[3248,124875,124876,124878],{},[409,124877,24189],{},[419,124879,1447],{},[409,124881,12413],{"mathvariant":678},[419,124883,9409],{},[412,124885,440],{"stretchy":435},[447,124887,124888],{"encoding":449},"\\mathbb E_P[g]-\\mathbb E_Q[g+g^2\u002F4]",[224,124890,124892,124956,125017],{"className":124891,"ariaHidden":455},[454],[224,124893,124895,124898,124938,124941,124944,124947,124950,124953],{"className":124894},[459],[224,124896],{"className":124897,"style":535},[463],[224,124899,124901,124904],{"className":124900},[468],[224,124902,20324],{"className":124903},[468,12389],[224,124905,124907],{"className":124906},[750],[224,124908,124910,124930],{"className":124909},[754,755],[224,124911,124913,124927],{"className":124912},[759],[224,124914,124916],{"className":124915,"style":764},[763],[224,124917,124918,124921],{"style":767},[224,124919],{"className":124920,"style":772},[771],[224,124922,124924],{"className":124923},[776,777,778,779],[224,124925,16018],{"className":124926,"style":653},[468,469,779],[224,124928,787],{"className":124929},[786],[224,124931,124933],{"className":124932},[759],[224,124934,124936],{"className":124935,"style":794},[763],[224,124937],{},[224,124939,268],{"className":124940},[542],[224,124942,24189],{"className":124943,"style":492},[468,469],[224,124945,440],{"className":124946},[549],[224,124948],{"className":124949,"style":474},[473],[224,124951,5268],{"className":124952},[478],[224,124954],{"className":124955,"style":474},[473],[224,124957,124959,124962,125002,125005,125008,125011,125014],{"className":124958},[459],[224,124960],{"className":124961,"style":20368},[463],[224,124963,124965,124968],{"className":124964},[468],[224,124966,20324],{"className":124967},[468,12389],[224,124969,124971],{"className":124970},[750],[224,124972,124974,124994],{"className":124973},[754,755],[224,124975,124977,124991],{"className":124976},[759],[224,124978,124980],{"className":124979,"style":764},[763],[224,124981,124982,124985],{"style":767},[224,124983],{"className":124984,"style":772},[771],[224,124986,124988],{"className":124987},[776,777,778,779],[224,124989,20315],{"className":124990},[468,469,779],[224,124992,787],{"className":124993},[786],[224,124995,124997],{"className":124996},[759],[224,124998,125000],{"className":124999,"style":20409},[763],[224,125001],{},[224,125003,268],{"className":125004},[542],[224,125006,24189],{"className":125007,"style":492},[468,469],[224,125009],{"className":125010,"style":474},[473],[224,125012,319],{"className":125013},[478],[224,125015],{"className":125016,"style":474},[473],[224,125018,125020,125023,125052,125055],{"className":125019},[459],[224,125021],{"className":125022,"style":39943},[463],[224,125024,125026,125029],{"className":125025},[468],[224,125027,24189],{"className":125028,"style":492},[468,469],[224,125030,125032],{"className":125031},[750],[224,125033,125035],{"className":125034},[754],[224,125036,125038],{"className":125037},[759],[224,125039,125041],{"className":125040,"style":5671},[763],[224,125042,125043,125046],{"style":3288},[224,125044],{"className":125045,"style":772},[771],[224,125047,125049],{"className":125048},[776,777,778,779],[224,125050,1447],{"className":125051},[468,779],[224,125053,60286],{"className":125054},[468],[224,125056,440],{"className":125057},[549],"，即对判别器输出做",[378,125060,125061],{},"最小二乘","回归，梯度不会像 sigmoid 那样饱和。",[2677,125064,125065,12570,125068,7098,125191,3117],{},[378,125066,125067],{},"Hellinger²",[224,125069,125071,125105],{"className":125070,"translate":388},[392],[224,125072,125074],{"className":125073},[396],[398,125075,125076],{"xmlns":400},[403,125077,125078,125102],{},[406,125079,125080,125086,125088,125090,125092,125094,125096,125098,125100],{},[3248,125081,125082,125084],{},[409,125083,1705],{},[412,125085,27543],{},[412,125087,278],{},[409,125089,375],{},[409,125091,12413],{"mathvariant":678},[412,125093,239],{"stretchy":435},[419,125095,1438],{},[412,125097,5268],{},[409,125099,375],{},[412,125101,1716],{"stretchy":435},[447,125103,125104],{"encoding":449},"f^*=y\u002F(1-y)",[224,125106,125108,125152,125179],{"className":125107,"ariaHidden":455},[454],[224,125109,125111,125114,125143,125146,125149],{"className":125110},[459],[224,125112],{"className":125113,"style":2565},[463],[224,125115,125117,125120],{"className":125116},[468],[224,125118,1705],{"className":125119,"style":1740},[468,469],[224,125121,125123],{"className":125122},[750],[224,125124,125126],{"className":125125},[754],[224,125127,125129],{"className":125128},[759],[224,125130,125132],{"className":125131,"style":28465},[763],[224,125133,125134,125137],{"style":3288},[224,125135],{"className":125136,"style":772},[771],[224,125138,125140],{"className":125139},[776,777,778,779],[224,125141,27543],{"className":125142},[478,779],[224,125144],{"className":125145,"style":496},[473],[224,125147,278],{"className":125148},[500],[224,125150],{"className":125151,"style":496},[473],[224,125153,125155,125158,125161,125164,125167,125170,125173,125176],{"className":125154},[459],[224,125156],{"className":125157,"style":535},[463],[224,125159,375],{"className":125160,"style":492},[468,469],[224,125162,12413],{"className":125163},[468],[224,125165,239],{"className":125166},[542],[224,125168,1438],{"className":125169},[468],[224,125171],{"className":125172,"style":474},[473],[224,125174,5268],{"className":125175},[478],[224,125177],{"className":125178,"style":474},[473],[224,125180,125182,125185,125188],{"className":125181},[459],[224,125183],{"className":125184,"style":535},[463],[224,125186,375],{"className":125187,"style":492},[468,469],[224,125189,1716],{"className":125190},[549],[224,125192,125194,125211],{"className":125193,"translate":388},[392],[224,125195,125197],{"className":125196},[396],[398,125198,125199],{"xmlns":400},[403,125200,125201,125209],{},[406,125202,125203,125205,125207],{},[409,125204,375],{},[412,125206,1577],{},[419,125208,1438],{},[447,125210,86679],{"encoding":449},[224,125212,125214,125232],{"className":125213,"ariaHidden":455},[454],[224,125215,125217,125220,125223,125226,125229],{"className":125216},[459],[224,125218],{"className":125219,"style":488},[463],[224,125221,375],{"className":125222,"style":492},[468,469],[224,125224],{"className":125225,"style":496},[473],[224,125227,1577],{"className":125228},[500],[224,125230],{"className":125231,"style":496},[473],[224,125233,125235,125238],{"className":125234},[459],[224,125236],{"className":125237,"style":510},[463],[224,125239,1438],{"className":125240},[468],[613,125242,125244],{"id":125243},"_54-原始-gan-与-js-散度","5.4 原始 GAN 与 JS 散度",[11,125246,125247,125248,125430],{},"原始 GAN 的判别器输出经 sigmoid 得 ",[224,125249,125251,125303],{"className":125250,"translate":388},[392],[224,125252,125254],{"className":125253},[396],[398,125255,125256],{"xmlns":400},[403,125257,125258,125300],{},[406,125259,125260,125262,125264,125266,125268,125270,125272,125274,125276,125282,125284,125286,125288,125290,125292,125294,125296,125298],{},[409,125261,685],{},[412,125263,239],{"stretchy":435},[409,125265,20303],{},[412,125267,239],{"stretchy":435},[409,125269,371],{},[412,125271,1716],{"stretchy":435},[412,125273,1716],{"stretchy":435},[412,125275,278],{},[695,125277,125278,125280],{},[409,125279,20303],{},[409,125281,1858],{},[412,125283,239],{"stretchy":435},[409,125285,371],{},[412,125287,1716],{"stretchy":435},[412,125289,12248],{},[412,125291,239],{"stretchy":435},[419,125293,284],{},[412,125295,1877],{"separator":455},[419,125297,1438],{},[412,125299,1716],{"stretchy":435},[447,125301,125302],{"encoding":449},"\\sigma(D(x))=D_{s}(x)\\in(0,1)",[224,125304,125306,125339,125406],{"className":125305,"ariaHidden":455},[454],[224,125307,125309,125312,125315,125318,125321,125324,125327,125330,125333,125336],{"className":125308},[459],[224,125310],{"className":125311,"style":535},[463],[224,125313,685],{"className":125314,"style":492},[468,469],[224,125316,239],{"className":125317},[542],[224,125319,20303],{"className":125320,"style":2473},[468,469],[224,125322,239],{"className":125323},[542],[224,125325,371],{"className":125326},[468,469],[224,125328,38989],{"className":125329},[549],[224,125331],{"className":125332,"style":496},[473],[224,125334,278],{"className":125335},[500],[224,125337],{"className":125338,"style":496},[473],[224,125340,125342,125345,125388,125391,125394,125397,125400,125403],{"className":125341},[459],[224,125343],{"className":125344,"style":535},[463],[224,125346,125348,125351],{"className":125347},[468],[224,125349,20303],{"className":125350,"style":2473},[468,469],[224,125352,125354],{"className":125353},[750],[224,125355,125357,125380],{"className":125356},[754,755],[224,125358,125360,125377],{"className":125359},[759],[224,125361,125363],{"className":125362,"style":64841},[763],[224,125364,125365,125368],{"style":20390},[224,125366],{"className":125367,"style":772},[771],[224,125369,125371],{"className":125370},[776,777,778,779],[224,125372,125374],{"className":125373},[468,779],[224,125375,1858],{"className":125376},[468,469,779],[224,125378,787],{"className":125379},[786],[224,125381,125383],{"className":125382},[759],[224,125384,125386],{"className":125385,"style":794},[763],[224,125387],{},[224,125389,239],{"className":125390},[542],[224,125392,371],{"className":125393},[468,469],[224,125395,1716],{"className":125396},[549],[224,125398],{"className":125399,"style":496},[473],[224,125401,12248],{"className":125402},[500],[224,125404],{"className":125405,"style":496},[473],[224,125407,125409,125412,125415,125418,125421,125424,125427],{"className":125408},[459],[224,125410],{"className":125411,"style":535},[463],[224,125413,239],{"className":125414},[542],[224,125416,284],{"className":125417},[468],[224,125419,1877],{"className":125420},[1917],[224,125422],{"className":125423,"style":724},[473],[224,125425,1438],{"className":125426},[468],[224,125428,1716],{"className":125429},[549],"，其目标等价于最小化 JS 散度下的对抗博弈。判别器写成 logit",[224,125432,125434],{"className":125433,"translate":388},[387],[224,125435,125437,125515],{"className":125436,"translate":388},[392],[224,125438,125440],{"className":125439},[396],[398,125441,125442],{"xmlns":400,"display":401},[403,125443,125444,125512],{},[406,125445,125446,125448,125450,125452,125454,125456,125458,125460,125466,125468,125470,125472,125474,125476,125478,125480,125482,125484,125490,125492,125494,125496,125498,125500,125502,125504,125510],{},[409,125447,24189],{},[412,125449,239],{"stretchy":435},[409,125451,371],{},[412,125453,1716],{"stretchy":435},[412,125455,278],{},[409,125457,24131],{},[412,125459,24134],{},[695,125461,125462,125464],{},[409,125463,20303],{},[409,125465,1858],{},[412,125467,239],{"stretchy":435},[409,125469,371],{},[412,125471,1716],{"stretchy":435},[412,125473,5268],{},[409,125475,24131],{},[412,125477,24134],{},[412,125479,239],{"stretchy":435},[419,125481,1438],{},[412,125483,5268],{},[695,125485,125486,125488],{},[409,125487,20303],{},[409,125489,1858],{},[412,125491,239],{"stretchy":435},[409,125493,371],{},[412,125495,1716],{"stretchy":435},[412,125497,1716],{"stretchy":435},[412,125499,278],{},[409,125501,24131],{},[412,125503,24134],{},[20338,125505,125506,125508],{},[409,125507,11],{},[409,125509,2023],{},[412,125511,1877],{"separator":455},[447,125513,125514],{"encoding":449},"g(x)=\\log 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E_P[\\log D_s]+\\mathbb E_Q[\\log(1-D_s)]\n=2\\,D_{JS}\\!\\left(P\\ \\Big\\|\\ Q\\right)-2\\log 2 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E_P\\Big[\\log\\frac{2p}{p+q}\\Big]\n=\\mathbb E_P[\\log D_s]+\\log2,",[224,127551,127553,127589,127744,127853],{"className":127552,"ariaHidden":455},[454],[224,127554,127556,127559,127565,127568,127571,127574,127577,127580,127583,127586],{"className":127555},[459],[224,127557],{"className":127558,"style":535},[463],[224,127560,127562],{"className":127561},[468],[224,127563,52588],{"className":127564},[468,1906],[224,127566,239],{"className":127567},[542],[224,127569,16018],{"className":127570,"style":653},[468,469],[224,127572,20312],{"className":127573},[468],[224,127575,45500],{"className":127576,"style":45534},[468,469],[224,127578,1716],{"className":127579},[549],[224,127581],{"className":127582,"style":496},[473],[224,127584,278],{"className":127585},[500],[224,127587],{"className":127588,"style":496},[473],[224,127590,127592,127595,127635,127641,127644,127649,127652,127729,127735,127738,127741],{"className":127591},[459],[224,127593],{"className":127594,"style":51232},[463],[224,127596,127598,127601],{"className":127597},[468],[224,127599,20324],{"className":127600},[468,12389],[224,127602,127604],{"className":127603},[750],[224,127605,127607,127627],{"className":127606},[754,755],[224,127608,127610,127624],{"className":127609},[759],[224,127611,127613],{"className":127612,"style":764},[763],[224,127614,127615,127618],{"style":767},[224,127616],{"className":127617,"style":772},[771],[224,127619,127621],{"className":127620},[776,777,778,779],[224,127622,16018],{"className":127623,"style":653},[468,469,779],[224,127625,787],{"className":127626},[786],[224,127628,127630],{"className":127629},[759],[224,127631,127633],{"className":127632,"style":794},[763],[224,127634],{},[224,127636,127638],{"className":127637},[468],[224,127639,268],{"className":127640},[2363,23751],[224,127642],{"className":127643,"style":724},[473],[224,127645,24185,127647],{"className":127646},[21531],[224,127648,24189],{"style":24188},[224,127650],{"className":127651,"style":724},[473],[224,127653,127655,127658,127726],{"className":127654},[468],[224,127656],{"className":127657},[542,8479],[224,127659,127661],{"className":127660},[20338],[224,127662,127664,127718],{"className":127663},[754,755],[224,127665,127667,127715],{"className":127666},[759],[224,127668,127670,127693,127701],{"className":127669,"style":42827},[763],[224,127671,127672,127675],{"style":20531},[224,127673],{"className":127674,"style":2108},[771],[224,127676,127678,127681,127684,127687,127690],{"className":127677},[468],[224,127679,11],{"className":127680},[468,469],[224,127682],{"className":127683,"style":474},[473],[224,127685,319],{"className":127686},[478],[224,127688],{"className":127689,"style":474},[473],[224,127691,2023],{"className":127692,"style":492},[468,469],[224,127694,127695,127698],{"style":20546},[224,127696],{"className":127697,"style":2108},[771],[224,127699],{"className":127700,"style":20554},[20553],[224,127702,127703,127706],{"style":20557},[224,127704],{"className":127705,"style":2108},[771],[224,127707,127709,127712],{"className":127708},[468],[224,127710,1447],{"className":127711},[468],[224,127713,11],{"className":127714},[468,469],[224,127716,787],{"className":127717},[786],[224,127719,127721],{"className":127720},[759],[224,127722,127724],{"className":127723,"style":20579},[763],[224,127725],{},[224,127727],{"className":127728},[549,8479],[224,127730,127732],{"className":127731},[468],[224,127733,440],{"className":127734},[2363,23751],[224,127736],{"className":127737,"style":496},[473],[224,127739,278],{"className":127740},[500],[224,127742],{"className":127743,"style":496},[473],[224,127745,127747,127750,127790,127793,127798,127801,127841,127844,127847,127850],{"className":127746},[459],[224,127748],{"className":127749,"style":535},[463],[224,127751,127753,127756],{"className":127752},[468],[224,127754,20324],{"className":127755},[468,12389],[224,127757,127759],{"className":127758},[750],[224,127760,127762,127782],{"className":127761},[754,755],[224,127763,127765,127779],{"className":127764},[759],[224,127766,127768],{"className":127767,"style":764},[763],[224,127769,127770,127773],{"style":767},[224,127771],{"className":127772,"style":772},[771],[224,127774,127776],{"className":127775},[776,777,778,779],[224,127777,16018],{"className":127778,"style":653},[468,469,779],[224,127780,787],{"className":127781},[786],[224,127783,127785],{"className":127784},[759],[224,127786,127788],{"className":127787,"style":794},[763],[224,127789],{},[224,127791,268],{"className":127792},[542],[224,127794,24185,127796],{"className":127795},[21531],[224,127797,24189],{"style":24188},[224,127799],{"className":127800,"style":724},[473],[224,127802,127804,127807],{"className":127803},[468],[224,127805,20303],{"className":127806,"style":2473},[468,469],[224,127808,127810],{"className":127809},[750],[224,127811,127813,127833],{"className":127812},[754,755],[224,127814,127816,127830],{"className":127815},[759],[224,127817,127819],{"className":127818,"style":64841},[763],[224,127820,127821,127824],{"style":20390},[22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同理，两式相加即得。于是「最小化 JS」与「在最优判别器下最大化 GAN 目标」是同一件事，而常数 ",[224,127905,127907,127929],{"className":127906,"translate":388},[392],[224,127908,127910],{"className":127909},[396],[398,127911,127912],{"xmlns":400},[403,127913,127914,127926],{},[406,127915,127916,127918,127920,127922,127924],{},[412,127917,5268],{},[419,127919,1447],{},[409,127921,24131],{},[412,127923,24134],{},[419,127925,1447],{},[447,127927,127928],{"encoding":449},"-2\\log2",[224,127930,127932],{"className":127931,"ariaHidden":455},[454],[224,127933,127935,127938,127941,127944,127947,127952,127955],{"className":127934},[459],[224,127936],{"className":127937,"style":2565},[463],[224,127939,5268],{"className":127940},[468],[224,127942,1447],{"className":127943},[468],[224,127945],{"className":127946,"style":724},[473],[224,127948,24185,127950],{"className":127949},[21531],[224,127951,24189],{"style":24188},[224,127953],{"className":127954,"style":724},[473],[224,127956,1447],{"className":127957},[468]," 与生成器无关。这解释了「GAN 训练不收敛」的来源：JS 有界，一旦支撑集几近不重叠，",[224,127960,127962,127992],{"className":127961,"translate":388},[392],[224,127963,127965],{"className":127964},[396],[398,127966,127967],{"xmlns":400},[403,127968,127969,127989],{},[406,127970,127971,127981,127983,127985,127987],{},[695,127972,127973,127975],{},[409,127974,20303],{},[406,127976,127977,127979],{},[409,127978,43681],{},[409,127980,6924],{},[412,127982,1144],{},[409,127984,24131],{},[412,127986,24134],{},[419,127988,1447],{},[447,127990,127991],{"encoding":449},"D_{JS}\\to\\log2",[224,127993,127995,128056],{"className":127994,"ariaHidden":455},[454],[224,127996,127998,128001,128047,128050,128053],{"className":127997},[459],[224,127999],{"className":128000,"style":35905},[463],[224,128002,128004,128007],{"className":128003},[468],[224,128005,20303],{"className":128006,"style":2473},[468,469],[224,128008,128010],{"className":128009},[750],[224,128011,128013,128039],{"className":128012},[754,755],[224,128014,128016,128036],{"className":128015},[759],[224,128017,128019],{"className":128018,"style":764},[763],[224,128020,128021,128024],{"style":20390},[224,128022],{"className":128023,"style":772},[771],[224,128025,128027],{"className":128026},[776,777,778,779],[224,128028,128030,128033],{"className":128029},[468,779],[224,128031,43681],{"className":128032,"style":44091},[468,469,779],[224,128034,6924],{"className":128035,"style":6970},[468,469,779],[224,128037,787],{"className":128038},[786],[224,128040,128042],{"className":128041},[759],[224,128043,128045],{"className":128044,"style":794},[763],[224,128046],{},[224,128048],{"className":128049,"style":496},[473],[224,128051,1144],{"className":128052},[500],[224,128054],{"className":128055,"style":496},[473],[224,128057,128059,128062,128067,128070],{"className":128058},[459],[224,128060],{"className":128061,"style":2565},[463],[224,128063,24185,128065],{"className":128064},[21531],[224,128066,24189],{"style":24188},[224,128068],{"className":128069,"style":724},[473],[224,128071,1447],{"className":128072},[468],"，判别器梯度饱和，生成器梯度消失。反过来，换成 LSGAN（",[224,128075,128077,128094],{"className":128076,"translate":388},[392],[224,128078,128080],{"className":128079},[396],[398,128081,128082],{"xmlns":400},[403,128083,128084,128092],{},[406,128085,128086],{},[3248,128087,128088,128090],{},[409,128089,57370],{},[419,128091,1447],{},[447,128093,60308],{"encoding":449},[224,128095,128097],{"className":128096,"ariaHidden":455},[454],[224,128098,128100,128103],{"className":128099},[459],[224,128101],{"className":128102,"style":60318},[463],[224,128104,128106,128109],{"className":128105},[468],[224,128107,57370],{"className":128108},[468,469],[224,128110,128112],{"className":128111},[750],[224,128113,128115],{"className":128114},[754],[224,128116,128118],{"className":128117},[759],[224,128119,128121],{"className":128120,"style":5671},[763],[224,128122,128123,128126],{"style":3288},[224,128124],{"className":128125,"style":772},[771],[224,128127,128129],{"className":128128},[776,777,778,779],[224,128130,1447],{"className":128131},[468,779],"）、Wasserstein（非 f-散度）或 α-GAN 就能改变这一点。",[613,128134,128136],{"id":128135},"_55-估计性质","5.5 估计性质",[11,128138,128139],{},"用 5.1 的 Monte Carlo 下界做估计时：",[2674,128141,128142,128287,128442],{},[2677,128143,128144,128145,53748,128214,128216,128217,128286],{},"它是 ",[224,128146,128148,128165],{"className":128147,"translate":388},[392],[224,128149,128151],{"className":128150},[396],[398,128152,128153],{"xmlns":400},[403,128154,128155,128163],{},[406,128156,128157],{},[695,128158,128159,128161],{},[409,128160,20303],{},[409,128162,1705],{},[447,128164,24053],{"encoding":449},[224,128166,128168],{"className":128167,"ariaHidden":455},[454],[224,128169,128171,128174],{"className":128170},[459],[224,128172],{"className":128173,"style":24063},[463],[224,128175,128177,128180],{"className":128176},[468],[224,128178,20303],{"className":128179,"style":2473},[468,469],[224,128181,128183],{"className":128182},[750],[224,128184,128186,128206],{"className":128185},[754,755],[224,128187,128189,128203],{"className":128188},[759],[224,128190,128192],{"className":128191,"style":20387},[763],[224,128193,128194,128197],{"style":20390},[224,128195],{"className":128196,"style":772},[771],[224,128198,128200],{"className":128199},[776,777,778,779],[224,128201,1705],{"className":128202,"style":1740},[468,469,779],[224,128204,787],{"className":128205},[786],[224,128207,128209],{"className":128208},[759],[224,128210,128212],{"className":128211,"style":20409},[763],[224,128213],{},[378,128215,119737],{},"（偏差向下），且偏差来自 ",[224,128218,128220,128237],{"className":128219,"translate":388},[392],[224,128221,128223],{"className":128222},[396],[398,128224,128225],{"xmlns":400},[403,128226,128227,128235],{},[406,128228,128229],{},[695,128230,128231,128233],{},[409,128232,24189],{},[409,128234,119757],{},[447,128236,119760],{"encoding":449},[224,128238,128240],{"className":128239,"ariaHidden":455},[454],[224,128241,128243,128246],{"className":128242},[459],[224,128244],{"className":128245,"style":806},[463],[224,128247,128249,128252],{"className":128248},[468],[224,128250,24189],{"className":128251,"style":492},[468,469],[224,128253,128255],{"className":128254},[750],[224,128256,128258,128278],{"className":128257},[754,755],[224,128259,128261,128275],{"className":128260},[759],[224,128262,128264],{"className":128263,"style":64841},[763],[224,128265,128266,128269],{"style":3854},[224,128267],{"className":128268,"style":772},[771],[224,128270,128272],{"className":128271},[776,777,778,779],[224,128273,119757],{"className":128274,"style":492},[468,469,779],[224,128276,787],{"className":128277},[786],[224,128279,128281],{"className":128280},[759],[224,128282,128284],{"className":128283,"style":794},[763],[224,128285],{}," 未达到最优 + 样本有限；",[2677,128288,88038,128289,128319,128320,128378,128379,128441],{},[224,128290,128292,128307],{"className":128291,"translate":388},[392],[224,128293,128295],{"className":128294},[396],[398,128296,128297],{"xmlns":400},[403,128298,128299,128305],{},[406,128300,128301,128303],{},[409,128302,55354],{},[412,128304,24134],{},[447,128306,77053],{"encoding":449},[224,128308,128310],{"className":128309,"ariaHidden":455},[454],[224,128311,128313,128316],{"className":128312},[459],[224,128314],{"className":128315,"style":806},[463],[224,128317,55354],{"className":128318},[21531]," 取到极值时，估计量方差随 ",[224,128321,128323,128340],{"className":128322,"translate":388},[392],[224,128324,128326],{"className":128325},[396],[398,128327,128328],{"xmlns":400},[403,128329,128330,128338],{},[406,128331,128332],{},[3248,128333,128334,128336],{},[409,128335,1705],{},[412,128337,27543],{},[447,128339,80614],{"encoding":449},[224,128341,128343],{"className":128342,"ariaHidden":455},[454],[224,128344,128346,128349],{"className":128345},[459],[224,128347],{"className":128348,"style":2565},[463],[224,128350,128352,128355],{"className":128351},[468],[224,128353,1705],{"className":128354,"style":1740},[468,469],[224,128356,128358],{"className":128357},[750],[224,128359,128361],{"className":128360},[754],[224,128362,128364],{"className":128363},[759],[224,128365,128367],{"className":128366,"style":28465},[763],[224,128368,128369,128372],{"style":3288},[224,128370],{"className":128371,"style":772},[771],[224,128373,128375],{"className":128374},[776,777,778,779],[224,128376,27543],{"className":128377},[478,779]," 增长而增大（如 KL 的 ",[224,128380,128382,128400],{"className":128381,"translate":388},[392],[224,128383,128385],{"className":128384},[396],[398,128386,128387],{"xmlns":400},[403,128388,128389,128397],{},[406,128390,128391],{},[3248,128392,128393,128395],{},[409,128394,1861],{},[409,128396,24189],{},[447,128398,128399],{"encoding":449},"e^{g}",[224,128401,128403],{"className":128402,"ariaHidden":455},[454],[224,128404,128406,128409],{"className":128405},[459],[224,128407],{"className":128408,"style":3267},[463],[224,128410,128412,128415],{"className":128411},[468],[224,128413,1861],{"className":128414},[468,469],[224,128416,128418],{"className":128417},[750],[224,128419,128421],{"className":128420},[754],[224,128422,128424],{"className":128423},[759],[224,128425,128427],{"className":128426,"style":3267},[763],[224,128428,128429,128432],{"style":3288},[224,128430],{"className":128431,"style":772},[771],[224,128433,128435],{"className":128434},[776,777,778,779],[224,128436,128438],{"className":128437},[468,779],[224,128439,24189],{"className":128440,"style":492},[468,469,779]," 会重尾）；",[2677,128443,128444,128445,128473],{},"一个常用替代是「对称化」或「有界化」",[224,128446,128448,128461],{"className":128447,"translate":388},[392],[224,128449,128451],{"className":128450},[396],[398,128452,128453],{"xmlns":400},[403,128454,128455,128459],{},[406,128456,128457],{},[409,128458,1705],{},[447,128460,1705],{"encoding":449},[224,128462,128464],{"className":128463,"ariaHidden":455},[454],[224,128465,128467,128470],{"className":128466},[459],[224,128468],{"className":128469,"style":2565},[463],[224,128471,1705],{"className":128472,"style":1740},[468,469],"（如 JS、TV），牺牲敏感性换取稳定梯度。这与第 6 节讨论的「有界\u002F无界」准确性—稳健性权衡是同一件事。",[608,128475,128477],{"id":128476},"_6-与其他散度的比较","6. 与其他散度的比较",[613,128479,128481],{"id":128480},"_61-汇总","6.1 汇总",[72399,128483,128484,128614],{},[72402,128485,128486],{},[72405,128487,128488,128490,128535,128610,128612],{},[72408,128489,72410],{},[72408,128491,128492],{},[224,128493,128495,128514],{"className":128494,"translate":388},[392],[224,128496,128498],{"className":128497},[396],[398,128499,128500],{"xmlns":400},[403,128501,128502,128512],{},[406,128503,128504,128506,128508,128510],{},[409,128505,1705],{},[412,128507,239],{"stretchy":435},[409,128509,1435],{},[412,128511,1716],{"stretchy":435},[447,128513,72435],{"encoding":449},[224,128515,128517],{"className":128516,"ariaHidden":455},[454],[224,128518,128520,128523,128526,128529,128532],{"className":128519},[459],[224,128521],{"className":128522,"style":535},[463],[224,128524,1705],{"className":128525,"style":1740},[468,469],[224,128527,239],{"className":128528},[542],[224,128530,1435],{"className":128531},[468,469],[224,128533,1716],{"className":128534},[549],[72408,128536,128537],{},[224,128538,128540,128563],{"className":128539,"translate":388},[392],[224,128541,128543],{"className":128542},[396],[398,128544,128545],{"xmlns":400},[403,128546,128547,128561],{},[406,128548,128549,128555,128557,128559],{},[3248,128550,128551,128553],{},[409,128552,1705],{},[412,128554,27543],{},[412,128556,239],{"stretchy":435},[409,128558,375],{},[412,128560,1716],{"stretchy":435},[447,128562,84601],{"encoding":449},[224,128564,128566],{"className":128565,"ariaHidden":455},[454],[224,128567,128569,128572,128601,128604,128607],{"className":128568},[459],[224,128570],{"className":128571,"style":535},[463],[224,128573,128575,128578],{"className":128574},[468],[224,128576,1705],{"className":128577,"style":1740},[468,469],[224,128579,128581],{"className":128580},[750],[224,128582,128584],{"className":128583},[754],[224,128585,128587],{"className":128586},[759],[224,128588,128590],{"className":128589,"style":28465},[763],[224,128591,128592,128595],{"style":3288},[224,128593],{"className":128594,"style":772},[771],[224,128596,128598],{"className":128597},[776,777,778,779],[224,128599,27543],{"className":128600},[478,779],[224,128602,239],{"className":128603},[542],[224,128605,375],{"className":128606,"style":492},[468,469],[224,128608,1716],{"className":128609},[549],[72408,128611,49616],{},[72408,128613,72461],{},[72497,128615,128616,128749,128889,129379,129683,129992,130409,130720],{},[72405,128617,128618,128620,128670,128745,128747],{},[72502,128619,52588],{},[72502,128621,128622],{},[224,128623,128625,128644],{"className":128624,"translate":388},[392],[224,128626,128628],{"className":128627},[396],[398,128629,128630],{"xmlns":400},[403,128631,128632,128642],{},[406,128633,128634,128636,128638,128640],{},[409,128635,1435],{},[409,128637,24131],{},[412,128639,24134],{},[409,128641,1435],{},[447,128643,72633],{"encoding":449},[224,128645,128647],{"className":128646,"ariaHidden":455},[454],[224,128648,128650,128653,128656,128659,128664,128667],{"className":128649},[459],[224,128651],{"className":128652,"style":2565},[463],[224,128654,1435],{"className":128655},[468,469],[224,128657],{"className":128658,"style":724},[473],[224,128660,24185,128662],{"className":128661},[21531],[224,128663,24189],{"style":24188},[224,128665],{"className":128666,"style":724},[473],[224,128668,1435],{"className":128669},[468,469],[72502,128671,128672],{},[224,128673,128675,128698],{"className":128674,"translate":388},[392],[224,128676,128678],{"className":128677},[396],[398,128679,128680],{"xmlns":400},[403,128681,128682,128696],{},[406,128683,128684],{},[3248,128685,128686,128688],{},[409,128687,1861],{},[406,128689,128690,128692,128694],{},[409,128691,375],{},[412,128693,5268],{},[419,128695,1438],{},[447,128697,84808],{"encoding":449},[224,128699,128701],{"className":128700,"ariaHidden":455},[454],[224,128702,128704,128707],{"className":128703},[459],[224,128705],{"className":128706,"style":5671},[463],[224,128708,128710,128713],{"className":128709},[468],[224,128711,1861],{"className":128712},[468,469],[224,128714,128716],{"className":128715},[750],[224,128717,128719],{"className":128718},[754],[224,128720,128722],{"className":128721},[759],[224,128723,128725],{"className":128724,"style":5671},[763],[224,128726,128727,128730],{"style":3288},[224,128728],{"className":128729,"style":772},[771],[224,128731,128733],{"className":128732},[776,777,778,779],[224,128734,128736,128739,128742],{"className":128735},[468,779],[224,128737,375],{"className":128738,"style":492},[468,469,779],[224,128740,5268],{"className":128741},[478,779],[224,128743,1438],{"className":128744},[468,779],[72502,128746,72662],{},[72502,128748,72662],{},[72405,128750,128751,128754,128804,128885,128887],{},[72502,128752,128753],{},"反向 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e":132012},[542,8479],[224,132014,132016],{"className":132015},[20338],[224,132017,132019,132064],{"className":132018},[754,755],[224,132020,132022,132061],{"className":132021},[759],[224,132023,132025,132039,132047],{"className":132024,"style":22563},[763],[224,132026,132027,132030],{"style":22566},[224,132028],{"className":132029,"style":2108},[771],[224,132031,132033],{"className":132032},[776,777,778,779],[224,132034,132036],{"className":132035},[468,779],[224,132037,1435],{"className":132038},[468,469,779],[224,132040,132041,132044],{"style":20546},[224,132042],{"className":132043,"style":2108},[771],[224,132045],{"className":132046,"style":20554},[20553],[224,132048,132049,132052],{"style":22589},[224,132050],{"className":132051,"style":2108},[771],[224,132053,132055],{"className":132054},[776,777,778,779],[224,132056,132058],{"className":132057},[468,779],[224,132059,1438],{"className":132060},[468,779],[224,132062,787],{"className":132063},[786],[224,132065,132067],{"className":132066},[759],[224,132068,132070],{"className":132069,"style":22611},[763],[224,132071],{},[224,132073],{"className":132074},[549,8479],[224,132076,15077],{"className":132077},[549],"，驻点条件 ",[224,132080,132082,132116],{"className":132081,"translate":388},[392],[224,132083,132085],{"className":132084},[396],[398,132086,132087],{"xmlns":400},[403,132088,132089,132113],{},[406,132090,132091,132093,132095,132097,132099,132109,132111],{},[409,132092,375],{},[412,132094,5268],{},[419,132096,1438],{},[412,132098,319],{},[20338,132100,132101,132103],{},[419,132102,1438],{},[3248,132104,132105,132107],{},[409,132106,1435],{},[419,132108,1447],{},[412,132110,278],{},[419,132112,284],{},[447,132114,132115],{"encoding":449},"y-1+\\frac1{t^2}=0",[224,132117,132119,132137,132155,132264],{"className":132118,"ariaHidden":455},[454],[224,132120,132122,132125,132128,132131,132134],{"className":132121},[459],[224,132123],{"className":132124,"style":4846},[463],[224,132126,375],{"className":132127,"style":492},[468,469],[224,132129],{"className":132130,"style":474},[473],[224,132132,5268],{"className":132133},[478],[224,132135],{"className":132136,"style":474},[473],[224,132138,132140,132143,132146,132149,132152],{"className":132139},[459],[224,132141],{"className":132142,"style":7047},[463],[224,132144,1438],{"className":132145},[468],[224,132147],{"className":132148,"style":474},[473],[224,132150,319],{"className":132151},[478],[224,132153],{"className":132154,"style":474},[473],[224,132156,132158,132161,132255,132258,132261],{"className":132157},[459],[224,132159],{"className":132160,"style":22544},[463],[224,132162,132164,132167,132252],{"className":132163},[468],[224,132165],{"className":132166},[542,8479],[224,132168,132170],{"className":132169},[20338],[224,132171,132173,132244],{"className":132172},[754,755],[224,132174,132176,132241],{"className":132175},[759],[224,132177,132179,132219,132227],{"className":132178,"style":22563},[763],[224,132180,132181,132184],{"style":22566},[224,132182],{"className":132183,"style":2108},[771],[224,132185,132187],{"className":132186},[776,777,778,779],[224,132188,132190],{"className":132189},[468,779],[224,132191,132193,132196],{"className":132192},[468,779],[224,132194,1435],{"className":132195},[468,469,779],[224,132197,132199],{"className":132198},[750],[224,132200,132202],{"className":132201},[754],[224,132203,132205],{"className":132204},[759],[224,132206,132208],{"className":132207,"style":65370},[763],[224,132209,132210,132213],{"style":65373},[224,132211],{"className":132212,"style":50136},[771],[224,132214,132216],{"className":132215},[776,50140,2364,779],[224,132217,1447],{"className":132218},[468,779],[224,132220,132221,132224],{"style":20546},[224,132222],{"className":132223,"style":2108},[771],[224,132225],{"className":132226,"style":20554},[20553],[224,132228,132229,132232],{"style":22589},[224,132230],{"className":132231,"style":2108},[771],[224,132233,132235],{"className":132234},[776,777,778,779],[224,132236,132238],{"className":132237},[468,779],[224,132239,1438],{"className":132240},[468,779],[224,132242,787],{"className":132243},[786],[224,132245,132247],{"className":132246},[759],[224,132248,132250],{"className":132249,"style":22611},[763],[224,132251],{},[224,132253],{"className":132254},[549,8479],[224,132256],{"className":132257,"style":496},[473],[224,132259,278],{"className":132260},[500],[224,132262],{"className":132263,"style":496},[473],[224,132265,132267,132270],{"className":132266},[459],[224,132268],{"className":132269,"style":510},[463],[224,132271,284],{"className":132272},[468],[224,132274,132276,132306],{"className":132275,"translate":388},[392],[224,132277,132279],{"className":132278},[396],[398,132280,132281],{"xmlns":400},[403,132282,132283,132303],{},[406,132284,132285,132287,132289],{},[409,132286,1435],{},[412,132288,278],{},[20338,132290,132291,132293],{},[419,132292,1438],{},[54262,132294,132295],{},[406,132296,132297,132299,132301],{},[419,132298,1438],{},[412,132300,5268],{},[409,132302,375],{},[447,132304,132305],{"encoding":449},"t=\\frac{1}{\\sqrt{1-y}}",[224,132307,132309,132327],{"className":132308,"ariaHidden":455},[454],[224,132310,132312,132315,132318,132321,132324],{"className":132311},[459],[224,132313],{"className":132314,"style":25353},[463],[224,132316,1435],{"className":132317},[468,469],[224,132319],{"className":132320,"style":496},[473],[224,132322,278],{"className":132323},[500],[224,132325],{"className":132326,"style":496},[473],[224,132328,132330,132333],{"className":132329},[459],[224,132331],{"className":132332,"style":118041},[463],[224,132334,132336,132339,132452],{"className":132335},[468],[224,132337],{"className":132338},[542,8479],[224,132340,132342],{"className":132341},[20338],[224,132343,132345,132444],{"className":132344},[754,755],[224,132346,132348,132441],{"className":132347},[759],[224,132349,132351,132419,132427],{"className":132350,"style":22563},[763],[224,132352,132354,132357],{"style":132353},"top:-2.6191em;",[224,132355],{"className":132356,"style":2108},[771],[224,132358,132360],{"className":132359},[776,777,778,779],[224,132361,132363],{"className":132362},[468,779],[224,132364,132366],{"className":132365},[468,54290,779],[224,132367,132369,132410],{"className":132368},[754,755],[224,132370,132372,132407],{"className":132371},[759],[224,132373,132376,132394],{"className":132374,"style":132375},[763],"height:0.8156em;",[224,132377,132379,132382],{"className":132378,"style":2120},[2139],[224,132380],{"className":132381,"style":2108},[771],[224,132383,132385,132388,132391],{"className":132384,"style":54310},[468,779],[224,132386,1438],{"className":132387},[468,779],[224,132389,5268],{"className":132390},[478,779],[224,132392,375],{"className":132393,"style":492},[468,469,779],[224,132395,132397,132400],{"style":132396},"top:-2.7756em;",[224,132398],{"className":132399,"style":2108},[771],[224,132401,132403],{"className":132402,"style":54367},[54366,779],[2155,132404,132405],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,132406],{"d":54374},[224,132408,787],{"className":132409},[786],[224,132411,132413],{"className":132412},[759],[224,132414,132417],{"className":132415,"style":132416},[763],"height:0.2244em;",[224,132418],{},[224,132420,132421,132424],{"style":20546},[224,132422],{"className":132423,"style":2108},[771],[224,132425],{"className":132426,"style":20554},[20553],[224,132428,132429,132432],{"style":22589},[224,132430],{"className":132431,"style":2108},[771],[224,132433,132435],{"className":132434},[776,777,778,779],[224,132436,132438],{"className":132437},[468,779],[224,132439,1438],{"className":132440},[468,779],[224,132442,787],{"className":132443},[786],[224,132445,132447],{"className":132446},[759],[224,132448,132450],{"className":132449,"style":118150},[763],[224,132451],{},[224,132453],{"className":132454},[549,8479],[224,132456,132458,132475],{"className":132457,"translate":388},[392],[224,132459,132461],{"className":132460},[396],[398,132462,132463],{"xmlns":400},[403,132464,132465,132473],{},[406,132466,132467,132469,132471],{},[409,132468,375],{},[412,132470,1577],{},[419,132472,1438],{},[447,132474,86679],{"encoding":449},[224,132476,132478,132496],{"className":132477,"ariaHidden":455},[454],[224,132479,132481,132484,132487,132490,132493],{"className":132480},[459],[224,132482],{"className":132483,"style":488},[463],[224,132485,375],{"className":132486,"style":492},[468,469],[224,132488],{"className":132489,"style":496},[473],[224,132491,1577],{"className":132492},[500],[224,132494],{"className":132495,"style":496},[473],[224,132497,132499,132502],{"className":132498},[459],[224,132500],{"className":132501,"style":510},[463],[224,132503,1438],{"className":132504},[468],"），代回得 ",[224,132507,132509,132601],{"className":132508,"translate":388},[392],[224,132510,132512],{"className":132511},[396],[398,132513,132514],{"xmlns":400},[403,132515,132516,132598],{},[406,132517,132518,132524,132526,132528,132530,132532,132534,132536,132538,132540,132542,132544,132550,132552,132554,132564,132566,132568,132570,132580,132582,132584,132586,132588],{},[3248,132519,132520,132522],{},[409,132521,1705],{},[412,132523,27543],{},[412,132525,278],{},[409,132527,1435],{},[412,132529,239],{"stretchy":435},[409,132531,375],{},[412,132533,5268],{},[419,132535,1438],{},[412,132537,1716],{"stretchy":435},[412,132539,319],{},[419,132541,1447],{},[412,132543,5268],{},[20338,132545,132546,132548],{},[419,132547,1438],{},[409,132549,1435],{},[412,132551,278],{},[412,132553,5268],{},[54262,132555,132556],{},[406,132557,132558,132560,132562],{},[419,132559,1438],{},[412,132561,5268],{},[409,132563,375],{},[412,132565,319],{},[419,132567,1447],{},[412,132569,5268],{},[54262,132571,132572],{},[406,132573,132574,132576,132578],{},[419,132575,1438],{},[412,132577,5268],{},[409,132579,375],{},[412,132581,278],{},[419,132583,1447],{},[412,132585,5268],{},[419,132587,1447],{},[54262,132589,132590],{},[406,132591,132592,132594,132596],{},[419,132593,1438],{},[412,132595,5268],{},[409,132597,375],{},[447,132599,132600],{"encoding":449},"f^*=t(y-1)+2-\\frac1t=-\\sqrt{1-y}+2-\\sqrt{1-y}=2-2\\sqrt{1-y}",[224,132602,132604,132648,132672,132693,132711,132794,132871,132889,132963,132981],{"className":132603,"ariaHidden":455},[454],[224,132605,132607,132610,132639,132642,132645],{"className":132606},[459],[224,132608],{"className":132609,"style":2565},[463],[224,132611,132613,132616],{"className":132612},[468],[224,132614,1705],{"className":132615,"style":1740},[468,469],[224,132617,132619],{"className":132618},[750],[224,132620,132622],{"className":132621},[754],[224,132623,132625],{"className":132624},[759],[224,132626,132628],{"className":132627,"style":28465},[763],[224,132629,132630,132633],{"style":3288},[224,132631],{"className":132632,"style":772},[771],[224,132634,132636],{"className":132635},[776,777,778,779],[224,132637,27543],{"className":132638},[478,779],[224,132640],{"className":132641,"style":496},[473],[224,132643,278],{"className":132644},[500],[224,132646],{"className":132647,"style":496},[473],[224,132649,132651,132654,132657,132660,132663,132666,132669],{"className":132650},[459],[224,132652],{"className":132653,"style":535},[463],[224,132655,1435],{"className":132656},[468,469],[224,132658,239],{"className":132659},[542],[224,132661,375],{"className":132662,"style":492},[468,469],[224,132664],{"className":132665,"style":474},[473],[224,132667,5268],{"className":132668},[478],[224,132670],{"className":132671,"style":474},[473],[224,132673,132675,132678,132681,132684,132687,132690],{"className":132674},[459],[224,132676],{"className":132677,"style":535},[463],[224,132679,1438],{"className":132680},[468],[224,132682,1716],{"className":132683},[549],[224,132685],{"className":132686,"style":474},[473],[224,132688,319],{"className":132689},[478],[224,132691],{"className":132692,"style":474},[473],[224,132694,132696,132699,132702,132705,132708],{"className":132695},[459],[224,132697],{"className":132698,"style":7047},[463],[224,132700,1447],{"className":132701},[468],[224,132703],{"className":132704,"style":474},[473],[224,132706,5268],{"className":132707},[478],[224,132709],{"className":132710,"style":474},[473],[224,132712,132714,132717,132785,132788,132791],{"className":132713},[459],[224,132715],{"className":132716,"style":22544},[463],[224,132718,132720,132723,132782],{"className":132719},[468],[224,132721],{"className":132722},[542,8479],[224,132724,132726],{"className":132725},[20338],[224,132727,132729,132774],{"className":132728},[754,755],[224,132730,132732,132771],{"className":132731},[759],[224,132733,132735,132749,132757],{"className":132734,"style":22563},[763],[224,132736,132737,132740],{"style":22566},[224,132738],{"className":132739,"style":2108},[771],[224,132741,132743],{"className":132742},[776,777,778,779],[224,132744,132746],{"className":132745},[468,779],[224,132747,1435],{"className":132748},[468,469,779],[224,132750,132751,132754],{"style":20546},[224,132752],{"className":132753,"style":2108},[771],[224,132755],{"className":132756,"style":20554},[20553],[224,132758,132759,132762],{"style":22589},[224,132760],{"className":132761,"style":2108},[771],[224,132763,132765],{"className":132764},[776,777,778,779],[224,132766,132768],{"className":132767},[468,779],[224,132769,1438],{"className":132770},[468,779],[224,132772,787],{"className":132773},[786],[224,132775,132777],{"className":132776},[759],[224,132778,132780],{"className":132779,"style":22611},[763],[224,132781],{},[224,132783],{"className":132784},[549,8479],[224,132786],{"className":132787,"style":496},[473],[224,132789,278],{"className":132790},[500],[224,132792],{"className":132793,"style":496},[473],[224,132795,132797,132800,13280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一行取 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175},[468,779],[224,134177,787],{"className":134178},[786],[224,134180,134182],{"className":134181},[759],[224,134183,134185],{"className":134184,"style":22611},[763],[224,134186],{},[224,134188],{"className":134189},[549,8479],[224,134191,134193,134210],{"className":134192,"translate":388},[392],[224,134194,134196],{"className":134195},[396],[398,134197,134198],{"xmlns":400},[403,134199,134200,134208],{},[406,134201,134202,134204,134206],{},[409,134203,64666],{},[412,134205,278],{},[419,134207,284],{},[447,134209,66399],{"encoding":449},[224,134211,134213,134231],{"className":134212,"ariaHidden":455},[454],[224,134214,134216,134219,134222,134225,134228],{"className":134215},[459],[224,134217],{"className":134218,"style":839},[463],[224,134220,64666],{"className":134221,"style":64696},[468,469],[224,134223],{"className":134224,"style":496},[473],[224,134226,278],{"className":134227},[500],[224,134229],{"className":134230,"style":496},[473],[224,134232,134234,134237],{"className":134233},[459],[224,134235],{"className":134236,"style":510},[463],[224,134238,284],{"className":134239},[468],[224,134241,134243,134273],{"className":134242,"translate":388},[392],[224,134244,134246],{"className":134245},[396],[398,134247,134248],{"xmlns":400},[403,134249,134250,134270],{},[406,134251,134252,134254,134256,134258,134260,134262,134264,134268],{},[409,134253,1705],{},[412,134255,278],{},[419,134257,9409],{},[412,134259,239],{"stretchy":435},[419,134261,1438],{},[412,134263,5268],{},[54262,134265,134266],{},[409,134267,1435],{},[412,134269,1716],{"stretchy":435},[447,134271,134272],{"encoding":449},"f=4(1-\\sqrt t)",[224,134274,134276,134294,134318],{"className":134275,"ariaHidden":455},[454],[224,134277,134279,134282,134285,134288,134291],{"className":134278},[459],[224,134280],{"className":134281,"style":2565},[463],[224,134283,1705],{"className":134284,"style":1740},[468,469],[224,134286],{"className":134287,"style":496},[473],[224,134289,278],{"className":134290},[500],[224,134292],{"className":134293,"style":496},[473],[224,134295,134297,134300,134303,134306,134309,134312,134315],{"className":134296},[459],[224,134298],{"className":134299,"style":535},[463],[224,134301,9409],{"className":134302},[468],[224,134304,239],{"className":134305},[542],[224,134307,1438],{"className":134308},[468],[224,134310],{"className":134311,"style":474},[473],[224,134313,5268],{"className":134314},[478],[224,134316],{"className":134317,"style":474},[473],[224,134319,134321,134324,134368],{"className":134320},[459],[224,134322],{"className":134323,"style":62580},[463],[224,134325,134327],{"className":134326},[468,54290],[224,134328,134330,134360],{"className":134329},[754,755],[224,134331,134333,134357],{"className":134332},[759],[224,134334,134336,134345],{"className":134335,"style":62596},[763],[224,134337,134339,134342],{"className":134338,"style":2120},[2139],[224,134340],{"className":134341,"style":2108},[771],[224,134343,1435],{"className":134344,"style":54310},[468,469],[224,134346,134347,134350],{"style":62608},[224,134348],{"className":134349,"style":2108},[771],[224,134351,134353],{"className":134352,"style":54367},[54366],[2155,134354,134355],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,134356],{"d":54374},[224,134358,787],{"className":134359},[786],[224,134361,134363],{"className":134362},[759],[224,134364,134366],{"className":134365,"style":62628},[763],[224,134367],{},[224,134369,1716],{"className":134370},[549],"）时公式给出 ",[224,134373,134375,134401],{"className":134374,"translate":388},[392],[224,134376,134378],{"className":134377},[396],[398,134379,134380],{"xmlns":400},[403,134381,134382,134398],{},[406,134383,134384,134394,134396],{},[20338,134385,134386,134388],{},[419,134387,9409],{},[406,134389,134390,134392],{},[412,134391,5268],{},[409,134393,375],{},[412,134395,5268],{},[419,134397,9409],{},[447,134399,134400],{"encoding":449},"\\frac{4}{-y}-4",[224,134402,134404,134491],{"className":134403,"ariaHidden":455},[454],[224,134405,134407,134411,134482,134485,134488],{"className":134406},[459],[224,134408],{"className":134409,"style":134410},[463],"height:1.3262em;vertical-align:-0.4811em;",[224,134412,134414,134417,134479],{"className":134413},[468],[224,134415],{"className":134416},[542,8479],[224,134418,134420],{"className":134419},[20338],[224,134421,134423,134471],{"className":134422},[754,755],[224,134424,134426,134468],{"className":134425},[759],[224,134427,134429,134446,134454],{"className":134428,"style":22563},[763],[224,134430,134431,134434],{"style":22566},[224,134432],{"className":134433,"style":2108},[771],[224,134435,134437],{"className":134436},[776,777,778,779],[224,134438,134440,134443],{"className":134439},[468,779],[224,134441,5268],{"className":134442},[468,779],[224,134444,375],{"className":134445,"style":492},[468,469,779],[224,134447,134448,134451],{"style":20546},[224,134449],{"className":134450,"style":2108},[771],[224,134452],{"className":134453,"style":20554},[20553],[224,134455,134456,134459],{"style":22589},[224,134457],{"className":134458,"style":2108},[771],[224,134460,134462],{"className":134461},[776,777,778,779],[224,134463,134465],{"className":134464},[468,779],[224,134466,9409],{"className":134467},[468,779],[224,134469,787],{"className":134470},[786],[224,134472,134474],{"className":134473},[759],[224,134475,134477],{"className":134476,"style":22776},[763],[224,134478],{},[224,134480],{"className":134481},[549,8479],[224,134483],{"className":134484,"style":474},[473],[224,134486,5268],{"className":134487},[478],[224,134489],{"className":134490,"style":474},[473],[224,134492,134494,134497],{"className":134493},[459],[224,134495],{"className":134496,"style":510},[463],[224,134498,9409],{"className":134499},[468],"，可与直接展开互相验证。）",[11,134502,134503],{},"可见，f-散度族统一了「不对称\u002F对称」「有界\u002F无界」的多种散度。",[613,134505,134507],{"id":134506},"_62-积分概率度量ipm","6.2 积分概率度量（IPM）",[11,134509,134510,134511,134560],{},"f-散度只依赖似然比 ",[224,134512,134514,134536],{"className":134513,"translate":388},[392],[224,134515,134517],{"className":134516},[396],[398,134518,134519],{"xmlns":400},[403,134520,134521,134533],{},[406,134522,134523,134525,134527,134529,134531],{},[409,134524,13182],{},[409,134526,16018],{},[409,134528,12413],{"mathvariant":678},[409,134530,13182],{},[409,134532,20315],{},[447,134534,134535],{"encoding":449},"dP\u002FdQ",[224,134537,134539],{"className":134538,"ariaHidden":455},[454],[224,134540,134542,134545,134548,134551,134554,134557],{"className":134541},[459],[224,134543],{"className":134544,"style":535},[463],[224,134546,13182],{"className":134547},[468,469],[224,134549,16018],{"className":134550,"style":653},[468,469],[224,134552,12413],{"className":134553},[468],[224,134555,13182],{"className":134556},[468,469],[224,134558,20315],{"className":134559},[468,469],"，因此把两个分布同时「搬到别处」值不变。这带来一个根本局限：",[224,134562,134564],{"className":134563,"translate":388},[387],[224,134565,134567,134622],{"className":134566,"translate":388},[392],[224,134568,134570],{"className":134569},[396],[398,134571,134572],{"xmlns":400,"display":401},[403,134573,134574,134619],{},[406,134575,134576,134578,134580,134582,134584,134586,134588,134591,134593,134595,134601,134603,134605,134607,134609,134615,134617],{},[423,134577,24364],{},[409,134579,16018],{},[412,134581,1088],{},[409,134583,20315],{},[423,134585,15054],{},[412,134587,239],{"stretchy":435},[423,134589,134590],{},"支撑集不重叠，如 ",[409,134592,16018],{},[412,134594,278],{},[695,134596,134597,134599],{},[409,134598,35657],{},[419,134600,284],{},[412,134602,1877],{"separator":455},[423,134604,15054],{},[409,134606,20315],{},[412,134608,278],{},[695,134610,134611,134613],{},[409,134612,35657],{},[419,134614,1438],{},[412,134616,1716],{"stretchy":435},[412,134618,1877],{"separator":455},[447,134620,134621],{"encoding":449},"\\text{若 } P\\perp Q\\\n(\\text{支撑集不重叠，如 } P=\\delta_0,\\ Q=\\delta_1),",[224,134623,134625,134652,134689,134756],{"className":134624,"ariaHidden":455},[454],[224,134626,134628,134631,134640,134643,134646,134649],{"className":134627},[459],[224,134629],{"className":134630,"style":7642},[463],[224,134632,134634,134637],{"className":134633},[468,11373],[224,134635,24661],{"className":134636},[468,16159],[224,134638,15054],{"className":134639},[468],[224,134641,16018],{"className":134642,"style":653},[468,469],[224,134644],{"className":134645,"style":496},[473],[224,134647,1088],{"className":134648},[500],[224,134650],{"className":134651,"style":496},[473],[224,134653,134655,134658,134661,134664,134667,134677,134680,134683,134686],{"className":134654},[459],[224,134656],{"className":134657,"style":535},[463],[224,134659,20315],{"className":134660},[468,469],[224,134662,15054],{"className":134663},[473],[224,134665,239],{"className":134666},[542],[224,134668,134670,134674],{"className":134669},[468,11373],[224,134671,134673],{"className":134672},[468,16159],"支撑集不重叠，如",[224,134675,15054],{"className":134676},[468],[224,134678,16018],{"className":134679,"style":653},[468,469],[224,134681],{"className":134682,"style":496},[473],[224,134684,278],{"className":134685},[500],[224,134687],{"className":134688,"style":496},[473],[224,134690,134692,134695,134735,134738,134741,134744,134747,134750,134753],{"className":134691},[459],[224,134693],{"className":134694,"style":2565},[463],[224,134696,134698,134701],{"className":134697},[468],[224,134699,35657],{"className":134700,"style":35706},[468,469],[224,134702,134704],{"className":134703},[750],[224,134705,134707,134727],{"className":134706},[754,755],[224,134708,134710,134724],{"className":134709},[759],[224,134711,134713],{"className":134712,"style":1479},[763],[224,134714,134715,134718],{"style":114641},[224,134716],{"className":134717,"style":772},[771],[224,134719,134721],{"className":134720},[776,777,778,779],[224,134722,284],{"className":134723},[468,779],[224,134725,787],{"className":134726},[786],[224,134728,134730],{"className":134729},[759],[224,134731,134733],{"className":134732,"style":794},[763],[224,134734],{},[224,134736,1877],{"className":134737},[1917],[224,134739,15054],{"className":134740},[473],[224,134742],{"className":134743,"style":724},[473],[224,134745,20315],{"className":134746},[468,469],[224,134748],{"className":134749,"style":496},[473],[224,134751,278],{"className":134752},[500],[224,134754],{"className":134755,"style":496},[473],[224,134757,134759,134762,134802,134805],{"className":134758},[459],[224,134760],{"className":134761,"style":535},[463],[224,134763,134765,134768],{"className":134764},[468],[224,134766,35657],{"className":134767,"style":35706},[468,469],[224,134769,134771],{"className":134770},[750],[224,134772,134774,134794],{"className":134773},[754,755],[224,134775,134777,134791],{"className":134776},[759],[224,134778,134780],{"className":134779,"style":1479},[763],[224,134781,134782,134785],{"style":114641},[224,134783],{"className":134784,"style":772},[771],[224,134786,134788],{"className":134787},[776,777,778,779],[224,134789,1438],{"className":134790},[468,779],[224,134792,787],{"className":134793},[786],[224,134795,134797],{"className":134796},[759],[224,134798,134800],{"className":134799,"style":794},[763],[224,134801],{},[224,134803,1716],{"className":134804},[549],[224,134806,1877],{"className":134807},[1917],[224,134809,134811],{"className":134810,"translate":388},[387],[224,134812,134814,134864],{"className":134813,"translate":388},[392],[224,134815,134817],{"className":134816},[396],[398,134818,134819],{"xmlns":400,"display":401},[403,134820,134821,134861],{},[406,134822,134823,134833,134835,134837,134839,134841,134843,134845,134855,134857,134859],{},[695,134824,134825,134827],{},[409,134826,20303],{},[406,134828,134829,134831],{},[409,134830,43681],{},[409,134832,6924],{},[412,134834,278],{},[409,134836,24131],{},[412,134838,24134],{},[419,134840,1447],{},[412,134842,1877],{"separator":455},[473,134844],{"width":691},[695,134846,134847,134849],{},[409,134848,20303],{},[406,134850,134851,134853],{},[409,134852,637],{},[409,134854,16059],{},[412,134856,278],{},[419,134858,1438],{},[412,134860,1877],{"separator":455},[447,134862,134863],{"encoding":449},"D_{JS}=\\log2,\\quad D_{TV}=1,",[224,134865,134867,134928,135009],{"className":134866,"ariaHidden":455},[454],[224,134868,134870,134873,134919,134922,134925],{"className":134869},[459],[224,134871],{"className":134872,"style":35905},[463],[224,134874,134876,134879],{"className":134875},[468],[224,134877,20303],{"className":134878,"style":2473},[468,469],[224,134880,134882],{"className":134881},[750],[224,134883,134885,134911],{"className":134884},[754,755],[224,134886,134888,134908],{"className":134887},[759],[224,134889,134891],{"className":134890,"style":764},[763],[224,134892,134893,134896],{"style":20390},[224,134894],{"className":134895,"style":772},[771],[224,134897,134899],{"className":134898},[776,777,778,779],[224,134900,134902,134905],{"className":134901},[468,779],[224,134903,43681],{"className":134904,"style":44091},[468,469,779],[224,134906,6924],{"className":134907,"style":6970},[468,469,779],[224,134909,787],{"className":134910},[786],[224,134912,134914],{"className":134913},[759],[224,134915,134917],{"className":134916,"style":794},[763],[224,134918],{},[224,134920],{"className":134921,"style":496},[473],[224,134923,278],{"className":134924},[500],[224,134926],{"className":134927,"style":496},[473],[224,134929,134931,134934,134939,134942,134945,134948,134951,134954,135000,135003,135006],{"className":134930},[459],[224,134932],{"className":134933,"style":2565},[463],[224,134935,24185,134937],{"className":134936},[21531],[224,134938,24189],{"style":24188},[224,134940],{"className":134941,"style":724},[473],[224,134943,1447],{"className":134944},[468],[224,134946,1877],{"className":134947},[1917],[224,134949],{"className":134950,"style":734},[473],[224,134952],{"className":134953,"style":724},[473],[224,134955,134957,134960],{"className":134956},[468],[224,134958,20303],{"className":134959,"style":2473},[468,469],[224,134961,134963],{"className":134962},[750],[224,134964,134966,134992],{"className":134965},[754,755],[224,134967,134969,134989],{"className":134968},[759],[224,134970,134972],{"className":134971,"style":764},[763],[224,134973,134974,134977],{"style":20390},[224,134975],{"className":134976,"style":772},[771],[224,134978,134980],{"className":134979},[776,777,778,779],[224,134981,134983,134986],{"className":134982},[468,779],[224,134984,637],{"className":134985,"style":653},[468,469,779],[224,134987,16059],{"className":134988,"style":474},[468,469,779],[224,134990,787],{"className":134991},[786],[224,134993,134995],{"className":134994},[759],[224,134996,134998],{"className":134997,"style":794},[763],[224,134999],{},[224,135001],{"className":135002,"style":496},[473],[224,135004,278],{"className":135005},[500],[224,135007],{"className":135008,"style":496},[473],[224,135010,135012,135015,135018],{"className":135011},[459],[224,135013],{"className":135014,"style":8688},[463],[224,135016,1438],{"className":135017},[468],[224,135019,1877],{"className":135020},[1917],[11,135022,135023,135024,135027,135028,135092,135093,135143,135144,12570],{},"也就是说，f-散度在支撑分离时",[378,135025,135026],{},"饱和到常数上界，对生成器参数的微小改动失去梯度","：因为被积分的量 ",[224,135029,135031,135059],{"className":135030,"translate":388},[392],[224,135032,135034],{"className":135033},[396],[398,135035,135036],{"xmlns":400},[403,135037,135038,135056],{},[406,135039,135040,135042,135044,135046,135048,135050,135052,135054],{},[409,135041,2023],{},[423,135043,682],{},[409,135045,1705],{},[412,135047,239],{"stretchy":435},[409,135049,11],{},[409,135051,12413],{"mathvariant":678},[409,135053,2023],{},[412,135055,1716],{"stretchy":435},[447,135057,135058],{"encoding":449},"q\\,f(p\u002Fq)",[224,135060,135062],{"className":135061,"ariaHidden":455},[454],[224,135063,135065,135068,135071,135074,135077,135080,135083,135086,135089],{"className":135064},[459],[224,135066],{"className":135067,"style":535},[463],[224,135069,2023],{"className":135070,"style":492},[468,469],[224,135072],{"className":135073,"style":724},[473],[224,135075,1705],{"className":135076,"style":1740},[468,469],[224,135078,239],{"className":135079},[542],[224,135081,11],{"className":135082},[468,469],[224,135084,12413],{"className":135085},[468],[224,135087,2023],{"className":135088,"style":492},[468,469],[224,135090,1716],{"className":135091},[549]," 只在 ",[224,135094,135096,135113],{"className":135095,"translate":388},[392],[224,135097,135099],{"className":135098},[396],[398,135100,135101],{"xmlns":400},[403,135102,135103,135111],{},[406,135104,135105,135107,135109],{},[409,135106,2023],{},[412,135108,325],{},[419,135110,284],{},[447,135112,51610],{"encoding":449},[224,135114,135116,135134],{"className":135115,"ariaHidden":455},[454],[224,135117,135119,135122,135125,135128,135131],{"className":135118},[459],[224,135120],{"className":135121,"style":488},[463],[224,135123,2023],{"className":135124,"style":492},[468,469],[224,135126],{"className":135127,"style":496},[473],[224,135129,325],{"className":135130},[500],[224,135132],{"className":135133,"style":496},[473],[224,135135,135137,135140],{"className":135136},[459],[224,135138],{"className":135139,"style":510},[463],[224,135141,284],{"className":135142},[468]," 处有定义，当两个支撑几乎不重叠时，极小化散度的博弈只剩\"常数项 − 常数项\"，梯度信息全部丢失（这正是原 GAN 训练崩溃的理论根源）。Wasserstein 距离与 MMD 属于另一族——",[378,135145,135146],{},"积分概率度量",[224,135148,135150],{"className":135149,"translate":388},[387],[224,135151,135153,135228],{"className":135152,"translate":388},[392],[224,135154,135156],{"className":135155},[396],[398,135157,135158],{"xmlns":400,"display":401},[403,135159,135160,135225],{},[406,135161,135162,135169,135171,135173,135175,135177,135179,135181,135197,135199,135201,135203,135205,135207,135209,135211,135213,135215,135217,135219,135221,135223],{},[695,135163,135164,135167],{},[423,135165,135166],{},"IPM",[409,135168,56805],{"mathvariant":20667},[412,135170,239],{"stretchy":435},[409,135172,16018],{},[412,135174,1877],{"separator":455},[409,135176,20315],{},[412,135178,1716],{"stretchy":435},[412,135180,278],{},[1965,135182,135183,135189],{},[406,135184,135185,135187],{},[409,135186,55354],{},[412,135188,24134],{},[406,135190,135191,135193,135195],{},[409,135192,58586],{},[412,135194,12248],{},[409,135196,56805],{"mathvariant":20667},[412,135198,1083],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[412,135200,21338],{},[409,135202,58586],{},[423,135204,682],{},[409,135206,13182],{},[409,135208,16018],{},[412,135210,5268],{},[412,135212,21338],{},[409,135214,58586],{},[423,135216,682],{},[409,135218,13182],{},[409,135220,20315],{},[412,135222,1083],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[409,135224,688],{"mathvariant":678},[447,135226,135227],{"encoding":449},"\\text{IPM}_{\\mathcal F}(P,Q)\n=\\sup_{h\\in\\mathcal F}\\Big|\\int h\\,dP\n-\\int h\\,dQ\\Big| .",[224,135229,135231,135311,135444],{"className":135230,"ariaHidden":455},[454],[224,135232,135234,135237,135284,135287,135290,135293,135296,135299,135302,135305,135308],{"className":135233},[459],[224,135235],{"className":135236,"style":535},[463],[224,135238,135240,135246],{"className":135239},[468],[224,135241,135243],{"className":135242},[468,11373],[224,135244,135166],{"className":135245},[468],[224,135247,135249],{"className":135248},[750],[224,135250,135252,135276],{"className":135251},[754,755],[224,135253,135255,135273],{"className":135254},[759],[224,135256,135258],{"className":135257,"style":764},[763],[224,135259,135260,135263],{"style":72353},[224,135261],{"className":135262,"style":772},[771],[224,135264,135266],{"className":135265},[776,777,778,779],[224,135267,135269],{"className":135268},[468,779],[224,135270,56805],{"className":135271,"style":135272},[468,20693,779],"margin-right:0.0993em;",[224,135274,787],{"className":135275},[786],[224,135277,135279],{"className":135278},[759],[224,135280,135282],{"className":135281,"style":794},[763],[224,135283],{},[224,135285,239],{"className":135286},[542],[224,135288,16018],{"className":135289,"style":653},[468,469],[224,135291,1877],{"className":135292},[1917],[224,135294],{"className":135295,"style":724},[473],[224,135297,20315],{"className":135298},[468,469],[224,135300,1716],{"className":135301},[549],[224,135303],{"className":135304,"style":496},[473],[224,135306,278],{"className":135307},[500],[224,135309],{"className":135310,"style":496},[473],[224,135312,135314,135318,135371,135374,135414,135417,135420,135423,135426,135429,135432,135435,135438,135441],{"className":135313},[459],[224,135315],{"className":135316,"style":135317},[463],"height:2.3339em;vertical-align:-0.9739em;",[224,135319,135321],{"className":135320},[21531,33665],[224,135322,135324,135363],{"className":135323},[754,755],[224,135325,135327,135360],{"className":135326},[759],[224,135328,135330,135350],{"className":135329,"style":839},[763],[224,135331,135332,135335],{"style":71613},[224,135333],{"className":135334,"style":2108},[771],[224,135336,135338],{"className":135337},[776,777,778,779],[224,135339,135341,135344,135347],{"className":135340},[468,779],[224,135342,58586],{"className":135343},[468,469,779],[224,135345,12248],{"className":135346},[500,779],[224,135348,56805],{"className":135349,"style":135272},[468,20693,779],[224,135351,135352,135355],{"style":2120},[224,135353],{"className":135354,"style":2108},[771],[224,135356,135357],{},[224,135358,55354],{"className":135359},[21531],[224,135361,787],{"className":135362},[786],[224,135364,135366],{"className":135365},[759],[224,135367,135369],{"className":135368,"style":71657},[763],[224,135370],{},[224,135372],{"className":135373,"style":724},[473],[224,135375,135377],{"className":135376},[468],[224,135378,135380],{"className":135379},[2363,34778],[224,135381,135383,135406],{"className":135382},[754,755],[224,135384,135386,135403],{"className":135385},[759],[224,135387,135389],{"className":135388,"style":105417},[763],[224,135390,135391,135394],{"style":93430},[224,135392],{"className":135393,"style":102490},[771],[224,135395,135397],{"style":135396},"width:0.333em;height:1.8em;",[2155,135398,135400],{"xmlns":2157,"width":54973,"height":37382,"viewBox":135399},"0 0 333 1800",[2163,135401],{"d":135402},"M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z",[224,135404,787],{"className":135405},[786],[224,135407,135409],{"className":135408},[759],[224,135410,135412],{"className":135411,"style":105441},[763],[224,135413],{},[224,135415],{"className":135416,"style":724},[473],[224,135418,21338],{"className":135419,"style":21534},[21531,21532,21533],[224,135421],{"className":135422,"style":724},[473],[224,135424,58586],{"className":135425},[468,469],[224,135427],{"className":135428,"style":724},[473],[224,135430,13182],{"className":135431},[468,469],[224,135433,16018],{"className":135434,"style":653},[468,469],[224,135436],{"className":135437,"style":474},[473],[224,135439,5268],{"className":135440},[478],[224,135442],{"className":135443,"style":474},[473],[224,135445,135447,135450,135453,135456,135459,135462,135465,135468,135505],{"className":135446},[459],[224,135448],{"className":135449,"style":25785},[463],[224,135451,21338],{"className":135452,"style":21534},[21531,21532,21533],[224,135454],{"className":135455,"style":724},[473],[224,135457,58586],{"className":135458},[468,469],[224,135460],{"className":135461,"style":724},[473],[224,135463,13182],{"className":135464},[468,469],[224,135466,20315],{"className":135467},[468,469],[224,135469,135471],{"className":135470},[468],[224,135472,135474],{"className":135473},[2363,34778],[224,135475,135477,135497],{"className":135476},[754,755],[224,135478,135480,135494],{"className":135479},[759],[224,135481,135483],{"className":135482,"style":105417},[763],[224,135484,135485,135488],{"style":93430},[224,135486],{"className":135487,"style":102490},[771],[224,135489,135490],{"style":135396},[2155,135491,135492],{"xmlns":2157,"width":54973,"height":37382,"viewBox":135399},[2163,135493],{"d":135402},[224,135495,787],{"className":135496},[786],[224,135498,135500],{"className":135499},[759],[224,135501,135503],{"className":135502,"style":105441},[763],[224,135504],{},[224,135506,688],{"className":135507},[468],[11,135509,135510,135511,135540,135541,135592,135593,135622,135623,135673,135674,135702,135703,135753,135754,135782],{},"它们比较的是「在函数类 ",[224,135512,135514,135528],{"className":135513,"translate":388},[392],[224,135515,135517],{"className":135516},[396],[398,135518,135519],{"xmlns":400},[403,135520,135521,135525],{},[406,135522,135523],{},[409,135524,56805],{"mathvariant":20667},[447,135526,135527],{"encoding":449},"\\mathcal F",[224,135529,135531],{"className":135530,"ariaHidden":455},[454],[224,135532,135534,135537],{"className":135533},[459],[224,135535],{"className":135536,"style":649},[463],[224,135538,56805],{"className":135539,"style":135272},[468,20693]," 上积分的差」，而不是似然比，因此利用了样本空间的度量\u002F几何结构。KS 距离（",[224,135542,135544,135562],{"className":135543,"translate":388},[392],[224,135545,135547],{"className":135546},[396],[398,135548,135549],{"xmlns":400},[403,135550,135551,135559],{},[406,135552,135553,135555,135557],{},[409,135554,56805],{"mathvariant":20667},[412,135556,278],{},[412,135558,8110],{"stretchy":435},[447,135560,135561],{"encoding":449},"\\mathcal F=\\{",[224,135563,135565,135583],{"className":135564,"ariaHidden":455},[454],[224,135566,135568,135571,135574,135577,135580],{"className":135567},[459],[224,135569],{"className":135570,"style":649},[463],[224,135572,56805],{"className":135573,"style":135272},[468,20693],[224,135575],{"className":135576,"style":496},[473],[224,135578,278],{"className":135579},[500],[224,135581],{"className":135582,"style":496},[473],[224,135584,135586,135589],{"className":135585},[459],[224,135587],{"className":135588,"style":535},[463],[224,135590,8110],{"className":135591},[542],"一维区间指示函数",[224,135594,135596,135610],{"className":135595,"translate":388},[392],[224,135597,135599],{"className":135598},[396],[398,135600,135601],{"xmlns":400},[403,135602,135603,135607],{},[406,135604,135605],{},[412,135606,15077],{"stretchy":435},[447,135608,135609],{"encoding":449},"\\}",[224,135611,135613],{"className":135612,"ariaHidden":455},[454],[224,135614,135616,135619],{"className":135615},[459],[224,135617],{"className":135618,"style":535},[463],[224,135620,15077],{"className":135621},[549],"）、Wasserstein（",[224,135624,135626,135643],{"className":135625,"translate":388},[392],[224,135627,135629],{"className":135628},[396],[398,135630,135631],{"xmlns":400},[403,135632,135633,135641],{},[406,135634,135635,135637,135639],{},[409,135636,56805],{"mathvariant":20667},[412,135638,278],{},[412,135640,8110],{"stretchy":435},[447,135642,135561],{"encoding":449},[224,135644,135646,135664],{"className":135645,"ariaHidden":455},[454],[224,135647,135649,135652,135655,135658,135661],{"className":135648},[459],[224,135650],{"className":135651,"style":649},[463],[224,135653,56805],{"className":135654,"style":135272},[468,20693],[224,135656],{"className":135657,"style":496},[473],[224,135659,278],{"className":135660},[500],[224,135662],{"className":135663,"style":496},[473],[224,135665,135667,135670],{"className":135666},[459],[224,135668],{"className":135669,"style":535},[463],[224,135671,8110],{"className":135672},[542],"1-Lipschitz 函数",[224,135675,135677,135690],{"className":135676,"translate":388},[392],[224,135678,135680],{"className":135679},[396],[398,135681,135682],{"xmlns":400},[403,135683,135684,135688],{},[406,135685,135686],{},[412,135687,15077],{"stretchy":435},[447,135689,135609],{"encoding":449},[224,135691,135693],{"className":135692,"ariaHidden":455},[454],[224,135694,135696,135699],{"className":135695},[459],[224,135697],{"className":135698,"style":535},[463],[224,135700,15077],{"className":135701},[549],"）、MMD（",[224,135704,135706,135723],{"className":135705,"translate":388},[392],[224,135707,135709],{"className":135708},[396],[398,135710,135711],{"xmlns":400},[403,135712,135713,135721],{},[406,135714,135715,135717,135719],{},[409,135716,56805],{"mathvariant":20667},[412,135718,278],{},[412,135720,8110],{"stretchy":435},[447,135722,135561],{"encoding":449},[224,135724,135726,135744],{"className":135725,"ariaHidden":455},[454],[224,135727,135729,135732,135735,135738,135741],{"className":135728},[459],[224,135730],{"className":135731,"style":649},[463],[224,135733,56805],{"className":135734,"style":135272},[468,20693],[224,135736],{"className":135737,"style":496},[473],[224,135739,278],{"className":135740},[500],[224,135742],{"className":135743,"style":496},[473],[224,135745,135747,135750],{"className":135746},[459],[224,135748],{"className":135749,"style":535},[463],[224,135751,8110],{"className":135752},[542],"RKHS 单位球",[224,135755,135757,135770],{"className":135756,"translate":388},[392],[224,135758,135760],{"className":135759},[396],[398,135761,135762],{"xmlns":400},[403,135763,135764,135768],{},[406,135765,135766],{},[412,135767,15077],{"stretchy":435},[447,135769,135609],{"encoding":449},[224,135771,135773],{"className":135772,"ariaHidden":455},[454],[224,135774,135776,135779],{"className":135775},[459],[224,135777],{"className":135778,"style":535},[463],[224,135780,15077],{"className":135781},[549],"）都是 IPM。上面对 f-散度的变分表示其实是「带惩罚的 IPM」：",[224,135784,135786],{"className":135785,"translate":388},[387],[224,135787,135789,135871],{"className":135788,"translate":388},[392],[224,135790,135792],{"className":135791},[396],[398,135793,135794],{"xmlns":400,"display":401},[403,135795,135796,135868],{},[406,135797,135798,135804,135806,135808,135810,135812,135814,135816,135826,135828,135834,135836,135838,135840,135842,135848,135850,135856,135858,135860,135862,135864,135866],{},[695,135799,135800,135802],{},[409,135801,20303],{},[409,135803,1705],{},[412,135805,239],{"stretchy":435},[409,135807,16018],{},[409,135809,20312],{"mathvariant":678},[409,135811,20315],{},[412,135813,1716],{"stretchy":435},[412,135815,278],{},[1965,135817,135818,135824],{},[406,135819,135820,135822],{},[409,135821,55354],{},[412,135823,24134],{},[409,135825,24189],{},[412,135827,8110],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[695,135829,135830,135832],{},[409,135831,20324],{"mathvariant":12251},[409,135833,16018],{},[412,135835,268],{"stretchy":435},[409,135837,24189],{},[412,135839,440],{"stretchy":435},[412,135841,5268],{},[695,135843,135844,135846],{},[409,135845,20324],{"mathvariant":12251},[409,135847,20315],{},[412,135849,268],{"stretchy":435},[3248,135851,135852,135854],{},[409,135853,1705],{},[412,135855,27543],{},[412,135857,239],{"stretchy":435},[409,135859,24189],{},[412,135861,1716],{"stretchy":435},[412,135863,440],{"stretchy":435},[412,135865,15077],{"fence":435,"stretchy":455,"minsize":37382,"maxsize":37382},[412,135867,1877],{"separator":455},[447,135869,135870],{"encoding":449},"D_f(P\\|Q)\n=\\sup_{g}\\Big\\{\\mathbb E_P[g]\n-\\mathbb E_Q[f^*(g)]\\Big\\},",[224,135872,135874,135944,136064],{"className":135873,"ariaHidden":455},[454],[224,135875,135877,135880,135920,135923,135926,135929,135932,135935,135938,135941],{"className":135876},[459],[224,135878],{"className":135879,"style":20368},[463],[224,135881,135883,135886],{"className":135882},[468],[224,135884,20303],{"className":135885,"style":2473},[468,469],[224,135887,135889],{"className":135888},[750],[224,135890,135892,135912],{"className":135891},[754,755],[224,135893,135895,135909],{"className":135894},[759],[224,135896,135898],{"className":135897,"style":20387},[763],[224,135899,135900,135903],{"style":20390},[224,135901],{"className":135902,"style":772},[771],[224,135904,135906],{"className":135905},[776,777,778,779],[224,135907,1705],{"className":135908,"style":1740},[468,469,779],[224,135910,787],{"className":135911},[786],[224,135913,135915],{"className":135914},[759],[224,135916,135918],{"className":135917,"style":20409},[763],[224,135919],{},[224,135921,239],{"className":135922},[542],[224,135924,16018],{"className":135925,"style":653},[468,469],[224,135927,20312],{"className":135928},[468],[224,135930,20315],{"className":135931},[468,469],[224,135933,1716],{"className":135934},[549],[224,135936],{"className":135937,"style":496},[473],[224,135939,278],{"className":135940},[500],[224,135942],{"className":135943,"style":496},[473],[224,135945,135947,135950,135997,136000,136006,136046,136049,136052,136055,136058,136061],{"className":135946},[459],[224,135948],{"className":135949,"style":119381},[463],[224,135951,135953],{"className":135952},[21531,33665],[224,135954,135956,135989],{"className":135955},[754,755],[224,135957,135959,135986],{"className":135958},[759],[224,135960,135962,135976],{"className":135961,"style":839},[763],[224,135963,135964,135967],{"style":81096},[224,135965],{"className":135966,"style":2108},[771],[224,135968,135970],{"className":135969},[776,777,778,779],[224,135971,135973],{"className":135972},[468,779],[224,135974,24189],{"className":135975,"style":492},[468,469,779],[224,135977,135978,135981],{"style":2120},[224,135979],{"className":135980,"style":2108},[771],[224,135982,135983],{},[224,135984,55354],{"className":135985},[21531],[224,135987,787],{"className":135988},[786],[224,135990,135992],{"className":135991},[759],[224,135993,135995],{"className":135994,"style":81128},[763],[224,135996],{},[224,135998],{"className":135999,"style":724},[473],[224,136001,136003],{"className":136002},[468],[224,136004,8110],{"className":136005},[2363,23751],[224,136007,136009,136012],{"className":136008},[468],[224,136010,20324],{"className":136011},[468,12389],[224,136013,136015],{"className":136014},[750],[224,136016,136018,136038],{"className":136017},[754,755],[224,136019,136021,136035],{"className":136020},[759],[224,136022,136024],{"className":136023,"style":764},[763],[224,136025,136026,136029],{"style":767},[224,136027],{"className":136028,"style":772},[771],[224,136030,136032],{"className":136031},[776,777,778,779],[224,136033,16018],{"className":136034,"style":653},[468,469,779],[224,136036,787],{"className":136037},[786],[224,136039,136041],{"className":136040},[759],[224,136042,136044],{"className":136043,"style":794},[763],[224,136045],{},[224,136047,268],{"className":136048},[542],[224,136050,24189],{"className":136051,"style":492},[468,469],[224,136053,440],{"className":136054},[549],[224,136056],{"className":136057,"style":474},[473],[224,136059,5268],{"className":136060},[478],[224,136062],{"className":136063,"style":474},[473],[224,136065,136067,136070,136110,136113,136142,136145,136148,136151,136157],{"className":136066},[459],[224,136068],{"className":136069,"style":88023},[463],[224,136071,136073,136076],{"className":136072},[468],[224,136074,20324],{"className":136075},[468,12389],[224,136077,136079],{"className":136078},[750],[224,136080,136082,136102],{"className":136081},[754,755],[224,136083,136085,136099],{"className":136084},[759],[224,136086,136088],{"className":136087,"style":764},[763],[224,136089,136090,136093],{"style":767},[224,136091],{"className":136092,"style":772},[771],[224,136094,136096],{"className":136095},[776,777,778,779],[224,136097,20315],{"className":136098},[468,469,779],[224,136100,787],{"className":136101},[786],[224,136103,136105],{"className":136104},[759],[224,136106,136108],{"className":136107,"style":20409},[763],[224,136109],{},[224,136111,268],{"className":136112},[542],[224,136114,136116,136119],{"className":136115},[468],[224,136117,1705],{"className":136118,"style":1740},[468,469],[224,136120,136122],{"className":136121},[750],[224,136123,136125],{"className":136124},[754],[224,136126,136128],{"className":136127},[759],[224,136129,136131],{"className":136130,"style":28217},[763],[224,136132,136133,136136],{"style":5074},[224,136134],{"className":136135,"style":772},[771],[224,136137,136139],{"className":136138},[776,777,778,779],[224,136140,27543],{"className":136141},[478,779],[224,136143,239],{"className":136144},[542],[224,136146,24189],{"className":136147,"style":492},[468,469],[224,136149,40329],{"className":136150},[549],[224,136152,136154],{"className":136153},[468],[224,136155,15077],{"className":136156},[2363,23751],[224,136158,1877],{"className":136159},[1917],[11,136161,136162,136163,136237,136238,136280],{},"它的被估计量是 ",[224,136164,136166,136190],{"className":136165,"translate":388},[392],[224,136167,136169],{"className":136168},[396],[398,136170,136171],{"xmlns":400},[403,136172,136173,136187],{},[406,136174,136175,136181,136183,136185],{},[3248,136176,136177,136179],{},[409,136178,1705],{},[412,136180,27543],{},[412,136182,239],{"stretchy":435},[409,136184,24189],{},[412,136186,1716],{"stretchy":435},[447,136188,136189],{"encoding":449},"f^*(g)",[224,136191,136193],{"className":136192,"ariaHidden":455},[454],[224,136194,136196,136199,136228,136231,136234],{"className":136195},[459],[224,136197],{"className":136198,"style":535},[463],[224,136200,136202,136205],{"className":136201},[468],[224,136203,1705],{"className":136204,"style":1740},[468,469],[224,136206,136208],{"className":136207},[750],[224,136209,136211],{"className":136210},[754],[224,136212,136214],{"className":136213},[759],[224,136215,136217],{"className":136216,"style":28465},[763],[224,136218,136219,136222],{"style":3288},[224,136220],{"className":136221,"style":772},[771],[224,136223,136225],{"className":136224},[776,777,778,779],[224,136226,27543],{"className":136227},[478,779],[224,136229,239],{"className":136230},[542],[224,136232,24189],{"className":136233,"style":492},[468,469],[224,136235,1716],{"className":136236},[549],"，而不是 ",[224,136239,136241,136259],{"className":136240,"translate":388},[392],[224,136242,136244],{"className":136243},[396],[398,136245,136246],{"xmlns":400},[403,136247,136248,136256],{},[406,136249,136250,136252,136254],{},[409,136251,1083],{"mathvariant":678},[412,136253,23331],{"lspace":5213,"rspace":5213},[409,136255,1083],{"mathvariant":678},[447,136257,136258],{"encoding":449},"|{\\cdot}|",[224,136260,136262],{"className":136261,"ariaHidden":455},[454],[224,136263,136265,136268,136271,136277],{"className":136264},[459],[224,136266],{"className":136267,"style":535},[463],[224,136269,1083],{"className":136270},[468],[224,136272,136274],{"className":136273},[468],[224,136275,23331],{"className":136276},[468],[224,136278,1083],{"className":136279},[468],"，所以是「非对称 IPM」。",[11,136282,136283,136285],{},[378,136284,26477],{}," Wasserstein\u002FMMD 不是 f-散度，也不能写成 f-散度；它们的关键区别在于是否利用几何、是否在支撑分离时仍给梯度、以及是否满足数据处理不等式（Wasserstein 一般不满足数据处理不等式，因为传输方案可以随条件分布\"重新安排\"质量；MMD 由核特征映射的条件期望是收缩算子，沿核的具体结构可证其满足数据处理不等式）。",[613,136287,136289],{"id":136288},"_63-支撑不重叠的极限","6.3 支撑不重叠的极限",[72399,136291,136292,136372],{},[72402,136293,136294],{},[72405,136295,136296,136299,136301,136303,136305,136307,136367,136369],{},[72408,136297,136298],{},"情形",[72408,136300,52588],{},[72408,136302,128753],{},[72408,136304,73031],{},[72408,136306,106664],{},[72408,136308,136309],{},[224,136310,136312,136329],{"className":136311,"translate":388},[392],[224,136313,136315],{"className":136314},[396],[398,136316,136317],{"xmlns":400},[403,136318,136319,136327],{},[406,136320,136321],{},[3248,136322,136323,136325],{},[409,136324,57370],{},[419,136326,1447],{},[447,136328,60308],{"encoding":449},[224,136330,136332],{"className":136331,"ariaHidden":455},[454],[224,136333,136335,136338],{"className":136334},[459],[224,136336],{"className":136337,"style":60318},[463],[224,136339,136341,136344],{"className":136340},[468],[224,136342,57370],{"className":136343},[468,469],[224,136345,136347],{"className":136346},[750],[224,136348,136350],{"className":136349},[754],[224,136351,136353],{"className":136352},[759],[224,136354,136356],{"className":136355,"style":5671},[763],[224,136357,136358,136361],{"style":3288},[224,136359],{"className":136360,"style":772},[771],[224,136362,136364],{"className":136363},[776,777,778,779],[224,136365,1447],{"className":136366},[468,779],[72408,136368,125067],{},[72408,136370,136371],{},"Wasserstein",[72497,136373,136374,136784],{},[72405,136375,136376,136547,136582,136617,136659,136689,136724,136754],{},[72502,136377,136378,136546],{},[224,136379,136381,136415],{"className":136380,"translate":388},[392],[224,136382,136384],{"className":136383},[396],[398,136385,136386],{"xmlns":400},[403,136387,136388,136412],{},[406,136389,136390,136392,136394,136400,136402,136404,136406],{},[409,136391,16018],{},[412,136393,278],{},[695,136395,136396,136398],{},[409,136397,35657],{},[419,136399,284],{},[412,136401,1877],{"separator":455},[409,136403,20315],{},[412,136405,278],{},[695,136407,136408,136410],{},[409,136409,35657],{},[419,136411,1438],{},[447,136413,136414],{"encoding":449},"P=\\delta_0,Q=\\delta_1",[224,136416,136418,136436,136500],{"className":136417,"ariaHidden":455},[454],[224,136419,136421,136424,136427,136430,136433],{"className":136420},[459],[224,136422],{"className":136423,"style":649},[463],[224,136425,16018],{"className":136426,"style":653},[468,469],[224,136428],{"className":136429,"style":496},[473],[224,136431,278],{"className":136432},[500],[224,136434],{"className":136435,"style":496},[473],[224,136437,136439,136442,136482,136485,136488,136491,136494,136497],{"className":136438},[459],[224,136440],{"className":136441,"style":2565},[463],[224,136443,136445,136448],{"className":136444},[468],[224,136446,35657],{"className":136447,"style":35706},[468,469],[224,136449,136451],{"className":136450},[750],[224,136452,136454,136474],{"className":136453},[754,755],[224,136455,136457,136471],{"className":136456},[759],[224,136458,136460],{"className":136459,"style":1479},[763],[224,136461,136462,136465],{"style":114641},[224,136463],{"className":136464,"style":772},[771],[224,136466,136468],{"className":136467},[776,777,778,779],[224,136469,284],{"className":136470},[468,779],[224,136472,787],{"className":136473},[786],[224,136475,136477],{"className":136476},[759],[224,136478,136480],{"className":136479,"style":794},[763],[224,136481],{},[224,136483,1877],{"className":136484},[1917],[224,136486],{"className":136487,"style":724},[473],[224,136489,20315],{"className":136490},[468,469],[224,136492],{"className":136493,"style":496},[473],[224,136495,278],{"className":136496},[500],[224,136498],{"className":136499,"style":496},[473],[224,136501,136503,136506],{"className":136502},[459],[224,136504],{"className":136505,"style":31429},[463],[224,136507,136509,136512],{"className":136508},[468],[224,136510,35657],{"className":136511,"style":35706},[468,469],[224,136513,136515],{"className":136514},[750],[224,136516,136518,136538],{"className":136517},[754,755],[224,136519,136521,136535],{"className":136520},[759],[224,136522,136524],{"className":136523,"style":1479},[763],[224,136525,136526,136529],{"style":114641},[224,136527],{"className":136528,"style":772},[771],[224,136530,136532],{"className":136531},[776,777,778,779],[224,136533,1438],{"className":136534},[468,779],[224,136536,787],{"className":136537},[786],[224,136539,136541],{"className":136540},[759],[224,136542,136544],{"className":136543,"style":794},[763],[224,136545],{},"（约）",[72502,136548,136549],{},[224,136550,136552,136567],{"className":136551,"translate":388},[392],[224,136553,136555],{"className":136554},[396],[398,136556,136557],{"xmlns":400},[403,136558,136559,136565],{},[406,136560,136561,136563],{},[412,136562,319],{},[409,136564,21085],{"mathvariant":678},[447,136566,24272],{"encoding":449},[224,136568,136570],{"className":136569,"ariaHidden":455},[454],[224,136571,136573,136576,136579],{"className":136572},[459],[224,136574],{"className":136575,"style":464},[463],[224,136577,319],{"className":136578},[468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e^{10^4}",[224,137153,137155,137167],{"className":137154,"ariaHidden":455},[454],[224,137156,137158,137161,137164],{"className":137157},[459],[224,137159],{"className":137160,"style":137054},[463],[224,137162,52884],{"className":137163},[500],[224,137165],{"className":137166,"style":496},[473],[224,137168,137170,137174],{"className":137169},[459],[224,137171],{"className":137172,"style":137173},[463],"height:0.9869em;",[224,137175,137177,137180],{"className":137176},[468],[224,137178,1861],{"className":137179},[468,469],[224,137181,137183],{"className":137182},[750],[224,137184,137186],{"className":137185},[754],[224,137187,137189],{"className":137188},[759],[224,137190,137192],{"className":137191,"style":137173},[763],[224,137193,137194,137197],{"style":3288},[224,137195],{"className":137196,"style":772},[771],[224,137198,137200],{"className":137199},[776,777,778,779],[224,137201,137203,137206],{"className":137202},[468,779],[224,137204,1438],{"className":137205},[468,779],[224,137207,137209,137212],{"className":137208},[468,779],[224,137210,284],{"className":137211},[468,779],[224,137213,137215],{"className":137214},[750],[224,137216,137218],{"className":137217},[754],[224,137219,137221],{"className":137220},[759],[224,137222,137224],{"className":137223,"style":58433},[763],[224,137225,137226,137229],{"style":58436},[224,137227],{"className":137228,"style":50136},[771],[224,137230,137232],{"className":137231},[776,50140,2364,779],[224,137233,9409],{"className":137234},[468,779],[72502,137236,137237],{},[224,137238,137240,137256],{"className":137239,"translate":388},[392],[224,137241,137243],{"className":137242},[396],[398,137244,137245],{"xmlns":400},[403,137246,137247,137253],{},[406,137248,137249,137251],{},[412,137250,52884],{},[419,137252,1447],{},[447,137254,137255],{"encoding":449},"\\approx2",[224,137257,137259,137271],{"className":137258,"ariaHidden":455},[454],[224,137260,137262,137265,137268],{"className":137261},[459],[224,137263],{"className":137264,"style":137054},[463],[224,137266,52884],{"className":137267},[500],[224,137269],{"className":137270,"style":496},[473],[224,137272,137274,137277],{"className":137273},[459],[224,137275],{"className":137276,"style":510},[463],[224,137278,1447],{"className":137279},[468],[72502,137281,137282],{},[224,137283,137285,137301],{"className":137284,"translate":388},[392],[224,137286,137288],{"className":137287},[396],[398,137289,137290],{"xmlns":400},[403,137291,137292,137298],{},[406,137293,137294,137296],{},[412,137295,278],{},[419,137297,272],{},[447,137299,137300],{"encoding":449},"=100",[224,137302,137304,137316],{"className":137303,"ariaHidden":455},[454],[224,137305,137307,137310,137313],{"className":137306},[459],[224,137308],{"className":137309,"style":30960},[463],[224,137311,278],{"className":137312},[500],[224,137314],{"className":137315,"style":496},[473],[224,137317,137319,137322],{"className":137318},[459],[224,137320],{"className":137321,"style":510},[463],[224,137323,272],{"className":137324},[468],[11,137326,137327],{},"f-散度在第二行全部「饱和到下界\u002F上界」，而 Wasserstein 随距离线性增长——这正是词嵌入、最优传输、Wasserstein GAN 的动机。",[608,137329,137331],{"id":137330},"_7-机器学习应用","7. 机器学习应用",[11,137333,137334],{},"f-散度族之所以重要，是因为：",[9997,137336,137337,137343,137353,137388],{},[2677,137338,137339,137342],{},[378,137340,137341],{},"统一框架","：许多生成模型的目标函数可以写成 f-散度。",[2677,137344,137345,137348,137349,137352],{},[378,137346,137347],{},"变分估计","：通过凸共轭，可以用神经网络估计 f-散度，这就是 ",[378,137350,137351],{},"f-GAN"," 的核心思想。",[2677,137354,137355,137358,137359,137387],{},[378,137356,137357],{},"灵活选择","：不同 ",[224,137360,137362,137375],{"className":137361,"translate":388},[392],[224,137363,137365],{"className":137364},[396],[398,137366,137367],{"xmlns":400},[403,137368,137369,137373],{},[406,137370,137371],{},[409,137372,1705],{},[447,137374,1705],{"encoding":449},[224,137376,137378],{"className":137377,"ariaHidden":455},[454],[224,137379,137381,137384],{"className":137380},[459],[224,137382],{"className":137383,"style":2565},[463],[224,137385,1705],{"className":137386,"style":1740},[468,469]," 对应不同训练性质。例如 JS 有界但可能梯度消失，KL 无界但敏感。",[2677,137389,137390,137393],{},[378,137391,137392],{},"理论分析","：数据处理不等式、凸性、对偶表示等为收敛性和泛化分析提供工具。",[11,137395,137396],{},"下面把这几条落到具体场景。",[613,137398,137400],{"id":137399},"_71-f-gan","7.1 f-GAN",[11,137402,137403,137404,137532,137533,137561,137562,137590],{},"见第 5 节。核心是：",[224,137405,137407,137439],{"className":137406,"translate":388},[392],[224,137408,137410],{"className":137409},[396],[398,137411,137412],{"xmlns":400},[403,137413,137414,137436],{},[406,137415,137416,137426],{},[695,137417,137418,137424],{},[406,137419,137420,137422],{},[409,137421,71515],{},[412,137423,24134],{},[409,137425,120653],{},[695,137427,137428,137434],{},[406,137429,137430,137432],{},[409,137431,72294],{},[412,137433,24134],{},[409,137435,119757],{},[447,137437,137438],{"encoding":449},"\\min_\\theta\\max_\\omega",[224,137440,137442],{"className":137441,"ariaHidden":455},[454],[224,137443,137445,137449,137489,137492],{"className":137444},[459],[224,137446],{"className":137447,"style":137448},[463],"height:0.8179em;vertical-align:-0.15em;",[224,137450,137452,137455],{"className":137451},[21531],[224,137453,71515],{"className":137454},[21531],[224,137456,137458],{"className":137457},[750],[224,137459,137461,137481],{"className":137460},[754,755],[224,137462,137464,137478],{"className":137463},[759],[224,137465,137467],{"className":137466,"style":20387},[763],[224,137468,137469,137472],{"style":72353},[224,137470],{"className":137471,"style":772},[771],[224,137473,137475],{"className":137474},[776,777,778,779],[224,137476,120653],{"className":137477,"style":2473},[468,469,779],[224,137479,787],{"className":137480},[786],[224,137482,137484],{"className":137483},[759],[224,137485,137487],{"className":137486,"style":794},[763],[224,137488],{},[224,137490],{"className":137491,"style":724},[473],[224,137493,137495,137498],{"className":137494},[21531],[224,137496,72294],{"className":137497},[21531],[224,137499,137501],{"className":137500},[750],[224,137502,137504,137524],{"className":137503},[754,755],[224,137505,137507,137521],{"className":137506},[759],[224,137508,137510],{"className":137509,"style":64841},[763],[224,137511,137512,137515],{"style":72353},[224,137513],{"className":137514,"style":772},[771],[224,137516,137518],{"className":137517},[776,777,778,779],[224,137519,119757],{"className":137520,"style":492},[468,469,779],[224,137522,787],{"className":137523},[786],[224,137525,137527],{"className":137526},[759],[224,137528,137530],{"className":137529,"style":794},[763],[224,137531],{}," 的对抗目标 = 对 ",[224,137534,137536,137549],{"className":137535,"translate":388},[392],[224,137537,137539],{"className":137538},[396],[398,137540,137541],{"xmlns":400},[403,137542,137543,137547],{},[406,137544,137545],{},[409,137546,1705],{},[447,137548,1705],{"encoding":449},[224,137550,137552],{"className":137551,"ariaHidden":455},[454],[224,137553,137555,137558],{"className":137554},[459],[224,137556],{"className":137557,"style":2565},[463],[224,137559,1705],{"className":137560,"style":1740},[468,469]," 做对偶后的 f-散度估计。选择 ",[224,137563,137565,137578],{"className":137564,"translate":388},[392],[224,137566,137568],{"className":137567},[396],[398,137569,137570],{"xmlns":400},[403,137571,137572,137576],{},[406,137573,137574],{},[409,137575,1705],{},[447,137577,1705],{"encoding":449},[224,137579,137581],{"className":137580,"ariaHidden":455},[454],[224,137582,137584,137587],{"className":137583},[459],[224,137585],{"className":137586,"style":2565},[463],[224,137588,1705],{"className":137589,"style":1740},[468,469]," 就是选择「用哪种散度训练生成器」，这直接决定模式崩塌、梯度消失与训练稳定性的权衡。",[613,137592,137594],{"id":137593},"_72-变分推断与-elbo","7.2 变分推断与 ELBO",[11,137596,137597],{},"变分推断里最大化证据下界",[224,137599,137601],{"className":137600,"translate":388},[387],[224,137602,137604,137762],{"className":137603,"translate":388},[392],[224,137605,137607],{"className":137606},[396],[398,137608,137609],{"xmlns":400,"display":401},[403,137610,137611,137759],{},[5209,137612,137613,137689],{"rowspacing":5211,"columnalign":5212,"columnspacing":5213},[5215,137614,137615,137629],{},[5218,137616,137617],{},[5221,137618,137619],{"scriptlevel":284,"displaystyle":455},[406,137620,137621,137623,137625,137627],{},[409,137622,41831],{"mathvariant":20667},[412,137624,239],{"stretchy":435},[409,137626,2023],{},[412,137628,1716],{"stretchy":435},[5218,137630,137631],{},[5221,137632,137633],{"scriptlevel":284,"displaystyle":455},[406,137634,137635,137637,137639,137645,137647,137649,137651,137653,137655,137657,137659,137661,137663,137665,137667,137673,137675,137677,137679,137681,137683,137685,137687],{},[406,137636],{},[412,137638,278],{},[695,137640,137641,137643],{},[409,137642,20324],{"mathvariant":12251},[409,137644,2023],{},[412,137646,268],{"stretchy":435},[409,137648,24131],{},[412,137650,24134],{},[409,137652,11],{},[412,137654,239],{"stretchy":435},[409,137656,371],{},[412,137658,1877],{"separator":455},[409,137660,5032],{},[412,137662,1716],{"stretchy":435},[412,137664,440],{"stretchy":435},[412,137666,5268],{},[695,137668,137669,137671],{},[409,137670,20324],{"mathvariant":12251},[409,137672,2023],{},[412,137674,268],{"stretchy":435},[409,137676,24131],{},[412,137678,24134],{},[409,137680,2023],{},[412,137682,239],{"stretchy":435},[409,137684,5032],{},[412,137686,1716],{"stretchy":435},[412,137688,440],{"stretchy":435},[5215,137690,137691,137697],{},[5218,137692,137693],{},[5221,137694,137695],{"scriptlevel":284,"displaystyle":455},[406,137696],{},[5218,137698,137699],{},[5221,137700,137701],{"scriptlevel":284,"displaystyle":455},[406,137702,137703,137705,137707,137709,137711,137713,137715,137717,137719,137721,137727,137729,137731,137733,137735,137737,137739,137741,137743,137745,137747,137749,137751,137753,137755,137757],{},[406,137704],{},[412,137706,278],{},[409,137708,24131],{},[412,137710,24134],{},[409,137712,11],{},[412,137714,239],{"stretchy":435},[409,137716,371],{},[412,137718,1716],{"stretchy":435},[412,137720,5268],{},[406,137722,137723,137725],{},[409,137724,30652],{"mathvariant":678},[409,137726,41831],{"mathvariant":678},[412,137728,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,137730,2023],{},[412,137732,239],{"stretchy":435},[409,137734,5032],{},[412,137736,1716],{"stretchy":435},[423,137738,682],{},[409,137740,20312],{"mathvariant":678},[423,137742,682],{},[409,137744,11],{},[412,137746,239],{"stretchy":435},[409,137748,5032],{},[409,137750,1083],{"mathvariant":678},[409,137752,371],{},[412,137754,1716],{"stretchy":435},[412,137756,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[412,137758,1877],{"separator":455},[447,137760,137761],{"encoding":449},"\\begin{aligned}\n\\mathcal L(q)\n&=\\mathbb E_q[\\log p(x,z)]\n-\\mathbb E_q[\\log q(z)]\\\\\n&=\\log p(x)\n-\\mathrm{KL}\\big(q(z)\\,\\|\\,p(z|x)\\big),\n\\end{aligned}",[224,137763,137765],{"className":137764,"ariaHidden":455},[454],[224,137766,137768,137771],{"className":137767},[459],[224,137769],{"className":137770,"style":34535},[463],[224,137772,137774],{"className":137773},[468],[224,137775,137777,137829],{"className":137776},[5209],[224,137778,137780],{"className":137779},[5331],[224,137781,137783,137821],{"className":137782},[754,755],[224,137784,137786,137818],{"className":137785},[759],[224,137787,137789,137810],{"className":137788,"style":34554},[763],[224,137790,137792,137795],{"style":137791},"top:-3.765em;",[224,137793],{"className":137794,"style":2108},[771],[224,137796,137798,137801,137804,137807],{"className":137797},[468],[224,137799,41831],{"className":137800},[468,20693],[224,137802,239],{"className":137803},[542],[224,137805,2023],{"className":137806,"style":492},[468,469],[224,137808,1716],{"className":137809},[549],[224,137811,137812,137815],{"style":34941},[224,137813],{"className":137814,"style":2108},[771],[224,137816],{"className":137817},[468],[224,137819,787],{"className":137820},[786],[224,137822,137824],{"className":137823},[759],[224,137825,137827],{"className":137826,"style":34583},[763],[224,137828],{},[224,137830,137832],{"className":137831},[5399],[224,137833,137835,138117],{"className":137834},[754,755],[224,137836,137838,138114],{"className":137837},[759],[224,137839,137841,138005],{"className":137840,"style":34554},[763],[224,137842,137843,137846],{"style":137791},[224,137844],{"className":137845,"style":2108},[771],[224,137847,137849,137852,137855,137858,137861,137901,137904,137909,137912,137915,137918,137921,137924,137927,137930,137933,137936,137939,137942,137982,137985,137990,137993,137996,137999,138002],{"className":137848},[468],[224,137850],{"className":137851},[468],[224,137853],{"className":137854,"style":496},[473],[224,137856,278],{"className":137857},[500],[224,137859],{"className":137860,"style":496},[473],[224,137862,137864,137867],{"className":137863},[468],[224,137865,20324],{"className":137866},[468,12389],[224,137868,137870],{"className":137869},[750],[224,137871,137873,137893],{"className":137872},[754,755],[224,137874,137876,137890],{"className":137875},[759],[224,137877,137879],{"className":137878,"style":64841},[763],[224,137880,137881,137884],{"style":767},[224,137882],{"className":137883,"style":772},[771],[224,137885,137887],{"className":137886},[776,777,778,779],[224,137888,2023],{"className":137889,"style":492},[468,469,779],[224,137891,787],{"className":137892},[786],[224,137894,137896],{"className":137895},[759],[224,137897,137899],{"className":137898,"style":20409},[763],[224,137900],{},[224,137902,268],{"className":137903},[542],[224,137905,24185,137907],{"className":137906},[21531],[224,137908,24189],{"style":24188},[224,137910],{"className":137911,"style":724},[473],[224,137913,11],{"className":137914},[468,469],[224,137916,239],{"className":137917},[542],[224,137919,371],{"className":137920},[468,469],[224,137922,1877],{"className":137923},[1917],[224,137925],{"className":137926,"style":724},[473],[224,137928,5032],{"className":137929,"style":5150},[468,469],[224,137931,40329],{"className":137932},[549],[224,137934],{"className":137935,"style":474},[473],[224,137937,5268],{"className":137938},[478],[224,137940],{"className":137941,"style":474},[473],[224,137943,137945,137948],{"className":137944},[468],[224,137946,20324],{"className":137947},[468,12389],[224,137949,137951],{"className":137950},[750],[224,137952,137954,137974],{"className":137953},[754,755],[224,137955,137957,137971],{"className":137956},[759],[224,137958,137960],{"className":137959,"style":64841},[763],[224,137961,137962,137965],{"style":767},[224,137963],{"className":137964,"style":772},[771],[224,137966,137968],{"className":137967},[776,777,778,779],[224,137969,2023],{"className":137970,"style":492},[468,469,779],[224,137972,787],{"className":137973},[786],[224,137975,137977],{"className":137976},[759],[224,137978,137980],{"className":137979,"style":20409},[763],[224,137981],{},[224,137983,268],{"className":137984},[542],[224,137986,24185,137988],{"className":137987},[21531],[224,137989,24189],{"style":24188},[224,137991],{"className":137992,"style":724},[473],[224,137994,2023],{"className":137995,"style":492},[468,469],[224,137997,239],{"className":137998},[542],[224,138000,5032],{"className":138001,"style":5150},[468,469],[224,138003,40329],{"className":138004},[549],[224,138006,138007,138010],{"style":34941},[224,138008],{"className":138009,"style":2108},[771],[224,138011,138013,138016,138019,138022,138025,138030,138033,138036,138039,138042,138045,138048,138051,138054,138060,138066,138069,138072,138075,138078,138081,138084,138087,138090,138093,138096,138099,138102,138105,138111],{"className":138012},[468],[224,138014],{"className":138015},[468],[224,138017],{"className":138018,"style":496},[473],[224,138020,278],{"className":138021},[500],[224,138023],{"className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是 f-散度（",[224,138190,138192,138216],{"className":138191,"translate":388},[392],[224,138193,138195],{"className":138194},[396],[398,138196,138197],{"xmlns":400},[403,138198,138199,138213],{},[406,138200,138201,138203,138205,138207,138209,138211],{},[409,138202,1705],{},[412,138204,278],{},[409,138206,1435],{},[409,138208,24131],{},[412,138210,24134],{},[409,138212,1435],{},[447,138214,138215],{"encoding":449},"f=t\\log t",[224,138217,138219,138237],{"className":138218,"ariaHidden":455},[454],[224,138220,138222,138225,138228,138231,138234],{"className":138221},[459],[224,138223],{"className":138224,"style":2565},[463],[224,138226,1705],{"className":138227,"style":1740},[468,469],[224,138229],{"className":138230,"style":496},[473],[224,138232,278],{"className":138233},[500],[224,138235],{"className":138236,"style":496},[473],[224,138238,138240,138243,138246,138249,138254,138257],{"className":138239},[459],[224,138241],{"className":138242,"style":2565},[463],[224,138244,1435],{"className":138245},[468,469],[224,138247],{"className":138248,"style":724},[473],[224,138250,24185,138252],{"className":138251},[21531],[224,138253,24189],{"style":24188},[224,138255],{"className":138256,"style":724},[473],[224,138258,1435],{"className":138259},[468,469],"）。反过来，任何以 ",[224,138262,138264,138296],{"className":138263,"translate":388},[392],[224,138265,138267],{"className":138266},[396],[398,138268,138269],{"xmlns":400},[403,138270,138271,138293],{},[406,138272,138273,138279,138281,138283,138285,138287,138289,138291],{},[695,138274,138275,138277],{},[409,138276,20324],{"mathvariant":12251},[409,138278,2023],{},[412,138280,268],{"stretchy":435},[409,138282,24131],{},[412,138284,24134],{},[412,138286,239],{"stretchy":435},[412,138288,23331],{},[412,138290,1716],{"stretchy":435},[412,138292,440],{"stretchy":435},[447,138294,138295],{"encoding":449},"\\mathbb E_q[\\log(\\cdot)]",[224,138297,138299],{"className":138298,"ariaHidden":455},[454],[224,138300,138302,138305,138345,138348,138353,138356,138359],{"className":138301},[459],[224,138303],{"className":138304,"style":20368},[463],[224,138306,138308,138311],{"className":138307},[468],[224,138309,20324],{"className":138310},[468,12389],[224,138312,138314],{"className":138313},[750],[224,138315,138317,138337],{"className":138316},[754,755],[224,138318,138320,138334],{"className":138319},[759],[224,138321,138323],{"className":138322,"style":64841},[763],[224,138324,138325,138328],{"style":767},[224,138326],{"className":138327,"style":772},[771],[224,138329,138331],{"className":138330},[776,777,778,779],[224,138332,2023],{"className":138333,"style":492},[468,469,779],[224,138335,787],{"className":138336},[786],[224,138338,138340],{"className":138339},[759],[224,138341,138343],{"className":138342,"style":20409},[763],[224,138344],{},[224,138346,268],{"className":138347},[542],[224,138349,24185,138351],{"className":138350},[21531],[224,138352,24189],{"style":24188},[224,138354,239],{"className":138355},[542],[224,138357,23331],{"className":138358},[468],[224,138360,40329],{"className":138361},[549]," 为组成部分的目标都可以用 f-散度的语言重写；",[224,138364,138366,138379],{"className":138365,"translate":388},[392],[224,138367,138369],{"className":138368},[396],[398,138370,138371],{"xmlns":400},[403,138372,138373,138377],{},[406,138374,138375],{},[409,138376,64666],{},[447,138378,68999],{"encoding":449},[224,138380,138382],{"className":138381,"ariaHidden":455},[454],[224,138383,138385,138388],{"className":138384},[459],[224,138386],{"className":138387,"style":839},[463],[224,138389,64666],{"className":138390,"style":64696},[468,469],"-VI、",[224,138393,138395,138412],{"className":138394,"translate":388},[392],[224,138396,138398],{"className":138397},[396],[398,138399,138400],{"xmlns":400},[403,138401,138402,138410],{},[406,138403,138404],{},[3248,138405,138406,138408],{},[409,138407,57370],{},[419,138409,1447],{},[447,138411,60308],{"encoding":449},[224,138413,138415],{"className":138414,"ariaHidden":455},[454],[224,138416,138418,138421],{"className":138417},[459],[224,138419],{"className":138420,"style":60318},[463],[224,138422,138424,138427],{"className":138423},[468],[224,138425,57370],{"className":138426},[468,469],[224,138428,138430],{"className":138429},[750],[224,138431,138433],{"className":138432},[754],[224,138434,138436],{"className":138435},[759],[224,138437,138439],{"className":138438,"style":5671},[763],[224,138440,138441,138444],{"style":3288},[224,138442],{"className":138443,"style":772},[771],[224,138445,138447],{"className":138446},[776,777,778,779],[224,138448,1447],{"className":138449},[468,779],"-VI 等就换用别的 ",[224,138452,138454,138467],{"className":138453,"translate":388},[392],[224,138455,138457],{"className":138456},[396],[398,138458,138459],{"xmlns":400},[403,138460,138461,138465],{},[406,138462,138463],{},[409,138464,1705],{},[447,138466,1705],{"encoding":449},[224,138468,138470],{"className":138469,"ariaHidden":455},[454],[224,138471,138473,138476],{"className":138472},[459],[224,138474],{"className":138475,"style":2565},[463],[224,138477,1705],{"className":138478,"style":1740},[468,469],"，在 3.1 节所说的「零回避」（mass-covering）与「零强制」（mode-seeking）两种极端行为之间取舍。",[613,138481,138483],{"id":138482},"_73-互信息估计","7.3 互信息估计",[11,138485,138486],{},"互信息可写成",[224,138488,138490],{"className":138489,"translate":388},[387],[224,138491,138493,138608],{"className":138492,"translate":388},[392],[224,138494,138496],{"className":138495},[396],[398,138497,138498],{"xmlns":400,"display":401},[403,138499,138500,138605],{},[406,138501,138502,138504,138506,138508,138510,138512,138514,138516,138522,138524,138534,138536,138538,138540,138546,138549,138555,138557,138559,138573,138575,138585,138587,138593,138595,138601,138603],{},[409,138503,53791],{},[412,138505,239],{"stretchy":435},[409,138507,20668],{},[412,138509,53798],{"separator":455},[409,138511,25410],{},[412,138513,1716],{"stretchy":435},[412,138515,278],{},[406,138517,138518,138520],{},[409,138519,30652],{"mathvariant":678},[409,138521,41831],{"mathvariant":678},[412,138523,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[695,138525,138526,138528],{},[409,138527,16018],{},[406,138529,138530,138532],{},[409,138531,20668],{},[409,138533,25410],{},[423,138535,682],{},[409,138537,20312],{"mathvariant":678},[423,138539,682],{},[695,138541,138542,138544],{},[409,138543,16018],{},[409,138545,20668],{},[412,138547,138548],{},"⊗",[695,138550,138551,138553],{},[409,138552,16018],{},[409,138554,25410],{},[412,138556,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[412,138558,278],{},[695,138560,138561,138563],{},[409,138562,20303],{},[406,138564,138565,138567,138569,138571],{},[409,138566,1435],{},[409,138568,24131],{},[412,138570,24134],{},[409,138572,1435],{},[412,138574,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[695,138576,138577,138579],{},[409,138578,16018],{},[406,138580,138581,138583],{},[409,138582,20668],{},[409,138584,25410],{},[409,138586,20312],{"mathvariant":678},[695,138588,138589,138591],{},[409,138590,16018],{},[409,138592,20668],{},[412,138594,138548],{},[695,138596,138597,138599],{},[409,138598,16018],{},[409,138600,25410],{},[412,138602,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[412,138604,1877],{"separator":455},[447,138606,138607],{"encoding":449},"I(X;Y)\n=\\mathrm{KL}\\big(P_{XY}\\,\\|\\,P_X\\otimes P_Y\\big)\n=D_{t\\log t}\\big(P_{XY}\\|P_X\\otimes P_Y\\big),",[224,138609,138611,138647,138769,138830,139004],{"className":138610,"ariaHidden":455},[454],[224,138612,138614,138617,138620,138623,138626,138629,138632,138635,138638,138641,138644],{"className":138613},[459],[224,138615],{"className":138616,"style":535},[463],[224,138618,53791],{"className":138619,"style":28852},[468,469],[224,138621,239],{"className":138622},[542],[224,138624,20668],{"className":138625,"style":28852},[468,469],[224,138627,53798],{"className":138628},[1917],[224,138630],{"className":138631,"style":724},[473],[224,138633,25410],{"className":138634,"style":474},[468,469],[224,138636,1716],{"className":138637},[549],[224,138639],{"className":138640,"style":496},[473],[224,138642,278],{"className":138643},[500],[224,138645],{"className":138646,"style":496},[473],[224,138648,138650,138653,138659,138665,138711,138714,138717,138720,138760,138763,138766],{"className":138649},[459],[224,138651],{"className":138652,"style":2663},[463],[224,138654,138656],{"className":138655},[468],[224,138657,52588],{"className":138658},[468,1906],[224,138660,138662],{"className":138661},[468],[224,138663,239],{"className":138664},[2363,2364],[224,138666,138668,138671],{"className":138667},[468],[224,138669,16018],{"className":138670,"style":653},[468,469],[224,138672,138674],{"className":138673},[750],[224,138675,138677,138703],{"className":138676},[754,755],[224,138678,138680,138700],{"className":138679},[759],[224,138681,138683],{"className":138682,"style":764},[763],[224,138684,138685,138688],{"style":31092},[224,138686],{"className":138687,"style":772},[771],[224,138689,138691],{"className":138690},[776,777,778,779],[224,138692,138694,138697],{"className":138693},[468,779],[224,138695,20668],{"className":138696,"style":28852},[468,469,779],[224,138698,25410],{"className":138699,"style":474},[468,469,779],[224,138701,787],{"className":138702},[786],[224,138704,138706],{"className":138705},[759],[224,138707,138709],{"className":138708,"style":794},[763],[224,138710],{},[224,138712],{"className":138713,"style":724},[473],[224,138715,20312],{"className":138716},[468],[224,138718],{"className":138719,"style":724},[473],[224,138721,138723,138726],{"className":138722},[468],[224,138724,16018],{"className":138725,"style":653},[468,469],[224,138727,138729],{"className":138728},[750],[224,138730,138732,138752],{"className":138731},[754,755],[224,138733,138735,138749],{"className":138734},[759],[224,138736,138738],{"className":138737,"style":764},[763],[224,138739,138740,138743],{"style":31092},[224,138741],{"className":138742,"style":772},[771],[224,138744,138746],{"className":138745},[776,777,778,779],[224,138747,20668],{"className":138748,"style":28852},[468,469,779],[224,138750,787],{"className":138751},[786],[224,138753,138755],{"className":138754},[759],[224,138756,138758],{"className":138757,"style":794},[763],[224,138759],{},[224,138761],{"className":138762,"style":474},[473],[224,138764,138548],{"className":138765},[478],[224,138767],{"className":138768,"style":474},[473],[224,138770,138772,138775,138815,138821,138824,138827],{"className":138771},[459],[224,138773],{"className":138774,"style":2663},[463],[224,138776,138778,138781],{"className":138777},[468],[224,138779,16018],{"className":138780,"style":653},[468,469],[224,138782,138784],{"className":138783},[750],[224,138785,138787,138807],{"className":138786},[754,755],[224,138788,138790,138804],{"className":138789},[759],[224,138791,138793],{"className":138792,"style":764},[763],[224,138794,138795,138798],{"style":31092},[224,138796],{"className":138797,"style":772},[771],[224,138799,138801],{"className":138800},[776,777,778,779],[224,138802,25410],{"className":138803,"style":474},[468,469,779],[224,138805,787],{"className":138806},[786],[224,138808,138810],{"className":138809},[759],[224,138811,138813],{"className":138812,"style":794},[763],[224,138814],{},[224,138816,138818],{"className":138817},[468],[224,138819,1716],{"className":138820},[2363,2364],[224,138822],{"className":138823,"style":496},[473],[224,138825,278],{"className":138826},[500],[224,138828],{"className":138829,"style":496},[473],[224,138831,138833,138836,138900,138906,138952,138955,138995,138998,139001],{"className":138832},[459],[224,138834],{"className":138835,"style":2663},[463],[224,138837,138839,138842],{"className":138838},[468],[224,138840,20303],{"className":138841,"style":2473},[468,469],[224,138843,138845],{"className":138844},[750],[224,138846,138848,138892],{"className":138847},[754,755],[224,138849,138851,138889],{"className":138850},[759],[224,138852,138854],{"className":138853,"style":20387},[763],[224,138855,138856,138859],{"style":20390},[224,138857],{"className":138858,"style":772},[771],[224,138860,138862],{"className":138861},[776,777,778,779],[224,138863,138865,138868,138871,138883,138886],{"className":138864},[468,779],[224,138866,1435],{"className":138867},[468,469,779],[224,138869],{"className":138870,"style":62279},[473,779],[224,138872,138874,138877,138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f-散度的特例。MINE 用 Donsker–Varadhan 对偶估计它，f-MINE 则对一般 ",[224,139063,139065,139078],{"className":139064,"translate":388},[392],[224,139066,139068],{"className":139067},[396],[398,139069,139070],{"xmlns":400},[403,139071,139072,139076],{},[406,139073,139074],{},[409,139075,1705],{},[447,139077,1705],{"encoding":449},[224,139079,139081],{"className":139080,"ariaHidden":455},[454],[224,139082,139084,139087],{"className":139083},[459],[224,139085],{"className":139086,"style":2565},[463],[224,139088,1705],{"className":139089,"style":1740},[468,469]," 用 4.4 节的对偶；不同 ",[224,139092,139094,139107],{"className":139093,"translate":388},[392],[224,139095,139097],{"className":139096},[396],[398,139098,139099],{"xmlns":400},[403,139100,139101,139105],{},[406,139102,139103],{},[409,139104,1705],{},[447,139106,1705],{"encoding":449},[224,139108,139110],{"className":139109,"ariaHidden":455},[454],[224,139111,139113,139116],{"className":139112},[459],[224,139114],{"className":139115,"style":2565},[463],[224,139117,1705],{"className":139118,"style":1740},[468,469],"（KL、JS、",[224,139121,139123,139140],{"className":139122,"translate":388},[392],[224,139124,139126],{"className":139125},[396],[398,139127,139128],{"xmlns":400},[403,139129,139130,139138],{},[406,139131,139132],{},[3248,139133,139134,139136],{},[409,139135,57370],{},[419,139137,1447],{},[447,139139,60308],{"encoding":449},[224,139141,139143],{"className":139142,"ariaHidden":455},[454],[224,139144,139146,139149],{"className":139145},[459],[224,139147],{"className":139148,"style":60318},[463],[224,139150,139152,139155],{"className":139151},[468],[224,139153,57370],{"className":139154},[468,469],[224,139156,139158],{"className":139157},[750],[224,139159,139161],{"className":139160},[754],[224,139162,139164],{"className":139163},[759],[224,139165,139167],{"className":139166,"style":5671},[763],[224,139168,139169,139172],{"style":3288},[224,139170],{"className":139171,"style":772},[771],[224,139173,139175],{"className":139174},[776,777,778,779],[224,139176,1447],{"className":139177},[468,779],"）给出不同的「MI 下界」和不同的方差—偏差折衷。对比式学习的 InfoNCE 也是 KL 下界的一种变体。",[613,139180,139182],{"id":139181},"_74-表示学习","7.4 表示学习",[11,139184,139185,139186,139230],{},"任何表示 ",[224,139187,139189,139209],{"className":139188,"translate":388},[392],[224,139190,139192],{"className":139191},[396],[398,139193,139194],{"xmlns":400},[403,139195,139196,139206],{},[406,139197,139198,139200,139202,139204],{},[409,139199,637],{},[412,139201,239],{"stretchy":435},[409,139203,20668],{},[412,139205,1716],{"stretchy":435},[447,139207,139208],{"encoding":449},"T(X)",[224,139210,139212],{"className":139211,"ariaHidden":455},[454],[224,139213,139215,139218,139221,139224,139227],{"className":139214},[459],[224,139216],{"className":139217,"style":535},[463],[224,139219,637],{"className":139220,"style":653},[468,469],[224,139222,239],{"className":139223},[542],[224,139225,20668],{"className":139226,"style":28852},[468,469],[224,139228,1716],{"className":139229},[549]," 都满足（4.5 节）",[224,139232,139234],{"className":139233,"translate":388},[387],[224,139235,139237,139317],{"className":139236,"translate":388},[392],[224,139238,139240],{"className":139239},[396],[398,139241,139242],{"xmlns":400,"display":401},[403,139243,139244,139314],{},[406,139245,139246,139252,139254,139268,139270,139284,139286,139288,139294,139296,139302,139304,139310,139312],{},[695,139247,139248,139250],{},[409,139249,20303],{},[409,139251,1705],{},[412,139253,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[695,139255,139256,139258],{},[409,139257,16018],{},[406,139259,139260,139262,139264,139266],{},[409,139261,637],{},[412,139263,239],{"stretchy":435},[409,139265,20668],{},[412,139267,1716],{"stretchy":435},[409,139269,20312],{"mathvariant":678},[695,139271,139272,139274],{},[409,139273,20315],{},[406,139275,139276,139278,139280,139282],{},[409,139277,637],{},[412,139279,239],{"stretchy":435},[409,139281,20668],{},[412,139283,1716],{"stretchy":435},[412,139285,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[412,139287,2070],{},[695,139289,139290,139292],{},[409,139291,20303],{},[409,139293,1705],{},[412,139295,239],{"stretchy":435},[695,139297,139298,139300],{},[409,139299,16018],{},[409,139301,20668],{},[409,139303,20312],{"mathvariant":678},[695,139305,139306,139308],{},[409,139307,20315],{},[409,139309,20668],{},[412,139311,1716],{"stretchy":435},[412,139313,1877],{"separator":455},[447,139315,139316],{"encoding":449},"D_f\\big(P_{T(X)}\\|Q_{T(X)}\\big)\n\\le D_f(P_X\\|Q_X),",[224,139318,139320,139496],{"className":139319,"ariaHidden":455},[454],[224,139321,139323,139327,139367,139373,139426,139429,139481,139487,139490,139493],{"className":139322},[459],[224,139324],{"className":139325,"style":139326},[463],"height:1.2052em;vertical-align:-0.3552em;",[224,139328,139330,139333],{"className":139329},[468],[224,139331,20303],{"className":139332,"style":2473},[468,469],[224,139334,139336],{"className":139335},[750],[224,139337,139339,139359],{"className":139338},[754,755],[224,139340,139342,139356],{"className":139341},[759],[224,139343,139345],{"className":139344,"style":20387},[763],[224,139346,139347,139350],{"style":20390},[224,139348],{"className":139349,"style":772},[771],[224,139351,139353],{"className":139352},[776,777,778,779],[224,139354,1705],{"className":139355,"style":1740},[468,469,779],[224,139357,787],{"className":139358},[786],[224,139360,139362],{"className":139361},[759],[224,139363,139365],{"className":139364,"style":20409},[763],[224,139366],{},[224,139368,139370],{"className":139369},[468],[224,139371,239],{"className":139372},[2363,2364],[224,139374,139376,139379],{"className":139375},[468],[224,139377,16018],{"className":139378,"style":653},[468,469],[224,139380,139382],{"className":139381},[750],[224,139383,139385,139418],{"className":139384},[754,755],[224,139386,139388,139415],{"className":139387},[759],[224,139389,139391],{"className":139390,"style":26101},[763],[224,139392,139394,139397],{"style":139393},"top:-2.5198em;margin-left:-0.1389em;margin-right:0.05em;",[224,139395],{"className":139396,"style":772},[771],[224,139398,139400],{"className":139399},[776,777,778,779],[224,139401,139403,139406,139409,139412],{"className":139402},[468,779],[224,139404,637],{"className":139405,"style":653},[468,469,779],[224,139407,239],{"className":139408},[542,779],[224,139410,20668],{"className":139411,"style":28852},[468,469,779],[224,139413,1716],{"className":139414},[549,779],[224,139416,787],{"className":139417},[786],[224,139419,139421],{"className":139420},[759],[224,139422,139424],{"className":139423,"style":26147},[763],[224,139425],{},[224,139427,20312],{"className":139428},[468],[224,139430,139432,139435],{"className":139431},[468],[224,139433,20315],{"className":139434},[468,469],[224,139436,139438],{"className":139437},[750],[224,139439,139441,139473],{"className":139440},[754,755],[224,139442,139444,139470],{"className":139443},[759],[224,139445,139447],{"className":139446,"style":26101},[763],[224,139448,139449,139452],{"style":95478},[224,139450],{"className":139451,"style":772},[771],[224,139453,139455],{"className":139454},[776,777,778,779],[224,139456,139458,139461,139464,139467],{"className":139457},[468,779],[224,139459,637],{"className":139460,"style":653},[468,469,779],[224,139462,239],{"className":139463},[542,779],[224,139465,20668],{"className":139466,"style":28852},[468,469,779],[224,139468,1716],{"className":139469},[549,779],[224,139471,787],{"className":139472},[786],[224,139474,139476],{"className":139475},[759],[224,139477,139479],{"className":139478,"style":26147},[763],[224,139480],{},[224,139482,139484],{"className":139483},[468],[224,139485,1716],{"className":139486},[2363,2364],[224,139488],{"className":139489,"style":496},[473],[224,139491,2070],{"className":139492},[500],[224,139494],{"className":139495,"style":496},[473],[224,139497,139499,139502,139542,139545,139585,139588,139628,139631],{"className":139498},[459],[224,139500],{"className":139501,"style":20368},[463],[224,139503,139505,139508],{"className":139504},[468],[224,139506,20303],{"className":139507,"style":2473},[468,469],[224,139509,139511],{"className":139510},[750],[224,139512,139514,139534],{"className":139513},[754,755],[224,139515,139517,139531],{"className":139516},[759],[224,139518,139520],{"className":139519,"style":20387},[763],[224,139521,139522,139525],{"style":20390},[224,139523],{"className":139524,"style":772},[771],[224,139526,139528],{"className":139527},[776,777,778,779],[224,139529,1705],{"className":139530,"style":1740},[468,469,779],[224,139532,787],{"className":139533},[786],[224,139535,139537],{"className":139536},[759],[224,139538,139540],{"className":139539,"style":20409},[763],[224,139541],{},[224,139543,239],{"className":139544},[542],[224,139546,139548,139551],{"className":139547},[468],[224,139549,16018],{"className":139550,"style":653},[468,469],[224,139552,139554],{"className":139553},[750],[224,139555,139557,139577],{"className":139556},[754,755],[224,139558,139560,139574],{"className":139559},[759],[224,139561,139563],{"className":139562,"style":764},[763],[224,139564,139565,139568],{"style":31092},[224,139566],{"className":139567,"style":772},[771],[224,139569,139571],{"className":139570},[776,777,778,779],[224,139572,20668],{"className":139573,"style":28852},[468,469,779],[224,139575,787],{"className":139576},[786],[224,139578,139580],{"className":139579},[759],[224,139581,139583],{"className":139582,"style":794},[763],[224,139584],{},[224,139586,20312],{"className":139587},[468],[224,139589,139591,139594],{"className":139590},[468],[224,139592,20315],{"className":139593},[468,469],[224,139595,139597],{"className":139596},[750],[224,139598,139600,139620],{"className":139599},[754,755],[224,139601,139603,139617],{"className":139602},[759],[224,139604,139606],{"className":139605,"style":764},[763],[224,139607,139608,139611],{"style":767},[224,139609],{"className":139610,"style":772},[771],[224,139612,139614],{"className":139613},[776,777,778,779],[224,139615,20668],{"className":139616,"style":28852},[468,469,779],[224,139618,787],{"className":139619},[786],[224,139621,139623],{"className":139622},[759],[224,139624,139626],{"className":139625,"style":794},[763],[224,139627],{},[224,139629,1716],{"className":139630},[549],[224,139632,1877],{"className":139633},[1917],[11,139635,139636],{},"即「编码不会创造信息」。这为对比学习、域对齐、特征选择提供了可证伪的约束：任何声称「表示后差异更大」的方法都必然在某个环节违反了 DPI，需要解释为什么。",[613,139638,139640],{"id":139639},"_75-域适应","7.5 域适应",[11,139642,139643,139644,139674,139675,139705],{},"在源域 ",[224,139645,139647,139661],{"className":139646,"translate":388},[392],[224,139648,139650],{"className":139649},[396],[398,139651,139652],{"xmlns":400},[403,139653,139654,139658],{},[406,139655,139656],{},[409,139657,6924],{"mathvariant":20667},[447,139659,139660],{"encoding":449},"\\mathcal S",[224,139662,139664],{"className":139663,"ariaHidden":455},[454],[224,139665,139667,139670],{"className":139666},[459],[224,139668],{"className":139669,"style":649},[463],[224,139671,6924],{"className":139672,"style":139673},[468,20693],"margin-right:0.075em;"," 与目标域 ",[224,139676,139678,139692],{"className":139677,"translate":388},[392],[224,139679,139681],{"className":139680},[396],[398,139682,139683],{"xmlns":400},[403,139684,139685,139689],{},[406,139686,139687],{},[409,139688,637],{"mathvariant":20667},[447,139690,139691],{"encoding":449},"\\mathcal T",[224,139693,139695],{"className":139694,"ariaHidden":455},[454],[224,139696,139698,139701],{"className":139697},[459],[224,139699],{"className":139700,"style":649},[463],[224,139702,637],{"className":139703,"style":139704},[468,20693],"margin-right:0.2542em;"," 上，衡量分布漂移的现代方法（DANN、ADDA）本质上用 f-散度或 IPM 对齐边缘分布；其中用 JS 的（GAN 式）对齐与用 MMD\u002FWasserstein 的对齐，区别正是 6.2 节的几何与支撑问题。",[613,139707,139709],{"id":139708},"_76-假设检验","7.6 假设检验",[11,139711,139712],{},"f-散度是假设检验的错误指数：",[2674,139714,139715,139782],{},[2677,139716,139717,139720,139721,6095],{},[378,139718,139719],{},"Stein 引理","：在固定第一类错误下，第二类错误的最优指数为 ",[224,139722,139724,139752],{"className":139723,"translate":388},[392],[224,139725,139727],{"className":139726},[396],[398,139728,139729],{"xmlns":400},[403,139730,139731,139749],{},[406,139732,139733,139739,139741,139743,139745,139747],{},[406,139734,139735,139737],{},[409,139736,30652],{"mathvariant":678},[409,139738,41831],{"mathvariant":678},[412,139740,239],{"stretchy":435},[409,139742,16018],{},[409,139744,20312],{"mathvariant":678},[409,139746,20315],{},[412,139748,1716],{"stretchy":435},[447,139750,139751],{"encoding":449},"\\mathrm{KL}(P\\|Q)",[224,139753,139755],{"className":139754,"ariaHidden":455},[454],[224,139756,139758,139761,139767,139770,139773,139776,139779],{"className":139757},[459],[224,139759],{"className":139760,"style":535},[463],[224,139762,139764],{"className":139763},[468],[224,139765,52588],{"className":139766},[468,1906],[224,139768,239],{"className":139769},[542],[224,139771,16018],{"className":139772,"style":653},[468,469],[224,139774,20312],{"className":139775},[468],[224,139777,20315],{"className":139778},[468,469],[224,139780,1716],{"className":139781},[549],[2677,139783,139784,12570,139787,140066],{},[378,139785,139786],{},"Chernoff 信息",[224,139788,139790,139864],{"className":139789,"translate":388},[392],[224,139791,139793],{"className":139792},[396],[398,139794,139795],{"xmlns":400},[403,139796,139797,139861],{},[406,139798,139799,139801,139803,139805,139807,139809,139811,139813,139833,139835,139837,139839,139841,139843,139851,139853,139855,139857,139859],{},[409,139800,16021],{},[412,139802,239],{"stretchy":435},[409,139804,16018],{},[412,139806,1877],{"separator":455},[409,139808,20315],{},[412,139810,1716],{"stretchy":435},[412,139812,278],{},[695,139814,139815,139821],{},[406,139816,139817,139819],{},[409,139818,55354],{},[412,139820,24134],{},[406,139822,139823,139825,139827,139829,139831],{},[419,139824,284],{},[412,139826,1577],{},[409,139828,31295],{},[412,139830,1577],{},[419,139832,1438],{},[412,139834,239],{"stretchy":435},[419,139836,1438],{},[412,139838,5268],{},[409,139840,31295],{},[412,139842,1716],{"stretchy":435},[27527,139844,139845,139847,139849],{},[409,139846,20303],{},[409,139848,31295],{},[409,139850,21092],{},[412,139852,239],{"stretchy":435},[409,139854,16018],{},[409,139856,20312],{"mathvariant":678},[409,139858,20315],{},[412,139860,1716],{"stretchy":435},[447,139862,139863],{"encoding":449},"C(P,Q)=\\sup_{0\u003C\\lambda\u003C1}(1-\\lambda)D^{R}_{\\lambda}(P\\|Q)",[224,139865,139867,139903,139980],{"className":139866,"ariaHidden":455},[454],[224,139868,139870,139873,139876,139879,139882,139885,139888,139891,139894,139897,139900],{"className":139869},[459],[224,139871],{"className":139872,"style":535},[463],[224,139874,16021],{"className":139875,"style":30740},[468,469],[224,139877,239],{"className":139878},[542],[224,139880,16018],{"className":139881,"style":653},[468,469],[224,139883,1877],{"className":139884},[1917],[224,139886],{"className":139887,"style":724},[473],[224,139889,20315],{"className":139890},[468,469],[224,139892,1716],{"className":139893},[549],[224,139895],{"className":139896,"style":496},[473],[224,139898,278],{"className":139899},[500],[224,139901],{"className":139902,"style":496},[473],[224,139904,139906,139909,139965,139968,139971,139974,139977],{"className":139905},[459],[224,139907],{"className":139908,"style":55478},[463],[224,139910,139912,139915],{"className":139911},[21531],[224,139913,55354],{"className":139914},[21531],[224,139916,139918],{"className":139917},[750],[224,139919,139921,139957],{"className":139920},[754,755],[224,139922,139924,139954],{"className":139923},[759],[224,139925,139928],{"className":139926,"style":139927},[763],"height:0.242em;",[224,139929,139930,139933],{"style":55500},[224,139931],{"className":139932,"style":772},[771],[224,139934,139936],{"className":139935},[776,777,778,779],[224,139937,139939,139942,139945,139948,139951],{"className":139938},[468,779],[224,139940,284],{"className":139941},[468,779],[224,139943,1577],{"className":139944},[500,779],[224,139946,31295],{"className":139947},[468,469,779],[224,139949,1577],{"className":139950},[500,779],[224,139952,1438],{"className":139953},[468,779],[224,139955,787],{"className":139956},[786],[224,139958,139960],{"className":139959},[759],[224,139961,139963],{"className":139962,"style":55528},[763],[224,139964],{},[224,139966,239],{"className":139967},[542],[224,139969,1438],{"className":139970},[468],[224,139972],{"className":139973,"style":474},[473],[224,139975,5268],{"className":139976},[478],[224,139978],{"className":139979,"style":474},[473],[224,139981,139983,139987,139990,139993,140051,140054,140057,140060,140063],{"className":139982},[459],[224,139984],{"className":139985,"style":139986},[463],"height:1.1244em;vertical-align:-0.2831em;",[224,139988,31295],{"className":139989},[468,469],[224,139991,1716],{"className":139992},[549],[224,139994,139996,139999],{"className":139995},[468],[224,139997,20303],{"className":139998,"style":2473},[468,469],[224,140000,140002],{"className":140001},[750],[224,140003,140005,140043],{"className":140004},[754,755],[224,140006,140008,140040],{"className":140007},[759],[224,140009,140011,140026],{"className":140010,"style":71992},[763],[224,140012,140014,140017],{"style":140013},"top:-2.4169em;margin-left:-0.0278em;margin-right:0.05em;",[224,140015],{"className":140016,"style":772},[771],[224,140018,140020],{"className":140019},[776,777,778,779],[224,140021,140023],{"className":140022},[468,779],[224,140024,31295],{"className":140025},[468,469,779],[224,140027,140028,140031],{"style":3288},[224,140029],{"className":140030,"style":772},[771],[224,140032,140034],{"className":140033},[776,777,778,779],[224,140035,140037],{"className":140036},[468,779],[224,140038,21092],{"className":140039,"style":69507},[468,469,779],[224,140041,787],{"className":140042},[786],[224,140044,140046],{"className":140045},[759],[224,140047,140049],{"className":140048,"style":28513},[763],[224,140050],{},[224,140052,239],{"className":140053},[542],[224,140055,16018],{"className":140056,"style":653},[468,469],[224,140058,20312],{"className":140059},[468],[224,140061,20315],{"className":140062},[468,469],[224,140064,1716],{"className":140065},[549],"，即 Rényi\u002Fα-族中的最优点；对称检验的指数由它决定。",[11,140068,140069],{},"因此 f-散度不只是「训练目标」，也是统计判决的极限性能。",[608,140071,140073],{"id":140072},"_8-差分隐私f-dp","8. 差分隐私（f-DP）",[11,140075,140076,140077,140080,140081,26478,140084,140112,140113,140154],{},"一个较新的应用是把 f-散度的语言搬到隐私保护。差分隐私（DP）要求相邻数据集的输出分布「难以区分」。Dong–Roth–Su 提出 ",[378,140078,140079],{},"f-差分隐私（f-DP）","：用一个",[378,140082,140083],{},"权衡函数（tradeoff function）",[224,140085,140087,140100],{"className":140086,"translate":388},[392],[224,140088,140090],{"className":140089},[396],[398,140091,140092],{"xmlns":400},[403,140093,140094,140098],{},[406,140095,140096],{},[409,140097,1705],{},[447,140099,1705],{"encoding":449},[224,140101,140103],{"className":140102,"ariaHidden":455},[454],[224,140104,140106,140109],{"className":140105},[459],[224,140107],{"className":140108,"style":2565},[463],[224,140110,1705],{"className":140111,"style":1740},[468,469]," 刻画两个输出分布 ",[224,140114,140116,140133],{"className":140115,"translate":388},[392],[224,140117,140119],{"className":140118},[396],[398,140120,140121],{"xmlns":400},[403,140122,140123,140131],{},[406,140124,140125,140127,140129],{},[409,140126,16018],{},[412,140128,1877],{"separator":455},[409,140130,20315],{},[447,140132,20627],{"encoding":449},[224,140134,140136],{"className":140135,"ariaHidden":455},[454],[224,140137,140139,140142,140145,140148,140151],{"className":140138},[459],[224,140140],{"className":140141,"style":20637},[463],[224,140143,16018],{"className":140144,"style":653},[468,469],[224,140146,1877],{"className":140147},[1917],[224,140149],{"className":140150,"style":724},[473],[224,140152,20315],{"className":140153},[468,469]," 上「漏报率—误报率」的权衡曲线：",[224,140156,140158],{"className":140157,"translate":388},[387],[224,140159,140161,140258],{"className":140160,"translate":388},[392],[224,140162,140164],{"className":140163},[396],[398,140165,140166],{"xmlns":400,"display":401},[403,140167,140168,140255],{},[406,140169,140170,140172,140174,140176,140178,140180,140182,140184,140186,140188,140190,140201,140203,140205,140207,140209,140211,140213,140215,140217,140219,140221,140223,140225,140227,140229,140231,140233,140235,140237,140239,140241,140243,140245,140247,140249,140251,140253],{},[409,140171,637],{},[412,140173,239],{"stretchy":435},[409,140175,16018],{},[412,140177,1877],{"separator":455},[409,140179,20315],{},[412,140181,1716],{"stretchy":435},[412,140183,239],{"stretchy":435},[409,140185,64666],{},[412,140187,1716],{"stretchy":435},[412,140189,278],{},[1965,140191,140192,140199],{},[406,140193,140194,140197],{},[409,140195,140196],{},"inf",[412,140198,24134],{},[409,140200,1097],{},[412,140202,8110],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,140204,16018],{},[412,140206,239],{"stretchy":435},[409,140208,1097],{},[412,140210,1716],{"stretchy":435},[423,140212,15054],{},[412,140214,1071],{},[423,140216,15054],{},[409,140218,20315],{},[412,140220,239],{"stretchy":435},[409,140222,1097],{},[412,140224,1716],{"stretchy":435},[412,140226,9766],{},[419,140228,1438],{},[412,140230,5268],{},[409,140232,64666],{},[412,140234,15077],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[412,140236,1877],{"separator":455},[473,140238],{"width":4247},[409,140240,64666],{},[412,140242,12248],{},[412,140244,268],{"stretchy":435},[419,140246,284],{},[412,140248,1877],{"separator":455},[419,140250,1438],{},[412,140252,440],{"stretchy":435},[409,140254,688],{"mathvariant":678},[447,140256,140257],{"encoding":449},"T(P,Q)(\\alpha)\n=\\inf_{A}\\big\\{P(A)\\ :\\ Q(A)\\ge1-\\alpha\\big\\},\n\\qquad \\alpha\\in[0,1] .",[224,140259,140261,140306,140400,140427,140445,140481],{"className":140260,"ariaHidden":455},[454],[224,140262,140264,140267,140270,140273,140276,140279,140282,140285,140288,140291,140294,140297,140300,140303],{"className":140263},[459],[224,140265],{"className":140266,"style":535},[463],[224,140268,637],{"className":140269,"style":653},[468,469],[224,140271,239],{"className":140272},[542],[224,140274,16018],{"className":140275,"style":653},[468,469],[224,140277,1877],{"className":140278},[1917],[224,140280],{"className":140281,"style":724},[473],[224,140283,20315],{"className":140284},[468,469],[224,140286,1716],{"className":140287},[549],[224,140289,239],{"className":140290},[542],[224,140292,64666],{"className":140293,"style":64696},[468,469],[224,140295,1716],{"className":140296},[549],[224,140298],{"className":140299,"style":496},[473],[224,140301,278],{"className":140302},[500],[224,140304],{"className":140305,"style":496},[473],[224,140307,140309,140313,140364,140367,140373,140376,140379,140382,140385,140388,140391,140394,140397],{"className":140308},[459],[224,140310],{"className":140311,"style":140312},[463],"height:1.5943em;vertical-align:-0.7443em;",[224,140314,140316],{"className":140315},[21531,33665],[224,140317,140319,140355],{"className":140318},[754,755],[224,140320,140322,140352],{"className":140321},[759],[224,140323,140325,140340],{"className":140324,"style":7642},[763],[224,140326,140328,140331],{"style":140327},"top:-2.3557em;margin-left:0em;",[224,140329],{"className":140330,"style":2108},[771],[224,140332,140334],{"className":140333},[776,777,778,779],[224,140335,140337],{"className":140336},[468,779],[224,140338,1097],{"className":140339},[468,469,779],[224,140341,140342,140345],{"style":2120},[224,140343],{"className":140344,"style":2108},[771],[224,140346,140347],{},[224,140348,4090,140350],{"className":140349},[21531],[224,140351,1705],{"style":114732},[224,140353,787],{"className":140354},[786],[224,140356,140358],{"className":140357},[759],[224,140359,140362],{"className":140360,"style":140361},[763],"height:0.7443em;",[224,140363],{},[224,140365],{"className":140366,"style":724},[473],[224,140368,140370],{"className":140369},[468],[224,140371,8110],{"className":140372},[2363,2364],[224,140374,16018],{"className":140375,"style":653},[468,469],[224,140377,239],{"className":140378},[542],[224,140380,1097],{"className":140381},[468,469],[224,140383,1716],{"className":140384},[549],[224,140386,15054],{"className":140387},[473],[224,140389],{"className":140390,"style":496},[473],[224,140392,1071],{"className":140393},[500],[224,140395,15054],{"className":140396},[473],[224,140398],{"className":140399,"style":496},[473],[224,140401,140403,140406,140409,140412,140415,140418,140421,140424],{"className":140402},[459],[224,140404],{"className":140405,"style":535},[463],[224,140407,20315],{"className":140408},[468,469],[224,140410,239],{"className":140411},[542],[224,140413,1097],{"className":140414},[468,469],[224,140416,1716],{"className":140417},[549],[224,140419],{"className":140420,"style":496},[473],[224,140422,9766],{"className":140423},[500],[224,140425],{"className":140426,"style":496},[473],[224,140428,140430,140433,140436,140439,140442],{"className":140429},[459],[224,140431],{"className":140432,"style":7047},[463],[224,140434,1438],{"className":140435},[468],[224,140437],{"className":140438,"style":474},[473],[224,140440,5268],{"className":140441},[478],[224,140443],{"className":140444,"style":474},[473],[224,140446,140448,140451,140454,140460,140463,140466,140469,140472,140475,140478],{"className":140447},[459],[224,140449],{"className":140450,"style":2663},[463],[224,140452,64666],{"className":140453,"style":64696},[468,469],[224,140455,140457],{"className":140456},[468],[224,140458,15077],{"className":140459},[2363,2364],[224,140461,1877],{"className":140462},[1917],[224,140464],{"className":140465,"style":4416},[473],[224,140467],{"className":140468,"style":724},[473],[224,140470,64666],{"className":140471,"style":64696},[468,469],[224,140473],{"className":140474,"style":496},[473],[224,140476,12248],{"className":140477},[500],[224,140479],{"className":140480,"style":496},[473],[224,140482,140484,140487,140490,140493,140496,140499,140502,140505],{"className":140483},[459],[224,140485],{"className":140486,"style":535},[463],[224,140488,268],{"className":140489},[542],[224,140491,284],{"className":140492},[468],[224,140494,1877],{"className":140495},[1917],[224,140497],{"className":140498,"style":724},[473],[224,140500,1438],{"className":140501},[468],[224,140503,440],{"className":140504},[549],[224,140506,688],{"className":140507},[468],[11,140509,140510,140511,97971,140539,140567,140568,140643,140644,140783],{},"机制 ",[224,140512,140514,140527],{"className":140513,"translate":388},[392],[224,140515,140517],{"className":140516},[396],[398,140518,140519],{"xmlns":400},[403,140520,140521,140525],{},[406,140522,140523],{},[409,140524,45500],{},[447,140526,45500],{"encoding":449},[224,140528,140530],{"className":140529,"ariaHidden":455},[454],[224,140531,140533,140536],{"className":140532},[459],[224,140534],{"className":140535,"style":649},[463],[224,140537,45500],{"className":140538,"style":45534},[468,469],[224,140540,140542,140555],{"className":140541,"translate":388},[392],[224,140543,140545],{"className":140544},[396],[398,140546,140547],{"xmlns":400},[403,140548,140549,140553],{},[406,140550,140551],{},[409,140552,1705],{},[447,140554,1705],{"encoding":449},[224,140556,140558],{"className":140557,"ariaHidden":455},[454],[224,140559,140561,140564],{"className":140560},[459],[224,140562],{"className":140563,"style":2565},[463],[224,140565,1705],{"className":140566,"style":1740},[468,469],"-DP，若对所有相邻 ",[224,140569,140571,140593],{"className":140570,"translate":388},[392],[224,140572,140574],{"className":140573},[396],[398,140575,140576],{"xmlns":400},[403,140577,140578,140590],{},[406,140579,140580,140582,140584],{},[409,140581,20303],{},[412,140583,1877],{"separator":455},[3248,140585,140586,140588],{},[409,140587,20303],{},[412,140589,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[447,140591,140592],{"encoding":449},"D,D'",[224,140594,140596],{"className":140595,"ariaHidden":455},[454],[224,140597,140599,140602,140605,140608,140611],{"className":140598},[459],[224,140600],{"className":140601,"style":84067},[463],[224,140603,20303],{"className":140604,"style":2473},[468,469],[224,140606,1877],{"className":140607},[1917],[224,140609],{"className":140610,"style":724},[473],[224,140612,140614,140617],{"className":140613},[468],[224,140615,20303],{"className":140616,"style":2473},[468,469],[224,140618,140620],{"className":140619},[750],[224,140621,140623],{"className":140622},[754],[224,140624,140626],{"className":140625},[759],[224,140627,140629],{"className":140628,"style":6992},[763],[224,140630,140631,140634],{"style":3288},[224,140632],{"className":140633,"style":772},[771],[224,140635,140637],{"className":140636},[776,777,778,779],[224,140638,140640],{"className":140639},[468,779],[224,140641,6933],{"className":140642},[468,779],"，恒有 ",[224,140645,140647,140691],{"className":140646,"translate":388},[392],[224,140648,140650],{"className":140649},[396],[398,140651,140652],{"xmlns":400},[403,140653,140654,140688],{},[406,140655,140656,140658,140660,140662,140664,140666,140668,140670,140672,140674,140680,140682,140684,140686],{},[409,140657,637],{},[412,140659,239],{"stretchy":435},[409,140661,45500],{},[412,140663,239],{"stretchy":435},[409,140665,20303],{},[412,140667,1716],{"stretchy":435},[412,140669,1877],{"separator":455},[409,140671,45500],{},[412,140673,239],{"stretchy":435},[3248,140675,140676,140678],{},[409,140677,20303],{},[412,140679,6933],{"mathvariant":678,"lspace":5213,"rspace":5213},[412,140681,1716],{"stretchy":435},[412,140683,1716],{"stretchy":435},[412,140685,9766],{},[409,140687,1705],{},[447,140689,140690],{"encoding":449},"T(M(D),M(D'))\\ge f",[224,140692,140694,140774],{"className":140693,"ariaHidden":455},[454],[224,140695,140697,140700,140703,140706,140709,140712,140715,140718,140721,140724,140727,140730,140762,140765,140768,140771],{"className":140696},[459],[224,140698],{"className":140699,"style":6966},[463],[224,140701,637],{"className":140702,"style":653},[468,469],[224,140704,239],{"className":140705},[542],[224,140707,45500],{"className":140708,"style":45534},[468,469],[224,140710,239],{"className":140711},[542],[224,140713,20303],{"className":140714,"style":2473},[468,469],[224,140716,1716],{"className":140717},[549],[224,140719,1877],{"className":140720},[1917],[224,140722],{"className":140723,"style":724},[473],[224,140725,45500],{"className":140726,"style":45534},[468,469],[224,140728,239],{"className":140729},[542],[224,140731,140733,140736],{"className":140732},[468],[224,140734,20303],{"className":140735,"style":2473},[468,469],[224,140737,140739],{"className":140738},[750],[224,140740,140742],{"className":140741},[754],[224,140743,140745],{"className":140744},[759],[224,140746,140748],{"className":140747,"style":6992},[763],[224,140749,140750,140753],{"style":3288},[224,140751],{"className":140752,"style":772},[771],[224,140754,140756],{"className":140755},[776,777,778,779],[224,140757,140759],{"className":140758},[468,779],[224,140760,6933],{"className":140761},[468,779],[224,140763,38989],{"className":140764},[549],[224,140766],{"className":140767,"style":496},[473],[224,140769,9766],{"className":140770},[500],[224,140772],{"className":140773,"style":496},[473],[224,140775,140777,140780],{"className":140776},[459],[224,140778],{"className":140779,"style":2565},[463],[224,140781,1705],{"className":140782,"style":1740},[468,469],"。可以证明：",[2674,140785,140786],{},[2677,140787,140788,52016,140791,140863,140864,140912,140913,140941],{},[378,140789,140790],{},"权衡曲线与 f-散度互相决定。",[224,140792,140794,140824],{"className":140793,"translate":388},[392],[224,140795,140797],{"className":140796},[396],[398,140798,140799],{"xmlns":400},[403,140800,140801,140821],{},[406,140802,140803,140805,140807,140809,140811,140813,140815,140817,140819],{},[409,140804,637],{},[412,140806,239],{"stretchy":435},[409,140808,16018],{},[412,140810,1877],{"separator":455},[409,140812,20315],{},[412,140814,1716],{"stretchy":435},[412,140816,239],{"stretchy":435},[409,140818,64666],{},[412,140820,1716],{"stretchy":435},[447,140822,140823],{"encoding":449},"T(P,Q)(\\alpha)",[224,140825,140827],{"className":140826,"ariaHidden":455},[454],[224,140828,140830,140833,140836,140839,140842,140845,140848,140851,140854,140857,140860],{"className":140829},[459],[224,140831],{"className":140832,"style":535},[463],[224,140834,637],{"className":140835,"style":653},[468,469],[224,140837,239],{"className":140838},[542],[224,140840,16018],{"className":140841,"style":653},[468,469],[224,140843,1877],{"className":140844},[1917],[224,140846],{"className":140847,"style":724},[473],[224,140849,20315],{"className":140850},[468,469],[224,140852,1716],{"className":140853},[549],[224,140855,239],{"className":140856},[542],[224,140858,64666],{"className":140859,"style":64696},[468,469],[224,140861,1716],{"className":140862},[549]," 只依赖似然比 ",[224,140865,140867,140888],{"className":140866,"translate":388},[392],[224,140868,140870],{"className":140869},[396],[398,140871,140872],{"xmlns":400},[403,140873,140874,140886],{},[406,140875,140876,140878,140880,140882,140884],{},[409,140877,13182],{},[409,140879,16018],{},[409,140881,12413],{"mathvariant":678},[409,140883,13182],{},[409,140885,20315],{},[447,140887,134535],{"encoding":449},[224,140889,140891],{"className":140890,"ariaHidden":455},[454],[224,140892,140894,140897,140900,140903,140906,140909],{"className":140893},[459],[224,140895],{"className":140896,"style":535},[463],[224,140898,13182],{"className":140899},[468,469],[224,140901,16018],{"className":140902,"style":653},[468,469],[224,140904,12413],{"className":140905},[468],[224,140907,13182],{"className":140908},[468,469],[224,140910,20315],{"className":140911},[468,469]," 的分布，因此 ",[224,140914,140916,140929],{"className":140915,"translate":388},[392],[224,140917,140919],{"className":140918},[396],[398,140920,140921],{"xmlns":400},[403,140922,140923,140927],{},[406,140924,140925],{},[409,140926,637],{},[447,140928,637],{"encoding":449},[224,140930,140932],{"className":140931,"ariaHidden":455},[454],[224,140933,140935,140938],{"className":140934},[459],[224,140936],{"className":140937,"style":649},[463],[224,140939,637],{"className":140940,"style":653},[468,469]," 与\"全体 f-散度\"携带同样的信息。对 hockey-stick 生成元族",[224,140943,140945],{"className":140944,"translate":388},[387],[224,140946,140948,141015],{"className":140947,"translate":388},[392],[224,140949,140951],{"className":140950},[396],[398,140952,140953],{"xmlns":400,"display":401},[403,140954,140955,141012],{},[406,140956,140957,140964,140966,140968,140970,140972,140974,140976,140978,140980,140982,140984,140986,140988,140990,140992,140994,140996,140998,141000,141002,141004,141006,141008,141010],{},[695,140958,140959,140961],{},[409,140960,24189],{},[409,140962,140963],{},"γ",[412,140965,239],{"stretchy":435},[409,140967,1435],{},[412,140969,1716],{"stretchy":435},[412,140971,278],{},[409,140973,72294],{},[412,140975,24134],{},[412,140977,8110],{"stretchy":435},[419,140979,284],{},[412,140981,1877],{"separator":455},[423,140983,682],{},[409,140985,1435],{},[412,140987,5268],{},[409,140989,140963],{},[412,140991,15077],{"stretchy":435},[412,140993,1877],{"separator":455},[473,140995],{"width":4247},[409,140997,140963],{},[412,140999,12248],{},[412,141001,268],{"stretchy":435},[419,141003,284],{},[412,141005,1877],{"separator":455},[419,141007,1438],{},[412,141009,440],{"stretchy":435},[412,141011,1877],{"separator":455},[447,141013,141014],{"encoding":449},"g_\\gamma(t)=\\max\\{0,\\,t-\\gamma\\},\n\\qquad \\gamma\\in[0,1],",[224,141016,141018,141083,141119,141152],{"className":141017,"ariaHidden":455},[454],[224,141019,141021,141024,141065,141068,141071,141074,141077,141080],{"className":141020},[459],[224,141022],{"className":141023,"style":20368},[463],[224,141025,141027,141030],{"className":141026},[468],[224,141028,24189],{"className":141029,"style":492},[468,469],[224,141031,141033],{"className":141032},[750],[224,141034,141036,141057],{"className":141035},[754,755],[224,141037,141039,141054],{"className":141038},[759],[224,141040,141042],{"className":141041,"style":64841},[763],[224,141043,141044,141047],{"style":3854},[224,141045],{"className":141046,"style":772},[771],[224,141048,141050],{"className":141049},[776,777,778,779],[224,141051,140963],{"className":141052,"style":141053},[468,469,779],"margin-right:0.0556em;",[224,141055,787],{"className":141056},[786],[224,141058,141060],{"className":141059},[759],[224,141061,141063],{"className":141062,"style":20409},[763],[224,141064],{},[224,141066,239],{"className":141067},[542],[224,141069,1435],{"className":141070},[468,469],[224,141072,1716],{"className":141073},[549],[224,141075],{"className":141076,"style":496},[473],[224,141078,278],{"className":141079},[500],[224,141081],{"className":141082,"style":496},[473],[224,141084,141086,141089,141092,141095,141098,141101,141104,141107,141110,141113,141116],{"className":141085},[459],[224,141087],{"className":141088,"style":535},[463],[224,141090,72294],{"className":141091},[21531],[224,141093,8110],{"className":141094},[542],[224,141096,284],{"className":141097},[468],[224,141099,1877],{"className":141100},[1917],[224,141102],{"className":141103,"style":724},[473],[224,141105],{"className":141106,"style":724},[473],[224,141108,1435],{"className":141109},[468,469],[224,141111],{"className":141112,"style":474},[473],[224,141114,5268],{"className":141115},[478],[224,141117],{"className":141118,"style":474},[473],[224,141120,141122,141125,141128,141131,141134,141137,141140,141143,141146,141149],{"className":141121},[459],[224,141123],{"className":141124,"style":535},[463],[224,141126,140963],{"className":141127,"style":141053},[468,469],[224,141129,15077],{"className":141130},[549],[224,141132,1877],{"className":141133},[1917],[224,141135],{"className":141136,"style":4416},[473],[224,141138],{"className":141139,"style":724},[473],[224,141141,140963],{"className":141142,"style":141053},[468,469],[224,141144],{"className":141145,"style":496},[473],[224,141147,12248],{"className":141148},[500],[224,141150],{"className":141151,"style":496},[473],[224,141153,141155,141158,141161,141164,141167,141170,141173,141176],{"className":141154},[459],[224,141156],{"className":141157,"style":535},[463],[224,141159,268],{"className":141160},[542],[224,141162,284],{"className":141163},[468],[224,141165,1877],{"className":141166},[1917],[224,141168],{"className":141169,"style":724},[473],[224,141171,1438],{"className":141172},[468],[224,141174,440],{"className":141175},[549],[224,141177,1877],{"className":141178},[1917],[11,141180,141181,141182,141363,141364,7396,141393,141444],{},"（Dong–Roth–Su 的 ",[224,141183,141185,141233],{"className":141184,"translate":388},[392],[224,141186,141188],{"className":141187},[396],[398,141189,141190],{"xmlns":400},[403,141191,141192,141230],{},[406,141193,141194,141200,141202,141204,141206,141208,141210,141212,141214,141216,141218,141220,141222,141224,141226,141228],{},[695,141195,141196,141198],{},[409,141197,58586],{},[409,141199,140963],{},[412,141201,239],{"stretchy":435},[409,141203,1435],{},[412,141205,1716],{"stretchy":435},[412,141207,278],{},[409,141209,72294],{},[412,141211,24134],{},[412,141213,8110],{"stretchy":435},[419,141215,284],{},[412,141217,1877],{"separator":455},[409,141219,1435],{},[412,141221,5268],{},[419,141223,1438],{},[412,141225,319],{},[409,141227,140963],{},[412,141229,15077],{"stretchy":435},[447,141231,141232],{"encoding":449},"h_\\gamma(t)=\\max\\{0,t-1+\\gamma\\}",[224,141234,141236,141300,141333,141351],{"className":141235,"ariaHidden":455},[454],[224,141237,141239,141242,141282,141285,141288,141291,141294,141297],{"className":141238},[459],[224,141240],{"className":141241,"style":20368},[463],[224,141243,141245,141248],{"className":141244},[468],[224,141246,58586],{"className":141247},[468,469],[224,141249,141251],{"className":141250},[750],[224,141252,141254,141274],{"className":141253},[754,755],[224,141255,141257,141271],{"className":141256},[759],[224,141258,141260],{"className":141259,"style":64841},[763],[224,141261,141262,141265],{"style":767},[224,141263],{"className":141264,"style":772},[771],[224,141266,141268],{"className":141267},[776,777,778,779],[224,141269,140963],{"className":141270,"style":141053},[468,469,779],[224,141272,787],{"className":141273},[786],[224,141275,141277],{"className":141276},[759],[224,141278,141280],{"className":141279,"style":20409},[763],[224,141281],{},[224,141283,239],{"className":141284},[542],[224,141286,1435],{"className":141287},[468,469],[224,141289,1716],{"className":141290},[549],[224,141292],{"className":141293,"style":496},[473],[224,141295,278],{"className":141296},[500],[224,141298],{"className":141299,"style":496},[473],[224,141301,141303,141306,141309,141312,141315,141318,141321,141324,141327,141330],{"className":141302},[459],[224,141304],{"className":141305,"style":535},[463],[224,141307,72294],{"className":141308},[21531],[224,141310,8110],{"className":141311},[542],[224,141313,284],{"className":141314},[468],[224,141316,1877],{"className":141317},[1917],[224,141319],{"className":141320,"style":724},[473],[224,141322,1435],{"className":141323},[468,469],[224,141325],{"className":141326,"style":474},[473],[224,141328,5268],{"className":141329},[478],[224,141331],{"className":141332,"style":474},[473],[224,141334,141336,141339,141342,141345,141348],{"className":141335},[459],[224,141337],{"className":141338,"style":7047},[463],[224,141340,1438],{"className":141341},[468],[224,141343],{"className":141344,"style":474},[473],[224,141346,319],{"className":141347},[478],[224,141349],{"className":141350,"style":474},[473],[224,141352,141354,141357,141360],{"className":141353},[459],[224,141355],{"className":141356,"style":535},[463],[224,141358,140963],{"className":141359,"style":141053},[468,469],[224,141361,15077],{"className":141362},[549]," 就是把 ",[224,141365,141367,141381],{"className":141366,"translate":388},[392],[224,141368,141370],{"className":141369},[396],[398,141371,141372],{"xmlns":400},[403,141373,141374,141378],{},[406,141375,141376],{},[409,141377,140963],{},[447,141379,141380],{"encoding":449},"\\gamma",[224,141382,141384],{"className":141383,"ariaHidden":455},[454],[224,141385,141387,141390],{"className":141386},[459],[224,141388],{"className":141389,"style":806},[463],[224,141391,140963],{"className":141392,"style":141053},[468,469],[224,141394,141396,141414],{"className":141395,"translate":388},[392],[224,141397,141399],{"className":141398},[396],[398,141400,141401],{"xmlns":400},[403,141402,141403,141411],{},[406,141404,141405,141407,141409],{},[419,141406,1438],{},[412,141408,5268],{},[409,141410,140963],{},[447,141412,141413],{"encoding":449},"1-\\gamma",[224,141415,141417,141435],{"className":141416,"ariaHidden":455},[454],[224,141418,141420,141423,141426,141429,141432],{"className":141419},[459],[224,141421],{"className":141422,"style":7047},[463],[224,141424,1438],{"className":141425},[468],[224,141427],{"className":141428,"style":474},[473],[224,141430,5268],{"className":141431},[478],[224,141433],{"className":141434,"style":474},[473],[224,141436,141438,141441],{"className":141437},[459],[224,141439],{"className":141440,"style":806},[463],[224,141442,140963],{"className":141443,"style":141053},[468,469]," 的同一族），变分表示给出干净的检验集形式：",[224,141446,141448],{"className":141447,"translate":388},[387],[224,141449,141451,141529],{"className":141450,"translate":388},[392],[224,141452,141454],{"className":141453},[396],[398,141455,141456],{"xmlns":400,"display":401},[403,141457,141458,141526],{},[406,141459,141460,141470,141472,141474,141476,141478,141480,141482,141498,141500,141502,141504,141506,141508,141510,141512,141514,141516,141518,141520,141522,141524],{},[695,141461,141462,141464],{},[409,141463,20303],{},[695,141465,141466,141468],{},[409,141467,24189],{},[409,141469,140963],{},[412,141471,239],{"stretchy":435},[409,141473,16018],{},[409,141475,20312],{"mathvariant":678},[409,141477,20315],{},[412,141479,1716],{"stretchy":435},[412,141481,278],{},[1965,141483,141484,141490],{},[406,141485,141486,141488],{},[409,141487,55354],{},[412,141489,24134],{},[406,141491,141492,141494,141496],{},[409,141493,1097],{},[412,141495,12248],{},[409,141497,1097],{"mathvariant":20667},[412,141499,8110],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,141501,16018],{},[412,141503,239],{"stretchy":435},[409,141505,1097],{},[412,141507,1716],{"stretchy":435},[412,141509,5268],{},[409,141511,140963],{},[423,141513,682],{},[409,141515,20315],{},[412,141517,239],{"stretchy":435},[409,141519,1097],{},[412,141521,1716],{"stretchy":435},[412,141523,15077],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[409,141525,688],{"mathvariant":678},[447,141527,141528],{"encoding":449},"D_{g_\\gamma}(P\\|Q)\n=\\sup_{A\\in\\mathcal A}\\big\\{P(A)\n-\\gamma\\,Q(A)\\big\\}.",[224,141530,141532,141646,141735],{"className":141531,"ariaHidden":455},[454],[224,141533,141535,141539,141622,141625,141628,141631,141634,141637,141640,141643],{"className":141534},[459],[224,141536],{"className":141537,"style":141538},[463],"height:1.0973em;vertical-align:-0.3473em;",[224,141540,141542,141545],{"className":141541},[468],[224,141543,20303],{"className":141544,"style":2473},[468,469],[224,141546,141548],{"className":141547},[750],[224,141549,141551,141613],{"className":141550},[754,755],[224,141552,141554,141610],{"className":141553},[759],[224,141555,141557],{"className":141556,"style":64841},[763],[224,141558,141559,141562],{"style":20390},[224,141560],{"className":141561,"style":772},[771],[224,141563,141565],{"className":141564},[776,777,778,779],[224,141566,141568],{"className":141567},[468,779],[224,141569,141571,141574],{"className":141570},[468,779],[224,141572,24189],{"className":141573,"style":492},[468,469,779],[224,141575,141577],{"className":141576},[750],[224,141578,141580,141601],{"className":141579},[754,755],[224,141581,141583,141598],{"className":141582},[759],[224,141584,141586],{"className":141585,"style":94168},[763],[224,141587,141589,141592],{"style":141588},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[224,141590],{"className":141591,"style":50136},[771],[224,141593,141595],{"className":141594},[776,50140,2364,779],[224,141596,140963],{"className":141597,"style":141053},[468,469,779],[224,141599,787],{"className":141600},[786],[224,141602,141604],{"className":141603},[759],[224,141605,141608],{"className":141606,"style":141607},[763],"height:0.2819em;",[224,141609],{},[224,141611,787],{"className":141612},[786],[224,141614,141616],{"className":141615},[759],[224,141617,141620],{"className":141618,"style":141619},[763],"height:0.3473em;",[224,141621],{},[224,141623,239],{"className":141624},[542],[224,141626,16018],{"className":141627,"style":653},[468,469],[224,141629,20312],{"className":141630},[468],[224,141632,20315],{"className":141633},[468,469],[224,141635,1716],{"className":141636},[549],[224,141638],{"className":141639,"style":496},[473],[224,141641,278],{"className":141642},[500],[224,141644],{"className":141645,"style":496},[473],[224,141647,141649,141652,141705,141708,141714,141717,141720,141723,141726,141729,141732],{"className":141648},[459],[224,141650],{"className":141651,"style":56275},[463],[224,141653,141655],{"className":141654},[21531,33665],[224,141656,141658,141697],{"className":141657},[754,755],[224,141659,141661,141694],{"className":141660},[759],[224,141662,141664,141684],{"className":141663,"style":839},[763],[224,141665,141666,141669],{"style":56290},[224,141667],{"className":141668,"style":2108},[771],[224,141670,141672],{"className":141671},[776,777,778,779],[224,141673,141675,141678,141681],{"className":141674},[468,779],[224,141676,1097],{"className":141677},[468,469,779],[224,141679,12248],{"className":141680},[500,779],[224,141682,1097],{"className":141683},[468,20693,779],[224,141685,141686,141689],{"style":2120},[224,141687],{"className":141688,"style":2108},[771],[224,141690,141691],{},[224,141692,55354],{"className":141693},[21531],[224,141695,787],{"className":141696},[786],[224,141698,141700],{"className":141699},[759],[224,141701,141703],{"className":141702,"style":56328},[763],[224,141704],{},[224,141706],{"className":141707,"style":724},[473],[224,141709,141711],{"className":141710},[468],[224,141712,8110],{"className":141713},[2363,2364],[224,141715,16018],{"className":141716,"style":653},[468,469],[224,141718,239],{"className":141719},[542],[224,141721,1097],{"className":141722},[468,469],[224,141724,1716],{"className":141725},[549],[224,141727],{"className":141728,"style":474},[473],[224,141730,5268],{"className":141731},[478],[224,141733],{"className":141734,"style":474},[473],[224,141736,141738,141741,141744,141747,141750,141753,141756,141759,141765],{"className":141737},[459],[224,141739],{"className":141740,"style":2663},[463],[224,141742,140963],{"className":141743,"style":141053},[468,469],[224,141745],{"className":141746,"style":724},[473],[224,141748,20315],{"className":141749},[468,469],[224,141751,239],{"className":141752},[542],[224,141754,1097],{"className":141755},[468,469],[224,141757,1716],{"className":141758},[549],[224,141760,141762],{"className":141761},[468],[224,141763,15077],{"className":141764},[2363,2364],[224,141766,688],{"className":141767},[468],[11,141769,141770,141772,141773,141846,141847,22325,142016,142044,142045,142138],{},[378,141771,127171],{}," 一方面，任意 ",[224,141774,141776,141798],{"className":141775,"translate":388},[392],[224,141777,141779],{"className":141778},[396],[398,141780,141781],{"xmlns":400},[403,141782,141783,141795],{},[406,141784,141785,141787,141789,141791,141793],{},[419,141786,284],{},[412,141788,2070],{},[409,141790,24189],{},[412,141792,2070],{},[419,141794,1438],{},[447,141796,141797],{"encoding":449},"0\\le g\\le1",[224,141799,141801,141819,141837],{"className":141800,"ariaHidden":455},[454],[224,141802,141804,141807,141810,141813,141816],{"className":141803},[459],[224,141805],{"className":141806,"style":18160},[463],[224,141808,284],{"className":141809},[468],[224,141811],{"className":141812,"style":496},[473],[224,141814,2070],{"className":141815},[500],[224,141817],{"className":141818,"style":496},[473],[224,141820,141822,141825,141828,141831,141834],{"className":141821},[459],[224,141823],{"className":141824,"style":2644},[463],[224,141826,24189],{"className":141827,"style":492},[468,469],[224,141829],{"className":141830,"style":496},[473],[224,141832,2070],{"className":141833},[500],[224,141835],{"className":141836,"style":496},[473],[224,141838,141840,141843],{"className":141839},[459],[224,141841],{"className":141842,"style":510},[463],[224,141844,1438],{"className":141845},[468]," 可写为指示函数的线性组合（层叠公式），而 ",[224,141848,141850,141890],{"className":141849,"translate":388},[392],[224,141851,141853],{"className":141852},[396],[398,141854,141855],{"xmlns":400},[403,141856,141857,141887],{},[406,141858,141859,141865,141867,141869,141871,141873,141875,141881,141883,141885],{},[695,141860,141861,141863],{},[409,141862,20324],{},[409,141864,16018],{},[412,141866,268],{"stretchy":435},[409,141868,24189],{},[412,141870,440],{"stretchy":435},[412,141872,5268],{},[409,141874,140963],{},[695,141876,141877,141879],{},[409,141878,20324],{},[409,141880,20315],{},[412,141882,268],{"stretchy":435},[409,141884,24189],{},[412,141886,440],{"stretchy":435},[447,141888,141889],{"encoding":449},"E_P[g]-\\gamma E_Q[g]",[224,141891,141893,141958],{"className":141892,"ariaHidden":455},[454],[224,141894,141896,141899,141940,141943,141946,141949,141952,141955],{"className":141895},[459],[224,141897],{"className":141898,"style":535},[463],[224,141900,141902,141905],{"className":141901},[468],[224,141903,20324],{"className":141904,"style":6970},[468,469],[224,141906,141908],{"className":141907},[750],[224,141909,141911,141932],{"className":141910},[754,755],[224,141912,141914,141929],{"className":141913},[759],[224,141915,141917],{"className":141916,"style":764},[763],[224,141918,141920,141923],{"style":141919},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[224,141921],{"className":141922,"style":772},[771],[224,141924,141926],{"className":141925},[776,777,778,779],[224,141927,16018],{"className":141928,"style":653},[468,469,779],[224,141930,787],{"className":141931},[786],[224,141933,141935],{"className":141934},[759],[224,141936,141938],{"className":141937,"style":794},[763],[224,141939],{},[224,141941,268],{"className":141942},[542],[224,141944,24189],{"className":141945,"style":492},[468,469],[224,141947,440],{"className":141948},[549],[224,141950],{"className":141951,"style":474},[473],[224,141953,5268],{"className":141954},[478],[224,141956],{"className":141957,"style":474},[473],[224,141959,141961,141964,141967,142007,142010,142013],{"className":141960},[459],[224,141962],{"className":141963,"style":20368},[463],[224,141965,140963],{"className":141966,"style":141053},[468,469],[224,141968,141970,141973],{"className":141969},[468],[224,141971,20324],{"className":141972,"style":6970},[468,469],[224,141974,141976],{"className":141975},[750],[224,141977,141979,141999],{"className":141978},[754,755],[224,141980,141982,141996],{"className":141981},[759],[224,141983,141985],{"className":141984,"style":764},[763],[224,141986,141987,141990],{"style":141919},[224,141988],{"className":141989,"style":772},[771],[224,141991,141993],{"className":141992},[776,777,778,779],[224,141994,20315],{"className":141995},[468,469,779],[224,141997,787],{"className":141998},[786],[224,142000,142002],{"className":142001},[759],[224,142003,142005],{"className":142004,"style":20409},[763],[224,142006],{},[224,142008,268],{"className":142009},[542],[224,142011,24189],{"className":142012,"style":492},[468,469],[224,142014,440],{"className":142015},[549],[224,142017,142019,142032],{"className":142018,"translate":388},[392],[224,142020,142022],{"className":142021},[396],[398,142023,142024],{"xmlns":400},[403,142025,142026,142030],{},[406,142027,142028],{},[409,142029,24189],{},[447,142031,24189],{"encoding":449},[224,142033,142035],{"className":142034,"ariaHidden":455},[454],[224,142036,142038,142041],{"className":142037},[459],[224,142039],{"className":142040,"style":806},[463],[224,142042,24189],{"className":142043,"style":492},[468,469]," 的线性泛函，故其上确界在指示函数 ",[224,142046,142048,142070],{"className":142047,"translate":388},[392],[224,142049,142051],{"className":142050},[396],[398,142052,142053],{"xmlns":400},[403,142054,142055,142067],{},[406,142056,142057,142059,142061],{},[409,142058,24189],{},[412,142060,278],{},[695,142062,142063,142065],{},[419,142064,1438],{"mathvariant":15754},[409,142066,1097],{},[447,142068,142069],{"encoding":449},"g=\\mathbf1_A",[224,142071,142073,142091],{"className":142072,"ariaHidden":455},[454],[224,142074,142076,142079,142082,142085,142088],{"className":142075},[459],[224,142077],{"className":142078,"style":806},[463],[224,142080,24189],{"className":142081,"style":492},[468,469],[224,142083],{"className":142084,"style":496},[473],[224,142086,278],{"className":142087},[500],[224,142089],{"className":142090,"style":496},[473],[224,142092,142094,142098],{"className":142093},[459],[224,142095],{"className":142096,"style":142097},[463],"height:0.7944em;vertical-align:-0.15em;",[224,142099,142101,142104],{"className":142100},[468],[224,142102,1438],{"className":142103},[468,95463],[224,142105,142107],{"className":142106},[750],[224,142108,142110,142130],{"className":142109},[754,755],[224,142111,142113,142127],{"className":142112},[759],[224,142114,142116],{"className":142115,"style":764},[763],[224,142117,142118,142121],{"style":767},[224,142119],{"className":142120,"style":772},[771],[224,142122,142124],{"className":142123},[776,777,778,779],[224,142125,1097],{"className":142126},[468,469,779],[224,142128,787],{"className":142129},[786],[224,142131,142133],{"className":142132},[759],[224,142134,142136],{"className":142135,"style":794},[763],[224,142137],{}," 上达到：",[224,142140,142142],{"className":142141,"translate":388},[387],[224,142143,142145,142247],{"className":142144,"translate":388},[392],[224,142146,142148],{"className":142147},[396],[398,142149,142150],{"xmlns":400,"display":401},[403,142151,142152,142244],{},[406,142153,142154,142174,142176,142182,142184,142186,142188,142190,142192,142198,142200,142202,142204,142206,142208,142218,142220,142222,142224,142226,142228,142230,142232,142234,142236,142238,142240,142242],{},[1965,142155,142156,142162],{},[406,142157,142158,142160],{},[409,142159,55354],{},[412,142161,24134],{},[406,142163,142164,142166,142168,142170,142172],{},[419,142165,284],{},[412,142167,2070],{},[409,142169,24189],{},[412,142171,2070],{},[419,142173,1438],{},[412,142175,8110],{"stretchy":435},[695,142177,142178,142180],{},[409,142179,20324],{},[409,142181,16018],{},[412,142183,268],{"stretchy":435},[409,142185,24189],{},[412,142187,440],{"stretchy":435},[412,142189,5268],{},[409,142191,140963],{},[695,142193,142194,142196],{},[409,142195,20324],{},[409,142197,20315],{},[412,142199,268],{"stretchy":435},[409,142201,24189],{},[412,142203,440],{"stretchy":435},[412,142205,15077],{"stretchy":435},[412,142207,278],{},[1965,142209,142210,142216],{},[406,142211,142212,142214],{},[409,142213,55354],{},[412,142215,24134],{},[409,142217,1097],{},[412,142219,8110],{"stretchy":435},[409,142221,16018],{},[412,142223,239],{"stretchy":435},[409,142225,1097],{},[412,142227,1716],{"stretchy":435},[412,142229,5268],{},[409,142231,140963],{},[409,142233,20315],{},[412,142235,239],{"stretchy":435},[409,142237,1097],{},[412,142239,1716],{"stretchy":435},[412,142241,15077],{"stretchy":435},[412,142243,1877],{"separator":455},[447,142245,142246],{"encoding":449},"\\sup_{0\\le g\\le1}\\{E_P[g]-\\gamma E_Q[g]\\}\n=\\sup_{A}\\{P(A)-\\gamma Q(A)\\},",[224,142248,142250,142378,142446,142525],{"className":142249,"ariaHidden":455},[454],[224,142251,142253,142257,142317,142320,142360,142363,142366,142369,142372,142375],{"className":142252},[459],[224,142254],{"className":142255,"style":142256},[463],"height:1.7977em;vertical-align:-1.0477em;",[224,142258,142260],{"className":142259},[21531,33665],[224,142261,142263,142308],{"className":142262},[754,755],[224,142264,142266,142305],{"className":142265},[759],[224,142267,142269,142295],{"className":142268,"style":839},[763],[224,142270,142271,142274],{"style":80828},[224,142272],{"className":142273,"style":2108},[771],[224,142275,142277],{"className":142276},[776,777,778,779],[224,142278,142280,142283,142286,142289,142292],{"className":142279},[468,779],[224,142281,284],{"className":142282},[468,779],[224,142284,2070],{"className":142285},[500,779],[224,142287,24189],{"className":142288,"style":492},[468,469,779],[224,142290,2070],{"className":142291},[500,779],[224,142293,1438],{"className":142294},[468,779],[224,142296,142297,142300],{"style":2120},[224,142298],{"className":142299,"style":2108},[771],[224,142301,142302],{},[224,142303,55354],{"className":142304},[21531],[224,142306,787],{"className":142307},[786],[224,142309,142311],{"className":142310},[759],[224,142312,142315],{"className":142313,"style":142314},[763],"height:1.0477em;",[224,142316],{},[224,142318,8110],{"className":142319},[542],[224,142321,142323,142326],{"className":142322},[468],[224,142324,20324],{"className":142325,"style":6970},[468,469],[224,142327,142329],{"className":142328},[750],[224,142330,142332,142352],{"className":142331},[754,755],[224,142333,142335,142349],{"className":142334},[759],[224,142336,142338],{"className":142337,"style":764},[763],[224,142339,142340,142343],{"style":141919},[224,142341],{"className":142342,"style":772},[771],[224,142344,142346],{"className":142345},[776,777,778,779],[224,142347,16018],{"className":142348,"style":653},[468,469,779],[224,142350,787],{"className":142351},[786],[224,142353,142355],{"className":142354},[759],[224,142356,142358],{"className":142357,"style":794},[763],[224,142359],{},[224,142361,268],{"className":142362},[542],[224,142364,24189],{"className":142365,"style":492},[468,469],[224,142367,440],{"className":142368},[549],[224,142370],{"className":142371,"style":474},[473],[224,142373,5268],{"className":142374},[478],[224,142376],{"className":142377,"style":474},[473],[224,142379,142381,142384,142387,142427,142430,142433,142437,142440,142443],{"className":142380},[459],[224,142382],{"className":142383,"style":20368},[463],[224,142385,140963],{"className":142386,"style":141053},[468,469],[224,142388,142390,142393],{"className":142389},[468],[224,142391,20324],{"className":142392,"style":6970},[468,469],[224,142394,142396],{"className":142395},[750],[224,142397,142399,142419],{"className":142398},[754,755],[224,142400,142402,142416],{"className":142401},[759],[224,142403,142405],{"className":142404,"style":764},[763],[224,142406,142407,142410],{"style":141919},[224,142408],{"className":142409,"style":772},[771],[224,142411,142413],{"className":142412},[776,777,778,779],[224,142414,20315],{"className":142415},[468,469,779],[224,142417,787],{"className":142418},[786],[224,142420,142422],{"className":142421},[759],[224,142423,142425],{"className":142424,"style":20409},[763],[224,142426],{},[224,142428,268],{"className":142429},[542],[224,142431,24189],{"className":142432,"style":492},[468,469],[224,142434,142436],{"className":142435},[549],"]}",[224,142438],{"className":142439,"style":496},[473],[224,142441,278],{"className":142442},[500],[224,142444],{"className":142445,"style":496},[473],[224,142447,142449,142453,142501,142504,142507,142510,142513,142516,142519,142522],{"className":142448},[459],[224,142450],{"className":142451,"style":142452},[463],"height:1.6888em;vertical-align:-0.9388em;",[224,142454,142456],{"className":142455},[21531,33665],[224,142457,142459,142492],{"className":142458},[754,755],[224,142460,142462,142489],{"className":142461},[759],[224,142463,142465,142479],{"className":142464,"style":839},[763],[224,142466,142467,142470],{"style":56290},[224,142468],{"className":142469,"style":2108},[771],[224,142471,142473],{"className":142472},[776,777,778,779],[224,142474,142476],{"className":142475},[468,779],[224,142477,1097],{"className":142478},[468,469,779],[224,142480,142481,142484],{"style":2120},[224,142482],{"className":142483,"style":2108},[771],[224,142485,142486],{},[224,142487,55354],{"className":142488},[21531],[224,142490,787],{"className":142491},[786],[224,142493,142495],{"className":142494},[759],[224,142496,142499],{"className":142497,"style":142498},[763],"height:0.9388em;",[224,142500],{},[224,142502,8110],{"className":142503},[542],[224,142505,16018],{"className":142506,"style":653},[468,469],[224,142508,239],{"className":142509},[542],[224,142511,1097],{"className":142512},[468,469],[224,142514,1716],{"className":142515},[549],[224,142517],{"className":142518,"style":474},[473],[224,142520,5268],{"className":142521},[478],[224,142523],{"className":142524,"style":474},[473],[224,142526,142528,142531,142534,142537,142540,142543,142546],{"className":142527},[459],[224,142529],{"className":142530,"style":535},[463],[224,142532,140963],{"className":142533,"style":141053},[468,469],[224,142535,20315],{"className":142536},[468,469],[224,142538,239],{"className":142539},[542],[224,142541,1097],{"className":142542},[468,469],[224,142544,80905],{"className":142545},[549],[224,142547,1877],{"className":142548},[1917],[11,142550,142551,142552,142725,142726,38651,142872,142987,142988,143136,143137,143423],{},"且由 ",[224,142553,142555,142597],{"className":142554,"translate":388},[392],[224,142556,142558],{"className":142557},[396],[398,142559,142560],{"xmlns":400},[403,142561,142562,142594],{},[406,142563,142564,142570,142572,142574,142576,142578,142584,142586,142588,142590,142592],{},[695,142565,142566,142568],{},[409,142567,20324],{},[409,142569,16018],{},[412,142571,268],{"stretchy":435},[409,142573,24189],{},[412,142575,440],{"stretchy":435},[412,142577,278],{},[695,142579,142580,142582],{},[409,142581,20324],{},[409,142583,20315],{},[412,142585,268],{"stretchy":435},[409,142587,24189],{},[423,142589,682],{},[409,142591,1435],{},[412,142593,440],{"stretchy":435},[447,142595,142596],{"encoding":449},"E_P[g]=E_Q[g\\,t]",[224,142598,142600,142664],{"className":142599,"ariaHidden":455},[454],[224,142601,142603,142606,142646,142649,142652,142655,142658,142661],{"className":142602},[459],[224,142604],{"className":142605,"style":535},[463],[224,142607,142609,142612],{"className":142608},[468],[224,142610,20324],{"className":142611,"style":6970},[468,469],[224,142613,142615],{"className":142614},[750],[224,142616,142618,142638],{"className":142617},[754,755],[224,142619,142621,142635],{"className":142620},[759],[224,142622,142624],{"className":142623,"style":764},[763],[224,142625,142626,142629],{"style":141919},[224,142627],{"className":142628,"style":772},[771],[224,142630,142632],{"className":142631},[776,777,778,779],[224,142633,16018],{"className":142634,"style":653},[468,469,779],[224,142636,787],{"className":142637},[786],[224,142639,142641],{"className":142640},[759],[224,142642,142644],{"className":142643,"style":794},[763],[224,142645],{},[224,142647,268],{"className":142648},[542],[224,142650,24189],{"className":142651,"style":492},[468,469],[224,142653,440],{"className":142654},[549],[224,142656],{"className":142657,"style":496},[473],[224,142659,278],{"className":142660},[500],[224,142662],{"className":142663,"style":496},[473],[224,142665,142667,142670,142710,142713,142716,142719,142722],{"className":142666},[459],[224,142668],{"className":142669,"style":20368},[463],[224,142671,142673,142676],{"className":142672},[468],[224,142674,20324],{"className":142675,"style":6970},[468,469],[224,142677,142679],{"className":142678},[750],[224,142680,142682,142702],{"className":142681},[754,755],[224,142683,142685,142699],{"className":142684},[759],[224,142686,142688],{"className":142687,"style":764},[763],[224,142689,142690,142693],{"style":141919},[224,142691],{"className":142692,"style":772},[771],[224,142694,142696],{"className":142695},[776,777,778,779],[224,142697,20315],{"className":142698},[468,469,779],[224,142700,787],{"className":142701},[786],[224,142703,142705],{"className":142704},[759],[224,142706,142708],{"className":142707,"style":20409},[763],[224,142709],{},[224,142711,268],{"className":142712},[542],[224,142714,24189],{"className":142715,"style":492},[468,469],[224,142717],{"className":142718,"style":724},[473],[224,142720,1435],{"className":142721},[468,469],[224,142723,440],{"className":142724},[549],"，左端点态上确界为 ",[224,142727,142729,142765],{"className":142728,"translate":388},[392],[224,142730,142732],{"className":142731},[396],[398,142733,142734],{"xmlns":400},[403,142735,142736,142762],{},[406,142737,142738,142744,142746,142748,142750,142752,142754,142760],{},[695,142739,142740,142742],{},[409,142741,20324],{},[409,142743,20315],{},[412,142745,268],{"stretchy":435},[412,142747,239],{"stretchy":435},[409,142749,1435],{},[412,142751,5268],{},[409,142753,140963],{},[3248,142755,142756,142758],{},[412,142757,1716],{"stretchy":435},[412,142759,319],{},[412,142761,440],{"stretchy":435},[447,142763,142764],{"encoding":449},"E_Q[(t-\\gamma)^+]",[224,142766,142768,142830],{"className":142767,"ariaHidden":455},[454],[224,142769,142771,142774,142814,142818,142821,142824,142827],{"className":142770},[459],[224,142772],{"className":142773,"style":20368},[463],[224,142775,142777,142780],{"className":142776},[468],[224,142778,20324],{"className":142779,"style":6970},[468,469],[224,142781,142783],{"className":142782},[750],[224,142784,142786,142806],{"className":142785},[754,755],[224,142787,142789,142803],{"className":142788},[759],[224,142790,142792],{"className":142791,"style":764},[763],[224,142793,142794,142797],{"style":141919},[224,142795],{"className":142796,"style":772},[771],[224,142798,142800],{"className":142799},[776,777,778,779],[224,142801,20315],{"className":142802},[468,469,779],[224,142804,787],{"className":142805},[786],[224,142807,142809],{"className":142808},[759],[224,142810,142812],{"className":142811,"style":20409},[763],[224,142813],{},[224,142815,142817],{"className":142816},[542],"[(",[224,142819,1435],{"className":142820},[468,469],[224,142822],{"className":142823,"style":474},[473],[224,142825,5268],{"className":142826},[478],[224,142828],{"className":142829,"style":474},[473],[224,142831,142833,142837,142840,142869],{"className":142832},[459],[224,142834],{"className":142835,"style":142836},[463],"height:1.0213em;vertical-align:-0.25em;",[224,142838,140963],{"className":142839,"style":141053},[468,469],[224,142841,142843,142846],{"className":142842},[549],[224,142844,1716],{"className":142845},[549],[224,142847,142849],{"className":142848},[750],[224,142850,142852],{"className":142851},[754],[224,142853,142855],{"className":142854},[759],[224,142856,142858],{"className":142857,"style":40947},[763],[224,142859,142860,142863],{"style":3288},[224,142861],{"className":142862,"style":772},[771],[224,142864,142866],{"className":142865},[776,777,778,779],[224,142867,319],{"className":142868},[478,779],[224,142870,440],{"className":142871},[549],[224,142873,142875,142897],{"className":142874,"translate":388},[392],[224,142876,142878],{"className":142877},[396],[398,142879,142880],{"xmlns":400},[403,142881,142882,142894],{},[406,142883,142884],{},[695,142885,142886,142888],{},[409,142887,20303],{},[695,142889,142890,142892],{},[409,142891,24189],{},[409,142893,140963],{},[447,142895,142896],{"encoding":449},"D_{g_\\gamma}",[224,142898,142900],{"className":142899,"ariaHidden":455},[454],[224,142901,142903,142907],{"className":142902},[459],[224,142904],{"className":142905,"style":142906},[463],"height:1.0307em;vertical-align:-0.3473em;",[224,142908,142910,142913],{"className":142909},[468],[224,142911,20303],{"className":142912,"style":2473},[468,469],[224,142914,142916],{"className":142915},[750],[224,142917,142919,142979],{"className":142918},[754,755],[224,142920,142922,142976],{"className":142921},[759],[224,142923,142925],{"className":142924,"style":64841},[763],[224,142926,142927,142930],{"style":20390},[224,142928],{"className":142929,"style":772},[771],[224,142931,142933],{"className":142932},[776,777,778,779],[224,142934,142936],{"className":142935},[468,779],[224,142937,142939,142942],{"className":142938},[468,779],[224,142940,24189],{"className":142941,"style":492},[468,469,779],[224,142943,142945],{"className":142944},[750],[224,142946,142948,142968],{"className":142947},[754,755],[224,142949,142951,142965],{"className":142950},[759],[224,142952,142954],{"className":142953,"style":94168},[763],[224,142955,142956,142959],{"style":141588},[224,142957],{"className":142958,"style":50136},[771],[224,142960,142962],{"className":142961},[776,50140,2364,779],[224,142963,140963],{"className":142964,"style":141053},[468,469,779],[224,142966,787],{"className":142967},[786],[224,142969,142971],{"className":142970},[759],[224,142972,142974],{"className":142973,"style":141607},[763],[224,142975],{},[224,142977,787],{"className":142978},[786],[224,142980,142982],{"className":142981},[759],[224,142983,142985],{"className":142984,"style":141619},[763],[224,142986],{},"。另一方面取 ",[224,142989,142991,143027],{"className":142990,"translate":388},[392],[224,142992,142994],{"className":142993},[396],[398,142995,142996],{"xmlns":400},[403,142997,142998,143024],{},[406,142999,143000,143006,143008],{},[3248,143001,143002,143004],{},[409,143003,24189],{},[412,143005,27543],{},[412,143007,278],{},[695,143009,143010,143012],{},[419,143011,1438],{"mathvariant":15754},[406,143013,143014,143016,143018,143020,143022],{},[412,143015,8110],{"stretchy":435},[409,143017,1435],{},[412,143019,325],{},[409,143021,140963],{},[412,143023,15077],{"stretchy":435},[447,143025,143026],{"encoding":449},"g^*=\\mathbf 1_{\\{t>\\gamma\\}}",[224,143028,143030,143074],{"className":143029,"ariaHidden":455},[454],[224,143031,143033,143036,143065,143068,143071],{"className":143032},[459],[224,143034],{"className":143035,"style":83921},[463],[224,143037,143039,143042],{"className":143038},[468],[224,143040,24189],{"className":143041,"style":492},[468,469],[224,143043,143045],{"className":143044},[750],[224,143046,143048],{"className":143047},[754],[224,143049,143051],{"className":143050},[759],[224,143052,143054],{"className":143053,"style":28465},[763],[224,143055,143056,143059],{"style":3288},[224,143057],{"className":143058,"style":772},[771],[224,143060,143062],{"className":143061},[776,777,778,779],[224,143063,27543],{"className":143064},[478,779],[224,143066],{"className":143067,"style":496},[473],[224,143069,278],{"className":143070},[500],[224,143072],{"className":143073,"style":496},[473],[224,143075,143077,143081],{"className":143076},[459],[224,143078],{"className":143079,"style":143080},[463],"height:0.9996em;vertical-align:-0.3552em;",[224,143082,143084,143087],{"className":143083},[468],[224,143085,1438],{"className":143086},[468,95463],[224,143088,143090],{"className":143089},[750],[224,143091,143093,143128],{"className":143092},[754,755],[224,143094,143096,143125],{"className":143095},[759],[224,143097,143099],{"className":143098,"style":26101},[763],[224,143100,143101,143104],{"style":95478},[224,143102],{"className":143103,"style":772},[771],[224,143105,143107],{"className":143106},[776,777,778,779],[224,143108,143110,143113,143116,143119,143122],{"className":143109},[468,779],[224,143111,8110],{"className":143112},[542,779],[224,143114,1435],{"className":143115},[468,469,779],[224,143117,325],{"className":143118},[500,779],[224,143120,140963],{"className":143121,"style":141053},[468,469,779],[224,143123,15077],{"className":143124},[549,779],[224,143126,787],{"className":143127},[786],[224,143129,143131],{"className":143130},[759],[224,143132,143134],{"className":143133,"style":26147},[763],[224,143135],{}," 恰有 ",[224,143138,143140,143204],{"className":143139,"translate":388},[392],[224,143141,143143],{"className":143142},[396],[398,143144,143145],{"xmlns":400},[403,143146,143147,143201],{},[406,143148,143149,143155,143157,143163,143165,143167,143169,143171,143173,143175,143177,143183,143185,143187,143189,143191,143193,143199],{},[695,143150,143151,143153],{},[409,143152,20324],{},[409,143154,20315],{},[412,143156,268],{"stretchy":435},[3248,143158,143159,143161],{},[409,143160,24189],{},[412,143162,27543],{},[412,143164,239],{"stretchy":435},[409,143166,1435],{},[412,143168,5268],{},[409,143170,140963],{},[412,143172,1716],{"stretchy":435},[412,143174,440],{"stretchy":435},[412,143176,278],{},[695,143178,143179,143181],{},[409,143180,20324],{},[409,143182,20315],{},[412,143184,268],{"stretchy":435},[412,143186,239],{"stretchy":435},[409,143188,1435],{},[412,143190,5268],{},[409,143192,140963],{},[3248,143194,143195,143197],{},[412,143196,1716],{"stretchy":435},[412,143198,319],{},[412,143200,440],{"stretchy":435},[447,143202,143203],{"encoding":449},"E_Q[g^*(t-\\gamma)]=E_Q[(t-\\gamma)^+]",[224,143205,143207,143300,143321,143382],{"className":143206,"ariaHidden":455},[454],[224,143208,143210,143213,143253,143256,143285,143288,143291,143294,143297],{"className":143209},[459],[224,143211],{"className":143212,"style":20368},[463],[224,143214,143216,143219],{"className":143215},[468],[224,143217,20324],{"className":143218,"style":6970},[468,469],[224,143220,143222],{"className":143221},[750],[224,143223,143225,143245],{"className":143224},[754,755],[224,143226,143228,143242],{"className":143227},[759],[224,143229,143231],{"className":143230,"style":764},[763],[224,143232,143233,143236],{"style":141919},[224,143234],{"className":143235,"style":772},[771],[224,143237,143239],{"className":143238},[776,777,778,779],[224,143240,20315],{"className":143241},[468,469,779],[224,143243,787],{"className":143244},[786],[224,143246,143248],{"className":143247},[759],[224,143249,143251],{"className":143250,"style":20409},[763],[224,143252],{},[224,143254,268],{"className":143255},[542],[224,143257,143259,143262],{"className":143258},[468],[224,143260,24189],{"className":143261,"style":492},[468,469],[224,143263,143265],{"className":143264},[750],[224,143266,143268],{"className":143267},[754],[224,143269,143271],{"className":143270},[759],[224,143272,143274],{"className":143273,"style":28465},[763],[224,143275,143276,143279],{"style":3288},[224,143277],{"className":143278,"style":772},[771],[224,143280,143282],{"className":143281},[776,777,778,779],[224,143283,27543],{"className":143284},[478,779],[224,143286,239],{"className":143287},[542],[224,143289,1435],{"className":143290},[468,469],[224,143292],{"className":143293,"style":474},[473],[224,143295,5268],{"className":143296},[478],[224,143298],{"className":143299,"style":474},[473],[224,143301,143303,143306,143309,143312,143315,143318],{"className":143302},[459],[224,143304],{"className":143305,"style":535},[463],[224,143307,140963],{"className":143308,"style":141053},[468,469],[224,143310,40329],{"className":143311},[549],[224,143313],{"className":143314,"style":496},[473],[224,143316,278],{"className":143317},[500],[224,143319],{"className":143320,"style":496},[473],[224,143322,143324,143327,143367,143370,143373,143376,143379],{"className":143323},[459],[224,143325],{"className":143326,"style":20368},[463],[224,143328,143330,143333],{"className":143329},[468],[224,143331,20324],{"className":143332,"style":6970},[468,469],[224,143334,143336],{"className":143335},[750],[224,143337,143339,143359],{"className":143338},[754,755],[224,143340,143342,143356],{"className":143341},[759],[224,143343,143345],{"className":143344,"style":764},[763],[224,143346,143347,143350],{"style":141919},[224,143348],{"className":143349,"style":772},[771],[224,143351,143353],{"className":143352},[776,777,778,779],[224,143354,20315],{"className":143355},[468,469,779],[224,143357,787],{"className":143358},[786],[224,143360,143362],{"className":143361},[759],[224,143363,143365],{"className":143364,"style":20409},[763],[224,143366],{},[224,143368,142817],{"className":143369},[542],[224,143371,1435],{"className":143372},[468,469],[224,143374],{"className":143375,"style":474},[473],[224,143377,5268],{"className":143378},[478],[224,143380],{"className":143381,"style":474},[473],[224,143383,143385,143388,143391,143420],{"className":143384},[459],[224,143386],{"className":143387,"style":142836},[463],[224,143389,140963],{"className":143390,"style":141053},[468,469],[224,143392,143394,143397],{"className":143393},[549],[224,143395,1716],{"className":143396},[549],[224,143398,143400],{"className":143399},[750],[224,143401,143403],{"className":143402},[754],[224,143404,143406],{"className":143405},[759],[224,143407,143409],{"className":143408,"style":40947},[763],[224,143410,143411,143414],{"style":3288},[224,143412],{"className":143413,"style":772},[771],[224,143415,143417],{"className":143416},[776,777,778,779],[224,143418,319],{"className":143419},[478,779],[224,143421,440],{"className":143422},[549],"，故等号可取。",[11,143425,143426,143427,143455,143456,143507,143508,143544,143545,35740,143648,143699,143700,143867,143868,143926,143927,143955],{},"几何解释：",[224,143428,143430,143443],{"className":143429,"translate":388},[392],[224,143431,143433],{"className":143432},[396],[398,143434,143435],{"xmlns":400},[403,143436,143437,143441],{},[406,143438,143439],{},[409,143440,140963],{},[447,143442,141380],{"encoding":449},[224,143444,143446],{"className":143445,"ariaHidden":455},[454],[224,143447,143449,143452],{"className":143448},[459],[224,143450],{"className":143451,"style":806},[463],[224,143453,140963],{"className":143454,"style":141053},[468,469]," 扫过 ",[224,143457,143459,143480],{"className":143458,"translate":388},[392],[224,143460,143462],{"className":143461},[396],[398,143463,143464],{"xmlns":400},[403,143465,143466,143478],{},[406,143467,143468,143470,143472,143474,143476],{},[412,143469,268],{"stretchy":435},[419,143471,284],{},[412,143473,1877],{"separator":455},[419,143475,1438],{},[412,143477,440],{"stretchy":435},[447,143479,56736],{"encoding":449},[224,143481,143483],{"className":143482,"ariaHidden":455},[454],[224,143484,143486,143489,143492,143495,143498,143501,143504],{"className":143485},[459],[224,143487],{"className":143488,"style":535},[463],[224,143490,268],{"className":143491},[542],[224,143493,284],{"className":143494},[468],[224,143496,1877],{"className":143497},[1917],[224,143499],{"className":143500,"style":724},[473],[224,143502,1438],{"className":143503},[468],[224,143505,440],{"className":143506},[549]," 时，右端就是权衡曲线（ROC 下边界）各条斜率为 ",[224,143509,143511,143529],{"className":143510,"translate":388},[392],[224,143512,143514],{"className":143513},[396],[398,143515,143516],{"xmlns":400},[403,143517,143518,143526],{},[406,143519,143520,143522,143524],{},[419,143521,1438],{},[409,143523,12413],{"mathvariant":678},[409,143525,140963],{},[447,143527,143528],{"encoding":449},"1\u002F\\gamma",[224,143530,143532],{"className":143531,"ariaHidden":455},[454],[224,143533,143535,143538,143541],{"className":143534},[459],[224,143536],{"className":143537,"style":535},[463],[224,143539,28555],{"className":143540},[468],[224,143542,140963],{"className":143543,"style":141053},[468,469]," 的支撑线的截距——",[378,143546,143547,143548,143576,143577,143647],{},"知道整条曲线 ",[224,143549,143551,143564],{"className":143550,"translate":388},[392],[224,143552,143554],{"className":143553},[396],[398,143555,143556],{"xmlns":400},[403,143557,143558,143562],{},[406,143559,143560],{},[409,143561,637],{},[447,143563,637],{"encoding":449},[224,143565,143567],{"className":143566,"ariaHidden":455},[454],[224,143568,143570,143573],{"className":143569},[459],[224,143571],{"className":143572,"style":649},[463],[224,143574,637],{"className":143575,"style":653},[468,469]," 就知道全部 ",[224,143578,143580,143598],{"className":143579,"translate":388},[392],[224,143581,143583],{"className":143582},[396],[398,143584,143585],{"xmlns":400},[403,143586,143587,143595],{},[406,143588,143589],{},[695,143590,143591,143593],{},[409,143592,24189],{},[409,143594,140963],{},[447,143596,143597],{"encoding":449},"g_\\gamma",[224,143599,143601],{"className":143600,"ariaHidden":455},[454],[224,143602,143604,143607],{"className":143603},[459],[224,143605],{"className":143606,"style":112858},[463],[224,143608,143610,143613],{"className":143609},[468],[224,143611,24189],{"className":143612,"style":492},[468,469],[224,143614,143616],{"className":143615},[750],[224,143617,143619,143639],{"className":143618},[754,755],[224,143620,143622,143636],{"className":143621},[759],[224,143623,143625],{"className":143624,"style":64841},[763],[224,143626,143627,143630],{"style":3854},[224,143628],{"className":143629,"style":772},[771],[224,143631,143633],{"className":143632},[776,777,778,779],[224,143634,140963],{"className":143635,"style":141053},[468,469,779],[224,143637,787],{"className":143638},[786],[224,143640,143642],{"className":143641},[759],[224,143643,143645],{"className":143644,"style":20409},[763],[224,143646],{},"-散度",[224,143649,143651,143669],{"className":143650,"translate":388},[392],[224,143652,143654],{"className":143653},[396],[398,143655,143656],{"xmlns":400},[403,143657,143658,143666],{},[406,143659,143660,143662,143664],{},[409,143661,140963],{},[412,143663,278],{},[419,143665,1438],{},[447,143667,143668],{"encoding":449},"\\gamma=1",[224,143670,143672,143690],{"className":143671,"ariaHidden":455},[454],[224,143673,143675,143678,143681,143684,143687],{"className":143674},[459],[224,143676],{"className":143677,"style":806},[463],[224,143679,140963],{"className":143680,"style":141053},[468,469],[224,143682],{"className":143683,"style":496},[473],[224,143685,278],{"className":143686},[500],[224,143688],{"className":143689,"style":496},[473],[224,143691,143693,143696],{"className":143692},[459],[224,143694],{"className":143695,"style":510},[463],[224,143697,1438],{"className":143698},[468]," 时立刻给出 ",[224,143701,143703,143755],{"className":143702,"translate":388},[392],[224,143704,143706],{"className":143705},[396],[398,143707,143708],{"xmlns":400},[403,143709,143710,143752],{},[406,143711,143712,143718,143720,143730,143732,143734,143736,143738,143740,143742,143744,143746,143748,143750],{},[406,143713,143714,143716],{},[409,143715,637],{"mathvariant":678},[409,143717,16059],{"mathvariant":678},[412,143719,278],{},[695,143721,143722,143728],{},[406,143723,143724,143726],{},[409,143725,55354],{},[412,143727,24134],{},[409,143729,1097],{},[412,143731,8110],{"stretchy":435},[409,143733,16018],{},[412,143735,239],{"stretchy":435},[409,143737,1097],{},[412,143739,1716],{"stretchy":435},[412,143741,5268],{},[409,143743,20315],{},[412,143745,239],{"stretchy":435},[409,143747,1097],{},[412,143749,1716],{"stretchy":435},[412,143751,15077],{"stretchy":435},[447,143753,143754],{"encoding":449},"\\mathrm{TV}=\\sup_A\\{P(A)-Q(A)\\}",[224,143756,143758,143779,143849],{"className":143757,"ariaHidden":455},[454],[224,143759,143761,143764,143770,143773,143776],{"className":143760},[459],[224,143762],{"className":143763,"style":649},[463],[224,143765,143767],{"className":143766},[468],[224,143768,106664],{"className":143769,"style":24188},[468,1906],[224,143771],{"className":143772,"style":496},[473],[224,143774,278],{"className":143775},[500],[224,143777],{"className":143778,"style":496},[473],[224,143780,143782,143785,143825,143828,143831,143834,143837,143840,143843,143846],{"className":143781},[459],[224,143783],{"className":143784,"style":535},[463],[224,143786,143788,143791],{"className":143787},[21531],[224,143789,55354],{"className":143790},[21531],[224,143792,143794],{"className":143793},[750],[224,143795,143797,143817],{"className":143796},[754,755],[224,143798,143800,143814],{"className":143799},[759],[224,143801,143803],{"className":143802,"style":55497},[763],[224,143804,143805,143808],{"style":55500},[224,143806],{"className":143807,"style":772},[771],[224,143809,143811],{"className":143810},[776,777,778,779],[224,143812,1097],{"className":143813},[468,469,779],[224,143815,787],{"className":143816},[786],[224,143818,143820],{"className":143819},[759],[224,143821,143823],{"className":143822,"style":56942},[763],[224,143824],{},[224,143826,8110],{"className":143827},[542],[224,143829,16018],{"className":143830,"style":653},[468,469],[224,143832,239],{"className":143833},[542],[224,143835,1097],{"className":143836},[468,469],[224,143838,1716],{"className":143839},[549],[224,143841],{"className":143842,"style":474},[473],[224,143844,5268],{"className":143845},[478],[224,143847],{"className":143848,"style":474},[473],[224,143850,143852,143855,143858,143861,143864],{"className":143851},[459],[224,143853],{"className":143854,"style":535},[463],[224,143856,20315],{"className":143857},[468,469],[224,143859,239],{"className":143860},[542],[224,143862,1097],{"className":143863},[468,469],[224,143865,80905],{"className":143866},[549],"）。反过来，知道单个标量散度（如 KL 或 ",[224,143869,143871,143888],{"className":143870,"translate":388},[392],[224,143872,143874],{"className":143873},[396],[398,143875,143876],{"xmlns":400},[403,143877,143878,143886],{},[406,143879,143880],{},[3248,143881,143882,143884],{},[409,143883,57370],{},[419,143885,1447],{},[447,143887,60308],{"encoding":449},[224,143889,143891],{"className":143890,"ariaHidden":455},[454],[224,143892,143894,143897],{"className":143893},[459],[224,143895],{"className":143896,"style":60318},[463],[224,143898,143900,143903],{"className":143899},[468],[224,143901,57370],{"className":143902},[468,469],[224,143904,143906],{"className":143905},[750],[224,143907,143909],{"className":143908},[754],[224,143910,143912],{"className":143911},[759],[224,143913,143915],{"className":143914,"style":5671},[763],[224,143916,143917,143920],{"style":3288},[224,143918],{"className":143919,"style":772},[771],[224,143921,143923],{"className":143922},[776,777,778,779],[224,143924,1447],{"className":143925},[468,779],"）只对 ",[224,143928,143930,143943],{"className":143929,"translate":388},[392],[224,143931,143933],{"className":143932},[396],[398,143934,143935],{"xmlns":400},[403,143936,143937,143941],{},[406,143938,143939],{},[409,143940,637],{},[447,143942,637],{"encoding":449},[224,143944,143946],{"className":143945,"ariaHidden":455},[454],[224,143947,143949,143952],{"className":143948},[459],[224,143950],{"className":143951,"style":649},[463],[224,143953,637],{"className":143954,"style":653},[468,469]," 施加一个切向约束（4.7 节的不等式网络），信息严格更少；",[2674,143957,143958,144190],{},[2677,143959,143960,143961,6095],{},"在「小隐私预算」下，所有 f-DP 都由**高斯差分隐私（GDP）**统一逼近，其权衡函数 ",[224,143962,143964,144022],{"className":143963,"translate":388},[392],[224,143965,143967],{"className":143966},[396],[398,143968,143969],{"xmlns":400},[403,143970,143971,144019],{},[406,143972,143973,143980,143982,143984,143986,143988,143991,143993,144003,144005,144007,144009,144011,144013,144015,144017],{},[695,143974,143975,143978],{},[409,143976,143977],{},"G",[409,143979,22405],{},[412,143981,239],{"stretchy":435},[409,143983,64666],{},[412,143985,1716],{"stretchy":435},[412,143987,278],{},[409,143989,143990],{"mathvariant":678},"Φ",[412,143992,239],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[3248,143994,143995,143997],{},[409,143996,143990],{"mathvariant":678},[406,143998,143999,144001],{},[412,144000,5268],{},[419,144002,1438],{},[412,144004,239],{"stretchy":435},[419,144006,1438],{},[412,144008,5268],{},[409,144010,64666],{},[412,144012,1716],{"stretchy":435},[412,144014,5268],{},[409,144016,22405],{},[412,144018,1716],{"fence":435,"stretchy":455,"minsize":2004,"maxsize":2004},[447,144020,144021],{"encoding":449},"G_\\mu(\\alpha)=\\Phi\\big(\\Phi^{-1}(1-\\alpha)-\\mu\\big)",[224,144023,144025,144089,144154,144175],{"className":144024,"ariaHidden":455},[454],[224,144026,144028,144031,144071,144074,144077,144080,144083,144086],{"className":144027},[459],[224,144029],{"className":144030,"style":20368},[463],[224,144032,144034,144037],{"className":144033},[468],[224,144035,143977],{"className":144036},[468,469],[224,144038,144040],{"className":144039},[750],[224,144041,144043,144063],{"className":144042},[754,755],[224,144044,144046,144060],{"className":144045},[759],[224,144047,144049],{"className":144048,"style":64841},[763],[224,144050,144051,144054],{"style":767},[224,144052],{"className":144053,"style":772},[771],[224,144055,144057],{"className":144056},[776,777,778,779],[224,144058,22405],{"className":144059},[468,469,779],[224,144061,787],{"className":144062},[786],[224,144064,144066],{"className":144065},[759],[224,144067,144069],{"className":144068,"style":20409},[763],[224,144070],{},[224,144072,239],{"className":144073},[542],[224,144075,64666],{"className":144076,"style":64696},[468,469],[224,144078,1716],{"className":144079},[549],[224,144081],{"className":144082,"style":496},[473],[224,144084,278],{"className":144085},[500],[224,144087],{"className":144088,"style":496},[473],[224,144090,144092,144095,144098,144104,144139,144142,144145,144148,144151],{"className":144091},[459],[224,144093],{"className":144094,"style":2663},[463],[224,144096,143990],{"className":144097},[468],[224,144099,144101],{"className":144100},[468],[224,144102,239],{"className":144103},[2363,2364],[224,144105,144107,144110],{"className":144106},[468],[224,144108,143990],{"className":144109},[468],[224,144111,144113],{"className":144112},[750],[224,144114,144116],{"className":144115},[754],[224,144117,144119],{"className":144118},[759],[224,144120,144122],{"className":144121,"style":5671},[763],[224,144123,144124,144127],{"style":3288},[224,144125],{"className":144126,"style":772},[771],[224,144128,144130],{"className":144129},[776,777,778,779],[224,144131,144133,144136],{"className":144132},[468,779],[224,144134,5268],{"className":144135},[468,779],[224,144137,1438],{"className":144138},[468,779],[224,144140,239],{"className":144141},[542],[224,144143,1438],{"className":144144},[468],[224,144146],{"className":144147,"style":474},[473],[224,144149,5268],{"className":144150},[478],[224,144152],{"className":144153,"style":474},[473],[224,144155,144157,144160,144163,144166,144169,144172],{"className":144156},[459],[224,144158],{"className":144159,"style":535},[463],[224,144161,64666],{"className":144162,"style":64696},[468,469],[224,144164,1716],{"className":144165},[549],[224,144167],{"className":144168,"style":474},[473],[224,144170,5268],{"className":144171},[478],[224,144173],{"className":144174,"style":474},[473],[224,144176,144178,144181,144184],{"className":144177},[459],[224,144179],{"className":144180,"style":2663},[463],[224,144182,22405],{"className":144183},[468,469],[224,144185,144187],{"className":144186},[468],[224,144188,1716],{"className":144189},[2363,2364],[2677,144191,144192,144193,144196,144197,144225],{},"权衡函数族在",[378,144194,144195],{},"组合","（post-processing、串联）下封闭，并装备了一个与 f-散度\"散度序\"同构的偏序——这与 f-散度族的\"",[224,144198,144200,144213],{"className":144199,"translate":388},[392],[224,144201,144203],{"className":144202},[396],[398,144204,144205],{"xmlns":400},[403,144206,144207,144211],{},[406,144208,144209],{},[412,144210,9766],{},[447,144212,75395],{"encoding":449},[224,144214,144216],{"className":144215,"ariaHidden":455},[454],[224,144217,144219,144222],{"className":144218},[459],[224,144220],{"className":144221,"style":3204},[463],[224,144223,9766],{"className":144224},[500]," 比较\"在结构上完全平行。",[11,144227,144228],{},"这说明 f-散度不仅统一了信息论度量，也统一了现代隐私的定义框架：用一条凸曲线代替一个标量数字。",[608,144230,144232],{"id":144231},"_9-计算实例","9. 计算实例",[613,144234,144236],{"id":144235},"_91-正态分布","9.1 正态分布",[11,144238,20606,144239,372,144414,3117],{},[224,144240,144242,144280],{"className":144241,"translate":388},[392],[224,144243,144245],{"className":144244},[396],[398,144246,144247],{"xmlns":400},[403,144248,144249,144277],{},[406,144250,144251,144253,144255,144257,144259,144265,144267,144275],{},[409,144252,16018],{},[412,144254,278],{},[409,144256,58205],{},[412,144258,239],{"stretchy":435},[695,144260,144261,144263],{},[409,144262,22405],{},[419,144264,1438],{},[412,144266,1877],{"separator":455},[27527,144268,144269,144271,144273],{},[409,144270,685],{},[419,144272,1438],{},[419,144274,1447],{},[412,144276,1716],{"stretchy":435},[447,144278,144279],{"encoding":449},"P=N(\\mu_1,\\sigma_1^2)",[224,144281,144283,144301],{"className":144282,"ariaHidden":455},[454],[224,144284,144286,144289,144292,144295,144298],{"className":144285},[459],[224,144287],{"className":144288,"style":649},[463],[224,144290,16018],{"className":144291,"style":653},[468,469],[224,144293],{"className":144294,"style":496},[473],[224,144296,278],{"className":144297},[500],[224,144299],{"className":144300,"style":496},[473],[224,144302,144304,144307,144310,144313,144353,144356,144359,144411],{"className":144303},[459],[224,144305],{"className":144306,"style":39943},[463],[224,144308,58205],{"className":144309,"style":45534},[468,469],[224,144311,239],{"className":144312},[542],[224,144314,144316,144319],{"className":144315},[468],[224,144317,22405],{"className":144318},[468,469],[224,144320,144322],{"className":144321},[750],[224,144323,144325,144345],{"className":144324},[754,755],[224,144326,144328,144342],{"className":144327},[759],[224,144329,144331],{"className":144330,"style":1479},[763],[224,144332,144333,144336],{"style":767},[224,144334],{"className":144335,"style":772},[771],[224,144337,144339],{"className":144338},[776,777,778,779],[224,144340,1438],{"className":144341},[468,779],[224,144343,787],{"className":144344},[786],[224,144346,144348],{"className":144347},[759],[224,144349,144351],{"className":144350,"style":794},[763],[224,144352],{},[224,144354,1877],{"className":144355},[1917],[224,144357],{"className":144358,"style":724},[473],[224,144360,144362,144365],{"className":144361},[468],[224,144363,685],{"className":144364,"style":492},[468,469],[224,144366,144368],{"className":144367},[750],[224,144369,144371,144403],{"className":144370},[754,755],[224,144372,144374,144400],{"className":144373},[759],[224,144375,144377,144389],{"className":144376,"style":5671},[763],[224,144378,144380,144383],{"style":144379},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[224,144381],{"className":144382,"style":772},[771],[224,144384,144386],{"className":144385},[776,777,778,779],[224,144387,1438],{"className":144388},[468,779],[224,144390,144391,144394],{"style":3288},[224,144392],{"className":144393,"style":772},[771],[224,144395,144397],{"className":144396},[776,777,778,779],[224,144398,1447],{"className":144399},[468,779],[224,144401,787],{"className":144402},[786],[224,144404,144406],{"className":144405},[759],[224,144407,144409],{"className":144408,"style":72031},[763],[224,144410],{},[224,144412,1716],{"className":144413},[549],[224,144415,144417,144455],{"className":144416,"translate":388},[392],[224,144418,144420],{"className":144419},[396],[398,144421,144422],{"xmlns":400},[403,144423,144424,144452],{},[406,144425,144426,144428,144430,144432,144434,144440,144442,144450],{},[409,144427,20315],{},[412,144429,278],{},[409,144431,58205],{},[412,144433,239],{"stretchy":435},[695,144435,144436,144438],{},[409,144437,22405],{},[419,144439,1447],{},[412,144441,1877],{"separator":455},[27527,144443,144444,144446,144448],{},[409,144445,685],{},[419,144447,1447],{},[419,144449,1447],{},[412,144451,1716],{"stretchy":435},[447,144453,144454],{"encoding":449},"Q=N(\\mu_2,\\sigma_2^2)",[224,144456,144458,144476],{"className":144457,"ariaHidden":455},[454],[224,144459,144461,144464,144467,144470,144473],{"className":144460},[459],[224,144462],{"className":144463,"style":20637},[463],[224,144465,20315],{"className":144466},[468,469],[224,144468],{"className":144469,"style":496},[473],[224,144471,278],{"className":144472},[500],[224,144474],{"className":144475,"style":496},[473],[224,144477,144479,144482,144485,144488,144528,144531,144534,144585],{"className":144478},[459],[224,144480],{"className":144481,"style":39943},[463],[224,144483,58205],{"className":144484,"style":45534},[468,469],[224,144486,239],{"className":144487},[542],[224,144489,144491,144494],{"className":144490},[468],[224,144492,22405],{"className":144493},[468,469],[224,144495,144497],{"className":144496},[750],[224,144498,144500,144520],{"className":144499},[754,755],[224,144501,144503,144517],{"className":144502},[759],[224,144504,144506],{"className":144505,"style":1479},[763],[224,144507,144508,144511],{"style":767},[224,144509],{"className":144510,"style":772},[771],[224,144512,144514],{"className":144513},[776,777,778,779],[224,144515,1447],{"className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",[224,145396,145398,145428],{"className":145397,"translate":388},[392],[224,145399,145401],{"className":145400},[396],[398,145402,145403],{"xmlns":400},[403,145404,145405,145425],{},[406,145406,145407,145413,145415,145421,145423],{},[695,145408,145409,145411],{},[409,145410,685],{},[419,145412,1438],{},[412,145414,278],{},[695,145416,145417,145419],{},[409,145418,685],{},[419,145420,1447],{},[412,145422,278],{},[409,145424,685],{},[447,145426,145427],{"encoding":449},"\\sigma_1=\\sigma_2=\\sigma",[224,145429,145431,145486,145541],{"className":145430,"ariaHidden":455},[454],[224,145432,145434,145437,145477,145480,145483],{"className":145433},[459],[224,145435],{"className":145436,"style":3625},[463],[224,145438,145440,145443],{"className":145439},[468],[224,145441,685],{"className":145442,"style":492},[468,469],[224,145444,145446],{"className":145445},[750],[224,145447,145449,145469],{"className":145448},[754,755],[224,145450,145452,145466],{"className":145451},[759],[224,145453,145455],{"className":145454,"style":1479},[763],[224,145456,145457,145460],{"style":3854},[224,145458],{"className":145459,"style":772},[771],[224,145461,145463],{"className":145462},[776,777,778,779],[224,145464,1438],{"className":145465},[468,779],[224,145467,787],{"className":145468},[786],[224,145470,145472],{"className":145471},[759],[224,145473,145475],{"className":145474,"style":794},[763],[224,145476],{},[224,145478],{"className":145479,"style":496},[473],[224,145481,278],{"className":145482},[500],[224,145484],{"className":145485,"style":496},[473],[224,145487,145489,145492,145532,145535,145538],{"className":145488},[459],[224,145490],{"className":145491,"style":3625},[463],[224,145493,145495,145498],{"className":145494},[468],[224,145496,685],{"className":145497,"style":492},[468,469],[224,145499,145501],{"className":145500},[750],[224,145502,145504,145524],{"className":145503},[754,755],[224,145505,145507,145521],{"className":145506},[759],[224,145508,145510],{"className":145509,"style":1479},[763],[224,145511,145512,145515],{"style":3854},[224,145513],{"className":145514,"style":772},[771],[224,145516,145518],{"className":145517},[776,777,778,779],[224,145519,1447],{"className":145520},[468,779],[224,145522,787],{"className":145523},[786],[224,145525,145527],{"className":145526},[759],[224,145528,145530],{"className":145529,"style":794},[763],[224,145531],{},[224,145533],{"className":145534,"style":496},[473],[224,145536,278],{"className":145537},[500],[224,145539],{"className":145540,"style":496},[473],[224,145542,145544,145547],{"className":145543},[459],[224,145545],{"className":145546,"style":839},[463],[224,145548,685],{"className":145549,"style":492},[468,469],[224,145551,145553,145584],{"className":145552,"translate":388},[392],[224,145554,145556],{"className":145555},[396],[398,145557,145558],{"xmlns":400},[403,145559,145560,145581],{},[406,145561,145562,145565,145567,145573,145575],{},[409,145563,145564],{"mathvariant":678},"Δ",[412,145566,278],{},[695,145568,145569,145571],{},[409,145570,22405],{},[419,145572,1438],{},[412,145574,5268],{},[695,145576,145577,145579],{},[409,145578,22405],{},[419,145580,1447],{},[447,145582,145583],{"encoding":449},"\\Delta=\\mu_1-\\mu_2",[224,145585,145587,145605,145660],{"className":145586,"ariaHidden":455},[454],[224,145588,145590,145593,145596,145599,145602],{"className":145589},[459],[224,145591],{"className":145592,"style":649},[463],[224,145594,145564],{"className":145595},[468],[224,145597],{"className":145598,"style":496},[473],[224,145600,278],{"className":145601},[500],[224,145603],{"className":145604,"style":496},[473],[224,145606,145608,145611,145651,145654,145657],{"className":145607},[459],[224,145609],{"className":145610,"style":4846},[463],[224,145612,145614,145617],{"className":145613},[468],[224,145615,22405],{"className":145616},[468,469],[224,145618,145620],{"className":145619},[750],[224,145621,145623,145643],{"className":145622},[754,755],[224,145624,145626,145640],{"className":145625},[759],[224,145627,145629],{"className":145628,"style":1479},[763],[224,145630,145631,145634],{"style":767},[224,145632],{"className":145633,"style":772},[771],[224,145635,145637],{"className":145636},[776,777,778,779],[224,145638,1438],{"className":145639},[468,779],[224,145641,787],{"className":145642},[786],[224,145644,145646],{"className":145645},[759],[224,145647,145649],{"className":145648,"style":794},[763],[224,145650],{},[224,145652],{"className":145653,"style":474},[473],[224,145655,5268],{"className":145656},[478],[224,145658],{"className":145659,"style":474},[473],[224,145661,145663,145666],{"className":145662},[459],[224,145664],{"className":145665,"style":806},[463],[224,145667,145669,145672],{"className":145668},[468],[224,145670,22405],{"className":145671},[468,469],[224,145673,145675],{"className":145674},[750],[224,145676,145678,145698],{"className":145677},[754,755],[224,145679,145681,145695],{"className":145680},[759],[224,145682,145684],{"className":145683,"style":1479},[763],[224,145685,145686,145689],{"style":767},[224,145687],{"className":145688,"style":772},[771],[224,145690,145692],{"className":145691},[776,777,778,779],[224,145693,1447],{"className":145694},[468,779],[224,145696,787],{"className":145697},[786],[224,145699,145701],{"className":145700},[759],[224,145702,145704],{"className":145703,"style":794},[763],[224,145705],{}," 计算 ",[224,145708,145710,145734],{"className":145709,"translate":388},[392],[224,145711,145713],{"className":145712},[396],[398,145714,145715],{"xmlns":400},[403,145716,145717,145731],{},[406,145718,145719,145721,145727,145729],{},[412,145720,21338],{},[3248,145722,145723,145725],{},[409,145724,11],{},[419,145726,1447],{},[409,145728,12413],{"mathvariant":678},[409,145730,2023],{},[447,145732,145733],{"encoding":449},"\\int p^2\u002Fq",[224,145735,145737],{"className":145736,"ariaHidden":455},[454],[224,145738,145740,145743,145746,145749,145778,145781],{"className":145739},[459],[224,145741],{"className":145742,"style":70236},[463],[224,145744,21338],{"className":145745,"style":25626},[21531,21532,25625],[224,145747],{"className":145748,"style":724},[473],[224,145750,145752,145755],{"className":145751},[468],[224,145753,11],{"className":145754},[468,469],[224,145756,145758],{"className":145757},[750],[224,145759,145761],{"className":145760},[754],[224,145762,145764],{"className":145763},[759],[224,145765,145767],{"className":145766,"style":5671},[763],[224,145768,145769,145772],{"style":3288},[224,145770],{"className":145771,"style":772},[771],[224,145773,145775],{"className":145774},[776,777,778,779],[224,145776,1447],{"className":145777},[468,779],[224,145779,12413],{"className":145780},[468],[224,145782,2023],{"className":145783,"style":492},[468,469],"：由配方可得",[224,145786,145788],{"className":145787,"translate":388},[387],[224,145789,145791,145852],{"className":145790,"translate":388},[392],[224,145792,145794],{"className":145793},[396],[398,145795,145796],{"xmlns":400,"display":401},[403,145797,145798,145849],{},[406,145799,145800,145802,145812,145814,145816,145818,145820,145823,145825,145827,145847],{},[412,145801,21338],{},[20338,145803,145804,145810],{},[3248,145805,145806,145808],{},[409,145807,11],{},[419,145809,1447],{},[409,145811,2023],{},[423,145813,682],{},[409,145815,13182],{},[409,145817,371],{},[412,145819,278],{},[409,145821,145822],{},"exp",[412,145824,24134],{},[423,145826,21353],{},[406,145828,145829,145831,145845],{},[412,145830,239],{"fence":455},[20338,145832,145833,145839],{},[3248,145834,145835,145837],{},[409,145836,145564],{"mathvariant":678},[419,145838,1447],{},[3248,145840,145841,145843],{},[409,145842,685],{},[419,145844,1447],{},[412,145846,1716],{"fence":455},[412,145848,1877],{"separator":455},[447,145850,145851],{"encoding":449},"\\int \\frac{p^2}{q}\\,dx\n=\\exp\\!\\left(\\frac{\\Delta^2}{\\sigma^2}\\right),",[224,145853,145855,145973],{"className":145854,"ariaHidden":455},[454],[224,145856,145858,145861,145864,145867,145955,145958,145961,145964,145967,145970],{"className":145857},[459],[224,145859],{"className":145860,"style":57619},[463],[224,145862,21338],{"className":145863,"style":21534},[21531,21532,21533],[224,145865],{"className":145866,"style":724},[473],[224,145868,145870,145873,145952],{"className":145869},[468],[224,145871],{"className":145872},[542,8479],[224,145874,145876],{"className":145875},[20338],[224,145877,145879,145944],{"className":145878},[754,755],[224,145880,145882,145941],{"className":145881},[759],[224,145883,145885,145896,145904],{"className":145884,"style":57644},[763],[224,145886,145887,145890],{"style":20531},[224,145888],{"className":145889,"style":2108},[771],[224,145891,145893],{"className":145892},[468],[224,145894,2023],{"className":145895,"style":492},[468,469],[224,145897,145898,145901],{"style":20546},[224,145899],{"className":145900,"style":2108},[771],[224,145902],{"className":145903,"style":20554},[20553],[224,145905,145906,145909],{"style":20557},[224,145907],{"className":145908,"style":2108},[771],[224,145910,145912],{"className":145911},[468],[224,145913,145915,145918],{"className":145914},[468],[224,145916,11],{"className":145917},[468,469],[224,145919,145921],{"className":145920},[750],[224,145922,145924],{"className":145923},[754],[224,145925,145927],{"className":145926},[759],[224,145928,145930],{"className":145929,"style":5671},[763],[224,145931,145932,145935],{"style":3288},[224,145933],{"className":145934,"style":772},[771],[224,145936,145938],{"className":145937},[776,777,778,779],[224,145939,1447],{"className":145940},[468,779],[224,145942,787],{"className":145943},[786],[224,145945,145947],{"className":145946},[759],[224,145948,145950],{"className":145949,"style":20579},[763],[224,145951],{},[224,145953],{"className":145954},[549,8479],[224,145956],{"className":145957,"style":724},[473],[224,145959,13182],{"className":145960},[468,469],[224,145962,371],{"className":145963},[468,469],[224,145965],{"className":145966,"style":496},[473],[224,145968,278],{"className":145969},[500],[224,145971],{"className":145972,"style":496},[473],[224,145974,145976,145979,145982,145985,145988,146117,146120],{"className":145975},[459],[224,145977],{"className":145978,"style":107159},[463],[224,145980,145822],{"className":145981},[21531],[224,145983],{"className":145984,"style":21559},[473],[224,145986],{"className":145987,"style":724},[473],[224,145989,145991,145997,146111],{"className":145990},[2091],[224,145992,145994],{"className":145993,"style":8281},[542,8280],[224,145995,239],{"className":145996},[2363,778],[224,145998,146000,146003,146108],{"className":145999},[468],[224,146001],{"className":146002},[542,8479],[224,146004,146006],{"className":146005},[20338],[224,146007,146009,146100],{"className":146008},[754,755],[224,146010,146012,146097],{"className":146011},[759],[224,146013,146015,146052,146060],{"className":146014,"style":57644},[763],[224,146016,146017,146020],{"style":20531},[224,146018],{"className":146019,"style":2108},[771],[224,146021,146023],{"className":146022},[468],[224,146024,146026,146029],{"className":146025},[468],[224,146027,685],{"className":146028,"style":492},[468,469],[224,146030,146032],{"className":146031},[750],[224,146033,146035],{"className":146034},[754],[224,146036,146038],{"className":146037},[759],[224,146039,146041],{"className":146040,"style":64944},[763],[224,146042,146043,146046],{"style":64947},[224,146044],{"className":146045,"style":772},[771],[224,146047,146049],{"className":146048},[776,777,778,779],[224,146050,1447],{"className":146051},[468,779],[224,146053,146054,146057],{"style":20546},[224,146055],{"className":146056,"style":2108},[771],[224,146058],{"className":146059,"style":20554},[20553],[224,146061,146062,146065],{"style":20557},[224,146063],{"className":146064,"style":2108},[771],[224,146066,146068],{"className":146067},[468],[224,146069,146071,146074],{"className":146070},[468],[224,146072,145564],{"className":146073},[468],[224,146075,146077],{"className":146076},[750],[224,146078,146080],{"className":146079},[754],[224,146081,146083],{"className":146082},[759],[224,146084,146086],{"className":146085,"style":5671},[763],[224,146087,146088,146091],{"style":3288},[224,146089],{"className":146090,"style":772},[771],[224,146092,146094],{"className":146093},[776,777,778,779],[224,146095,1447],{"className":146096},[468,779],[224,146098,787],{"className":146099},[786],[224,146101,146103],{"className":146102},[759],[224,146104,146106],{"className":146105,"style":23809},[763],[224,146107],{},[224,146109],{"className":146110},[549,8479],[224,146112,146114],{"className":146113,"style":8281},[549,8280],[224,146115,1716],{"className":146116},[2363,778],[224,146118],{"className":146119,"style":724},[473],[224,146121,1877],{"className":146122},[1917],[11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.",[224,146240,146242,146301,146437,146546,146564,146717],{"className":146241,"ariaHidden":455},[454],[224,146243,146245,146248,146277,146280,146283,146286,146289,146292,146295,146298],{"className":146244},[459],[224,146246],{"className":146247,"style":13273},[463],[224,146249,146251,146254],{"className":146250},[468],[224,146252,57370],{"className":146253},[468,469],[224,146255,146257],{"className":146256},[750],[224,146258,146260],{"className":146259},[754],[224,146261,146263],{"className":146262},[759],[224,146264,146266],{"className":146265,"style":5071},[763],[224,146267,146268,146271],{"style":5074},[224,146269],{"className":146270,"style":772},[771],[224,146272,146274],{"className":146273},[776,777,778,779],[224,146275,1447],{"className":146276},[468,779],[224,146278,239],{"className":146279},[542],[224,146281,16018],{"className":146282,"style":653},[468,469],[224,146284,20312],{"className":146285},[468],[224,146287,20315],{"className":146288},[468,469],[224,146290,1716],{"className":146291},[549],[224,146293],{"className":146294,"style":496},[473],[224,146296,278],{"className":146297},[500],[224,146299],{"className":146300,"style":496},[473],[224,146302,146304,146307,146310,146313,146419,146422,146425,146428,146431,146434],{"className":146303},[459],[224,146305],{"className":146306,"style":57619},[463],[224,146308,21338],{"className":146309,"style":21534},[21531,21532,21533],[224,146311],{"className":146312,"style":724},[473],[224,146314,146316,146319,146416],{"className":146315},[468],[224,146317],{"className":146318},[542,8479],[224,146320,146322],{"className":146321},[20338],[224,146323,146325,146408],{"className":146324},[754,755],[224,146326,146328,146405],{"className":146327},[759],[224,146329,146331,146342,146350],{"className":146330,"style":57644},[763],[224,146332,146333,146336],{"style":20531},[224,146334],{"className":146335,"style":2108},[771],[224,146337,146339],{"className":146338},[468],[224,146340,2023],{"className":146341,"style":492},[468,469],[224,146343,146344,146347],{"style":20546},[224,146345],{"className":146346,"style":2108},[771],[224,146348],{"className":146349,"style":20554},[20553],[224,146351,146352,146355],{"style":20557},[224,146353],{"className":146354,"style":2108},[771],[224,146356,146358,146361,146364,146367,146370,146373,146376],{"className":146357},[468],[224,146359,239],{"className":146360},[542],[224,146362,11],{"className":146363},[468,469],[224,146365],{"className":146366,"style":474},[473],[224,146368,5268],{"className":146369},[478],[224,146371],{"className":146372,"style":474},[473],[224,146374,2023],{"className":146375,"style":492},[468,469],[224,146377,146379,146382],{"className":146378},[549],[224,146380,1716],{"className":146381},[549],[224,146383,146385],{"className":146384},[750],[224,146386,146388],{"className":146387},[754],[224,146389,146391],{"className":146390},[759],[224,146392,146394],{"className":146393,"style":5671},[763],[224,146395,146396,146399],{"style":3288},[224,146397],{"className":146398,"style":772},[771],[224,146400,146402],{"className":146401},[776,777,778,779],[224,146403,1447],{"className":146404},[468,779],[224,146406,787],{"className":146407},[786],[224,146409,146411],{"className":146410},[759],[224,146412,146414],{"className":146413,"style":20579},[763],[224,146415],{},[224,146417],{"className":146418},[549,8479],[224,146420],{"className":146421,"style":724},[473],[224,146423,13182],{"className":146424},[468,469],[224,146426,371],{"className":146427},[468,469],[224,146429],{"className":146430,"style":496},[473],[224,146432,278],{"className":146433},[500],[224,146435],{"className":146436,"style":496},[473],[224,146438,146440,146443,146446,146449,146537,146540,146543],{"className":146439},[459],[224,146441],{"className":146442,"style":57619},[463],[224,146444,21338],{"className":146445,"style":21534},[21531,21532,21533],[224,146447],{"className":146448,"style":724},[473],[224,146450,146452,146455,146534],{"className":146451},[468],[224,146453],{"className":146454},[542,8479],[224,146456,146458],{"className":146457},[20338],[224,146459,146461,146526],{"className":146460},[754,755],[224,146462,146464,146523],{"className":146463},[759],[224,146465,146467,146478,146486],{"className":146466,"style":57644},[763],[224,146468,146469,146472],{"style":20531},[224,146470],{"className":146471,"style":2108},[771],[224,146473,146475],{"className":146474},[468],[224,146476,2023],{"className":146477,"style":492},[468,469],[224,146479,146480,146483],{"style":20546},[224,146481],{"className":146482,"style":2108},[771],[224,146484],{"className":146485,"style":20554},[20553],[224,146487,146488,146491],{"style":20557},[224,146489],{"className":146490,"style":2108},[771],[224,146492,146494],{"className":146493},[468],[224,146495,146497,146500],{"className":146496},[468],[224,146498,11],{"className":146499},[468,469],[224,146501,146503],{"className":146502},[750],[224,146504,146506],{"className":146505},[754],[224,146507,146509],{"className":146508},[759],[224,146510,146512],{"className":146511,"style":5671},[763],[224,146513,146514,146517],{"style":3288},[224,146515],{"className":146516,"style":772},[771],[224,146518,146520],{"className":146519},[776,777,778,779],[224,146521,1447],{"className":146522},[468,779],[224,146524,787],{"className":146525},[786],[224,146527,146529],{"className":146528},[759],[224,146530,146532],{"className":146531,"style":20579},[763],[224,146533],{},[224,146535],{"className":146536},[549,8479],[224,146538],{"className":146539,"style":474},[473],[224,146541,5268],{"className":146542},[478],[224,146544],{"className":146545,"style":474},[473],[224,146547,146549,146552,146555,146558,146561],{"className":146548},[459],[224,146550],{"className":146551,"style":510},[463],[224,146553,1438],{"className":146554},[468],[224,146556],{"className":146557,"style":496},[473],[224,146559,278],{"className":146560},[500],[224,146562],{"className":146563,"style":496},[473],[224,146565,146567,146570,146573,146576,146579,146708,146711,146714],{"className":146566},[459],[224,146568],{"className":146569,"style":107159},[463],[224,146571,145822],{"className":146572},[21531],[224,146574],{"className":146575,"style":21559},[473],[224,146577],{"className":146578,"style":724},[473],[224,146580,146582,146588,146702],{"className":146581},[2091],[224,146583,146585],{"className":146584,"style":8281},[542,8280],[224,146586,239],{"className":146587},[2363,778],[224,146589,146591,146594,146699],{"className":146590},[468],[224,146592],{"className":146593},[542,8479],[224,146595,146597],{"className":146596},[20338],[224,146598,146600,146691],{"className":146599},[754,755],[224,146601,146603,146688],{"className":146602},[759],[224,146604,146606,146643,146651],{"className":146605,"style":57644},[763],[224,146607,146608,146611],{"style":20531},[224,146609],{"className":146610,"style":2108},[771],[224,146612,146614],{"className":146613},[468],[224,146615,146617,146620],{"className":146616},[468],[224,146618,685],{"className":146619,"style":492},[468,469],[224,146621,146623],{"className":146622},[750],[224,146624,146626],{"className":146625},[754],[224,146627,146629],{"className":146628},[759],[224,146630,146632],{"className":146631,"style":64944},[763],[224,146633,146634,146637],{"style":64947},[224,146635],{"className":146636,"style":772},[771],[224,146638,146640],{"className":146639},[776,777,778,779],[224,146641,1447],{"className":146642},[468,779],[224,146644,146645,146648],{"style":20546},[224,146646],{"className":146647,"style":2108},[771],[224,146649],{"className":146650,"style":20554},[20553],[224,146652,146653,146656],{"style":20557},[224,146654],{"className":146655,"style":2108},[771],[224,146657,146659],{"className":146658},[468],[224,146660,146662,146665],{"className":146661},[468],[224,146663,145564],{"className":146664},[468],[224,146666,146668],{"className":146667},[750],[224,146669,146671],{"className":146670},[754],[224,146672,146674],{"className":146673},[759],[224,146675,146677],{"className":146676,"style":5671},[763],[224,146678,146679,146682],{"style":3288},[224,146680],{"className":146681,"style":772},[771],[224,146683,146685],{"className":146684},[776,777,778,779],[224,146686,1447],{"className":146687},[468,779],[224,146689,787],{"className":146690},[786],[224,146692,146694],{"className":146693},[759],[224,146695,146697],{"className":146696,"style":23809},[763],[224,146698],{},[224,146700],{"className":146701},[549,8479],[224,146703,146705],{"className":146704,"style":8281},[549,8280],[224,146706,1716],{"className":146707},[2363,778],[224,146709],{"className":146710,"style":474},[473],[224,146712,5268],{"className":146713},[478],[224,146715],{"className":146716,"style":474},[473],[224,146718,146720,146723],{"className":146719},[459],[224,146721],{"className":146722,"style":510},[463],[224,146724,69418],{"className":146725},[468],[11,146727,146728,146729,146787,146788,146887],{},"这印证了「无界」：两个虽很接近但支撑重叠的正态，",[224,146730,146732,146749],{"className":146731,"translate":388},[392],[224,146733,146735],{"className":146734},[396],[398,146736,146737],{"xmlns":400},[403,146738,146739,146747],{},[406,146740,146741],{},[3248,146742,146743,146745],{},[409,146744,57370],{},[419,146746,1447],{},[447,146748,60308],{"encoding":449},[224,146750,146752],{"className":146751,"ariaHidden":455},[454],[224,146753,146755,146758],{"className":146754},[459],[224,146756],{"className":146757,"style":60318},[463],[224,146759,146761,146764],{"className":146760},[468],[224,146762,57370],{"className":146763},[468,469],[224,146765,146767],{"className":146766},[750],[224,146768,146770],{"className":146769},[754],[224,146771,146773],{"className":146772},[759],[224,146774,146776],{"className":146775,"style":5671},[763],[224,146777,146778,146781],{"style":3288},[224,146779],{"className":146780,"style":772},[771],[224,146782,146784],{"className":146783},[776,777,778,779],[224,146785,1447],{"className":146786},[468,779]," 也会随 ",[224,146789,146791,146817],{"className":146790,"translate":388},[392],[224,146792,146794],{"className":146793},[396],[398,146795,146796],{"xmlns":400},[403,146797,146798,146814],{},[406,146799,146800,146806,146808],{},[3248,146801,146802,146804],{},[409,146803,145564],{"mathvariant":678},[419,146805,1447],{},[409,146807,12413],{"mathvariant":678},[3248,146809,146810,146812],{},[409,146811,685],{},[419,146813,1447],{},[447,146815,146816],{"encoding":449},"\\Delta^2\u002F\\sigma^2",[224,146818,146820],{"className":146819,"ariaHidden":455},[454],[224,146821,146823,146826,146855,146858],{"className":146822},[459],[224,146824],{"className":146825,"style":39943},[463],[224,146827,146829,146832],{"className":146828},[468],[224,146830,145564],{"className":146831},[468],[224,146833,146835],{"className":146834},[750],[224,146836,146838],{"className":146837},[754],[224,146839,146841],{"className":146840},[759],[224,146842,146844],{"className":146843,"style":5671},[763],[224,146845,146846,146849],{"style":3288},[224,146847],{"className":146848,"style":772},[771],[224,146850,146852],{"className":146851},[776,777,778,779],[224,146853,1447],{"className":146854},[468,779],[224,146856,12413],{"className":146857},[468],[224,146859,146861,146864],{"className":146860},[468],[224,146862,685],{"className":146863,"style":492},[468,469],[224,146865,146867],{"className":146866},[750],[224,146868,146870],{"className":146869},[754],[224,146871,146873],{"className":146872},[759],[224,146874,146876],{"className":146875,"style":5671},[763],[224,146877,146878,146881],{"style":3288},[224,146879],{"className":146880,"style":772},[771],[224,146882,146884],{"className":146883},[776,777,778,779],[224,146885,1447],{"className":146886},[468,779]," 指数爆炸。",[11,146889,146890,146893],{},[378,146891,146892],{},"Hellinger²（等方差）。"," Bhattacharyya 系数",[224,146895,146897],{"className":146896,"translate":388},[387],[224,146898,146900,146974],{"className":146899,"translate":388},[392],[224,146901,146903],{"className":146902},[396],[398,146904,146905],{"xmlns":400,"display":401},[403,146906,146907,146971],{},[406,146908,146909,146911,146913,146915,146917,146919,146921,146923,146925,146927,146935,146937,146939,146941,146943,146969],{},[409,146910,9889],{},[409,146912,16021],{},[412,146914,239],{"stretchy":435},[409,146916,16018],{},[412,146918,1877],{"separator":455},[409,146920,20315],{},[412,146922,1716],{"stretchy":435},[412,146924,278],{},[412,146926,21338],{},[54262,146928,146929],{},[406,146930,146931,146933],{},[409,146932,11],{},[409,146934,2023],{},[412,146936,278],{},[409,146938,145822],{},[412,146940,24134],{},[423,146942,21353],{},[406,146944,146945,146947,146949,146967],{},[412,146946,239],{"fence":455},[412,146948,5268],{},[20338,146950,146951,146957],{},[3248,146952,146953,146955],{},[409,146954,145564],{"mathvariant":678},[419,146956,1447],{},[406,146958,146959,146961],{},[419,146960,5430],{},[3248,146962,146963,146965],{},[409,146964,685],{},[419,146966,1447],{},[412,146968,1716],{"fence":455},[412,146970,1877],{"separator":455},[447,146972,146973],{"encoding":449},"BC(P,Q)=\\int\\sqrt{pq}\n=\\exp\\!\\left(-\\frac{\\Delta^2}{8\\sigma^2}\\right),",[224,146975,146977,147016,147084],{"className":146976,"ariaHidden":455},[454],[224,146978,146980,146983,146986,146989,146992,146995,146998,147001,147004,147007,147010,147013],{"className":146979},[459],[224,146981],{"className":146982,"style":535},[463],[224,146984,9889],{"className":146985,"style":9923},[468,469],[224,146987,16021],{"className":146988,"style":30740},[468,469],[224,146990,239],{"className":146991},[542],[224,146993,16018],{"className":146994,"style":653},[468,469],[224,146996,1877],{"className":146997},[1917],[224,146999],{"className":147000,"style":724},[473],[224,147002,20315],{"className":147003},[468,469],[224,147005,1716],{"className":147006},[549],[224,147008],{"className":147009,"style":496},[473],[224,147011,278],{"className":147012},[500],[224,147014],{"className":147015,"style":496},[473],[224,147017,147019,147022,147025,147028,147075,147078,147081],{"className":147018},[459],[224,147020],{"className":147021,"style":25785},[463],[224,147023,21338],{"className":147024,"style":21534},[21531,21532,21533],[224,147026],{"className":147027,"style":724},[473],[224,147029,147031],{"className":147030},[468,54290],[224,147032,147034,147067],{"className":147033},[754,755],[224,147035,147037,147064],{"className":147036},[759],[224,147038,147040,147052],{"className":147039,"style":6992},[763],[224,147041,147043,147046],{"className":147042,"style":2120},[2139],[224,147044],{"className":147045,"style":2108},[771],[224,147047,147049],{"className":147048,"style":54310},[468],[224,147050,61518],{"className":147051,"style":492},[468,469],[224,147053,147054,147057],{"style":61263},[224,147055],{"className":147056,"style":2108},[771],[224,147058,147060],{"className":147059,"style":54367},[54366],[2155,147061,147062],{"xmlns":2157,"width":2158,"height":54370,"viewBox":54371,"preserveAspectRatio":2161},[2163,147063],{"d":54374},[224,147065,787],{"className":147066},[786],[224,147068,147070],{"className":147069},[759],[224,147071,147073],{"className":147072,"style":61283},[763],[224,147074],{},[224,147076],{"className":147077,"style":496},[473],[224,147079,278],{"className":147080},[500],[224,147082],{"className":147083,"style":496},[473],[224,147085,147087,147090,147093,147096,147099,147234,147237],{"className":147086},[459],[224,147088],{"className":147089,"style":107159},[463],[224,147091,145822],{"className":147092},[21531],[224,147094],{"className":147095,"style":21559},[473],[224,147097],{"className":147098,"style":724},[473],[224,147100,147102,147108,147111,147228],{"className":147101},[2091],[224,147103,147105],{"className":147104,"style":8281},[542,8280],[224,147106,239],{"className":147107},[2363,778],[224,147109,5268],{"className":147110},[468],[224,147112,147114,147117,147225],{"className":147113},[468],[224,147115],{"className":147116},[542,8479],[224,147118,147120],{"className":147119},[20338],[224,147121,147123,147217],{"className":147122},[754,755],[224,147124,147126,147214],{"className":147125},[759],[224,147127,147129,147169,147177],{"className":147128,"style":57644},[763],[224,147130,147131,147134],{"style":20531},[224,147132],{"className":147133,"style":2108},[771],[224,147135,147137,147140],{"className":147136},[468],[224,147138,5430],{"className":147139},[468],[224,147141,147143,147146],{"className":147142},[468],[224,147144,685],{"className":147145,"style":492},[468,469],[224,147147,147149],{"className":147148},[750],[224,147150,147152],{"className":147151},[754],[224,147153,147155],{"className":147154},[759],[224,147156,147158],{"className":147157,"style":64944},[763],[224,147159,147160,147163],{"style":64947},[224,147161],{"className":147162,"style":772},[771],[224,147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由 3.6 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a=\\frac{1+\\alpha}{2},",[224,147877,147879,147934,148041,148253],{"className":147878,"ariaHidden":455},[454],[224,147880,147882,147885,147925,147928,147931],{"className":147881},[459],[224,147883],{"className":147884,"style":35905},[463],[224,147886,147888,147891],{"className":147887},[468],[224,147889,20303],{"className":147890,"style":2473},[468,469],[224,147892,147894],{"className":147893},[750],[224,147895,147897,147917],{"className":147896},[754,755],[224,147898,147900,147914],{"className":147899},[759],[224,147901,147903],{"className":147902,"style":64841},[763],[224,147904,147905,147908],{"style":20390},[224,147906],{"className":147907,"style":772},[771],[224,147909,147911],{"className":147910},[776,777,778,779],[224,147912,64666],{"className":147913,"style":64696},[468,469,779],[224,147915,787],{"className":147916},[786],[224,147918,147920],{"className":147919},[759],[224,147921,147923],{"className":147922,"style":794},[763],[224,147924],{},[224,147926],{"className":147927,"style":496},[473],[224,147929,278],{"className":147930},[500],[224,147932],{"className":147933,"style":496},[473],[224,147935,147937,147940,148023,148029,148032,148035,148038],{"className":147936},[459],[224,147938],{"className":147939,"style":88475},[463],[224,147941,147943,147946,148020],{"className":147942},[468],[224,147944],{"className":147945},[542,8479],[224,147947,147949],{"className":147948},[20338],[224,147950,147952,148012],{"className":147951},[754,755],[224,147953,147955,148009],{"className":147954},[759],[224,147956,147958,147990,147998],{"className":147957,"style":42827},[763],[224,147959,147960,147963],{"style":20531},[224,147961],{"className":147962,"style":2108},[771],[224,147964,147966,147969,147972,147975,147978,147981,147984,147987],{"className":147965},[468],[224,147967,15],{"className":147968},[468,469],[224,147970,239],{"className":147971},[542],[224,147973,1438],{"className":147974},[468],[224,147976],{"className":147977,"style":474},[473],[224,147979,5268],{"className":147980},[478],[224,147982],{"className":147983,"style":474},[473],[224,147985,15],{"className":147986},[468,469],[224,147988,1716],{"className":147989},[549],[224,147991,147992,147995],{"style":20546},[224,147993],{"className":147994,"style":2108},[771],[224,147996],{"className":147997,"style":20554},[20553],[224,147999,148000,148003],{"style":20557},[224,148001],{"className":148002,"style":2108},[771],[224,148004,148006],{"className":148005},[468],[224,148007,1438],{"className":148008},[468],[224,148010,787],{"className":148011},[786],[224,148013,148015],{"className":148014},[759],[224,148016,148018],{"className":148017,"style":21648},[763],[224,148019],{},[224,148021],{"className":148022},[549,8479],[224,148024,148026],{"className":148025},[468],[224,148027,239],{"className":148028},[2363,23751],[224,148030,1438],{"className":148031},[468],[224,148033],{"className":148034,"style":474},[473],[224,148036,5268],{"className":148037},[478],[224,148039],{"className":148040,"style":474},[473],[224,148042,148044,148048,148226,148232,148235,148238,148241,148244,148247,148250],{"className":148043},[459],[224,148045],{"className":148046,"style":148047},[463],"height:1.9791em;vertical-align:-0.65em;",[224,148049,148051,148054],{"className":148050},[468],[224,148052,1861],{"className":148053},[468,469],[224,148055,148057],{"className":148056},[750],[224,148058,148060],{"className":148059},[754],[224,148061,148063],{"className":148062},[759],[224,148064,148067],{"className":148065,"style":148066},[763],"height:1.3291em;",[224,148068,148070,148073],{"style":148069},"top:-3.4534em;margin-right:0.05em;",[224,148071],{"className":148072,"style":2108},[771],[224,148074,148076],{"className":148075},[776,777,778,779],[224,148077,148079,148082],{"className":148078},[468,779],[224,148080,5268],{"className":148081},[468,779],[224,148083,148085,148088,148223],{"className":148084},[468,779],[224,148086],{"className":148087},[542,8479,776,50140,131217],[224,148089,148091],{"className":148090},[20338],[224,148092,148094,148214],{"className":148093},[754,755],[224,148095,148097,148211],{"className":148096},[759],[224,148098,148101,148145,148153],{"className":148099,"style":148100},[763],"height:1.251em;",[224,148102,148104,148107],{"style":148103},"top:-2.5062em;",[224,148105],{"className":148106,"style":2108},[771],[224,148108,148110],{"className":148109},[776,50140,2364,779],[224,148111,148113,148116],{"className":148112},[468,779],[224,148114,1447],{"className":148115},[468,779],[224,148117,148119,148122],{"className":148118},[468,779],[224,148120,685],{"className":148121,"style":492},[468,469,779],[224,148123,148125],{"className":148124},[750],[224,148126,148128],{"className":148127},[754],[224,148129,148131],{"className":148130},[759],[224,148132,148135],{"className":148133,"style":148134},[763],"height:0.9384em;",[224,148136,148138,148142],{"style":148137},"top:-2.9384em;margin-right:0.1em;",[224,148139],{"className":148140,"style":148141},[771],"height:2.6444em;",[224,148143,1447],{"className":148144},[468,779],[224,148146,148147,148150],{"style":131254},[224,148148],{"className":148149,"style":2108},[771],[224,148151],{"className":148152,"style":131261},[20553,779],[224,148154,148156,148159],{"style":148155},"top:-3.5021em;",[224,148157],{"className":148158,"style":2108},[771],[224,148160,148162],{"className":148161},[776,50140,2364,779],[224,148163,148165,148168,148171,148174,148177,148180,148183],{"className":148164},[468,779],[224,148166,15],{"className":148167},[468,469,779],[224,148169,239],{"className":148170},[542,779],[224,148172,1438],{"className":148173},[468,779],[224,148175,5268],{"className":148176},[478,779],[224,148178,15],{"className":148179},[468,469,779],[224,148181,1716],{"className":148182},[549,779],[224,148184,148186,148189],{"className":148185},[468,779],[224,148187,145564],{"className":148188},[468,779],[224,148190,148192],{"className":148191},[750],[224,148193,148195],{"className":148194},[754],[224,148196,148198],{"className":148197},[759],[224,148199,148202],{"className":148200,"style":148201},[763],"height:1.0484em;",[224,148203,148205,148208],{"style":148204},"top:-3.0484em;margin-right:0.1em;",[224,148206],{"className":148207,"style":148141},[771],[224,148209,1447],{"className":148210},[468,779],[224,148212,787],{"className":148213},[786],[224,148215,148217],{"className":148216},[759],[224,148218,148221],{"className":148219,"style":148220},[763],"height:0.4938em;",[224,148222],{},[224,148224],{"className":148225},[549,8479,776,50140,131217],[224,148227,148229],{"className":148228},[468],[224,148230,1716],{"className":148231},[2363,23751],[224,148233,1877],{"className":148234},[1917],[224,148236],{"className":148237,"style":4416},[473],[224,148239],{"className":148240,"style":724},[473],[224,148242,15],{"className":148243},[468,469],[224,148245],{"className":148246,"style":496},[473],[224,148248,278],{"className":148249},[500],[224,148251],{"className":148252,"style":496},[473],[224,148254,148256,148259,148333],{"className":148255},[459],[224,148257],{"className":148258,"style":45834},[463],[224,148260,148262,148265,148330],{"className":148261},[468],[224,148263],{"className":148264},[542,8479],[224,148266,148268],{"className":148267},[20338],[224,148269,148271,148322],{"className":148270},[754,755],[224,148272,148274,148319],{"className":148273},[759],[224,148275,148277,148288,148296],{"className":148276,"style":42827},[763],[224,148278,148279,148282],{"style":20531},[224,148280],{"className":148281,"style":2108},[771],[224,148283,148285],{"className":148284},[468],[224,148286,1447],{"className":148287},[468],[224,148289,148290,148293],{"style":20546},[224,148291],{"className":148292,"style":2108},[771],[224,148294],{"className":148295,"style":20554},[20553],[224,148297,148298,148301],{"style":20557},[224,148299],{"className":148300,"style":2108},[771],[224,148302,148304,148307,148310,148313,148316],{"className":148303},[468],[224,148305,1438],{"className":148306},[468],[224,148308],{"className":148309,"style":474},[473],[224,148311,319],{"className":148312},[478],[224,148314],{"className":148315,"style":474},[473],[224,148317,64666],{"className":148318,"style":64696},[468,469],[224,148320,787],{"className":148321},[786],[224,148323,148325],{"className":148324},[759],[224,148326,148328],{"className":148327,"style":23809},[763],[224,148329],{},[224,148331],{"className":148332},[549,8479],[224,148334,1877],{"className":148335},[1917],[11,148337,148338,148339,12708,148368,148548,148549,148577],{},"在小 ",[224,148340,148342,148356],{"className":148341,"translate":388},[392],[224,148343,148345],{"className":148344},[396],[398,148346,148347],{"xmlns":400},[403,148348,148349,148353],{},[406,148350,148351],{},[409,148352,145564],{"mathvariant":678},[447,148354,148355],{"encoding":449},"\\Delta",[224,148357,148359],{"className":148358,"ariaHidden":455},[454],[224,148360,148362,148365],{"className":148361},[459],[224,148363],{"className":148364,"style":649},[463],[224,148366,145564],{"className":148367},[468],[224,148369,148371,148403],{"className":148370,"translate":388},[392],[224,148372,148374],{"className":148373},[396],[398,148375,148376],{"xmlns":400},[403,148377,148378,148400],{},[406,148379,148380,148382],{},[412,148381,52884],{},[20338,148383,148384,148390],{},[3248,148385,148386,148388],{},[409,148387,145564],{"mathvariant":678},[419,148389,1447],{},[406,148391,148392,148394],{},[419,148393,1447],{},[3248,148395,148396,148398],{},[409,148397,685],{},[419,148399,1447],{},[447,148401,148402],{"encoding":449},"\\approx\\frac{\\Delta^2}{2\\sigma^2}",[224,148404,148406,148418],{"className":148405,"ariaHidden":455},[454],[224,148407,148409,148412,148415],{"className":148408},[459],[224,148410],{"className":148411,"style":137054},[463],[224,148413,52884],{"className":148414},[500],[224,148416],{"className":148417,"style":496},[473],[224,148419,148421,148425],{"className":148420},[459],[224,148422],{"className":148423,"style":148424},[463],"height:1.3629em;vertical-align:-0.345em;",[224,148426,148428,148431,148545],{"className":148427},[468],[224,148429],{"className":148430},[542,8479],[224,148432,148434],{"className":148433},[20338],[224,148435,148437,148537],{"className":148436},[754,755],[224,148438,148440,148534],{"className":148439},[759],[224,148441,148443,148486,148494],{"className":148442,"style":133666},[763],[224,148444,148445,148448],{"style":22566},[224,148446],{"className":148447,"style":2108},[771],[224,148449,148451],{"className":148450},[776,777,778,779],[224,148452,148454,148457],{"className":148453},[468,779],[224,148455,1447],{"className":148456},[468,779],[224,148458,148460,148463],{"className":148459},[468,779],[224,148461,685],{"className":148462,"style":492},[468,469,779],[224,148464,148466],{"className":148465},[750],[224,148467,148469],{"className":148468},[754],[224,148470,148472],{"className":148471},[759],[224,148473,148475],{"className":148474,"style":65370},[763],[224,148476,148477,148480],{"style":65373},[224,148478],{"className":148479,"style":50136},[771],[224,148481,148483],{"className":148482},[776,50140,2364,779],[224,148484,1447],{"className":148485},[468,779],[224,148487,148488,148491],{"style":20546},[224,148489],{"className":148490,"style":2108},[771],[224,148492],{"className":148493,"style":20554},[20553],[224,148495,148496,148499],{"style":22589},[224,148497],{"className":148498,"style":2108},[771],[224,148500,148502],{"className":148501},[776,777,778,779],[224,148503,148505],{"className":148504},[468,779],[224,148506,148508,148511],{"className":148507},[468,779],[224,148509,145564],{"className":148510},[468,779],[224,148512,148514],{"className":148513},[750],[224,148515,148517],{"className":148516},[754],[224,148518,148520],{"className":148519},[759],[224,148521,148523],{"className":148522,"style":58433},[763],[224,148524,148525,148528],{"style":58436},[224,148526],{"className":148527,"style":50136},[771],[224,148529,148531],{"className":148530},[776,50140,2364,779],[224,148532,1447],{"className":148533},[468,779],[224,148535,787],{"className":148536},[786],[224,148538,148540],{"className":148539},[759],[224,148541,148543],{"className":148542,"style":22611},[763],[224,148544],{},[224,148546],{"className":148547},[549,8479],"，与 4.6 节的局部展开一致（且此时与 ",[224,148550,148552,148565],{"className":148551,"translate":388},[392],[224,148553,148555],{"className":148554},[396],[398,148556,148557],{"xmlns":400},[403,148558,148559,148563],{},[406,148560,148561],{},[409,148562,64666],{},[447,148564,68999],{"encoding":449},[224,148566,148568],{"className":148567,"ariaHidden":455},[454],[224,148569,148571,148574],{"className":148570},[459],[224,148572],{"className":148573,"style":839},[463],[224,148575,64666],{"className":148576,"style":64696},[468,469]," 无关）。",[613,148579,148581],{"id":148580},"_92-伯努利分布","9.2 伯努利分布",[11,148583,20606,148584,372,148661,35740,148738,148825,148826,148928,148929,148957,148958,148928,149124,21053],{},[224,148585,148587,148619],{"className":148586,"translate":388},[392],[224,148588,148590],{"className":148589},[396],[398,148591,148592],{"xmlns":400},[403,148593,148594,148616],{},[406,148595,148596,148598,148600,148610,148612,148614],{},[409,148597,16018],{},[412,148599,278],{},[406,148601,148602,148604,148606,148608],{},[409,148603,9889],{"mathvariant":678},[409,148605,1861],{"mathvariant":678},[409,148607,2061],{"mathvariant":678},[409,148609,3254],{"mathvariant":678},[412,148611,239],{"stretchy":435},[409,148613,11],{},[412,148615,1716],{"stretchy":435},[447,148617,148618],{"encoding":449},"P=\\mathrm{Bern}(p)",[224,148620,148622,148640],{"className":148621,"ariaHidden":455},[454],[224,148623,148625,148628,148631,148634,148637],{"className":148624},[459],[224,148626],{"className":148627,"style":649},[463],[224,148629,16018],{"className":148630,"style":653},[468,469],[224,148632],{"className":148633,"style":496},[473],[224,148635,278],{"className":148636},[500],[224,148638],{"className":148639,"style":496},[473],[224,148641,148643,148646,148652,148655,148658],{"className":148642},[459],[224,148644],{"className":148645,"style":535},[463],[224,148647,148649],{"className":148648},[468],[224,148650,104210],{"className":148651},[468,1906],[224,148653,239],{"className":148654},[542],[224,148656,11],{"className":148657},[468,469],[224,148659,1716],{"className":148660},[549],[224,148662,148664,148696],{"className":148663,"translate":388},[392],[224,148665,148667],{"className":148666},[396],[398,148668,148669],{"xmlns":400},[403,148670,148671,148693],{},[406,148672,148673,148675,148677,148687,148689,148691],{},[409,148674,20315],{},[412,148676,278],{},[406,148678,148679,148681,148683,148685],{},[409,148680,9889],{"mathvariant":678},[409,148682,1861],{"mathvariant":678},[409,148684,2061],{"mathvariant":678},[409,148686,3254],{"mathvariant":678},[412,148688,239],{"stretchy":435},[409,148690,2023],{},[412,148692,1716],{"stretchy":435},[447,148694,148695],{"encoding":449},"Q=\\mathrm{Bern}(q)",[224,148697,148699,148717],{"className":148698,"ariaHidden":455},[454],[224,148700,148702,148705,148708,148711,148714],{"className":148701},[459],[224,148703],{"className":148704,"style":20637},[463],[224,148706,20315],{"className":148707},[468,469],[224,148709],{"className":148710,"style":496},[473],[224,148712,278],{"className":148713},[500],[224,148715],{"className":148716,"style":496},[473],[224,148718,148720,148723,148729,148732,148735],{"className":148719},[459],[224,148721],{"className":148722,"style":535},[463],[224,148724,148726],{"className":148725},[468],[224,148727,104210],{"className":148728},[468,1906],[224,148730,239],{"className":148731},[542],[224,148733,2023],{"className":148734,"style":492},[468,469],[224,148736,1716],{"className":148737},[549],[224,148739,148741,148771],{"className":148740,"translate":388},[392],[224,148742,148744],{"className":148743},[396],[398,148745,148746],{"xmlns":400},[403,148747,148748,148768],{},[406,148749,148750,148752,148754,148756,148758,148760,148762,148764,148766],{},[409,148751,11],{},[412,148753,1877],{"separator":455},[409,148755,2023],{},[412,148757,12248],{},[412,148759,239],{"stretchy":435},[419,148761,284],{},[412,148763,1877],{"separator":455},[419,148765,1438],{},[412,148767,1716],{"stretchy":435},[447,148769,148770],{"encoding":449},"p,q\\in(0,1)",[224,148772,148774,148801],{"className":148773,"ariaHidden":455},[454],[224,148775,148777,148780,148783,148786,148789,148792,148795,148798],{"className":148776},[459],[224,148778],{"className":148779,"style":488},[463],[224,148781,11],{"className":148782},[468,469],[224,148784,1877],{"className":148785},[1917],[224,148787],{"className":148788,"style":724},[473],[224,148790,2023],{"className":148791,"style":492},[468,469],[224,148793],{"className":148794,"style":496},[473],[224,148796,12248],{"className":148797},[500],[224,148799],{"className":148800,"style":496},[473],[224,148802,148804,148807,148810,148813,148816,148819,148822],{"className":148803},[459],[224,148805],{"className":148806,"style":535},[463],[224,148808,239],{"className":148809},[542],[224,148811,284],{"className":148812},[468],[224,148814,1877],{"className":148815},[1917],[224,148817],{"className":148818,"style":724},[473],[224,148820,1438],{"className":148821},[468],[224,148823,1716],{"className":148824},[549],"），似然比取值 ",[224,148827,148829,148855],{"className":148828,"translate":388},[392],[224,148830,148832],{"className":148831},[396],[398,148833,148834],{"xmlns":400},[403,148835,148836,148852],{},[406,148837,148838,148844,148846,148848,148850],{},[695,148839,148840,148842],{},[409,148841,1435],{},[419,148843,1438],{},[412,148845,278],{},[409,148847,11],{},[409,148849,12413],{"mathvariant":678},[409,148851,2023],{},[447,148853,148854],{"encoding":449},"t_1=p\u002Fq",[224,148856,148858,148913],{"className":148857,"ariaHidden":455},[454],[224,148859,148861,148864,148904,148907,148910],{"className":148860},[459],[224,148862],{"className":148863,"style":1460},[463],[224,148865,148867,148870],{"className":148866},[468],[224,148868,1435],{"className":148869},[468,469],[224,148871,148873],{"className":148872},[750],[224,148874,148876,148896],{"className":148875},[754,755],[224,148877,148879,148893],{"className":148878},[759],[224,148880,148882],{"className":148881,"style":1479},[763],[224,148883,148884,148887],{"style":767},[224,148885],{"className":148886,"style":772},[771],[224,148888,148890],{"className":148889},[776,777,778,779],[224,148891,1438],{"className":148892},[468,779],[224,148894,787],{"className":148895},[786],[224,148897,148899],{"className":148898},[759],[224,148900,148902],{"className":148901,"style":794},[763],[224,148903],{},[224,148905],{"className":148906,"style":496},[473],[224,148908,278],{"className":148909},[500],[224,148911],{"className":148912,"style":496},[473],[224,148914,148916,148919,148922,148925],{"className":148915},[459],[224,148917],{"className":148918,"style":535},[463],[224,148920,11],{"className":148921},[468,469],[224,148923,12413],{"className":148924},[468],[224,148926,2023],{"className":148927,"style":492},[468,469],"（概率 ",[224,148930,148932,148945],{"className":148931,"translate":388},[392],[224,148933,148935],{"className":148934},[396],[398,148936,148937],{"xmlns":400},[403,148938,148939,148943],{},[406,148940,148941],{},[409,148942,2023],{},[447,148944,2023],{"encoding":449},[224,148946,148948],{"className":148947,"ariaHidden":455},[454],[224,148949,148951,148954],{"className":148950},[459],[224,148952],{"className":148953,"style":806},[463],[224,148955,2023],{"className":148956,"style":492},[468,469],"）与 ",[224,148959,148961,149003],{"className":148960,"translate":388},[392],[224,148962,148964],{"className":148963},[396],[398,148965,148966],{"xmlns":400},[403,148967,148968,149000],{},[406,148969,148970,148976,148978,148980,148982,148984,148986,148988,148990,148992,148994,148996,148998],{},[695,148971,148972,148974],{},[409,148973,1435],{},[419,148975,284],{},[412,148977,278],{},[412,148979,239],{"stretchy":435},[419,148981,1438],{},[412,148983,5268],{},[409,148985,11],{},[412,148987,1716],{"stretchy":435},[409,148989,12413],{"mathvariant":678},[412,148991,239],{"stretchy":435},[419,148993,1438],{},[412,148995,5268],{},[409,148997,2023],{},[412,148999,1716],{"stretchy":435},[447,149001,149002],{"encoding":449},"t_0=(1-p)\u002F(1-q)",[224,149004,149006,149061,149082,149112],{"className":149005,"ariaHidden":455},[454],[224,149007,149009,149012,149052,149055,149058],{"className":149008},[459],[224,149010],{"className":149011,"style":1460},[463],[224,149013,149015,149018],{"className":149014},[468],[224,149016,1435],{"className":149017},[468,469],[224,149019,149021],{"className":149020},[750],[224,149022,149024,149044],{"className":149023},[754,755],[224,149025,149027,149041],{"className":149026},[759],[224,149028,149030],{"className":149029,"style":1479},[763],[224,149031,149032,149035],{"style":767},[224,149033],{"className":149034,"style":772},[771],[224,149036,149038],{"className":149037},[776,777,778,779],[224,149039,284],{"className":149040},[468,779],[224,149042,787],{"className":149043},[786],[224,149045,149047],{"className":149046},[759],[224,149048,149050],{"className":149049,"style":794},[763],[224,149051],{},[224,149053],{"className":149054,"style":496},[473],[224,149056,278],{"className":149057},[500],[224,149059],{"className":149060,"style":496},[473],[224,149062,149064,149067,149070,149073,149076,149079],{"className":149063},[459],[224,149065],{"className":149066,"style":535},[463],[224,149068,239],{"className":149069},[542],[224,149071,1438],{"className":149072},[468],[224,149074],{"className":149075,"style":474},[473],[224,149077,5268],{"className":149078},[478],[224,149080],{"className":149081,"style":474},[473],[224,149083,149085,149088,149091,149094,149097,149100,149103,149106,149109],{"className":149084},[459],[224,149086],{"className":149087,"style":535},[463],[224,149089,11],{"className":149090},[468,469],[224,149092,1716],{"className":149093},[549],[224,149095,12413],{"className":149096},[468],[224,149098,239],{"className":149099},[542],[224,149101,1438],{"className":149102},[468],[224,149104],{"className":149105,"style":474},[473],[224,149107,5268],{"className":149108},[478],[224,149110],{"className":149111,"style":474},[473],[224,149113,149115,149118,149121],{"className":149114},[459],[224,149116],{"className":149117,"style":535},[463],[224,149119,2023],{"className":149120,"style":492},[468,469],[224,149122,1716],{"className":149123},[549],[224,149125,149127,149145],{"className":149126,"translate":388},[392],[224,149128,149130],{"className":149129},[396],[398,149131,149132],{"xmlns":400},[403,149133,149134,149142],{},[406,149135,149136,149138,149140],{},[419,149137,1438],{},[412,149139,5268],{},[409,149141,2023],{},[447,149143,149144],{"encoding":449},"1-q",[224,149146,149148,149166],{"className":149147,"ariaHidden":455},[454],[224,149149,149151,149154,149157,149160,149163],{"className":149150},[459],[224,149152],{"className":149153,"style":7047},[463],[224,149155,1438],{"className":149156},[468],[224,149158],{"className":149159,"style":474},[473],[224,149161,5268],{"className":149162},[478],[224,149164],{"className":149165,"style":474},[473],[224,149167,149169,149172],{"className":149168},[459],[224,149170],{"className":149171,"style":806},[463],[224,149173,2023],{"className":149174,"style":492},[468,469],[224,149176,149178],{"className":149177,"translate":388},[387],[224,149179,149181,149263],{"className":149180,"translate":388},[392],[224,149182,149184],{"className":149183},[396],[398,149185,149186],{"xmlns":400,"display":401},[403,149187,149188,149260],{},[406,149189,149190,149200,149202,149204,149206,149208,149210,149212,149214,149216,149218,149224,149226,149228,149230,149232,149234,149236,149238,149240,149258],{},[695,149191,149192,149194],{},[409,149193,20303],{},[406,149195,149196,149198],{},[409,149197,30652],{},[409,149199,41831],{},[412,149201,239],{"stretchy":435},[409,149203,16018],{},[409,149205,20312],{"mathvariant":678},[409,149207,20315],{},[412,149209,1716],{"stretchy":435},[412,149211,278],{},[409,149213,11],{},[409,149215,24131],{},[412,149217,24134],{},[20338,149219,149220,149222],{},[409,149221,11],{},[409,149223,2023],{},[412,149225,319],{},[412,149227,239],{"stretchy":435},[419,149229,1438],{},[412,149231,5268],{},[409,149233,11],{},[412,149235,1716],{"stretchy":435},[409,149237,24131],{},[412,149239,24134],{},[20338,149241,149242,149250],{},[406,149243,149244,149246,149248],{},[419,149245,1438],{},[412,149247,5268],{},[409,149249,11],{},[406,149251,149252,149254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",[224,150338,150340,150362],{"className":150339,"translate":388},[392],[224,150341,150343],{"className":150342},[396],[398,150344,150345],{"xmlns":400},[403,150346,150347,150359],{},[406,150348,150349],{},[695,150350,150351,150353],{},[409,150352,20303],{},[406,150354,150355,150357],{},[409,150356,30652],{},[409,150358,41831],{},[447,150360,150361],{"encoding":449},"D_{KL}",[224,150363,150365],{"className":150364,"ariaHidden":455},[454],[224,150366,150368,150371],{"className":150367},[459],[224,150369],{"className":150370,"style":35905},[463],[224,150372,150374,150377],{"className":150373},[468],[224,150375,20303],{"className":150376,"style":2473},[468,469],[224,150378,150380],{"className":150379},[750],[224,150381,150383,150409],{"className":150382},[754,755],[224,150384,150386,150406],{"className":150385},[759],[224,150387,150389],{"className":150388,"style":764},[763],[224,150390,150391,150394],{"style":20390},[224,150392],{"className":150393,"style":772},[771],[224,150395,150397],{"className":150396},[776,777,778,779],[224,150398,150400,150403],{"className":150399},[468,779],[224,150401,30652],{"className":150402,"style":30740},[468,469,779],[224,150404,41831],{"className":150405},[468,469,779],[224,150407,787],{"className":150408},[786],[224,150410,150412],{"className":150411},[759],[224,150413,150415],{"className":150414,"style":794},[763],[224,150416],{}," 对 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处做二阶展开，应回到 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4.6 节吻合。",[613,150809,150811],{"id":150810},"_93-用对偶表示反推-kl","9.3 用对偶表示反推 KL",[11,150813,15644,150814,372,150899,151040,151041,12570],{},[224,150815,150817,150846],{"className":150816,"translate":388},[392],[224,150818,150820],{"className":150819},[396],[398,150821,150822],{"xmlns":400},[403,150823,150824,150844],{},[406,150825,150826,150828,150830,150832,150834,150836,150838,150840,150842],{},[409,150827,1705],{},[412,150829,239],{"stretchy":435},[409,150831,1435],{},[412,150833,1716],{"stretchy":435},[412,150835,278],{},[409,150837,1435],{},[409,150839,24131],{},[412,150841,24134],{},[409,150843,1435],{},[447,150845,24139],{"encoding":449},[224,150847,150849,150876],{"className":150848,"ariaHidden":455},[454],[224,150850,150852,150855,150858,150861,150864,150867,150870,150873],{"className":150851},[459],[224,150853],{"className":150854,"style":535},[463],[224,150856,1705],{"className":150857,"style":1740},[468,469],[224,150859,239],{"className":150860},[542],[224,150862,1435],{"className":150863},[468,469],[224,150865,1716],{"className":150866},[549],[224,150868],{"className":150869,"style":496},[473],[224,150871,278],{"className":150872},[500],[224,150874],{"className":150875,"style":496},[473],[224,150877,150879,150882,150885,150888,150893,150896],{"className":150878},[459],[224,150880],{"className":150881,"style":2565},[463],[224,150883,1435],{"className":150884},[468,469],[224,150886],{"className":150887,"style":724},[473],[224,150889,24185,150891],{"className":150890},[21531],[224,150892,24189],{"style":24188},[224,150894],{"className":150895,"style":724},[473],[224,150897,1435],{"className":150898},[468,469],[224,150900,150902,150940],{"className":150901,"translate":388},[392],[224,150903,150905],{"className":150904},[396],[398,150906,150907],{"xmlns":400},[403,150908,150909,150937],{},[406,150910,150911,150917,150919,150921,150923,150925],{},[3248,150912,150913,150915],{},[409,150914,1705],{},[412,150916,27543],{},[412,150918,239],{"stretchy":435},[409,150920,375],{},[412,150922,1716],{"stretchy":435},[412,150924,278],{},[3248,150926,150927,150929],{},[409,150928,1861],{},[406,150930,150931,150933,150935],{},[409,150932,375],{},[412,150934,5268],{},[419,150936,1438],{},[447,150938,150939],{"encoding":449},"f^*(y)=e^{y-1}",[224,150941,150943,150996],{"className":150942,"ariaHidden":455},[454],[224,150944,150946,150949,150978,150981,150984,150987,150990,150993],{"className":150945},[459],[224,150947],{"className":150948,"style":535},[463],[224,150950,150952,150955],{"className":150951},[468],[224,150953,1705],{"className":150954,"style":1740},[468,469],[224,150956,150958],{"className":150957},[750],[224,150959,150961],{"className":150960},[754],[224,150962,150964],{"className":150963},[759],[224,150965,150967],{"className":150966,"style":28465},[763],[224,150968,150969,150972],{"style":3288},[224,150970],{"className":150971,"style":772},[771],[224,150973,150975],{"className":150974},[776,777,778,779],[224,150976,27543],{"className":150977},[478,779],[224,150979,239],{"className":150980},[542],[224,150982,375],{"className":150983,"style":492},[468,469],[224,150985,1716],{"className":150986},[549],[224,150988],{"className":150989,"style":496},[473],[224,150991,278],{"className":150992},[500],[224,150994],{"className":150995,"style":496},[473],[224,150997,150999,151002],{"className":150998},[459],[224,151000],{"className":151001,"style":5671},[463],[224,151003,151005,151008],{"className":151004},[468],[224,151006,1861],{"className":151007},[468,469],[224,151009,151011],{"className":151010},[750],[224,151012,151014],{"className":151013},[754],[224,151015,151017],{"className":151016},[759],[224,151018,151020],{"className":151019,"style":5671},[763],[224,151021,151022,151025],{"style":3288},[224,151023],{"className":151024,"style":772},[771],[224,151026,151028],{"className":151027},[776,777,778,779],[224,151029,151031,151034,151037],{"className":151030},[468,779],[224,151032,375],{"className":151033,"style":492},[468,469,779],[224,151035,5268],{"className":151036},[478,779],[224,151038,1438],{"className":151039},[468,779],"，则对任意 ",[224,151042,151044,151057],{"className":151043,"translate":388},[392],[224,151045,151047],{"className":151046},[396],[398,151048,151049],{"xmlns":400},[403,151050,151051,151055],{},[406,151052,151053],{},[409,151054,24189],{},[447,151056,24189],{"encoding":449},[224,151058,151060],{"className":151059,"ariaHidden":455},[454],[224,151061,151063,151066],{"className":151062},[459],[224,151064],{"className":151065,"style":806},[463],[224,151067,24189],{"className":151068,"style":492},[468,469],[224,151070,151072],{"className":151071,"translate":388},[387],[224,151073,151075,151147],{"className":151074,"translate":388},[392],[224,151076,151078],{"className":151077},[396],[398,151079,151080],{"xmlns":400,"display":401},[403,151081,151082,151144],{},[406,151083,151084,151094,151096,151098,151100,151102,151104,151106,151112,151114,151116,151118,151120,151126,151128,151140,151142],{},[695,151085,151086,151088],{},[409,151087,20303],{},[406,151089,151090,151092],{},[409,151091,30652],{},[409,151093,41831],{},[412,151095,239],{"stretchy":435},[409,151097,16018],{},[409,151099,20312],{"mathvariant":678},[409,151101,20315],{},[412,151103,1716],{"stretchy":435},[412,151105,9766],{},[695,151107,151108,151110],{},[409,151109,20324],{"mathvariant":12251},[409,151111,16018],{},[412,151113,268],{"stretchy":435},[409,151115,24189],{},[412,151117,440],{"stretchy":435},[412,151119,5268],{},[695,151121,151122,151124],{},[409,151123,20324],{"mathvariant":12251},[409,151125,20315],{},[412,151127,268],{"stretchy":435},[3248,151129,151130,151132],{},[409,151131,1861],{},[406,151133,151134,151136,151138],{},[409,151135,24189],{},[412,151137,5268],{},[419,151139,1438],{},[412,151141,440],{"stretchy":435},[412,151143,1877],{"separator":455},[447,151145,151146],{"encoding":449},"D_{KL}(P\\|Q)\\ge \\mathbb E_P[g]\n-\\mathbb E_Q[e^{g-1}],",[224,151148,151150,151226,151290],{"className":151149,"ariaHidden":455},[454],[224,151151,151153,151156,151202,151205,151208,151211,151214,151217,151220,151223],{"className":151152},[459],[224,151154],{"className":151155,"style":535},[463],[224,151157,151159,151162],{"className":151158},[468],[224,151160,20303],{"className":151161,"style":2473},[468,469],[224,151163,151165],{"className":151164},[750],[224,151166,151168,151194],{"className":151167},[754,755],[224,151169,151171,151191],{"className":151170},[759],[224,151172,151174],{"className":151173,"style":764},[763],[224,151175,151176,151179],{"style":20390},[224,151177],{"className":151178,"style":772},[771],[224,151180,151182],{"className":151181},[776,777,778,779],[224,151183,151185,151188],{"className":151184},[468,779],[224,151186,30652],{"className":151187,"style":30740},[468,469,779],[224,151189,41831],{"className":151190},[468,469,779],[224,151192,787],{"className":151193},[786],[224,151195,151197],{"className":151196},[759],[224,151198,151200],{"className":151199,"style":794},[763],[224,151201],{},[224,151203,239],{"className":151204},[542],[224,151206,16018],{"className":151207,"style":653},[468,469],[224,151209,20312],{"className":151210},[468],[224,151212,20315],{"className":151213},[468,469],[224,151215,1716],{"className":151216},[549],[224,151218],{"className":151219,"style":496},[473],[224,151221,9766],{"className":151222},[500],[224,151224],{"className":151225,"style":496},[473],[224,151227,151229,151232,151272,151275,151278,151281,151284,151287],{"className":151228},[459],[224,151230],{"className":151231,"style":535},[463],[224,151233,151235,151238],{"className":151234},[468],[224,151236,20324],{"className":151237},[468,12389],[224,151239,151241],{"className":151240},[750],[224,151242,151244,151264],{"className":151243},[754,755],[224,151245,151247,151261],{"className":151246},[759],[224,151248,151250],{"className":151249,"style":764},[763],[224,151251,151252,151255],{"style":767},[224,151253],{"className":151254,"style":772},[771],[224,151256,151258],{"className":151257},[776,777,778,779],[224,151259,16018],{"className":151260,"style":653},[468,469,779],[224,151262,787],{"className":151263},[786],[224,151265,151267],{"className":151266},[759],[224,151268,151270],{"className":151269,"style":794},[763],[224,151271],{},[224,151273,268],{"className":151274},[542],[224,151276,24189],{"className":151277,"style":492},[468,469],[224,151279,440],{"className":151280},[549],[224,151282],{"className":151283,"style":474},[473],[224,151285,5268],{"className":151286},[478],[224,151288],{"className":151289,"style":474},[473],[224,151291,151293,151297,151337,151340,151378,151381],{"className":151292},[459],[224,151294],{"className":151295,"style":151296},[463],"height:1.1502em;vertical-align:-0.2861em;",[224,151298,151300,151303],{"className":151299},[468],[224,151301,20324],{"className":151302},[468,12389],[224,151304,151306],{"className":151305},[750],[224,151307,151309,151329],{"className":151308},[754,755],[224,151310,151312,151326],{"className":151311},[759],[224,151313,151315],{"className":151314,"style":764},[763],[224,151316,151317,151320],{"style":767},[224,151318],{"className":151319,"style":772},[771],[224,151321,151323],{"className":151322},[776,777,778,779],[224,151324,20315],{"className":151325},[468,469,779],[224,151327,787],{"className":151328},[786],[224,151330,151332],{"className":151331},[759],[224,151333,151335],{"className":151334,"style":20409},[763],[224,151336],{},[224,151338,268],{"className":151339},[542],[224,151341,151343,151346],{"className":151342},[468],[224,151344,1861],{"className":151345},[468,469],[224,151347,151349],{"className":151348},[750],[224,151350,151352],{"className":151351},[754],[224,151353,151355],{"className":151354},[759],[224,151356,151358],{"className":151357,"style":5071},[763],[224,151359,151360,151363],{"style":5074},[224,151361],{"className":151362,"style":772},[771],[224,151364,151366],{"className":151365},[776,777,778,779],[224,151367,151369,151372,151375],{"className":151368},[468,779],[224,151370,24189],{"className":151371,"style":492},[468,469,779],[224,151373,5268],{"className":151374},[478,779],[224,151376,1438],{"className":151377},[468,779],[224,151379,440],{"className":151380},[549],[224,151382,1877],{"className":151383},[1917],[11,151385,151386,151387,151489],{},"等号在 ",[224,151388,151390,151424],{"className":151389,"translate":388},[392],[224,151391,151393],{"className":151392},[396],[398,151394,151395],{"xmlns":400},[403,151396,151397,151421],{},[406,151398,151399,151401,151403,151405,151407,151409,151411,151413,151415,151417,151419],{},[409,151400,24189],{},[412,151402,278],{},[409,151404,24131],{},[412,151406,24134],{},[412,151408,239],{"stretchy":435},[409,151410,11],{},[409,151412,12413],{"mathvariant":678},[409,151414,2023],{},[412,151416,1716],{"stretchy":435},[412,151418,319],{},[419,151420,1438],{},[447,151422,151423],{"encoding":449},"g=\\log(p\u002Fq)+1",[224,151425,151427,151445,151480],{"className":151426,"ariaHidden":455},[454],[224,151428,151430,151433,151436,151439,151442],{"className":151429},[459],[224,151431],{"className":151432,"style":806},[463],[224,151434,24189],{"className":151435,"style":492},[468,469],[224,151437],{"className":151438,"style":496},[473],[224,151440,278],{"className":151441},[500],[224,151443],{"className":151444,"style":496},[473],[224,151446,151448,151451,151456,151459,151462,151465,151468,151471,151474,151477],{"className":151447},[459],[224,151449],{"className":151450,"style":535},[463],[224,151452,24185,151454],{"className":151453},[21531],[224,151455,24189],{"style":24188},[224,151457,239],{"className":151458},[542],[224,151460,11],{"className":151461},[468,469],[224,151463,12413],{"className":151464},[468],[224,151466,2023],{"className":151467,"style":492},[468,469],[224,151469,1716],{"className":151470},[549],[224,151472],{"className":151473,"style":474},[473],[224,151475,319],{"className":151476},[478],[224,151478],{"className":151479,"style":474},[473],[224,151481,151483,151486],{"className":151482},[459],[224,151484],{"className":151485,"style":510},[463],[224,151487,1438],{"className":151488},[468]," 处取到。代回：",[224,151491,151493],{"className":151492,"translate":388},[387],[224,151494,151496,151622],{"className":151495,"translate":388},[392],[224,151497,151499],{"className":151498},[396],[398,151500,151501],{"xmlns":400,"display":401},[403,151502,151503,151619],{},[406,151504,151505,151511,151513,151515,151517,151519,151521,151523,151525,151527,151529,151531,151533,151535,151541,151543,151563,151565,151567,151577,151579,151581,151583,151585,151587,151589,151591,151593,151595,151597,151607,151609,151611,151613,151615,151617],{},[695,151506,151507,151509],{},[409,151508,20324],{"mathvariant":12251},[409,151510,16018],{},[412,151512,268],{"stretchy":435},[409,151514,24131],{},[412,151516,24134],{},[412,151518,239],{"stretchy":435},[409,151520,11],{},[409,151522,12413],{"mathvariant":678},[409,151524,2023],{},[412,151526,1716],{"stretchy":435},[412,151528,319],{},[419,151530,1438],{},[412,151532,440],{"stretchy":435},[412,151534,5268],{},[695,151536,151537,151539],{},[409,151538,20324],{"mathvariant":12251},[409,151540,20315],{},[412,151542,268],{"stretchy":435},[3248,151544,151545,151547],{},[409,151546,1861],{},[406,151548,151549,151551,151553,151555,151557,151559,151561],{},[409,151550,24131],{},[412,151552,24134],{},[412,151554,239],{"stretchy":435},[409,151556,11],{},[409,151558,12413],{"mathvariant":678},[409,151560,2023],{},[412,151562,1716],{"stretchy":435},[412,151564,440],{"stretchy":435},[412,151566,278],{},[695,151568,151569,151571],{},[409,151570,20303],{},[406,151572,151573,151575],{},[409,151574,30652],{},[409,151576,41831],{},[412,151578,239],{"stretchy":435},[409,151580,16018],{},[409,151582,20312],{"mathvariant":678},[409,151584,20315],{},[412,151586,1716],{"stretchy":435},[412,151588,319],{},[419,151590,1438],{},[412,151592,5268],{},[419,151594,1438],{},[412,151596,278],{},[695,151598,151599,151601],{},[409,151600,20303],{},[406,151602,151603,151605],{},[409,151604,30652],{},[409,151606,41831],{},[412,151608,239],{"stretchy":435},[409,151610,16018],{},[409,151612,20312],{"mathvariant":678},[409,151614,20315],{},[412,151616,1716],{"stretchy":435},[412,151618,1877],{"separator":455},[447,151620,151621],{"encoding":449},"\\mathbb E_P[\\log(p\u002Fq)+1]\n-\\mathbb E_Q[e^{\\log(p\u002Fq)}]\n=D_{KL}(P\\|Q)+1-1=D_{KL}(P\\|Q),",[224,151623,151625,151703,151724,151842,151918,151936,151954],{"className":151624,"ariaHidden":455},[454],[224,151626,151628,151631,151671,151674,151679,151682,151685,151688,151691,151694,151697,151700],{"className":151627},[459],[224,151629],{"className":151630,"style":535},[463],[224,151632,151634,151637],{"className":151633},[468],[224,151635,20324],{"className":151636},[468,12389],[224,151638,151640],{"className":151639},[750],[224,151641,151643,151663],{"className":151642},[754,755],[224,151644,151646,151660],{"className":151645},[759],[224,151647,151649],{"className":151648,"style":764},[763],[224,151650,151651,151654],{"style":767},[224,151652],{"className":151653,"style":772},[771],[224,151655,151657],{"className":151656},[776,777,778,779],[224,151658,16018],{"className":151659,"style":653},[468,469,779],[224,151661,787],{"className":151662},[786],[224,151664,151666],{"className":151665},[759],[224,151667,151669],{"className":151668,"style":794},[763],[224,151670],{},[224,151672,268],{"className":151673},[542],[224,151675,24185,151677],{"className":151676},[21531],[224,151678,24189],{"style":24188},[224,151680,239],{"className":151681},[542],[224,151683,11],{"className":151684},[468,469],[224,151686,12413],{"className":151687},[468],[224,151689,2023],{"className":151690,"style":492},[468,469],[224,151692,1716],{"className":151693},[549],[224,151695],{"className":151696,"style":474},[473],[224,151698,319],{"className":151699},[478],[224,151701],{"className":151702,"style":474},[473],[224,151704,151706,151709,151712,151715,151718,151721],{"className":151705},[459],[224,151707],{"className":151708,"style":535},[463],[224,151710,1438],{"className":151711},[468],[224,151713,440],{"className":151714},[549],[224,151716],{"className":151717,"style":474},[473],[224,151719,5268],{"className":151720},[478],[224,151722],{"className":151723,"style":474},[473],[224,151725,151727,151731,151771,151774,151830,151833,151836,151839],{"className":151726},[459],[224,151728],{"className":151729,"style":151730},[463],"height:1.2241em;vertical-align:-0.2861em;",[224,151732,151734,151737],{"className":151733},[468],[224,151735,20324],{"className":151736},[468,12389],[224,151738,151740],{"className":151739},[750],[224,151741,151743,151763],{"className":151742},[754,755],[224,151744,151746,151760],{"className":151745},[759],[224,151747,151749],{"className":151748,"style":764},[763],[224,151750,151751,151754],{"style":767},[224,151752],{"className":151753,"style":772},[771],[224,151755,151757],{"className":151756},[776,777,778,779],[224,151758,20315],{"className":151759},[468,469,779],[224,151761,787],{"className":151762},[786],[224,151764,151766],{"className":151765},[759],[224,151767,151769],{"className":151768,"style":20409},[763],[224,151770],{},[224,151772,268],{"className":151773},[542],[224,151775,151777,151780],{"className":151776},[468],[224,151778,1861],{"className":151779},[468,469],[224,151781,151783],{"className":151782},[750],[224,151784,151786],{"className":151785},[754],[224,151787,151789],{"className":151788},[759],[224,151790,151792],{"className":151791,"style":65032},[763],[224,151793,151794,151797],{"style":5074},[224,151795],{"className":151796,"style":772},[771],[224,151798,151800],{"className":151799},[776,777,778,779],[224,151801,151803,151815,151818,151821,151824,151827],{"className":151802},[468,779],[224,151804,151806,151809,151812],{"className":151805},[21531,779],[224,151807,1864],{"className":151808},[779],[224,151810,4127],{"className":151811},[779],[224,151813,24189],{"className":151814,"style":24188},[779],[224,151816,239],{"className":151817},[542,779],[224,151819,11],{"className":151820},[468,469,779],[224,151822,12413],{"className":151823},[468,779],[224,151825,2023],{"className":151826,"style":492},[468,469,779],[224,151828,1716],{"className":151829},[549,779],[224,151831,440],{"className":151832},[549],[224,151834],{"className":151835,"style":496},[473],[224,151837,278],{"className":151838},[500],[224,151840],{"className":151841,"style":496},[473],[224,151843,151845,151848,151894,151897,151900,151903,151906,151909,151912,151915],{"className":151844},[459],[224,151846],{"className":151847,"style":535},[463],[224,151849,151851,151854],{"className":151850},[468],[224,151852,20303],{"className":151853,"style":2473},[468,469],[224,151855,151857],{"className":151856},[750],[224,151858,151860,151886],{"className":151859},[754,755],[224,151861,151863,151883],{"className":151862},[759],[224,151864,151866],{"className":151865,"style":764},[763],[224,151867,151868,151871],{"style":20390},[224,151869],{"className":151870,"style":772},[771],[224,151872,151874],{"className":151873},[776,777,778,779],[224,151875,151877,151880],{"className":151876},[468,779],[224,151878,30652],{"className":151879,"style":30740},[468,469,779],[224,151881,41831],{"className":151882},[468,469,779],[224,151884,787],{"className":151885},[786],[224,151887,151889],{"className":151888},[759],[224,151890,151892],{"className":151891,"style":794},[763],[224,151893],{},[224,151895,239],{"className":151896},[542],[224,151898,16018],{"className":151899,"style":653},[468,469],[224,151901,20312],{"className":151902},[468],[224,151904,20315],{"className":151905},[468,469],[224,151907,1716],{"className":151908},[549],[224,151910],{"className":151911,"style":474},[473],[224,151913,319],{"className":151914},[478],[224,151916],{"className":151917,"style":474},[473],[224,151919,151921,151924,151927,151930,151933],{"className":151920},[459],[224,151922],{"className":151923,"style":7047},[463],[224,151925,1438],{"className":151926},[468],[224,151928],{"className":151929,"style":474},[473],[224,151931,5268],{"className":151932},[478],[224,151934],{"className":151935,"style":474},[473],[224,151937,151939,151942,151945,151948,151951],{"className":151938},[459],[224,151940],{"className":151941,"style":510},[463],[224,151943,1438],{"className":151944},[468],[224,151946],{"className":151947,"style":496},[473],[224,151949,278],{"className":151950},[500],[224,151952],{"className":151953,"style":496},[473],[224,151955,151957,151960,152006,152009,152012,152015,152018,152021],{"className":151956},[459],[224,151958],{"className":151959,"style":535},[463],[224,151961,151963,151966],{"className":151962},[468],[224,151964,20303],{"className":151965,"style":2473},[468,469],[224,151967,151969],{"className":151968},[750],[224,151970,151972,151998],{"className":151971},[754,755],[224,151973,151975,151995],{"className":151974},[759],[224,151976,151978],{"className":151977,"style":764},[763],[224,151979,151980,151983],{"style":20390},[224,151981],{"className":151982,"style":772},[771],[224,151984,151986],{"className":151985},[776,777,778,779],[224,151987,151989,151992],{"className":151988},[468,779],[224,151990,30652],{"className":151991,"style":30740},[468,469,779],[224,151993,41831],{"className":151994},[468,469,779],[224,151996,787],{"className":151997},[786],[224,151999,152001],{"className":152000},[759],[224,152002,152004],{"className":152003,"style":794},[763],[224,152005],{},[224,152007,239],{"className":152008},[542],[224,152010,16018],{"className":152011,"style":653},[468,469],[224,152013,20312],{"className":152014},[468],[224,152016,20315],{"className":152017},[468,469],[224,152019,1716],{"className":152020},[549],[224,152022,1877],{"className":152023},[1917],[11,152025,152026],{},"完全自洽。这给出了一个「可验证」的对偶表示练习。",[608,152028,152030],{"id":152029},"_10-结论","10. 结论",[11,152032,152033],{},"f-散度族的故事可以浓缩成三句话：",[9997,152035,152036,152339,152505],{},[2677,152037,152038,12570,152041,7098,152246,112369,152274,3117],{},[378,152039,152040],{},"一个模板",[224,152042,152044,152100],{"className":152043,"translate":388},[392],[224,152045,152047],{"className":152046},[396],[398,152048,152049],{"xmlns":400},[403,152050,152051,152097],{},[406,152052,152053,152059,152061,152063,152065,152067,152069,152071,152077,152079,152081,152083,152085,152087,152089,152091,152093,152095],{},[695,152054,152055,152057],{},[409,152056,20303],{},[409,152058,1705],{},[412,152060,239],{"stretchy":435},[409,152062,16018],{},[409,152064,20312],{"mathvariant":678},[409,152066,20315],{},[412,152068,1716],{"stretchy":435},[412,152070,278],{},[695,152072,152073,152075],{},[409,152074,20324],{"mathvariant":12251},[409,152076,20315],{},[412,152078,268],{"stretchy":435},[409,152080,1705],{},[412,152082,239],{"stretchy":435},[409,152084,13182],{},[409,152086,16018],{},[409,152088,12413],{"mathvariant":678},[409,152090,13182],{},[409,152092,20315],{},[412,152094,1716],{"stretchy":435},[412,152096,440],{"stretchy":435},[447,152098,152099],{"encoding":449},"D_f(P\\|Q)=\\mathbb E_Q[f(dP\u002FdQ)]",[224,152101,152103,152173],{"className":152102,"ariaHidden":455},[454],[224,152104,152106,152109,152149,152152,152155,152158,152161,152164,152167,152170],{"className":152105},[459],[224,152107],{"className":152108,"style":20368},[463],[224,152110,152112,152115],{"className":152111},[468],[224,152113,20303],{"className":152114,"style":2473},[468,469],[224,152116,152118],{"className":152117},[750],[224,152119,152121,152141],{"className":152120},[754,755],[224,152122,152124,152138],{"className":152123},[759],[224,152125,152127],{"className":152126,"style":20387},[763],[224,152128,152129,152132],{"style":20390},[224,152130],{"className":152131,"style":772},[771],[224,152133,152135],{"className":152134},[776,777,778,779],[224,152136,1705],{"className":152137,"style":1740},[468,469,779],[224,152139,787],{"className":152140},[786],[224,152142,152144],{"className":152143},[759],[224,152145,152147],{"className":152146,"style":20409},[763],[224,152148],{},[224,152150,239],{"className":152151},[542],[224,152153,16018],{"className":152154,"style":653},[468,469],[224,152156,20312],{"className":152157},[468],[224,152159,20315],{"className":152160},[468,469],[224,152162,1716],{"className":152163},[549],[224,152165],{"className":152166,"style":496},[473],[224,152168,278],{"className":152169},[500],[224,152171],{"className":152172,"style":496},[473],[224,152174,152176,152179,152219,152222,152225,152228,152231,152234,152237,152240,152243],{"className":152175},[459],[224,152177],{"className":152178,"style":20368},[463],[224,152180,152182,152185],{"className":152181},[468],[224,152183,20324],{"className":152184},[468,12389],[224,152186,152188],{"className":152187},[750],[224,152189,152191,152211],{"className":152190},[754,755],[224,152192,152194,152208],{"className":152193},[759],[224,152195,152197],{"className":152196,"style":764},[763],[224,152198,152199,152202],{"style":767},[224,152200],{"className":152201,"style":772},[771],[224,152203,152205],{"className":152204},[776,777,778,779],[224,152206,20315],{"className":152207},[468,469,779],[224,152209,787],{"className":152210},[786],[224,152212,152214],{"className":152213},[759],[224,152215,152217],{"className":152216,"style":20409},[763],[224,152218],{},[224,152220,268],{"className":152221},[542],[224,152223,1705],{"className":152224,"style":1740},[468,469],[224,152226,239],{"className":152227},[542],[224,152229,13182],{"className":152230},[468,469],[224,152232,16018],{"className":152233,"style":653},[468,469],[224,152235,12413],{"className":152236},[468],[224,152238,13182],{"className":152239},[468,469],[224,152241,20315],{"className":152242},[468,469],[224,152244,40329],{"className":152245},[549],[224,152247,152249,152262],{"className":152248,"translate":388},[392],[224,152250,152252],{"className":152251},[396],[398,152253,152254],{"xmlns":400},[403,152255,152256,152260],{},[406,152257,152258],{},[409,152259,1705],{},[447,152261,1705],{"encoding":449},[224,152263,152265],{"className":152264,"ariaHidden":455},[454],[224,152266,152268,152271],{"className":152267},[459],[224,152269],{"className":152270,"style":2565},[463],[224,152272,1705],{"className":152273,"style":1740},[468,469],[224,152275,152277,152300],{"className":152276,"translate":388},[392],[224,152278,152280],{"className":152279},[396],[398,152281,152282],{"xmlns":400},[403,152283,152284,152298],{},[406,152285,152286,152288,152290,152292,152294,152296],{},[409,152287,1705],{},[412,152289,239],{"stretchy":435},[419,152291,1438],{},[412,152293,1716],{"stretchy":435},[412,152295,278],{},[419,152297,284],{},[447,152299,20243],{"encoding":449},[224,152301,152303,152330],{"className":152302,"ariaHidden":455},[454],[224,152304,152306,152309,152312,152315,152318,152321,152324,152327],{"className":152305},[459],[224,152307],{"className":152308,"style":535},[463],[224,152310,1705],{"className":152311,"style":1740},[468,469],[224,152313,239],{"className":152314},[542],[224,152316,1438],{"className":152317},[468],[224,152319,1716],{"className":152320},[549],[224,152322],{"className":152323,"style":496},[473],[224,152325,278],{"className":152326},[500],[224,152328],{"className":152329,"style":496},[473],[224,152331,152333,152336],{"className":152332},[459],[224,152334],{"className":152335,"style":510},[463],[224,152337,284],{"className":152338},[468],[2677,152340,152341,12570,152344,152425,152426,152504],{},[378,152342,152343],{},"两组对偶",[224,152345,152347,152369],{"className":152346,"translate":388},[392],[224,152348,152350],{"className":152349},[396],[398,152351,152352],{"xmlns":400},[403,152353,152354,152366],{},[406,152355,152356,152358,152360],{},[409,152357,1705],{},[412,152359,41297],{},[3248,152361,152362,152364],{},[409,152363,1705],{},[412,152365,27543],{},[447,152367,152368],{"encoding":449},"f\\leftrightarrow f^*",[224,152370,152372,152390],{"className":152371,"ariaHidden":455},[454],[224,152373,152375,152378,152381,152384,152387],{"className":152374},[459],[224,152376],{"className":152377,"style":2565},[463],[224,152379,1705],{"className":152380,"style":1740},[468,469],[224,152382],{"className":152383,"style":496},[473],[224,152385,41297],{"className":152386},[500],[224,152388],{"className":152389,"style":496},[473],[224,152391,152393,152396],{"className":152392},[459],[224,152394],{"className":152395,"style":2565},[463],[224,152397,152399,152402],{"className":152398},[468],[224,152400,1705],{"className":152401,"style":1740},[468,469],[224,152403,152405],{"className":152404},[750],[224,152406,152408],{"className":152407},[754],[224,152409,152411],{"className":152410},[759],[224,152412,152414],{"className":152413,"style":28465},[763],[224,152415,152416,152419],{"style":3288},[224,152417],{"className":152418,"style":772},[771],[224,152420,152422],{"className":152421},[776,777,778,779],[224,152423,27543],{"className":152424},[478,779]," 给出变分表示（可用于估计与对抗训练）；",[224,152427,152429,152459],{"className":152428,"translate":388},[392],[224,152430,152432],{"className":152431},[396],[398,152433,152434],{"xmlns":400},[403,152435,152436,152456],{},[406,152437,152438,152440,152442,152444,152446,152448,152450,152452,152454],{},[409,152439,1705],{},[412,152441,41297],{},[409,152443,1435],{},[409,152445,1705],{},[412,152447,239],{"stretchy":435},[419,152449,1438],{},[409,152451,12413],{"mathvariant":678},[409,152453,1435],{},[412,152455,1716],{"stretchy":435},[447,152457,152458],{"encoding":449},"f\\leftrightarrow t f(1\u002Ft)",[224,152460,152462,152480],{"className":152461,"ariaHidden":455},[454],[224,152463,152465,152468,152471,152474,152477],{"className":152464},[459],[224,152466],{"className":152467,"style":2565},[463],[224,152469,1705],{"className":152470,"style":1740},[468,469],[224,152472],{"className":152473,"style":496},[473],[224,152475,41297],{"className":152476},[500],[224,152478],{"className":152479,"style":496},[473],[224,152481,152483,152486,152489,152492,152495,152498,152501],{"className":152482},[459],[224,152484],{"className":152485,"style":535},[463],[224,152487,1435],{"className":152488},[468,469],[224,152490,1705],{"className":152491,"style":1740},[468,469],[224,152493,239],{"className":152494},[542],[224,152496,28555],{"className":152497},[468],[224,152499,1435],{"className":152500},[468,469],[224,152502,1716],{"className":152503},[549]," 给出方向对称性（可用于构造对称散度）。",[2677,152506,152507,152510],{},[378,152508,152509],{},"一个不变量","：数据处理不等式，说明它是「信息量」而非「几何量」。正因如此，它与 Wasserstein\u002FMMD 在支撑分离时行为迥异。",[11,152512,152513],{},"经典文献可参考：Csiszár (1967)、Ali–Silvey (1966) 的原始定义；Liese–Vajda 的专著对 f-散度的对偶、拓扑与充分性做了系统处理；Amari 的信息几何给出 α-族与 Fisher 度量；Nowozin 等 (2016) 的 f-GAN、Belghazi 等 (2018) 的 MINE 把对偶表示变成可训练算法；Dong–Roth–Su (2022) 说明 f-散度的语言如何统一差分隐私。把这几条线索接起来，读者就能在「定义—性质—估计—应用」四个层面上自由穿梭。",{"title":186,"searchDepth":187,"depth":187,"links":152515},[152516,152522,152527,152537,152547,152554,152559,152567,152568,152573],{"id":20602,"depth":187,"text":20603,"children":152517},[152518,152519,152520,152521],{"id":22357,"depth":290,"text":22358},{"id":25214,"depth":290,"text":25215},{"id":25582,"depth":290,"text":25583},{"id":27335,"depth":290,"text":27336},{"id":28716,"depth":187,"text":152523,"children":152524},"2. 为何 fff 凸且 f(1)=0f(1)=0f(1)=0",[152525,152526],{"id":28814,"depth":290,"text":28815},{"id":30518,"depth":290,"text":30519},{"id":41345,"depth":187,"text":41346,"children":152528},[152529,152530,152531,152532,152533,152534,152535,152536],{"id":41407,"depth":290,"text":41408},{"id":45481,"depth":290,"text":45482},{"id":54432,"depth":290,"text":54433},{"id":57070,"depth":290,"text":57071},{"id":60351,"depth":290,"text":60352},{"id":64648,"depth":290,"text":64649},{"id":69284,"depth":290,"text":69285},{"id":72396,"depth":290,"text":72397},{"id":74818,"depth":187,"text":74819,"children":152538},[152539,152540,152541,152542,152543,152544,152545,152546],{"id":75117,"depth":290,"text":75118},{"id":76042,"depth":290,"text":76043},{"id":78196,"depth":290,"text":78197},{"id":80587,"depth":290,"text":80588},{"id":94356,"depth":290,"text":94357},{"id":97936,"depth":290,"text":97937},{"id":101962,"depth":290,"text":101963},{"id":116846,"depth":290,"text":116847},{"id":119193,"depth":187,"text":119194,"children":152548},[152549,152550,152551,152552,152553],{"id":119197,"depth":290,"text":119198},{"id":121239,"depth":290,"text":121240},{"id":123081,"depth":290,"text":123082},{"id":125243,"depth":290,"text":125244},{"id":128135,"depth":290,"text":128136},{"id":128476,"depth":187,"text":128477,"children":152555},[152556,152557,152558],{"id":128480,"depth":290,"text":128481},{"id":134506,"depth":290,"text":134507},{"id":136288,"depth":290,"text":136289},{"id":137330,"depth":187,"text":137331,"children":152560},[152561,152562,152563,152564,152565,152566],{"id":137399,"depth":290,"text":137400},{"id":137593,"depth":290,"text":137594},{"id":138482,"depth":290,"text":138483},{"id":139181,"depth":290,"text":139182},{"id":139639,"depth":290,"text":139640},{"id":139708,"depth":290,"text":139709},{"id":140072,"depth":187,"text":140073},{"id":144231,"depth":187,"text":144232,"children":152569},[152570,152571,152572],{"id":144235,"depth":290,"text":144236},{"id":148580,"depth":290,"text":148581},{"id":150810,"depth":290,"text":150811},{"id":152029,"depth":187,"text":152030},"f-散度族是概率分布之间差异的一大类度量，它用一个凸函数 fff 来统一描述 KL 散度、JS 散度、全变差、卡方散度等许多常见散度。可以把它理解为「散度」的通用模板。",{},"\u002Fblog\u002F2026\u002F0011-about-the-f-divergence",{"title":20150,"description":152574},"blog\u002F2026\u002F0011-about-the-f-divergence","f-散度族（f-divergence）用一个凸函数 f 统一刻画两个概率分布之间的差异，KL、JS、全变差、卡方、Hellinger、α-散度都是它的特例。",[152581,152582],"statistics","mathematics","2026-10-17T04:47:08+08:00","iwqHXZqjOaalRKrTl4b8l9DoWDfn8fPQY2GEv_zt_Xw",1794483690792]