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年的论文《",[19,20,24],"a",{"href":21,"rel":22},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2304.15004",[23],"nofollow","大语言模型的涌现能力是幻象吗？","》发表以来获得了广泛支持。这个论证的吸引力在于它的简洁：选一个不连续的指标，你就会在一个底层数据生成过程中不存在不连续的地方制造出一个不连续。本文考察这一论证的力量与局限，把它置于物理与生物系统相变研究的更广阔文献之中，并提出一个更具体的问题：当代机器学习理论——尺度定律、损失地形几何、grokking 与机制可解释性——究竟告诉我们，经过训练的网络中出现的涌现，是测量假象、真实的动力学相变，还是根本抗拒这种二分法？",[15,27,28],{},"我在下文详细论证的结论是：「幻象」批判作为对某一类基准报告式涌现能力的一个局部、方法论层面的论断是正确的，但它并不能推广为对涌现作为科学概念的彻底否定。过去三年的机器学习研究，已经从训练损失相变、grokking 动力学和电路层级的可解释性等方面独立积累了证据，证明神经网络行为中的某些不连续性是优化动力学自身的属性，与读出指标的选择无关。区分这两种情形需要特定的方法论纪律，本文试图把它明确化。",[11,30,31],{"id":31},"幻象论证的形式化",[15,33,34,35,82,83,112,113,144,145,310],{},"Schaeffer 等人的核心技术主张建立在一个简单的概率分解之上。考虑一个需要 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平滑、连续地提高（比方说，按照尺度定律理论的标准形式遵循幂律，",[36,146,148,196],{"className":147,"translate":40},[39],[36,149,151],{"className":150},[44],[46,152,153],{"xmlns":48},[50,154,155,193],{},[53,156,157,159,164,166,169,172,176,179,181],{},[56,158,15],{},[160,161,163],"mo",{"stretchy":162},"false","(",[56,165,127],{},[160,167,168],{"stretchy":162},")",[160,170,171],{},"≈",[173,174,175],"mn",{},"1",[160,177,178],{},"−",[56,180,19],{},[182,183,184,186],"msup",{},[56,185,127],{},[53,187,188,190],{},[160,189,178],{},[56,191,192],{},"α",[60,194,195],{"encoding":62},"p(N) \\approx 1 - 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1 的量的幂，",[36,494,496,509],{"className":495,"translate":40},[39],[36,497,499],{"className":498},[44],[46,500,501],{"xmlns":48},[50,502,503,507],{},[53,504,505],{},[56,506,15],{},[60,508,15],{"encoding":62},[36,510,512],{"className":511,"ariaHidden":67},[66],[36,513,515,518],{"className":514},[71],[36,516],{"className":517,"style":108},[75],[36,519,15],{"className":520},[80,81]," 在接近上限处的微小绝对增长，会引起 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不成比例的巨大相对增长。结果是出现一条先平后陡的曲线——正是 BIG-Bench 上几十个任务报告的那种标志性的「涌现」S 型曲线——尽管 ",[36,601,603,623],{"className":602,"translate":40},[39],[36,604,606],{"className":605},[44],[46,607,608],{"xmlns":48},[50,609,610,620],{},[53,611,612,614,616,618],{},[56,613,15],{},[160,615,163],{"stretchy":162},[56,617,127],{},[160,619,168],{"stretchy":162},[60,621,622],{"encoding":62},"p(N)",[36,624,626],{"className":625,"ariaHidden":67},[66],[36,627,629,632,635,638,641],{"className":628},[71],[36,630],{"className":631,"style":205},[75],[36,633,15],{"className":634},[80,81],[36,636,163],{"className":637},[212],[36,639,127],{"className":640,"style":143},[80,81],[36,642,168],{"className":643},[219]," 本身完全平滑。关键在于，作者还表明：换成某种连续的指标（词元级对数似然、布莱尔分数或部分得分），就能在同一批模型、同一批任务上恢复平滑性。这之所以是一个有力的结果，恰恰因为它是一个受控比较：同样的系统、同样的数据，不同的读出函数，得到不同的定性结论。",[15,646,647],{},"这推广了统计学与心理物理学中一个早已熟知的观点：连续变量的阶跃函数可以通过取阈值轻而易举地构造出来，而取阈值带来的不连续，告诉你的只是阈值本身，而不是那个变量。同样的逻辑也能解释洞察式问题解决研究中出现的「恍然大悟时刻」——主观的、二值的自我报告掩盖了连续的、低于阈值的信息积累，后者要等眼动追踪等连续仪器才能揭示。它也能解释那些表面上的生态「崩塌」，一旦提高采样频率，便消解为渐进的衰退。这些情形的共同结构缺陷是：把观测的离散性误当成了过程的离散性。",[11,649,650],{"id":650},"论证的越界",[15,652,653],{},"从「某些报告的涌现能力是指标假象」推出「涌现是伪概念」，犯了我所谓反例消除谬误的错误：对某一子类主张的一次具体、受控的证伪，被当成了对整个范畴的证伪。有三点考虑限制了幻象论证的适用范围。",[655,656,658],"h3",{"id":657},"组合式任务结构是世界的真实属性而非仅指标的属性","组合式任务结构是世界的真实属性，而非仅指标的属性",[15,660,661,662,690,691,849],{},"如果某个下游应用确实要求全部 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就不是测量假象，而是对用户相关的成功概率的正确描述。其机制完全可以解释、毫无神秘可言（这是初等概率，不是隐藏的相变），但现象本身——在一个狭窄的能力区间内可用输出占比的急剧上升——是真实且有后果的。幻象批判正确地揭开了机制的谜团；它并没有消除现象。这个重要的区分在论证的通俗转述中被抹平了：「无法解释」与「虚假」不是同义词。",[655,851,853],{"id":852},"复杂系统中确实存在真实相变具有指标无关的签名","复杂系统中确实存在真实相变，具有指标无关的签名",[15,855,856,857,889,890,889,921,952],{},"统计物理提供了一个严谨的反例类。铁磁体系中的居里点是真正意义上的二级相变：重整化群理论预言了临界指数（",[36,858,860,875],{"className":859,"translate":40},[39],[36,861,863],{"className":862},[44],[46,864,865],{"xmlns":48},[50,866,867,872],{},[53,868,869],{},[56,870,871],{},"β",[60,873,874],{"encoding":62},"\\beta",[36,876,878],{"className":877,"ariaHidden":67},[66],[36,879,881,885],{"className":880},[71],[36,882],{"className":883,"style":884},[75],"height:0.8889em;vertical-align:-0.1944em;",[36,886,871],{"className":887,"style":888},[80,81],"margin-right:0.0528em;",", ",[36,891,893,908],{"className":892,"translate":40},[39],[36,894,896],{"className":895},[44],[46,897,898],{"xmlns":48},[50,899,900,905],{},[53,901,902],{},[56,903,904],{},"γ",[60,906,907],{"encoding":62},"\\gamma",[36,909,911],{"className":910,"ariaHidden":67},[66],[36,912,914,917],{"className":913},[71],[36,915],{"className":916,"style":108},[75],[36,918,904],{"className":919,"style":920},[80,81],"margin-right:0.0556em;",[36,922,924,939],{"className":923,"translate":40},[39],[36,925,927],{"className":926},[44],[46,928,929],{"xmlns":48},[50,930,931,936],{},[53,932,933],{},[56,934,935],{},"ν",[60,937,938],{"encoding":62},"\\nu",[36,940,942],{"className":941,"ariaHidden":67},[66],[36,943,945,948],{"className":944},[71],[36,946],{"className":947,"style":76},[75],[36,949,935],{"className":950,"style":951},[80,81],"margin-right:0.0637em;","），并在相互独立的可观测量——磁化强度、磁化率、关联长度、比热——上得到验证，所有这些量都在同一个临界温度处发散或消失，无论你选择测量哪一个。随机图上的渗流转变具有类似的普适性：当边密度跨越阈值时，连通性性质急剧转变，这可以从概率论的第一性原理出发证明，与任何特定的观测读出无关。在这些情形里，那个令 LLM「涌现能力」声名扫地的检验——转变在更换指标后是否依然存在？——被漂亮地通过了。一旦离开幻象论文所在的特定领域（LLM 基准评测），「涌现总是指标假象」这一全称论断在经验上就是错的。",[655,954,956],{"id":955},"涌现涵盖多种不同的现象其中只有一部分处于争议之中","「涌现」涵盖多种不同的现象，其中只有一部分处于争议之中",[15,958,959],{},"科学哲学文献（安德森 1972 年的《多即不同》是标准参考点）至少区分了：（a）认识论涌现——尽管原则上可由微观规则推导，却难以从微观规则预测的性质；（b）本体论\u002F弱涌现——在宏观尺度上新颖、但原则上可经由模拟还原的性质；以及（c）强涌现——被声称原则上也不可还原的性质，这是一个更有争议的概念。幻象论证在其恰当范围内，是一个关于特定认识论错觉——由指标选择诱发的表面上的不可预测性——的论证，而对弱涌现是否存在于神经网络中几乎不置一词。混淆这些含义，正是这场争论产生更多热量而非光亮的缘由之一。",[655,961,963],{"id":962},"镜像情形线性指标掩盖真实的非线性结构","镜像情形：线性指标掩盖真实的非线性结构",[15,965,966],{},"前面的讨论都把指标选择当作产生假象式不连续的来源——LLM 的情形、洞察式问题解决的情形、粗略生态普查的情形。三者共享同一个误差方向：非线性或取阈值的读出函数，从平滑的底层现实中制造出表面上的陡峭。但地震震级里氏标度的发展历史则展示了相反的误差方向，它之所以有教益，恰恰因为它不符合幻象模板。",[15,968,969],{},"地震学家最初用地面运动的原始振幅量化地震强度，这是一种线性的仪器读数，与地震仪记录到的物理位移成正比。结果发现，这个指标与研究者真正关心的构念——破坏潜力——匹配得很差，因为地震能量释放随振幅不是线性而是乘性地变化：振幅增大十倍，辐射能量约增大三十二倍，而破坏能力与能量、而非位移成正比。在原始的线性振幅指标下，现象的真实结构——能量释放的古登堡 - 里克特幂律分布，其中灾难性事件与轻微震颤相差许多个数量级——被压缩进一个狭窄的、视觉上波澜不惊的数值区间，系统地掩盖了 8 级事件与 5 级事件之间类别的天壤之别。查尔斯·里希特在 1935 年的解决方案，是把震级定义为振幅以 10 为底的对数（并作距离修正）。这是有意把非线性引入指标，使得新标度上的等量增量对应底层能量释放的等量比值——把一个乘性的、难以沟通的现实，变成一个加性的、可解读的。",[15,971,972],{},"地震情形是 LLM 幻象在结构上的镜像：在那里，非线性指标（组合式任务结构下的精确匹配准确率）从一个实际上平滑的底层现实中制造出虚假的不连续。在这里，线性指标（原始振幅）掩盖了一个真实的、性质刻画清楚的乘性结构，而纠正的做法恰恰是采用对数——即非线性——指标，正因为底层物理过程（由断层破裂面积与应力降控制的弹性能量释放）本身在所测量的量上是指数式的。关键在于，这并不是随意的选择：采用对数变换，是因为独立的物理理论（地震矩、破裂尺度与辐射能量之间的关系）预先预言了乘性关系，新标度也正是依据这一理论得到验证——这正是下文判别框架中将被形式化为检验 2 的那种「独立理论预言」标准。",[15,974,975],{},"把 LLM 与地震这两个案例放在一起读，可以三角定位出这场争论背后那个更深层的方法论原则：任何指标对现象施加的定性形状都不可能是中立的，而可能的失真方向是双向的。非线性\u002F取阈值的指标可以从平滑现实中制造出虚假的陡峭（LLM 基准）；线性指标同样可以掩盖真实的、有理论根基的非线性结构（里氏标度之前的振幅标度）。因此，「偏好连续指标」本身并不是从幻象论证中应该提取的正确方法论教训——正确的教训是：指标的函数形式必须与被测量的原始量与科学关注的构念之间真实关系的函数形式相匹配，而这种匹配必须独立地（通过理论，或通过与破坏性之类外部准则的验证）得到辩护，而不是在任一方向上默认为之。仅仅因为某个指标是连续的便选择它，却不追问那个特定的连续变换是否追踪了所关注的构念，将和幻象论文所批评的精确匹配错误一样没有原则。",[11,977,978],{"id":978},"机器学习理论的补充",[15,980,981],{},"最有趣的近期证据并不来自基准准确率曲线，而是来自神经网络的内部训练动力学——这是幻象论文未曾涉及的领域，因为它只研究跨模型规模的推理期尺度行为，而不研究单个模型在训练时间上的轨迹。",[655,983,985],{"id":984},"grokking真实的动力学转变","Grokking：真实的动力学转变",[15,987,988],{},"Power、Burns、Edwards、Babuschkin 与 Misra（2022）记录了「grokking」现象：在小算法任务（例如模算术训练）上，网络几乎立即记住训练集，随后长时间停留在接近零的测试准确率平台上，再急剧转变到接近完美的泛化——往往比训练损失首次收敛的时间晚好几个数量级。这不是 Schaeffer 意义上的基准指标假象：它在许多随机种子下的测试集准确率上都能观察到，在测试损失等连续指标读出下也成立，而且后来被赋予了一个机制解释——同一网络内部一个记忆电路与一个泛化电路相互竞争，在权重衰减压力下泛化电路最终胜出（Nanda 等，2023，《通过机制可解释性衡量 grokking 的进展指标》）。关键在于，Nanda 等人表明，连续的机制性进展指标——从学习到的嵌入的傅里叶结构导出、追踪用于实现模加法的三角表示的逐渐形成的量——在测试准确率看似平坦的平台上平滑、可预期地上升。这正是 LLM 基准故事在结构上的镜像：在这里，粗粒度指标（准确率）中貌似离散的转变，底层是良好选择的细粒度指标中的连续进展，而这种细粒度进展确实最终促成了网络内部计算一次真实的、在机制上得到刻画的重新组织——一次电路层级的相变，而不只是统计读出的效应。粗与细两幅图景可以同时为真；有趣的问题是细粒度进展指标做了什么工作，而 grokking 研究给出了异常清晰的回答。",[655,990,991],{"id":991},"归纳头与训练损失相变",[15,993,994,995,998,999,1002,1003,1005],{},"Olsson 等（2022，《上下文学习与归纳头》，Anthropic）报告了一个更有资格被称为真实相变的现象：transformer 语言模型的训练损失曲线上出现一个陡峭的隆起，发生在训练早期一个特定的、一致的时点，并在各模型规模与架构上，与「归纳头」的形成同步——归纳头是实现简单复制算法（[A]",[36,996,997],{},"B"," ... ",[36,1000,1001],{},"A"," → 预测 ",[36,1004,997],{},"）的注意力电路，后来被复用于更一般的上下文学习。这个转变在训练损失本身上可见——这是一个连续的、直接被优化的量，不涉及任何取阈值或精确匹配的离散化——这直接击破了幻象解释，因为幻象机制需要一个被人为离散化的读出才能制造出陡峭的外观。此外，损失隆起的时点（通过消融研究）与归纳电路的出现之间存在因果关系，这为转变为何发生提供了一个机制的说明，正如重整化群理论解释了为何居里点在磁化强度上产生陡峭行为一样。这是深度学习目前最接近物理式涌现的东西：一个指标不变的、有机制解释的、可复现的动力系统不连续。",[655,1007,1009],{"id":1008},"尺度定律损失地形与阈值附近方差的压缩","尺度定律、损失地形与阈值附近方差的压缩",[15,1011,1012],{},"还有一条互补的、更带怀疑色彩的 ML 理论脉络值得摆到台面上，因为它从另一个角度部分支持了幻象阵营。标准的神经网络尺度定律（Kaplan 等，2020；Hoffmann 等，2022）表明，损失本身——可用理论动机最充分的连续指标——随参数、数据与计算量按平滑幂律缩放，跨越多个数量级都没有不连续。如果损失平滑而下游任务准确率不平滑，那么在缺乏内部离散重组证据的情况下，最节俭的解释确实是：任务级不连续是从损失到任务表现的映射（组合式结构、取阈值、指标选择都从这儿进入）所产生的下游假象，而非模型底层能力的不连续。这正是 Schaeffer 等人的论点；在缺乏训练期机制性叙事来回应它时（正如大多数报告的「涌现」基准能力那样，与归纳头或 grokking 形成对照），幻象解释应当被视为默认的、更节俭的假设，符合奥卡姆剃刀式先验。",[11,1014,1015],{"id":1015},"判别框架",[15,1017,1018],{},"把这些线索汇聚起来，我提出三个可操作的检验，用于在任何复杂系统（包括机器学习）中区分真实的动力学涌现与指标诱发的错觉。",[15,1020,1021],{},"指标不变性，方向要正确。转变在更换指标后是否依然存在？该指标的函数形式必须被独立地证明确实追踪了所关注的构念，而不仅仅是连续。正如地震震级案例所示，「连续」并不等于「正确」——线性指标同样容易掩盖真实的乘性结构，正如取阈值指标容易制造虚假结构一样。相关的问题不是「读出是否连续？」而是「读出的函数形式是否与真实生成关系的函数形式相匹配？」通过这一恰当表述的检验（如归纳头的损失隆起、经由多个独立可观测量测得的铁磁临界指数、或对照地震能量理论验证的对数震级），是底层存在真实结构的强证据。未通过（如大多数 BIG-Bench「涌现能力」在对数似然读出下的表现），则是原始指标选择造成假象的强证据。",[15,1023,1024],{},"独立的理论预言对比事后的曲线拟合。转变的位置或存在，是由独立的理论框架（临界现象的重整化群理论；归纳头的电路形成理论）预言的，还是仅仅通过肉眼看出一条恰好弯曲的曲线而事后识别出来的？先验的预言比事后的模式识别具有强得多的认识论效力，后者极易受确认偏误与基准套件中几十个任务上的多重比较问题的影响。",[15,1026,1027],{},"不连续的所在：生成机制对比读出函数。不连续位于生成系统内部状态的过程之中（一个电路形成、一个序参量跨越临界值、一个竞争电路动力学趋于解决），还是位于把内部状态映射到外部报告的函数之中（精确匹配评分器、二值的「解决\u002F未解决」判断、粗糙的采样间隔）？第一类不连续是真实涌现的候选者；第二类不连续在没有反证之前应当按假象处理。",[15,1029,1030],{},"应用这个框架：归纳头的形成与 grokking 的电路竞争通过了全部三项检验，有资格被称为学习动力学中真实的——尽管规模不大——相变。而大语言模型的大多数基准报告的「涌现能力」正如 Schaeffer 等人直接展示的那样，在检验 1 上彻底失败，通常也在检验 2 上失败，因为它们是靠扫描许多任务找不连续形状的曲线识别出来的，而不是由独立的关于能力习得的理论预言出来的。",[11,1032,1033],{"id":1033},"结语",[15,1035,1036],{},"地震震级案例是一个有用的纠偏，可以和上面讨论的机器学习证据并置，因为它防止判别框架坍缩成一个简单启发式——「偏好平滑指标，警惕取阈值指标」。这个启发式是从 Schaeffer 等人那里应该得到的错误教训，因为它会劝告地震学家拒绝（非线性的、对数的）里氏标度，改用它所取代的（线性的、连续的）振幅读数——这恰恰反了，因为对数标度才是追踪物理现实的那个。一般原则与其说是关于线性本身，不如说是关于指标的函数形式是否已从关于底层生成过程的理论中独立推导出来、或依据该理论得到验证，而不是出于惯例或便利被采用。",[15,1038,1039],{},"幻象批判最好的理解方式，不是证明了涌现普遍是虚假的，而是证明了某一特定且流行的取证实践——把精确匹配准确率随模型规模的陡峭跳跃报告为质上崭新能力的证据——在统计上是有混淆的，应当退役或大幅限缩。这是对一处已然漂移的文献的一次真实而有价值的修正，那处文献在有些地方已滑向把基准曲线形状当作窥见模型认知的直接窗口。但机器学习理论本身，经由机制可解释性与训练动力学研究——它们与催生幻象论文的基准尺度文献基本正交——已经独立地提供了符合更严格、指标不变、有机制根基的真实动力学相变标准的例子：grokking 的电路竞争、归纳头的形成及其伴随的训练损失转变。这里的教训不是涌现作为一个笼统命题是真实的还是虚假的，而是这个词一直不加区分地被用来覆盖两类范畴，而这一领域如今第一次拥有了逐案区分二者的经验与理论工具。这种纪律——指标不变性、独立预言、以及对不连续所在之处的关注——才是对不加批判的涌现话语及其笼统否定两者的恰当继承者。",{"title":1041,"searchDepth":1042,"depth":1042,"links":1043},"",2,[1044,1045,1046,1053,1058,1059],{"id":13,"depth":1042,"text":13},{"id":31,"depth":1042,"text":31},{"id":650,"depth":1042,"text":650,"children":1047},[1048,1050,1051,1052],{"id":657,"depth":1049,"text":658},3,{"id":852,"depth":1049,"text":853},{"id":955,"depth":1049,"text":956},{"id":962,"depth":1049,"text":963},{"id":978,"depth":1042,"text":978,"children":1054},[1055,1056,1057],{"id":984,"depth":1049,"text":985},{"id":991,"depth":1049,"text":991},{"id":1008,"depth":1049,"text":1009},{"id":1015,"depth":1042,"text":1015},{"id":1033,"depth":1042,"text":1033},false,true,"md",null,{},"\u002Fblog\u002F2026\u002F2026-06-30-about-the-emergency",{"title":6,"description":1041},"blog\u002F2026\u002F2026-06-30-about-the-emergency","Schaeffer 等人表明，LLM 的「涌现能力」往往只是不连续的指标掩盖了平滑的能力增长。但 grokking、归纳头以及居里点式的转变都证明，真实的相变确实存在。",[1070,1071,1072],"ai","machine-learning","thoughts","2026-06-30T00:00:00+08:00","J2GOoTqHNfI--QpaxK0Ew-pGsAlVLh4epBhYTlovx48",{"id":1076,"title":1077,"body":1078,"description":1082,"draft":1060,"enableComment":1061,"extension":1062,"image":1041,"important":1060,"location":1063,"meta":1242,"navigation":1061,"ogImage":1063,"path":1243,"seo":1244,"stem":1245,"summary":1246,"tags":1247,"time":1248,"__hash__":1249},"blog\u002Fblog\u002F2026\u002F2026-07-30-what-agi-needs-may-not-be-a-larger-model-but-a-set-of-theories.md","在开始 AGI 之前，先定义它",{"type":8,"value":1079,"toc":1237},[1080,1083,1086,1089,1095,1098,1101,1104,1109,1112,1117,1120,1123,1126,1129,1145,1148,1151,1154,1157,1160,1163,1166,1169,1172,1180,1183,1188,1191,1194,1197,1200,1203,1208,1211,1214,1217,1222,1225,1228,1231,1234],[15,1081,1082],{},"最近，我看了梁文锋关于 AGI 的演讲，他指出人工智能的终极目标是通用人工智能（AGI）——一种能以类人方式解决各种问题的一般智能。",[15,1084,1085],{},"然而，我越来越觉得，当下 AI 领域围绕 AGI 的讨论仍停留在经验层面。",[15,1087,1088],{},"我们讨论模型规模、参数量、强化学习、智能体和多模态，却很少有人触及一个更根本的问题：",[1090,1091,1092],"blockquote",{},[15,1093,1094],{},"什么才是 AGI？",[15,1096,1097],{},"直到今天，仍然没有一个被广泛接受的正式定义。大多数讨论都在描述现象，而非把握本质。",[11,1099,1100],{"id":1100},"理论框架",[15,1102,1103],{},"历史上真正改变世界的，从来不是某台机器，而是它背后的理论。为了研究机械计算，图灵提出了图灵机模型。随后，丘奇 - 图灵论题指出：",[1090,1105,1106],{},[15,1107,1108],{},"任何能用有效程序计算的函数，都能被图灵机计算。",[15,1110,1111],{},"几乎所有现代计算机都是这个模型的工程实现：一个有限状态控制器加上外部存储。更重要的是，图灵进一步证明了停机问题不可判定。这意味着，人类第一次意识到：",[1090,1113,1114],{},[15,1115,1116],{},"并非所有问题都是可计算的。",[15,1118,1119],{},"换言之，计算机第一次拥有了明确定义的理论上限。这一发现的意义不亚于物理学中的热力学第二定律。",[11,1121,1122],{"id":1122},"理论的局限",[15,1124,1125],{},"今天的 AGI，本质上正处于相当于计算机发明之前的那个阶段。",[15,1127,1128],{},"每个人都在造越来越大的机器，但下面这些问题仍然没有答案：",[1130,1131,1132,1136,1139,1142],"ul",{},[1133,1134,1135],"li",{},"什么是智能？",[1133,1137,1138],{},"智能能被形式化吗？",[1133,1140,1141],{},"一般智能的必要条件是什么？",[1133,1143,1144],{},"智能有理论极限吗？",[15,1146,1147],{},"如果未来有人能提出一套 AGI 公理，类似于图灵机之于计算理论，其意义或许远远超过再训练一个更大的模型。",[15,1149,1150],{},"只有这样的理论建立起来之后，我们才能真正讨论：什么能做到，什么永远做不到。",[11,1152,1153],{"id":1153},"计算主义的内在边界",[15,1155,1156],{},"当今几乎所有主流 AI 方法，核心都建立在计算主义之上。",[15,1158,1159],{},"也就是说：智能就是计算。",[15,1161,1162],{},"如果这个前提成立，那么任何 AI 系统最终都属于可计算系统这一类。然而可计算性理论告诉我们，存在大量的不可判定问题（例如停机问题）。",[15,1164,1165],{},"于是，一个值得深思的问题浮现出来：如果 AGI 被定义为一种能够解决所有问题的智能，那么它必然要直面不可判定问题。",[15,1167,1168],{},"不可判定问题，按照丘奇 - 图灵框架，本质上是不可能被任何可计算系统解决的。",[15,1170,1171],{},"因此，如果采纳这种「全知全能」式的 AGI 定义，它与可计算性理论之间就出现了明显的张力。当然，这并不意味着 AGI 必然不可能；而是说明，「一般智能」到底意味着什么，需要更精确的定义。",[15,1173,1174,1179],{},[19,1175,1178],{"href":1176,"rel":1177},"https:\u002F\u002Farxiv.org\u002Fabs\u002Fcs\u002F0004001",[23],"AIXI 早已揭示了这一矛盾","。事实上，这个问题很早就出现了。Marcus Hutter 的 AIXI 是一般智能最著名的数学模型之一。",[15,1181,1182],{},"AIXI 把智能定义为：",[1090,1184,1185],{},[15,1186,1187],{},"在所有可计算环境中，最大化长期期望回报的最优智能体。",[15,1189,1190],{},"它拥有严谨的数学定义，因此常被视为 AGI 的理论上限。但与此同时，AIXI 本身却是不可计算的。原因在于，它依赖所罗门诺夫归纳，需要对所有可能的程序求和——这个操作天然涉及不可计算的对象，因而无法在真实的物理世界中实现。",[15,1192,1193],{},"换言之，我们已经有了一种「完美智能」的形式化定义，却证明了它无法被计算。",[15,1195,1196],{},"这不是工程上的困难，而是理论上的局限。",[15,1198,1199],{},"我认为有一个公理是缺失的。目前，包括 DeepMind 在内的许多机构都在探索元学习、自适应智能体、世界模型等统一框架。在某种意义上，这些努力都可以被理解为对 AGI「公理系统」的搜寻。",[15,1201,1202],{},"然而，我认为有一个关键公理可能是缺失的：",[1090,1204,1205],{},[15,1206,1207],{},"智能体必须与环境保持连续的因果耦合，而不是作为一个超然的、观察者式的求解器运转。",[15,1209,1210],{},"真正的智能不是对固定数据集的一次性推理，而是在与环境、行动、反馈与修正的持续互动中不断塑造自身。",[15,1212,1213],{},"没有这个因果闭环，即使再大的语言模型，也像是静态世界里运转的模板匹配系统，而不是在实质意义上发展智能。",[15,1215,1216],{},"也许，我们真正需要追寻的并不是一个万能 AI。我越来越倾向于认为，真正值得研究的问题不是如何造出一个「全能」的 AI，而是：",[1090,1218,1219],{},[15,1220,1221],{},"智能是否存在类似于热力学第二定律或香农极限那样的根本性约束原理？",[15,1223,1224],{},"如果未来有人能证明，任何位于有限物理时空中的系统都不可能实现绝对的一般智能，",[15,1226,1227],{},"那么这样工作的意义，或许不亚于图灵建立计算理论，也不亚于牛顿和爱因斯坦为物理学奠定的基础框架。",[15,1229,1230],{},"它不会宣告 AI 的失败。",[15,1232,1233],{},"恰恰相反，它将揭示：真正值得追求的并不是无限的智能，而是一种在理论极限内逐步逼近最优的有界智能。",[15,1235,1236],{},"也许，站在未来回望，人们记住的今天将不是 AGI 在 2026 年出现的那一年，而是我们开始重新思考这个根本问题的那一年：「智能，到底是什么？」",{"title":1041,"searchDepth":1042,"depth":1042,"links":1238},[1239,1240,1241],{"id":1100,"depth":1042,"text":1100},{"id":1122,"depth":1042,"text":1122},{"id":1153,"depth":1042,"text":1153},{},"\u002Fblog\u002F2026\u002F2026-07-30-what-agi-needs-may-not-be-a-larger-model-but-a-set-of-theories",{"title":1077,"description":1082},"blog\u002F2026\u002F2026-07-30-what-agi-needs-may-not-be-a-larger-model-but-a-set-of-theories","AGI 研究仍停留在经验层面，缺乏基础性的理论框架。计算主义与不可判定问题之间存在核心张力，AIXI 正是一个例证。作者提出，与环境保持连续的因果耦合可能是缺失的公理，并呼吁寻找能约束任何物理可实现智能的根本性约束原理。",[1070,1072],"2026-07-30T00:00:00+08:00","6b3H4Ls4HkUBe5xve0AixDdwujabBMiU5faXjBSLD4U",{"id":1251,"title":1252,"body":1253,"description":1041,"draft":1060,"enableComment":1061,"extension":1062,"image":1041,"important":1060,"location":1063,"meta":9935,"navigation":1061,"ogImage":1063,"path":9936,"seo":9937,"stem":9938,"summary":9939,"tags":9940,"time":9943,"__hash__":9944},"blog\u002Fblog\u002F2026\u002F2026-06-08-explanation-of-neural-network-from-maximum-likelihood-estimation.md","最大似然估计视角下的神经网络",{"type":8,"value":1254,"toc":9930},[1255,1258,1261,1264,1269,1301,1304,1361,1364,1532,1564,1620,1837,1919,1975,1978,2084,2087,2090,2093,2339,2372,2375,2470,2473,2773,2876,2879,3101,3104,3376,3408,3411,3740,3743,3746,3860,3863,3974,3977,4153,4321,4324,4546,4549,4857,4860,5224,5227,5593,5625,6068,6071,6411,6443,6832,7037,7129,7132,7232,7264,7675,7678,7846,7849,8464,8467,8993,9025,9028,9426,9429,9887,9919,9922,9927],[11,1256,1257],{"id":1257},"抛硬币",[15,1259,1260],{},"MLE（最大似然估计）是统计学中最核心的思想之一。",[15,1262,1263],{},"一句话总结：",[1090,1265,1266],{},[15,1267,1268],{},"MLE 从所有可能的模型参数中，选出那个「让观测数据最有可能出现」的参数。",[15,1270,1271,1272,1300],{},"下面，你拿到一枚纹理不均匀的残次硬币。所以你不知道它是否公平。也就是说，抛这枚硬币时，你不知道正面或反面的概率。你想知道：出现正面的概率 ",[36,1273,1275,1288],{"className":1274,"translate":40},[39],[36,1276,1278],{"className":1277},[44],[46,1279,1280],{"xmlns":48},[50,1281,1282,1286],{},[53,1283,1284],{},[56,1285,15],{},[60,1287,15],{"encoding":62},[36,1289,1291],{"className":1290,"ariaHidden":67},[66],[36,1292,1294,1297],{"className":1293},[71],[36,1295],{"className":1296,"style":108},[75],[36,1298,15],{"className":1299},[80,81]," 是多少。",[15,1302,1303],{},"于是你决定连抛 10 次，得到如下结果：正面 8 次，反面 2 次。",[15,1305,1306,1307,1360],{},"假设这是一枚公平硬币，那么设先验假设 ",[36,1308,1310,1329],{"className":1309,"translate":40},[39],[36,1311,1313],{"className":1312},[44],[46,1314,1315],{"xmlns":48},[50,1316,1317,1326],{},[53,1318,1319,1321,1323],{},[56,1320,15],{},[160,1322,346],{},[173,1324,1325],{},"0.5",[60,1327,1328],{"encoding":62},"p=0.5",[36,1330,1332,1350],{"className":1331,"ariaHidden":67},[66],[36,1333,1335,1338,1341,1344,1347],{"className":1334},[71],[36,1336],{"className":1337,"style":108},[75],[36,1339,15],{"className":1340},[80,81],[36,1342],{"className":1343,"style":224},[223],[36,1345,346],{"className":1346},[228],[36,1348],{"className":1349,"style":224},[223],[36,1351,1353,1357],{"className":1352},[71],[36,1354],{"className":1355,"style":1356},[75],"height:0.6444em;",[36,1358,1325],{"className":1359},[80],"。",[15,1362,1363],{},"在这一先验假设下，出现上述结果的概率为：",[36,1365,1367],{"className":1366,"translate":40},[314],[36,1368,1370,1415],{"className":1369,"translate":40},[39],[36,1371,1373],{"className":1372},[44],[46,1374,1375],{"xmlns":48,"display":323},[50,1376,1377,1412],{},[53,1378,1379,1381,1383,1386,1390,1392,1394,1396,1398,1400,1407,1409],{},[56,1380,333],{},[160,1382,163],{"stretchy":162},[56,1384,1385],{},"D",[56,1387,1389],{"mathvariant":1388},"normal","∣",[56,1391,15],{},[160,1393,346],{},[173,1395,1325],{},[160,1397,168],{"stretchy":162},[160,1399,346],{},[182,1401,1402,1404],{},[173,1403,1325],{},[173,1405,1406],{},"10",[160,1408,171],{},[173,1410,1411],{},"0.000976",[60,1413,1414],{"encoding":62},"P(D|p=0.5)\n=\n0.5^{10}\n\\approx 0.000976",[36,1416,1418,1449,1470,1523],{"className":1417,"ariaHidden":67},[66],[36,1419,1421,1424,1427,1430,1434,1437,1440,1443,1446],{"className":1420},[71],[36,1422],{"className":1423,"style":205},[75],[36,1425,333],{"className":1426,"style":377},[80,81],[36,1428,163],{"className":1429},[212],[36,1431,1385],{"className":1432,"style":1433},[80,81],"margin-right:0.0278em;",[36,1435,1389],{"className":1436},[80],[36,1438,15],{"className":1439},[80,81],[36,1441],{"className":1442,"style":224},[223],[36,1444,346],{"className":1445},[228],[36,1447],{"className":1448,"style":224},[223],[36,1450,1452,1455,1458,1461,1464,1467],{"className":1451},[71],[36,1453],{"className":1454,"style":205},[75],[36,1456,1325],{"className":1457},[80],[36,1459,168],{"className":1460},[219],[36,1462],{"className":1463,"style":224},[223],[36,1465,346],{"className":1466},[228],[36,1468],{"className":1469,"style":224},[223],[36,1471,1473,1477,1481,1514,1517,1520],{"className":1472},[71],[36,1474],{"className":1475,"style":1476},[75],"height:0.8641em;",[36,1478,1480],{"className":1479},[80],"0.",[36,1482,1484,1488],{"className":1483},[80],[36,1485,1487],{"className":1486},[80],"5",[36,1489,1491],{"className":1490},[272],[36,1492,1494],{"className":1493},[276],[36,1495,1497],{"className":1496},[280],[36,1498,1500],{"className":1499,"style":1476},[284],[36,1501,1502,1505],{"style":480},[36,1503],{"className":1504,"style":292},[291],[36,1506,1508],{"className":1507},[296,297,298,299],[36,1509,1511],{"className":1510},[80,299],[36,1512,1406],{"className":1513},[80,299],[36,1515],{"className":1516,"style":224},[223],[36,1518,171],{"className":1519},[228],[36,1521],{"className":1522,"style":224},[223],[36,1524,1526,1529],{"className":1525},[71],[36,1527],{"className":1528,"style":1356},[75],[36,1530,1411],{"className":1531},[80],[15,1533,1534,1535,1563],{},"也就是说，如果这是一枚公平硬币，那么抛 10 次得到 8 次正面、2 次反面的概率约为 ",[36,1536,1538,1551],{"className":1537,"translate":40},[39],[36,1539,1541],{"className":1540},[44],[46,1542,1543],{"xmlns":48},[50,1544,1545,1549],{},[53,1546,1547],{},[173,1548,1411],{},[60,1550,1411],{"encoding":62},[36,1552,1554],{"className":1553,"ariaHidden":67},[66],[36,1555,1557,1560],{"className":1556},[71],[36,1558],{"className":1559,"style":1356},[75],[36,1561,1411],{"className":1562},[80],"——在概率统计里，这个数显然被视为几乎不可能。所以你能清楚地感觉到这枚硬币不公平。",[15,1565,1566,1567,1619],{},"现在提出另一个假设：假设正面的概率是 ",[36,1568,1570,1589],{"className":1569,"translate":40},[39],[36,1571,1573],{"className":1572},[44],[46,1574,1575],{"xmlns":48},[50,1576,1577,1586],{},[53,1578,1579,1581,1583],{},[56,1580,15],{},[160,1582,346],{},[173,1584,1585],{},"0.8",[60,1587,1588],{"encoding":62},"p=0.8",[36,1590,1592,1610],{"className":1591,"ariaHidden":67},[66],[36,1593,1595,1598,1601,1604,1607],{"className":1594},[71],[36,1596],{"className":1597,"style":108},[75],[36,1599,15],{"className":1600},[80,81],[36,1602],{"className":1603,"style":224},[223],[36,1605,346],{"className":1606},[228],[36,1608],{"className":1609,"style":224},[223],[36,1611,1613,1616],{"className":1612},[71],[36,1614],{"className":1615,"style":1356},[75],[36,1617,1585],{"className":1618},[80],"。那么上述结果的概率为：",[36,1621,1623],{"className":1622,"translate":40},[314],[36,1624,1626,1679],{"className":1625,"translate":40},[39],[36,1627,1629],{"className":1628},[44],[46,1630,1631],{"xmlns":48,"display":323},[50,1632,1633,1676],{},[53,1634,1635,1637,1639,1641,1643,1645,1647,1649,1651,1653,1660,1663,1671,1673],{},[56,1636,333],{},[160,1638,163],{"stretchy":162},[56,1640,1385],{},[56,1642,1389],{"mathvariant":1388},[56,1644,15],{},[160,1646,346],{},[173,1648,1585],{},[160,1650,168],{"stretchy":162},[160,1652,346],{},[182,1654,1655,1657],{},[173,1656,1585],{},[173,1658,1659],{},"8",[160,1661,1662],{},"×",[182,1664,1665,1668],{},[173,1666,1667],{},"0.2",[173,1669,1670],{},"2",[160,1672,171],{},[173,1674,1675],{},"0.0067",[60,1677,1678],{"encoding":62},"P(D|p=0.8)\n=\n0.8^8\n\\times\n0.2^2 \\approx 0.0067",[36,1680,1682,1712,1733,1781,1828],{"className":1681,"ariaHidden":67},[66],[36,1683,1685,1688,1691,1694,1697,1700,1703,1706,1709],{"className":1684},[71],[36,1686],{"className":1687,"style":205},[75],[36,1689,333],{"className":1690,"style":377},[80,81],[36,1692,163],{"className":1693},[212],[36,1695,1385],{"className":1696,"style":1433},[80,81],[36,1698,1389],{"className":1699},[80],[36,1701,15],{"className":1702},[80,81],[36,1704],{"className":1705,"style":224},[223],[36,1707,346],{"className":1708},[228],[36,1710],{"className":1711,"style":224},[223],[36,1713,1715,1718,1721,1724,1727,1730],{"className":1714},[71],[36,1716],{"className":1717,"style":205},[75],[36,1719,1585],{"className":1720},[80],[36,1722,168],{"className":1723},[219],[36,1725],{"className":1726,"style":224},[223],[36,1728,346],{"className":1729},[228],[36,1731],{"className":1732,"style":224},[223],[36,1734,1736,1740,1743,1772,1775,1778],{"className":1735},[71],[36,1737],{"className":1738,"style":1739},[75],"height:0.9474em;vertical-align:-0.0833em;",[36,1741,1480],{"className":1742},[80],[36,1744,1746,1749],{"className":1745},[80],[36,1747,1659],{"className":1748},[80],[36,1750,1752],{"className":1751},[272],[36,1753,1755],{"className":1754},[276],[36,1756,1758],{"className":1757},[280],[36,1759,1761],{"className":1760,"style":1476},[284],[36,1762,1763,1766],{"style":480},[36,1764],{"className":1765,"style":292},[291],[36,1767,1769],{"className":1768},[296,297,298,299],[36,1770,1659],{"className":1771},[80,299],[36,1773],{"className":1774,"style":245},[223],[36,1776,1662],{"className":1777},[249],[36,1779],{"className":1780,"style":245},[223],[36,1782,1784,1787,1790,1819,1822,1825],{"className":1783},[71],[36,1785],{"className":1786,"style":1476},[75],[36,1788,1480],{"className":1789},[80],[36,1791,1793,1796],{"className":1792},[80],[36,1794,1670],{"className":1795},[80],[36,1797,1799],{"className":1798},[272],[36,1800,1802],{"className":1801},[276],[36,1803,1805],{"className":1804},[280],[36,1806,1808],{"className":1807,"style":1476},[284],[36,1809,1810,1813],{"style":480},[36,1811],{"className":1812,"style":292},[291],[36,1814,1816],{"className":1815},[296,297,298,299],[36,1817,1670],{"className":1818},[80,299],[36,1820],{"className":1821,"style":224},[223],[36,1823,171],{"className":1824},[228],[36,1826],{"className":1827,"style":224},[223],[36,1829,1831,1834],{"className":1830},[71],[36,1832],{"className":1833,"style":1356},[75],[36,1835,1675],{"className":1836},[80],[15,1838,1839,1840,1890,1891,1360],{},"我们还发现，当 ",[36,1841,1843,1860],{"className":1842,"translate":40},[39],[36,1844,1846],{"className":1845},[44],[46,1847,1848],{"xmlns":48},[50,1849,1850,1858],{},[53,1851,1852,1854,1856],{},[56,1853,15],{},[160,1855,346],{},[173,1857,1585],{},[60,1859,1588],{"encoding":62},[36,1861,1863,1881],{"className":1862,"ariaHidden":67},[66],[36,1864,1866,1869,1872,1875,1878],{"className":1865},[71],[36,1867],{"className":1868,"style":108},[75],[36,1870,15],{"className":1871},[80,81],[36,1873],{"className":1874,"style":224},[223],[36,1876,346],{"className":1877},[228],[36,1879],{"className":1880,"style":224},[223],[36,1882,1884,1887],{"className":1883},[71],[36,1885],{"className":1886,"style":1356},[75],[36,1888,1585],{"className":1889},[80]," 时，上述结果的概率最大，达到最大值 ",[36,1892,1894,1907],{"className":1893,"translate":40},[39],[36,1895,1897],{"className":1896},[44],[46,1898,1899],{"xmlns":48},[50,1900,1901,1905],{},[53,1902,1903],{},[173,1904,1675],{},[60,1906,1675],{"encoding":62},[36,1908,1910],{"className":1909,"ariaHidden":67},[66],[36,1911,1913,1916],{"className":1912},[71],[36,1914],{"className":1915,"style":1356},[75],[36,1917,1675],{"className":1918},[80],[36,1920,1922],{"className":1921,"translate":40},[314],[36,1923,1925,1944],{"className":1924,"translate":40},[39],[36,1926,1928],{"className":1927},[44],[46,1929,1930],{"xmlns":48,"display":323},[50,1931,1932,1941],{},[53,1933,1934,1936,1939],{},[173,1935,1675],{},[160,1937,1938],{},">",[173,1940,1411],{},[60,1942,1943],{"encoding":62},"0.0067\n>\n0.000976",[36,1945,1947,1966],{"className":1946,"ariaHidden":67},[66],[36,1948,1950,1954,1957,1960,1963],{"className":1949},[71],[36,1951],{"className":1952,"style":1953},[75],"height:0.6835em;vertical-align:-0.0391em;",[36,1955,1675],{"className":1956},[80],[36,1958],{"className":1959,"style":224},[223],[36,1961,1938],{"className":1962},[228],[36,1964],{"className":1965,"style":224},[223],[36,1967,1969,1972],{"className":1968},[71],[36,1970],{"className":1971,"style":1356},[75],[36,1973,1411],{"className":1974},[80],[15,1976,1977],{},"这说明：如果正面概率是 0.8，那么观察到这份数据的可能性最大。",[15,1979,1980,1981],{},"因此：",[36,1982,1984,2008],{"className":1983,"translate":40},[39],[36,1985,1987],{"className":1986},[44],[46,1988,1989],{"xmlns":48},[50,1990,1991,2005],{},[53,1992,1993,2001,2003],{},[1994,1995,1996,1998],"mover",{"accent":67},[56,1997,15],{},[160,1999,2000],{},"^",[160,2002,346],{},[173,2004,1585],{},[60,2006,2007],{"encoding":62},"\\hat p=0.8",[36,2009,2011,2075],{"className":2010,"ariaHidden":67},[66],[36,2012,2014,2017,2066,2069,2072],{"className":2013},[71],[36,2015],{"className":2016,"style":884},[75],[36,2018,2021],{"className":2019},[80,2020],"accent",[36,2022,2024,2057],{"className":2023},[276,384],[36,2025,2027,2054],{"className":2026},[280],[36,2028,2031,2041],{"className":2029,"style":2030},[284],"height:0.6944em;",[36,2032,2034,2038],{"style":2033},"top:-3em;",[36,2035],{"className":2036,"style":2037},[291],"height:3em;",[36,2039,15],{"className":2040},[80,81],[36,2042,2043,2046],{"style":2033},[36,2044],{"className":2045,"style":2037},[291],[36,2047,2051],{"className":2048,"style":2050},[2049],"accent-body","left:-0.1667em;",[36,2052,2000],{"className":2053},[80],[36,2055,416],{"className":2056},[415],[36,2058,2060],{"className":2059},[280],[36,2061,2064],{"className":2062,"style":2063},[284],"height:0.1944em;",[36,2065],{},[36,2067],{"className":2068,"style":224},[223],[36,2070,346],{"className":2071},[228],[36,2073],{"className":2074,"style":224},[223],[36,2076,2078,2081],{"className":2077},[71],[36,2079],{"className":2080,"style":1356},[75],[36,2082,1585],{"className":2083},[80],[15,2085,2086],{},"这就是最大似然估计。",[11,2088,2089],{"id":2089},"似然函数",[15,2091,2092],{},"给定数据：",[36,2094,2096],{"className":2095,"translate":40},[314],[36,2097,2099,2153],{"className":2098,"translate":40},[39],[36,2100,2102],{"className":2101},[44],[46,2103,2104],{"xmlns":48,"display":323},[50,2105,2106,2150],{},[53,2107,2108,2110,2112,2115,2122,2125,2131,2133,2136,2139,2141,2147],{},[56,2109,1385],{},[160,2111,346],{},[160,2113,2114],{"stretchy":162},"{",[329,2116,2117,2120],{},[56,2118,2119],{},"x",[173,2121,175],{},[160,2123,2124],{"separator":67},",",[329,2126,2127,2129],{},[56,2128,2119],{},[173,2130,1670],{},[160,2132,2124],{"separator":67},[160,2134,2135],{},"⋯",[335,2137,2138],{}," ",[160,2140,2124],{"separator":67},[329,2142,2143,2145],{},[56,2144,2119],{},[56,2146,58],{},[160,2148,2149],{"stretchy":162},"}",[60,2151,2152],{"encoding":62},"D = \\{x_1, x_2, \\cdots, x_n\\}",[36,2154,2156,2174],{"className":2155,"ariaHidden":67},[66],[36,2157,2159,2162,2165,2168,2171],{"className":2158},[71],[36,2160],{"className":2161,"style":139},[75],[36,2163,1385],{"className":2164,"style":1433},[80,81],[36,2166],{"className":2167,"style":224},[223],[36,2169,346],{"className":2170},[228],[36,2172],{"className":2173,"style":224},[223],[36,2175,2177,2180,2183,2225,2229,2233,2273,2276,2279,2283,2286,2289,2292,2295,2336],{"className":2176},[71],[36,2178],{"className":2179,"style":205},[75],[36,2181,2114],{"className":2182},[212],[36,2184,2186,2189],{"className":2185},[80],[36,2187,2119],{"className":2188},[80,81],[36,2190,2192],{"className":2191},[272],[36,2193,2195,2217],{"className":2194},[276,384],[36,2196,2198,2214],{"className":2197},[280],[36,2199,2202],{"className":2200,"style":2201},[284],"height:0.3011em;",[36,2203,2205,2208],{"style":2204},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[36,2206],{"className":2207,"style":292},[291],[36,2209,2211],{"className":2210},[296,297,298,299],[36,2212,175],{"className":2213},[80,299],[36,2215,416],{"className":2216},[415],[36,2218,2220],{"className":2219},[280],[36,2221,2223],{"className":2222,"style":423},[284],[36,2224],{},[36,2226,2124],{"className":2227},[2228],"mpunct",[36,2230],{"className":2231,"style":2232},[223],"margin-right:0.1667em;",[36,2234,2236,2239],{"className":2235},[80],[36,2237,2119],{"className":2238},[80,81],[36,2240,2242],{"className":2241},[272],[36,2243,2245,2265],{"className":2244},[276,384],[36,2246,2248,2262],{"className":2247},[280],[36,2249,2251],{"className":2250,"style":2201},[284],[36,2252,2253,2256],{"style":2204},[36,2254],{"className":2255,"style":292},[291],[36,2257,2259],{"className":2258},[296,297,298,299],[36,2260,1670],{"className":2261},[80,299],[36,2263,416],{"className":2264},[415],[36,2266,2268],{"className":2267},[280],[36,2269,2271],{"className":2270,"style":423},[284],[36,2272],{},[36,2274,2124],{"className":2275},[2228],[36,2277],{"className":2278,"style":2232},[223],[36,2280,2135],{"className":2281},[2282],"minner",[36,2284],{"className":2285,"style":2232},[223],[36,2287],{"className":2288,"style":2232},[223],[36,2290,2124],{"className":2291},[2228],[36,2293],{"className":2294,"style":2232},[223],[36,2296,2298,2301],{"className":2297},[80],[36,2299,2119],{"className":2300},[80,81],[36,2302,2304],{"className":2303},[272],[36,2305,2307,2328],{"className":2306},[276,384],[36,2308,2310,2325],{"className":2309},[280],[36,2311,2314],{"className":2312,"style":2313},[284],"height:0.1514em;",[36,2315,2316,2319],{"style":2204},[36,2317],{"className":2318,"style":292},[291],[36,2320,2322],{"className":2321},[296,297,298,299],[36,2323,58],{"className":2324},[80,81,299],[36,2326,416],{"className":2327},[415],[36,2329,2331],{"className":2330},[280],[36,2332,2334],{"className":2333,"style":423},[284],[36,2335],{},[36,2337,2149],{"className":2338},[219],[15,2340,2341,2342,1360],{},"假设我们有模型参数：",[36,2343,2345,2360],{"className":2344,"translate":40},[39],[36,2346,2348],{"className":2347},[44],[46,2349,2350],{"xmlns":48},[50,2351,2352,2357],{},[53,2353,2354],{},[56,2355,2356],{},"θ",[60,2358,2359],{"encoding":62},"\\theta",[36,2361,2363],{"className":2362,"ariaHidden":67},[66],[36,2364,2366,2369],{"className":2365},[71],[36,2367],{"className":2368,"style":2030},[75],[36,2370,2356],{"className":2371,"style":1433},[80,81],[15,2373,2374],{},"似然函数定义为：",[36,2376,2378],{"className":2377,"translate":40},[314],[36,2379,2381,2416],{"className":2380,"translate":40},[39],[36,2382,2384],{"className":2383},[44],[46,2385,2386],{"xmlns":48,"display":323},[50,2387,2388,2413],{},[53,2389,2390,2393,2395,2397,2399,2401,2403,2405,2407,2409,2411],{},[56,2391,2392],{},"L",[160,2394,163],{"stretchy":162},[56,2396,2356],{},[160,2398,168],{"stretchy":162},[160,2400,346],{},[56,2402,333],{},[160,2404,163],{"stretchy":162},[56,2406,1385],{},[56,2408,1389],{"mathvariant":1388},[56,2410,2356],{},[160,2412,168],{"stretchy":162},[60,2414,2415],{"encoding":62},"L(\\theta) = P(D|\\theta)",[36,2417,2419,2446],{"className":2418,"ariaHidden":67},[66],[36,2420,2422,2425,2428,2431,2434,2437,2440,2443],{"className":2421},[71],[36,2423],{"className":2424,"style":205},[75],[36,2426,2392],{"className":2427},[80,81],[36,2429,163],{"className":2430},[212],[36,2432,2356],{"className":2433,"style":1433},[80,81],[36,2435,168],{"className":2436},[219],[36,2438],{"className":2439,"style":224},[223],[36,2441,346],{"className":2442},[228],[36,2444],{"className":2445,"style":224},[223],[36,2447,2449,2452,2455,2458,2461,2464,2467],{"className":2448},[71],[36,2450],{"className":2451,"style":205},[75],[36,2453,333],{"className":2454,"style":377},[80,81],[36,2456,163],{"className":2457},[212],[36,2459,1385],{"className":2460,"style":1433},[80,81],[36,2462,1389],{"className":2463},[80],[36,2465,2356],{"className":2466,"style":1433},[80,81],[36,2468,168],{"className":2469},[219],[15,2471,2472],{},"最大似然估计（MLE）是：",[36,2474,2476],{"className":2475,"translate":40},[314],[36,2477,2479,2553],{"className":2478,"translate":40},[39],[36,2480,2482],{"className":2481},[44],[46,2483,2484],{"xmlns":48,"display":323},[50,2485,2486,2550],{},[53,2487,2488,2494,2496,2499,2502,2514,2516,2518,2520,2522,2524,2526,2528,2530,2532,2542,2544,2546,2548],{},[1994,2489,2490,2492],{"accent":67},[56,2491,2356],{},[160,2493,2000],{},[160,2495,346],{},[56,2497,2498],{},"arg",[160,2500,2501],{},"⁡",[2503,2504,2505,2512],"munder",{},[53,2506,2507,2510],{},[56,2508,2509],{},"max",[160,2511,2501],{},[56,2513,2356],{},[56,2515,333],{},[160,2517,163],{"stretchy":162},[56,2519,1385],{},[56,2521,1389],{"mathvariant":1388},[56,2523,2356],{},[160,2525,168],{"stretchy":162},[160,2527,346],{},[56,2529,2498],{},[160,2531,2501],{},[2503,2533,2534,2540],{},[53,2535,2536,2538],{},[56,2537,2509],{},[160,2539,2501],{},[56,2541,2356],{},[56,2543,2392],{},[160,2545,163],{"stretchy":162},[56,2547,2356],{},[160,2549,168],{"stretchy":162},[60,2551,2552],{"encoding":62},"\\hat\\theta = \\arg\\max_\\theta P(D|\\theta) = \\arg\\max_\\theta L(\\theta)",[36,2554,2556,2604,2700],{"className":2555,"ariaHidden":67},[66],[36,2557,2559,2563,2595,2598,2601],{"className":2558},[71],[36,2560],{"className":2561,"style":2562},[75],"height:0.9579em;",[36,2564,2566],{"className":2565},[80,2020],[36,2567,2569],{"className":2568},[276],[36,2570,2572],{"className":2571},[280],[36,2573,2575,2583],{"className":2574,"style":2562},[284],[36,2576,2577,2580],{"style":2033},[36,2578],{"className":2579,"style":2037},[291],[36,2581,2356],{"className":2582,"style":1433},[80,81],[36,2584,2586,2589],{"style":2585},"top:-3.2634em;",[36,2587],{"className":2588,"style":2037},[291],[36,2590,2592],{"className":2591,"style":2050},[2049],[36,2593,2000],{"className":2594},[80],[36,2596],{"className":2597,"style":224},[223],[36,2599,346],{"className":2600},[228],[36,2602],{"className":2603,"style":224},[223],[36,2605,2607,2611,2620,2623,2670,2673,2676,2679,2682,2685,2688,2691,2694,2697],{"className":2606},[71],[36,2608],{"className":2609,"style":2610},[75],"height:1.5021em;vertical-align:-0.7521em;",[36,2612,2615,2616],{"className":2613},[2614],"mop","ar",[36,2617,2619],{"style":2618},"margin-right:0.0139em;","g",[36,2621],{"className":2622,"style":2232},[223],[36,2624,2627],{"className":2625},[2614,2626],"op-limits",[36,2628,2630,2661],{"className":2629},[276,384],[36,2631,2633,2658],{"className":2632},[280],[36,2634,2636,2648],{"className":2635,"style":76},[284],[36,2637,2639,2642],{"style":2638},"top:-2.3479em;margin-left:0em;",[36,2640],{"className":2641,"style":2037},[291],[36,2643,2645],{"className":2644},[296,297,298,299],[36,2646,2356],{"className":2647,"style":1433},[80,81,299],[36,2649,2650,2653],{"style":2033},[36,2651],{"className":2652,"style":2037},[291],[36,2654,2655],{},[36,2656,2509],{"className":2657},[2614],[36,2659,416],{"className":2660},[415],[36,2662,2664],{"className":2663},[280],[36,2665,2668],{"className":2666,"style":2667},[284],"height:0.7521em;",[36,2669],{},[36,2671],{"className":2672,"style":2232},[223],[36,2674,333],{"className":2675,"style":377},[80,81],[36,2677,163],{"className":2678},[212],[36,2680,1385],{"className":2681,"style":1433},[80,81],[36,2683,1389],{"className":2684},[80],[36,2686,2356],{"className":2687,"style":1433},[80,81],[36,2689,168],{"className":2690},[219],[36,2692],{"className":2693,"style":224},[223],[36,2695,346],{"className":2696},[228],[36,2698],{"className":2699,"style":224},[223],[36,2701,2703,2706,2711,2714,2758,2761,2764,2767,2770],{"className":2702},[71],[36,2704],{"className":2705,"style":2610},[75],[36,2707,2615,2709],{"className":2708},[2614],[36,2710,2619],{"style":2618},[36,2712],{"className":2713,"style":2232},[223],[36,2715,2717],{"className":2716},[2614,2626],[36,2718,2720,2750],{"className":2719},[276,384],[36,2721,2723,2747],{"className":2722},[280],[36,2724,2726,2737],{"className":2725,"style":76},[284],[36,2727,2728,2731],{"style":2638},[36,2729],{"className":2730,"style":2037},[291],[36,2732,2734],{"className":2733},[296,297,298,299],[36,2735,2356],{"className":2736,"style":1433},[80,81,299],[36,2738,2739,2742],{"style":2033},[36,2740],{"className":2741,"style":2037},[291],[36,2743,2744],{},[36,2745,2509],{"className":2746},[2614],[36,2748,416],{"className":2749},[415],[36,2751,2753],{"className":2752},[280],[36,2754,2756],{"className":2755,"style":2667},[284],[36,2757],{},[36,2759],{"className":2760,"style":2232},[223],[36,2762,2392],{"className":2763},[80,81],[36,2765,163],{"className":2766},[212],[36,2768,2356],{"className":2769,"style":1433},[80,81],[36,2771,168],{"className":2772},[219],[15,2774,2775,2776,2830,2831,2875],{},"很多人初学时感到困惑：\n既然 ",[36,2777,2779,2803],{"className":2778,"translate":40},[39],[36,2780,2782],{"className":2781},[44],[46,2783,2784],{"xmlns":48},[50,2785,2786,2800],{},[53,2787,2788,2790,2792,2794,2796,2798],{},[56,2789,333],{},[160,2791,163],{"stretchy":162},[56,2793,1385],{},[56,2795,1389],{"mathvariant":1388},[56,2797,2356],{},[160,2799,168],{"stretchy":162},[60,2801,2802],{"encoding":62},"P(D|\\theta)",[36,2804,2806],{"className":2805,"ariaHidden":67},[66],[36,2807,2809,2812,2815,2818,2821,2824,2827],{"className":2808},[71],[36,2810],{"className":2811,"style":205},[75],[36,2813,333],{"className":2814,"style":377},[80,81],[36,2816,163],{"className":2817},[212],[36,2819,1385],{"className":2820,"style":1433},[80,81],[36,2822,1389],{"className":2823},[80],[36,2825,2356],{"className":2826,"style":1433},[80,81],[36,2828,168],{"className":2829},[219]," 和 ",[36,2832,2834,2854],{"className":2833,"translate":40},[39],[36,2835,2837],{"className":2836},[44],[46,2838,2839],{"xmlns":48},[50,2840,2841,2851],{},[53,2842,2843,2845,2847,2849],{},[56,2844,2392],{},[160,2846,163],{"stretchy":162},[56,2848,2356],{},[160,2850,168],{"stretchy":162},[60,2852,2853],{"encoding":62},"L(\\theta)",[36,2855,2857],{"className":2856,"ariaHidden":67},[66],[36,2858,2860,2863,2866,2869,2872],{"className":2859},[71],[36,2861],{"className":2862,"style":205},[75],[36,2864,2392],{"className":2865},[80,81],[36,2867,163],{"className":2868},[212],[36,2870,2356],{"className":2871,"style":1433},[80,81],[36,2873,168],{"className":2874},[219]," 的表达式完全一样，为什么要换一个名字？",[15,2877,2878],{},"关键在于视角不同：",[1130,2880,2881,2996],{},[1133,2882,2883,2884,2937,2938,2966,2967,2995],{},"概率 ",[36,2885,2887,2910],{"className":2886,"translate":40},[39],[36,2888,2890],{"className":2889},[44],[46,2891,2892],{"xmlns":48},[50,2893,2894,2908],{},[53,2895,2896,2898,2900,2902,2904,2906],{},[56,2897,333],{},[160,2899,163],{"stretchy":162},[56,2901,1385],{},[56,2903,1389],{"mathvariant":1388},[56,2905,2356],{},[160,2907,168],{"stretchy":162},[60,2909,2802],{"encoding":62},[36,2911,2913],{"className":2912,"ariaHidden":67},[66],[36,2914,2916,2919,2922,2925,2928,2931,2934],{"className":2915},[71],[36,2917],{"className":2918,"style":205},[75],[36,2920,333],{"className":2921,"style":377},[80,81],[36,2923,163],{"className":2924},[212],[36,2926,1385],{"className":2927,"style":1433},[80,81],[36,2929,1389],{"className":2930},[80],[36,2932,2356],{"className":2933,"style":1433},[80,81],[36,2935,168],{"className":2936},[219]," 的视角：\n参数 ",[36,2939,2941,2954],{"className":2940,"translate":40},[39],[36,2942,2944],{"className":2943},[44],[46,2945,2946],{"xmlns":48},[50,2947,2948,2952],{},[53,2949,2950],{},[56,2951,2356],{},[60,2953,2359],{"encoding":62},[36,2955,2957],{"className":2956,"ariaHidden":67},[66],[36,2958,2960,2963],{"className":2959},[71],[36,2961],{"className":2962,"style":2030},[75],[36,2964,2356],{"className":2965,"style":1433},[80,81]," 是固定的已知值，而数据 ",[36,2968,2970,2983],{"className":2969,"translate":40},[39],[36,2971,2973],{"className":2972},[44],[46,2974,2975],{"xmlns":48},[50,2976,2977,2981],{},[53,2978,2979],{},[56,2980,1385],{},[60,2982,1385],{"encoding":62},[36,2984,2986],{"className":2985,"ariaHidden":67},[66],[36,2987,2989,2992],{"className":2988},[71],[36,2990],{"className":2991,"style":139},[75],[36,2993,1385],{"className":2994,"style":1433},[80,81]," 是随机的。\n我们问：「在给定参数的情况下，不同的数据结果出现的可能性有多大？」",[1133,2997,2998,2999,3042,3043,3071,3072,3100],{},"似然 ",[36,3000,3002,3021],{"className":3001,"translate":40},[39],[36,3003,3005],{"className":3004},[44],[46,3006,3007],{"xmlns":48},[50,3008,3009,3019],{},[53,3010,3011,3013,3015,3017],{},[56,3012,2392],{},[160,3014,163],{"stretchy":162},[56,3016,2356],{},[160,3018,168],{"stretchy":162},[60,3020,2853],{"encoding":62},[36,3022,3024],{"className":3023,"ariaHidden":67},[66],[36,3025,3027,3030,3033,3036,3039],{"className":3026},[71],[36,3028],{"className":3029,"style":205},[75],[36,3031,2392],{"className":3032},[80,81],[36,3034,163],{"className":3035},[212],[36,3037,2356],{"className":3038,"style":1433},[80,81],[36,3040,168],{"className":3041},[219]," 的视角：\n数据 ",[36,3044,3046,3059],{"className":3045,"translate":40},[39],[36,3047,3049],{"className":3048},[44],[46,3050,3051],{"xmlns":48},[50,3052,3053,3057],{},[53,3054,3055],{},[56,3056,1385],{},[60,3058,1385],{"encoding":62},[36,3060,3062],{"className":3061,"ariaHidden":67},[66],[36,3063,3065,3068],{"className":3064},[71],[36,3066],{"className":3067,"style":139},[75],[36,3069,1385],{"className":3070,"style":1433},[80,81]," 是固定的观测值，而参数 ",[36,3073,3075,3088],{"className":3074,"translate":40},[39],[36,3076,3078],{"className":3077},[44],[46,3079,3080],{"xmlns":48},[50,3081,3082,3086],{},[53,3083,3084],{},[56,3085,2356],{},[60,3087,2359],{"encoding":62},[36,3089,3091],{"className":3090,"ariaHidden":67},[66],[36,3092,3094,3097],{"className":3093},[71],[36,3095],{"className":3096,"style":2030},[75],[36,3098,2356],{"className":3099,"style":1433},[80,81]," 是变化的未知量。\n我们问：「在不同的参数取值下，这份观测数据出现的可能性有多大？」",[15,3102,3103],{},"根据条件公式：",[36,3105,3107],{"className":3106,"translate":40},[314],[36,3108,3110,3171],{"className":3109,"translate":40},[39],[36,3111,3113],{"className":3112},[44],[46,3114,3115],{"xmlns":48,"display":323},[50,3116,3117,3168],{},[53,3118,3119,3121,3123,3125,3127,3129,3131,3133,3152,3154,3156,3162,3164,3166],{},[56,3120,333],{},[160,3122,163],{"stretchy":162},[56,3124,1385],{},[56,3126,1389],{"mathvariant":1388},[56,3128,2356],{},[160,3130,168],{"stretchy":162},[160,3132,346],{},[2503,3134,3135,3138],{},[160,3136,3137],{},"∏",[53,3139,3140,3147,3150],{},[329,3141,3142,3144],{},[56,3143,2119],{},[56,3145,3146],{},"i",[160,3148,3149],{},"∈",[56,3151,1385],{},[56,3153,333],{},[160,3155,163],{"stretchy":162},[329,3157,3158,3160],{},[56,3159,2119],{},[56,3161,3146],{},[56,3163,1389],{"mathvariant":1388},[56,3165,2356],{},[160,3167,168],{"stretchy":162},[60,3169,3170],{"encoding":62},"P(D|\\theta)\n=\n\\prod_{x_i \\in D}\nP(x_i|\\theta)",[36,3172,3174,3207],{"className":3173,"ariaHidden":67},[66],[36,3175,3177,3180,3183,3186,3189,3192,3195,3198,3201,3204],{"className":3176},[71],[36,3178],{"className":3179,"style":205},[75],[36,3181,333],{"className":3182,"style":377},[80,81],[36,3184,163],{"className":3185},[212],[36,3187,1385],{"className":3188,"style":1433},[80,81],[36,3190,1389],{"className":3191},[80],[36,3193,2356],{"className":3194,"style":1433},[80,81],[36,3196,168],{"className":3197},[219],[36,3199],{"className":3200,"style":224},[223],[36,3202,346],{"className":3203},[228],[36,3205],{"className":3206,"style":224},[223],[36,3208,3210,3214,3317,3320,3323,3326,3367,3370,3373],{"className":3209},[71],[36,3211],{"className":3212,"style":3213},[75],"height:2.4444em;vertical-align:-1.3944em;",[36,3215,3217],{"className":3216},[2614,2626],[36,3218,3220,3308],{"className":3219},[276,384],[36,3221,3223,3305],{"className":3222},[280],[36,3224,3227,3292],{"className":3225,"style":3226},[284],"height:1.05em;",[36,3228,3230,3234],{"style":3229},"top:-1.8557em;margin-left:0em;",[36,3231],{"className":3232,"style":3233},[291],"height:3.05em;",[36,3235,3237],{"className":3236},[296,297,298,299],[36,3238,3240,3286,3289],{"className":3239},[80,299],[36,3241,3243,3246],{"className":3242},[80,299],[36,3244,2119],{"className":3245},[80,81,299],[36,3247,3249],{"className":3248},[272],[36,3250,3252,3277],{"className":3251},[276,384],[36,3253,3255,3274],{"className":3254},[280],[36,3256,3259],{"className":3257,"style":3258},[284],"height:0.3281em;",[36,3260,3262,3266],{"style":3261},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[36,3263],{"className":3264,"style":3265},[291],"height:2.5em;",[36,3267,3271],{"className":3268},[296,3269,3270,299],"reset-size3","size1",[36,3272,3146],{"className":3273},[80,81,299],[36,3275,416],{"className":3276},[415],[36,3278,3280],{"className":3279},[280],[36,3281,3284],{"className":3282,"style":3283},[284],"height:0.143em;",[36,3285],{},[36,3287,3149],{"className":3288},[228,299],[36,3290,1385],{"className":3291,"style":1433},[80,81,299],[36,3293,3295,3298],{"style":3294},"top:-3.05em;",[36,3296],{"className":3297,"style":3233},[291],[36,3299,3300],{},[36,3301,3137],{"className":3302},[2614,3303,3304],"op-symbol","large-op",[36,3306,416],{"className":3307},[415],[36,3309,3311],{"className":3310},[280],[36,3312,3315],{"className":3313,"style":3314},[284],"height:1.3944em;",[36,3316],{},[36,3318],{"className":3319,"style":2232},[223],[36,3321,333],{"className":3322,"style":377},[80,81],[36,3324,163],{"className":3325},[212],[36,3327,3329,3332],{"className":3328},[80],[36,3330,2119],{"className":3331},[80,81],[36,3333,3335],{"className":3334},[272],[36,3336,3338,3359],{"className":3337},[276,384],[36,3339,3341,3356],{"className":3340},[280],[36,3342,3345],{"className":3343,"style":3344},[284],"height:0.3117em;",[36,3346,3347,3350],{"style":2204},[36,3348],{"className":3349,"style":292},[291],[36,3351,3353],{"className":3352},[296,297,298,299],[36,3354,3146],{"className":3355},[80,81,299],[36,3357,416],{"className":3358},[415],[36,3360,3362],{"className":3361},[280],[36,3363,3365],{"className":3364,"style":423},[284],[36,3366],{},[36,3368,1389],{"className":3369},[80],[36,3371,2356],{"className":3372,"style":1433},[80,81],[36,3374,168],{"className":3375},[219],[15,3377,3378,3379,3407],{},"如果数据 ",[36,3380,3382,3395],{"className":3381,"translate":40},[39],[36,3383,3385],{"className":3384},[44],[46,3386,3387],{"xmlns":48},[50,3388,3389,3393],{},[53,3390,3391],{},[56,3392,1385],{},[60,3394,1385],{"encoding":62},[36,3396,3398],{"className":3397,"ariaHidden":67},[66],[36,3399,3401,3404],{"className":3400},[71],[36,3402],{"className":3403,"style":139},[75],[36,3405,1385],{"className":3406,"style":1433},[80,81]," 包含很多样本，很多小数相乘会得到一个趋近于零的结果，在计算机里可能导致数值下溢。",[15,3409,3410],{},"于是我们取对数，把乘法变成加法：",[36,3412,3414],{"className":3413,"translate":40},[314],[36,3415,3417,3500],{"className":3416,"translate":40},[39],[36,3418,3420],{"className":3419},[44],[46,3421,3422],{"xmlns":48,"display":323},[50,3423,3424,3497],{},[53,3425,3426,3429,3431,3433,3435,3437,3439,3441,3444,3446,3448,3450,3452,3454,3456,3458,3460,3477,3479,3481,3483,3485,3491,3493,3495],{},[56,3427,3428],{"mathvariant":1388},"ℓ",[160,3430,163],{"stretchy":162},[56,3432,2356],{},[56,3434,1389],{"mathvariant":1388},[56,3436,1385],{},[160,3438,168],{"stretchy":162},[160,3440,346],{},[56,3442,3443],{},"log",[160,3445,2501],{},[56,3447,333],{},[160,3449,163],{"stretchy":162},[56,3451,1385],{},[56,3453,1389],{"mathvariant":1388},[56,3455,2356],{},[160,3457,168],{"stretchy":162},[160,3459,346],{},[2503,3461,3462,3465],{},[160,3463,3464],{},"∑",[53,3466,3467,3473,3475],{},[329,3468,3469,3471],{},[56,3470,2119],{},[56,3472,3146],{},[160,3474,3149],{},[56,3476,1385],{},[56,3478,3443],{},[160,3480,2501],{},[56,3482,333],{},[160,3484,163],{"stretchy":162},[329,3486,3487,3489],{},[56,3488,2119],{},[56,3490,3146],{},[56,3492,1389],{"mathvariant":1388},[56,3494,2356],{},[160,3496,168],{"stretchy":162},[60,3498,3499],{"encoding":62},"\\ell(\\theta|D)=\n\\log P(D|\\theta)\n=\n\\sum_{x_i \\in D}\n\\log P(x_i|\\theta)",[36,3501,3503,3536,3578],{"className":3502,"ariaHidden":67},[66],[36,3504,3506,3509,3512,3515,3518,3521,3524,3527,3530,3533],{"className":3505},[71],[36,3507],{"className":3508,"style":205},[75],[36,3510,3428],{"className":3511},[80],[36,3513,163],{"className":3514},[212],[36,3516,2356],{"className":3517,"style":1433},[80,81],[36,3519,1389],{"className":3520},[80],[36,3522,1385],{"className":3523,"style":1433},[80,81],[36,3525,168],{"className":3526},[219],[36,3528],{"className":3529,"style":224},[223],[36,3531,346],{"className":3532},[228],[36,3534],{"className":3535,"style":224},[223],[36,3537,3539,3542,3548,3551,3554,3557,3560,3563,3566,3569,3572,3575],{"className":3538},[71],[36,3540],{"className":3541,"style":205},[75],[36,3543,3545,3546],{"className":3544},[2614],"lo",[36,3547,2619],{"style":2618},[36,3549],{"className":3550,"style":2232},[223],[36,3552,333],{"className":3553,"style":377},[80,81],[36,3555,163],{"className":3556},[212],[36,3558,1385],{"className":3559,"style":1433},[80,81],[36,3561,1389],{"className":3562},[80],[36,3564,2356],{"className":3565,"style":1433},[80,81],[36,3567,168],{"className":3568},[219],[36,3570],{"className":3571,"style":224},[223],[36,3573,346],{"className":3574},[228],[36,3576],{"className":3577,"style":224},[223],[36,3579,3581,3584,3674,3677,3682,3685,3688,3691,3731,3734,3737],{"className":3580},[71],[36,3582],{"className":3583,"style":3213},[75],[36,3585,3587],{"className":3586},[2614,2626],[36,3588,3590,3666],{"className":3589},[276,384],[36,3591,3593,3663],{"className":3592},[280],[36,3594,3596,3653],{"className":3595,"style":3226},[284],[36,3597,3598,3601],{"style":3229},[36,3599],{"className":3600,"style":3233},[291],[36,3602,3604],{"className":3603},[296,297,298,299],[36,3605,3607,3647,3650],{"className":3606},[80,299],[36,3608,3610,3613],{"className":3609},[80,299],[36,3611,2119],{"className":3612},[80,81,299],[36,3614,3616],{"className":3615},[272],[36,3617,3619,3639],{"className":3618},[276,384],[36,3620,3622,3636],{"className":3621},[280],[36,3623,3625],{"className":3624,"style":3258},[284],[36,3626,3627,3630],{"style":3261},[36,3628],{"className":3629,"style":3265},[291],[36,3631,3633],{"className":3632},[296,3269,3270,299],[36,3634,3146],{"className":3635},[80,81,299],[36,3637,416],{"className":3638},[415],[36,3640,3642],{"className":3641},[280],[36,3643,3645],{"className":3644,"style":3283},[284],[36,3646],{},[36,3648,3149],{"className":3649},[228,299],[36,3651,1385],{"className":3652,"style":1433},[80,81,299],[36,3654,3655,3658],{"style":3294},[36,3656],{"className":3657,"style":3233},[291],[36,3659,3660],{},[36,3661,3464],{"className":3662},[2614,3303,3304],[36,3664,416],{"className":3665},[415],[36,3667,3669],{"className":3668},[280],[36,3670,3672],{"className":3671,"style":3314},[284],[36,3673],{},[36,3675],{"className":3676,"style":2232},[223],[36,3678,3545,3680],{"className":3679},[2614],[36,3681,2619],{"style":2618},[36,3683],{"className":3684,"style":2232},[223],[36,3686,333],{"className":3687,"style":377},[80,81],[36,3689,163],{"className":3690},[212],[36,3692,3694,3697],{"className":3693},[80],[36,3695,2119],{"className":3696},[80,81],[36,3698,3700],{"className":3699},[272],[36,3701,3703,3723],{"className":3702},[276,384],[36,3704,3706,3720],{"className":3705},[280],[36,3707,3709],{"className":3708,"style":3344},[284],[36,3710,3711,3714],{"style":2204},[36,3712],{"className":3713,"style":292},[291],[36,3715,3717],{"className":3716},[296,297,298,299],[36,3718,3146],{"className":3719},[80,81,299],[36,3721,416],{"className":3722},[415],[36,3724,3726],{"className":3725},[280],[36,3727,3729],{"className":3728,"style":423},[284],[36,3730],{},[36,3732,1389],{"className":3733},[80],[36,3735,2356],{"className":3736,"style":1433},[80,81],[36,3738,168],{"className":3739},[219],[15,3741,3742],{},"取对数后，数值保持在可接受的范围内，更利于优化。",[15,3744,3745],{},"因此，几乎所有实际训练都在优化：",[36,3747,3749],{"className":3748,"translate":40},[314],[36,3750,3752,3786],{"className":3751,"translate":40},[39],[36,3753,3755],{"className":3754},[44],[46,3756,3757],{"xmlns":48,"display":323},[50,3758,3759,3783],{},[53,3760,3761,3771,3773,3775,3777,3779,3781],{},[2503,3762,3763,3769],{},[53,3764,3765,3767],{},[56,3766,2509],{},[160,3768,2501],{},[56,3770,2356],{},[56,3772,3428],{"mathvariant":1388},[160,3774,163],{"stretchy":162},[56,3776,2356],{},[56,3778,1389],{"mathvariant":1388},[56,3780,1385],{},[160,3782,168],{"stretchy":162},[60,3784,3785],{"encoding":62},"\\max_\\theta\n\\ell(\\theta|D)",[36,3787,3789],{"className":3788,"ariaHidden":67},[66],[36,3790,3792,3795,3839,3842,3845,3848,3851,3854,3857],{"className":3791},[71],[36,3793],{"className":3794,"style":2610},[75],[36,3796,3798],{"className":3797},[2614,2626],[36,3799,3801,3831],{"className":3800},[276,384],[36,3802,3804,3828],{"className":3803},[280],[36,3805,3807,3818],{"className":3806,"style":76},[284],[36,3808,3809,3812],{"style":2638},[36,3810],{"className":3811,"style":2037},[291],[36,3813,3815],{"className":3814},[296,297,298,299],[36,3816,2356],{"className":3817,"style":1433},[80,81,299],[36,3819,3820,3823],{"style":2033},[36,3821],{"className":3822,"style":2037},[291],[36,3824,3825],{},[36,3826,2509],{"className":3827},[2614],[36,3829,416],{"className":3830},[415],[36,3832,3834],{"className":3833},[280],[36,3835,3837],{"className":3836,"style":2667},[284],[36,3838],{},[36,3840],{"className":3841,"style":2232},[223],[36,3843,3428],{"className":3844},[80],[36,3846,163],{"className":3847},[212],[36,3849,2356],{"className":3850,"style":1433},[80,81],[36,3852,1389],{"className":3853},[80],[36,3855,1385],{"className":3856,"style":1433},[80,81],[36,3858,168],{"className":3859},[219],[15,3861,3862],{},"这种形式叫作对数似然。",[15,3864,3865,3866,1360],{},"在似然函数中，我们要找到使似然最大的参数 ",[36,3867,3869,3892],{"className":3868,"translate":40},[39],[36,3870,3872],{"className":3871},[44],[46,3873,3874],{"xmlns":48},[50,3875,3876,3889],{},[53,3877,3878],{},[329,3879,3880,3886],{},[1994,3881,3882,3884],{"accent":67},[56,3883,2356],{},[160,3885,2000],{},[335,3887,3888],{},"MLE",[60,3890,3891],{"encoding":62},"\\hat{\\theta}_\\text{MLE}",[36,3893,3895],{"className":3894,"ariaHidden":67},[66],[36,3896,3898,3902],{"className":3897},[71],[36,3899],{"className":3900,"style":3901},[75],"height:1.1079em;vertical-align:-0.15em;",[36,3903,3905,3936],{"className":3904},[80],[36,3906,3908],{"className":3907},[80,2020],[36,3909,3911],{"className":3910},[276],[36,3912,3914],{"className":3913},[280],[36,3915,3917,3925],{"className":3916,"style":2562},[284],[36,3918,3919,3922],{"style":2033},[36,3920],{"className":3921,"style":2037},[291],[36,3923,2356],{"className":3924,"style":1433},[80,81],[36,3926,3927,3930],{"style":2585},[36,3928],{"className":3929,"style":2037},[291],[36,3931,3933],{"className":3932,"style":2050},[2049],[36,3934,2000],{"className":3935},[80],[36,3937,3939],{"className":3938},[272],[36,3940,3942,3966],{"className":3941},[276,384],[36,3943,3945,3963],{"className":3944},[280],[36,3946,3948],{"className":3947,"style":391},[284],[36,3949,3951,3954],{"style":3950},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[36,3952],{"className":3953,"style":292},[291],[36,3955,3957],{"className":3956},[296,297,298,299],[36,3958,3960],{"className":3959},[80,407,299],[36,3961,3888],{"className":3962},[80,299],[36,3964,416],{"className":3965},[415],[36,3967,3969],{"className":3968},[280],[36,3970,3972],{"className":3971,"style":423},[284],[36,3973],{},[11,3975,3976],{"id":3976},"求解参数",[15,3978,3979,3980,4034,4035,4152],{},"如果 ",[36,3981,3983,4007],{"className":3982,"translate":40},[39],[36,3984,3986],{"className":3985},[44],[46,3987,3988],{"xmlns":48},[50,3989,3990,4004],{},[53,3991,3992,3994,3996,3998,4000,4002],{},[56,3993,3428],{"mathvariant":1388},[160,3995,163],{"stretchy":162},[56,3997,2356],{},[56,3999,1389],{"mathvariant":1388},[56,4001,1385],{},[160,4003,168],{"stretchy":162},[60,4005,4006],{"encoding":62},"\\ell(\\theta|D)",[36,4008,4010],{"className":4009,"ariaHidden":67},[66],[36,4011,4013,4016,4019,4022,4025,4028,4031],{"className":4012},[71],[36,4014],{"className":4015,"style":205},[75],[36,4017,3428],{"className":4018},[80],[36,4020,163],{"className":4021},[212],[36,4023,2356],{"className":4024,"style":1433},[80,81],[36,4026,1389],{"className":4027},[80],[36,4029,1385],{"className":4030,"style":1433},[80,81],[36,4032,168],{"className":4033},[219]," 可微，",[36,4036,4038,4068],{"className":4037,"translate":40},[39],[36,4039,4041],{"className":4040},[44],[46,4042,4043],{"xmlns":48},[50,4044,4045,4065],{},[53,4046,4047],{},[329,4048,4049,4055],{},[1994,4050,4051,4053],{"accent":67},[56,4052,2356],{},[160,4054,2000],{},[53,4056,4057,4060,4062],{},[56,4058,4059],{"mathvariant":1388},"M",[56,4061,2392],{"mathvariant":1388},[56,4063,4064],{"mathvariant":1388},"E",[60,4066,4067],{"encoding":62},"\\hat{\\theta}_{\\rm MLE}",[36,4069,4071],{"className":4070,"ariaHidden":67},[66],[36,4072,4074,4077],{"className":4073},[71],[36,4075],{"className":4076,"style":3901},[75],[36,4078,4080,4111],{"className":4079},[80],[36,4081,4083],{"className":4082},[80,2020],[36,4084,4086],{"className":4085},[276],[36,4087,4089],{"className":4088},[280],[36,4090,4092,4100],{"className":4091,"style":2562},[284],[36,4093,4094,4097],{"style":2033},[36,4095],{"className":4096,"style":2037},[291],[36,4098,2356],{"className":4099,"style":1433},[80,81],[36,4101,4102,4105],{"style":2585},[36,4103],{"className":4104,"style":2037},[291],[36,4106,4108],{"className":4107,"style":2050},[2049],[36,4109,2000],{"className":4110},[80],[36,4112,4114],{"className":4113},[272],[36,4115,4117,4144],{"className":4116},[276,384],[36,4118,4120,4141],{"className":4119},[280],[36,4121,4123],{"className":4122,"style":391},[284],[36,4124,4125,4128],{"style":3950},[36,4126],{"className":4127,"style":292},[291],[36,4129,4131],{"className":4130},[296,297,298,299],[36,4132,4134],{"className":4133},[80,299],[36,4135,4137],{"className":4136},[80,299],[36,4138,3888],{"className":4139},[80,4140,299],"mathrm",[36,4142,416],{"className":4143},[415],[36,4145,4147],{"className":4146},[280],[36,4148,4150],{"className":4149,"style":423},[284],[36,4151],{}," 通常满足：",[36,4154,4156],{"className":4155,"translate":40},[314],[36,4157,4159,4202],{"className":4158,"translate":40},[39],[36,4160,4162],{"className":4161},[44],[46,4163,4164],{"xmlns":48,"display":323},[50,4165,4166,4199],{},[53,4167,4168,4194,4196],{},[4169,4170,4171,4188],"mfrac",{},[53,4172,4173,4176,4178,4180,4182,4184,4186],{},[56,4174,4175],{"mathvariant":1388},"∂",[56,4177,3428],{"mathvariant":1388},[160,4179,163],{"stretchy":162},[56,4181,2356],{},[56,4183,1389],{"mathvariant":1388},[56,4185,1385],{},[160,4187,168],{"stretchy":162},[53,4189,4190,4192],{},[56,4191,4175],{"mathvariant":1388},[56,4193,2356],{},[160,4195,346],{},[173,4197,4198],{},"0",[60,4200,4201],{"encoding":62},"\\frac{\\partial \\ell(\\theta|D)}{\\partial \\theta} = 0",[36,4203,4205,4312],{"className":4204,"ariaHidden":67},[66],[36,4206,4208,4212,4303,4306,4309],{"className":4207},[71],[36,4209],{"className":4210,"style":4211},[75],"height:2.113em;vertical-align:-0.686em;",[36,4213,4215,4219,4300],{"className":4214},[80],[36,4216],{"className":4217},[212,4218],"nulldelimiter",[36,4220,4222],{"className":4221},[4169],[36,4223,4225,4291],{"className":4224},[276,384],[36,4226,4228,4288],{"className":4227},[280],[36,4229,4232,4247,4258],{"className":4230,"style":4231},[284],"height:1.427em;",[36,4233,4235,4238],{"style":4234},"top:-2.314em;",[36,4236],{"className":4237,"style":2037},[291],[36,4239,4241,4244],{"className":4240},[80],[36,4242,4175],{"className":4243,"style":920},[80],[36,4245,2356],{"className":4246,"style":1433},[80,81],[36,4248,4250,4253],{"style":4249},"top:-3.23em;",[36,4251],{"className":4252,"style":2037},[291],[36,4254],{"className":4255,"style":4257},[4256],"frac-line","border-bottom-width:0.04em;",[36,4259,4261,4264],{"style":4260},"top:-3.677em;",[36,4262],{"className":4263,"style":2037},[291],[36,4265,4267,4270,4273,4276,4279,4282,4285],{"className":4266},[80],[36,4268,4175],{"className":4269,"style":920},[80],[36,4271,3428],{"className":4272},[80],[36,4274,163],{"className":4275},[212],[36,4277,2356],{"className":4278,"style":1433},[80,81],[36,4280,1389],{"className":4281},[80],[36,4283,1385],{"className":4284,"style":1433},[80,81],[36,4286,168],{"className":4287},[219],[36,4289,416],{"className":4290},[415],[36,4292,4294],{"className":4293},[280],[36,4295,4298],{"className":4296,"style":4297},[284],"height:0.686em;",[36,4299],{},[36,4301],{"className":4302},[219,4218],[36,4304],{"className":4305,"style":224},[223],[36,4307,346],{"className":4308},[228],[36,4310],{"className":4311,"style":224},[223],[36,4313,4315,4318],{"className":4314},[71],[36,4316],{"className":4317,"style":1356},[75],[36,4319,4198],{"className":4320},[80],[15,4322,4323],{},"二阶条件保证取到最大值：",[36,4325,4327],{"className":4326,"translate":40},[314],[36,4328,4330,4379],{"className":4329,"translate":40},[39],[36,4331,4333],{"className":4332},[44],[46,4334,4335],{"xmlns":48,"display":323},[50,4336,4337,4376],{},[53,4338,4339,4371,4374],{},[4169,4340,4341,4361],{},[53,4342,4343,4349,4351,4353,4355,4357,4359],{},[182,4344,4345,4347],{},[56,4346,4175],{"mathvariant":1388},[173,4348,1670],{},[56,4350,3428],{"mathvariant":1388},[160,4352,163],{"stretchy":162},[56,4354,2356],{},[56,4356,1389],{"mathvariant":1388},[56,4358,1385],{},[160,4360,168],{"stretchy":162},[53,4362,4363,4365],{},[56,4364,4175],{"mathvariant":1388},[182,4366,4367,4369],{},[56,4368,2356],{},[173,4370,1670],{},[160,4372,4373],{},"\u003C",[173,4375,4198],{},[60,4377,4378],{"encoding":62},"\\frac{\\partial^2 \\ell(\\theta|D)}{\\partial \\theta^2} \u003C 0",[36,4380,4382,4537],{"className":4381,"ariaHidden":67},[66],[36,4383,4385,4389,4528,4531,4534],{"className":4384},[71],[36,4386],{"className":4387,"style":4388},[75],"height:2.1771em;vertical-align:-0.686em;",[36,4390,4392,4395,4525],{"className":4391},[80],[36,4393],{"className":4394},[212,4218],[36,4396,4398],{"className":4397},[4169],[36,4399,4401,4517],{"className":4400},[276,384],[36,4402,4404,4514],{"className":4403},[280],[36,4405,4408,4450,4458],{"className":4406,"style":4407},[284],"height:1.4911em;",[36,4409,4410,4413],{"style":4234},[36,4411],{"className":4412,"style":2037},[291],[36,4414,4416,4419],{"className":4415},[80],[36,4417,4175],{"className":4418,"style":920},[80],[36,4420,4422,4425],{"className":4421},[80],[36,4423,2356],{"className":4424,"style":1433},[80,81],[36,4426,4428],{"className":4427},[272],[36,4429,4431],{"className":4430},[276],[36,4432,4434],{"className":4433},[280],[36,4435,4438],{"className":4436,"style":4437},[284],"height:0.7401em;",[36,4439,4441,4444],{"style":4440},"top:-2.989em;margin-right:0.05em;",[36,4442],{"className":4443,"style":292},[291],[36,4445,4447],{"className":4446},[296,297,298,299],[36,4448,1670],{"className":4449},[80,299],[36,4451,4452,4455],{"style":4249},[36,4453],{"className":4454,"style":2037},[291],[36,4456],{"className":4457,"style":4257},[4256],[36,4459,4460,4463],{"style":4260},[36,4461],{"className":4462,"style":2037},[291],[36,4464,4466,4496,4499,4502,4505,4508,4511],{"className":4465},[80],[36,4467,4469,4472],{"className":4468},[80],[36,4470,4175],{"className":4471,"style":920},[80],[36,4473,4475],{"className":4474},[272],[36,4476,4478],{"className":4477},[276],[36,4479,4481],{"className":4480},[280],[36,4482,4485],{"className":4483,"style":4484},[284],"height:0.8141em;",[36,4486,4487,4490],{"style":287},[36,4488],{"className":4489,"style":292},[291],[36,4491,4493],{"className":4492},[296,297,298,299],[36,4494,1670],{"className":4495},[80,299],[36,4497,3428],{"className":4498},[80],[36,4500,163],{"className":4501},[212],[36,4503,2356],{"className":4504,"style":1433},[80,81],[36,4506,1389],{"className":4507},[80],[36,4509,1385],{"className":4510,"style":1433},[80,81],[36,4512,168],{"className":4513},[219],[36,4515,416],{"className":4516},[415],[36,4518,4520],{"className":4519},[280],[36,4521,4523],{"className":4522,"style":4297},[284],[36,4524],{},[36,4526],{"className":4527},[219,4218],[36,4529],{"className":4530,"style":224},[223],[36,4532,4373],{"className":4533},[228],[36,4535],{"className":4536,"style":224},[223],[36,4538,4540,4543],{"className":4539},[71],[36,4541],{"className":4542,"style":1356},[75],[36,4544,4198],{"className":4545},[80],[15,4547,4548],{},"以最简单的伯努利分布为例：",[1130,4550,4551,4670],{},[1133,4552,4553,4554],{},"数据 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\\{0,1\\}",[36,4588,4590,4646],{"className":4589,"ariaHidden":67},[66],[36,4591,4593,4597,4637,4640,4643],{"className":4592},[71],[36,4594],{"className":4595,"style":4596},[75],"height:0.6891em;vertical-align:-0.15em;",[36,4598,4600,4603],{"className":4599},[80],[36,4601,2119],{"className":4602},[80,81],[36,4604,4606],{"className":4605},[272],[36,4607,4609,4629],{"className":4608},[276,384],[36,4610,4612,4626],{"className":4611},[280],[36,4613,4615],{"className":4614,"style":3344},[284],[36,4616,4617,4620],{"style":2204},[36,4618],{"className":4619,"style":292},[291],[36,4621,4623],{"className":4622},[296,297,298,299],[36,4624,3146],{"className":4625},[80,81,299],[36,4627,416],{"className":4628},[415],[36,4630,4632],{"className":4631},[280],[36,4633,4635],{"className":4634,"style":423},[284],[36,4636],{},[36,4638],{"className":4639,"style":224},[223],[36,4641,3149],{"className":4642},[228],[36,4644],{"className":4645,"style":224},[223],[36,4647,4649,4652,4655,4658,4661,4664,4667],{"className":4648},[71],[36,4650],{"className":4651,"style":205},[75],[36,4653,2114],{"className":4654},[212],[36,4656,4198],{"className":4657},[80],[36,4659,2124],{"className":4660},[2228],[36,4662],{"className":4663,"style":2232},[223],[36,4665,175],{"className":4666},[80],[36,4668,2149],{"className":4669},[219],[1133,4671,4672,4673],{},"模型：",[36,4674,4676,4727],{"className":4675,"translate":40},[39],[36,4677,4679],{"className":4678},[44],[46,4680,4681],{"xmlns":48},[50,4682,4683,4724],{},[53,4684,4685,4687,4689,4692,4694,4696,4698,4700,4702,4704,4706,4708,4710,4712,4714,4716,4718,4720,4722],{},[56,4686,333],{},[160,4688,163],{"stretchy":162},[56,4690,4691],{},"X",[160,4693,346],{},[173,4695,175],{},[160,4697,168],{"stretchy":162},[160,4699,346],{},[56,4701,15],{},[160,4703,2124],{"separator":67},[56,4705,333],{},[160,4707,163],{"stretchy":162},[56,4709,4691],{},[160,4711,346],{},[173,4713,4198],{},[160,4715,168],{"stretchy":162},[160,4717,346],{},[173,4719,175],{},[160,4721,178],{},[56,4723,15],{},[60,4725,4726],{"encoding":62},"P(X=1)=p, P(X=0)=1-p",[36,4728,4730,4755,4776,4809,4830,4848],{"className":4729,"ariaHidden":67},[66],[36,4731,4733,4736,4739,4742,4746,4749,4752],{"className":4732},[71],[36,4734],{"className":4735,"style":205},[75],[36,4737,333],{"className":4738,"style":377},[80,81],[36,4740,163],{"className":4741},[212],[36,4743,4691],{"className":4744,"style":4745},[80,81],"margin-right:0.0785em;",[36,4747],{"className":4748,"style":224},[223],[36,4750,346],{"className":4751},[228],[36,4753],{"className":4754,"style":224},[223],[36,4756,4758,4761,4764,4767,4770,4773],{"className":4757},[71],[36,4759],{"className":4760,"style":205},[75],[36,4762,175],{"className":4763},[80],[36,4765,168],{"className":4766},[219],[36,4768],{"className":4769,"style":224},[223],[36,4771,346],{"className":4772},[228],[36,4774],{"className":4775,"style":224},[223],[36,4777,4779,4782,4785,4788,4791,4794,4797,4800,4803,4806],{"className":4778},[71],[36,4780],{"className":4781,"style":205},[75],[36,4783,15],{"className":4784},[80,81],[36,4786,2124],{"className":4787},[2228],[36,4789],{"className":4790,"style":2232},[223],[36,4792,333],{"className":4793,"style":377},[80,81],[36,4795,163],{"className":4796},[212],[36,4798,4691],{"className":4799,"style":4745},[80,81],[36,4801],{"className":4802,"style":224},[223],[36,4804,346],{"className":4805},[228],[36,4807],{"className":4808,"style":224},[223],[36,4810,4812,4815,4818,4821,4824,4827],{"className":4811},[71],[36,4813],{"className":4814,"style":205},[75],[36,4816,4198],{"className":4817},[80],[36,4819,168],{"className":4820},[219],[36,4822],{"className":4823,"style":224},[223],[36,4825,346],{"className":4826},[228],[36,4828],{"className":4829,"style":224},[223],[36,4831,4833,4836,4839,4842,4845],{"className":4832},[71],[36,4834],{"className":4835,"style":238},[75],[36,4837,175],{"className":4838},[80],[36,4840],{"className":4841,"style":245},[223],[36,4843,178],{"className":4844},[249],[36,4846],{"className":4847,"style":245},[223],[36,4849,4851,4854],{"className":4850},[71],[36,4852],{"className":4853,"style":108},[75],[36,4855,15],{"className":4856},[80,81],[15,4858,4859],{},"我们有似然函数：",[36,4861,4863],{"className":4862,"translate":40},[314],[36,4864,4866,4941],{"className":4865,"translate":40},[39],[36,4867,4869],{"className":4868},[44],[46,4870,4871],{"xmlns":48,"display":323},[50,4872,4873,4938],{},[53,4874,4875,4877,4879,4881,4883,4885,4887,4889,4904,4914,4916,4918,4920,4922],{},[56,4876,2392],{},[160,4878,163],{"stretchy":162},[56,4880,15],{},[56,4882,1389],{"mathvariant":1388},[56,4884,1385],{},[160,4886,168],{"stretchy":162},[160,4888,346],{},[4890,4891,4892,4894,4902],"munderover",{},[160,4893,3137],{},[53,4895,4896,4898,4900],{},[56,4897,3146],{},[160,4899,346],{},[173,4901,175],{},[56,4903,58],{},[182,4905,4906,4908],{},[56,4907,15],{},[329,4909,4910,4912],{},[56,4911,2119],{},[56,4913,3146],{},[160,4915,163],{"stretchy":162},[173,4917,175],{},[160,4919,178],{},[56,4921,15],{},[182,4923,4924,4926],{},[160,4925,168],{"stretchy":162},[53,4927,4928,4930,4932],{},[173,4929,175],{},[160,4931,178],{},[329,4933,4934,4936],{},[56,4935,2119],{},[56,4937,3146],{},[60,4939,4940],{"encoding":62},"L(p|D) = \\prod_{i=1}^n p^{x_i} (1-p)^{1-x_i}",[36,4942,4944,4977,5139],{"className":4943,"ariaHidden":67},[66],[36,4945,4947,4950,4953,4956,4959,4962,4965,4968,4971,4974],{"className":4946},[71],[36,4948],{"className":4949,"style":205},[75],[36,4951,2392],{"className":4952},[80,81],[36,4954,163],{"className":4955},[212],[36,4957,15],{"className":4958},[80,81],[36,4960,1389],{"className":4961},[80],[36,4963,1385],{"className":4964,"style":1433},[80,81],[36,4966,168],{"className":4967},[219],[36,4969],{"className":4970,"style":224},[223],[36,4972,346],{"className":4973},[228],[36,4975],{"className":4976,"style":224},[223],[36,4978,4980,4984,5052,5055,5124,5127,5130,5133,5136],{"className":4979},[71],[36,4981],{"className":4982,"style":4983},[75],"height:2.9291em;vertical-align:-1.2777em;",[36,4985,4987],{"className":4986},[2614,2626],[36,4988,4990,5043],{"className":4989},[276,384],[36,4991,4993,5040],{"className":4992},[280],[36,4994,4997,5018,5028],{"className":4995,"style":4996},[284],"height:1.6514em;",[36,4998,5000,5003],{"style":4999},"top:-1.8723em;margin-left:0em;",[36,5001],{"className":5002,"style":3233},[291],[36,5004,5006],{"className":5005},[296,297,298,299],[36,5007,5009,5012,5015],{"className":5008},[80,299],[36,5010,3146],{"className":5011},[80,81,299],[36,5013,346],{"className":5014},[228,299],[36,5016,175],{"className":5017},[80,299],[36,5019,5020,5023],{"style":3294},[36,5021],{"className":5022,"style":3233},[291],[36,5024,5025],{},[36,5026,3137],{"className":5027},[2614,3303,3304],[36,5029,5031,5034],{"style":5030},"top:-4.3em;margin-left:0em;",[36,5032],{"className":5033,"style":3233},[291],[36,5035,5037],{"className":5036},[296,297,298,299],[36,5038,58],{"className":5039},[80,81,299],[36,5041,416],{"className":5042},[415],[36,5044,5046],{"className":5045},[280],[36,5047,5050],{"className":5048,"style":5049},[284],"height:1.2777em;",[36,5051],{},[36,5053],{"className":5054,"style":2232},[223],[36,5056,5058,5061],{"className":5057},[80],[36,5059,15],{"className":5060},[80,81],[36,5062,5064],{"className":5063},[272],[36,5065,5067],{"className":5066},[276],[36,5068,5070],{"className":5069},[280],[36,5071,5073],{"className":5072,"style":477},[284],[36,5074,5075,5078],{"style":480},[36,5076],{"className":5077,"style":292},[291],[36,5079,5081],{"className":5080},[296,297,298,299],[36,5082,5084],{"className":5083},[80,299],[36,5085,5087,5090],{"className":5086},[80,299],[36,5088,2119],{"className":5089},[80,81,299],[36,5091,5093],{"className":5092},[272],[36,5094,5096,5116],{"className":5095},[276,384],[36,5097,5099,5113],{"className":5098},[280],[36,5100,5102],{"className":5101,"style":3258},[284],[36,5103,5104,5107],{"style":3261},[36,5105],{"className":5106,"style":3265},[291],[36,5108,5110],{"className":5109},[296,3269,3270,299],[36,5111,3146],{"className":5112},[80,81,299],[36,5114,416],{"className":5115},[415],[36,5117,5119],{"className":5118},[280],[36,5120,5122],{"className":5121,"style":3283},[284],[36,5123],{},[36,5125,163],{"className":5126},[212],[36,5128,175],{"className":5129},[80],[36,5131],{"className":5132,"style":245},[223],[36,5134,178],{"className":5135},[249],[36,5137],{"className":5138,"style":245},[223],[36,5140,5142,5146,5149],{"className":5141},[71],[36,5143],{"className":5144,"style":5145},[75],"height:1.1141em;vertical-align:-0.25em;",[36,5147,15],{"className":5148},[80,81],[36,5150,5152,5155],{"className":5151},[219],[36,5153,168],{"className":5154},[219],[36,5156,5158],{"className":5157},[272],[36,5159,5161],{"className":5160},[276],[36,5162,5164],{"className":5163},[280],[36,5165,5167],{"className":5166,"style":1476},[284],[36,5168,5169,5172],{"style":480},[36,5170],{"className":5171,"style":292},[291],[36,5173,5175],{"className":5174},[296,297,298,299],[36,5176,5178,5181,5184],{"className":5177},[80,299],[36,5179,175],{"className":5180},[80,299],[36,5182,178],{"className":5183},[249,299],[36,5185,5187,5190],{"className":5186},[80,299],[36,5188,2119],{"className":5189},[80,81,299],[36,5191,5193],{"className":5192},[272],[36,5194,5196,5216],{"className":5195},[276,384],[36,5197,5199,5213],{"className":5198},[280],[36,5200,5202],{"className":5201,"style":3258},[284],[36,5203,5204,5207],{"style":3261},[36,5205],{"className":5206,"style":3265},[291],[36,5208,5210],{"className":5209},[296,3269,3270,299],[36,5211,3146],{"className":5212},[80,81,299],[36,5214,416],{"className":5215},[415],[36,5217,5219],{"className":5218},[280],[36,5220,5222],{"className":5221,"style":3283},[284],[36,5223],{},[15,5225,5226],{},"转化为对数似然：",[36,5228,5230],{"className":5229,"translate":40},[314],[36,5231,5233,5316],{"className":5232,"translate":40},[39],[36,5234,5236],{"className":5235},[44],[46,5237,5238],{"xmlns":48,"display":323},[50,5239,5240,5313],{},[53,5241,5242,5244,5246,5248,5250,5252,5254,5256,5270,5276,5278,5280,5282,5285,5287,5289,5291,5297,5299,5301,5303,5305,5307,5309,5311],{},[56,5243,3428],{"mathvariant":1388},[160,5245,163],{"stretchy":162},[56,5247,15],{},[56,5249,1389],{"mathvariant":1388},[56,5251,1385],{},[160,5253,168],{"stretchy":162},[160,5255,346],{},[4890,5257,5258,5260,5268],{},[160,5259,3464],{},[53,5261,5262,5264,5266],{},[56,5263,3146],{},[160,5265,346],{},[173,5267,175],{},[56,5269,58],{},[329,5271,5272,5274],{},[56,5273,2119],{},[56,5275,3146],{},[56,5277,3443],{},[160,5279,2501],{},[56,5281,15],{},[160,5283,5284],{},"+",[160,5286,163],{"stretchy":162},[173,5288,175],{},[160,5290,178],{},[329,5292,5293,5295],{},[56,5294,2119],{},[56,5296,3146],{},[160,5298,168],{"stretchy":162},[56,5300,3443],{},[160,5302,2501],{},[160,5304,163],{"stretchy":162},[173,5306,175],{},[160,5308,178],{},[56,5310,15],{},[160,5312,168],{"stretchy":162},[60,5314,5315],{"encoding":62},"\\ell(p|D) = \\sum_{i=1}^n x_i \\log p + (1-x_i) \\log (1-p)",[36,5317,5319,5352,5488,5509,5581],{"className":5318,"ariaHidden":67},[66],[36,5320,5322,5325,5328,5331,5334,5337,5340,5343,5346,5349],{"className":5321},[71],[36,5323],{"className":5324,"style":205},[75],[36,5326,3428],{"className":5327},[80],[36,5329,163],{"className":5330},[212],[36,5332,15],{"className":5333},[80,81],[36,5335,1389],{"className":5336},[80],[36,5338,1385],{"className":5339,"style":1433},[80,81],[36,5341,168],{"className":5342},[219],[36,5344],{"className":5345,"style":224},[223],[36,5347,346],{"className":5348},[228],[36,5350],{"className":5351,"style":224},[223],[36,5353,5355,5358,5422,5425,5465,5468,5473,5476,5479,5482,5485],{"className":5354},[71],[36,5356],{"className":5357,"style":4983},[75],[36,5359,5361],{"className":5360},[2614,2626],[36,5362,5364,5414],{"className":5363},[276,384],[36,5365,5367,5411],{"className":5366},[280],[36,5368,5370,5390,5400],{"className":5369,"style":4996},[284],[36,5371,5372,5375],{"style":4999},[36,5373],{"className":5374,"style":3233},[291],[36,5376,5378],{"className":5377},[296,297,298,299],[36,5379,5381,5384,5387],{"className":5380},[80,299],[36,5382,3146],{"className":5383},[80,81,299],[36,5385,346],{"className":5386},[228,299],[36,5388,175],{"className":5389},[80,299],[36,5391,5392,5395],{"style":3294},[36,5393],{"className":5394,"style":3233},[291],[36,5396,5397],{},[36,5398,3464],{"className":5399},[2614,3303,3304],[36,5401,5402,5405],{"style":5030},[36,5403],{"className":5404,"style":3233},[291],[36,5406,5408],{"className":5407},[296,297,298,299],[36,5409,58],{"className":5410},[80,81,299],[36,5412,416],{"className":5413},[415],[36,5415,5417],{"className":5416},[280],[36,5418,5420],{"className":5419,"style":5049},[284],[36,5421],{},[36,5423],{"className":5424,"style":2232},[223],[36,5426,5428,5431],{"className":5427},[80],[36,5429,2119],{"className":5430},[80,81],[36,5432,5434],{"className":5433},[272],[36,5435,5437,5457],{"className":5436},[276,384],[36,5438,5440,5454],{"className":5439},[280],[36,5441,5443],{"className":5442,"style":3344},[284],[36,5444,5445,5448],{"style":2204},[36,5446],{"className":5447,"style":292},[291],[36,5449,5451],{"className":5450},[296,297,298,299],[36,5452,3146],{"className":5453},[80,81,299],[36,5455,416],{"className":5456},[415],[36,5458,5460],{"className":5459},[280],[36,5461,5463],{"className":5462,"style":423},[284],[36,5464],{},[36,5466],{"className":5467,"style":2232},[223],[36,5469,3545,5471],{"className":5470},[2614],[36,5472,2619],{"style":2618},[36,5474],{"className":5475,"style":2232},[223],[36,5477,15],{"className":5478},[80,81],[36,5480],{"className":5481,"style":245},[223],[36,5483,5284],{"className":5484},[249],[36,5486],{"className":5487,"style":245},[223],[36,5489,5491,5494,5497,5500,5503,5506],{"className":5490},[71],[36,5492],{"className":5493,"style":205},[75],[36,5495,163],{"className":5496},[212],[36,5498,175],{"className":5499},[80],[36,5501],{"className":5502,"style":245},[223],[36,5504,178],{"className":5505},[249],[36,5507],{"className":5508,"style":245},[223],[36,5510,5512,5515,5555,5558,5561,5566,5569,5572,5575,5578],{"className":5511},[71],[36,5513],{"className":5514,"style":205},[75],[36,5516,5518,5521],{"className":5517},[80],[36,5519,2119],{"className":5520},[80,81],[36,5522,5524],{"className":5523},[272],[36,5525,5527,5547],{"className":5526},[276,384],[36,5528,5530,5544],{"className":5529},[280],[36,5531,5533],{"className":5532,"style":3344},[284],[36,5534,5535,5538],{"style":2204},[36,5536],{"className":5537,"style":292},[291],[36,5539,5541],{"className":5540},[296,297,298,299],[36,5542,3146],{"className":5543},[80,81,299],[36,5545,416],{"className":5546},[415],[36,5548,5550],{"className":5549},[280],[36,5551,5553],{"className":5552,"style":423},[284],[36,5554],{},[36,5556,168],{"className":5557},[219],[36,5559],{"className":5560,"style":2232},[223],[36,5562,3545,5564],{"className":5563},[2614],[36,5565,2619],{"style":2618},[36,5567,163],{"className":5568},[212],[36,5570,175],{"className":5571},[80],[36,5573],{"className":5574,"style":245},[223],[36,5576,178],{"className":5577},[249],[36,5579],{"className":5580,"style":245},[223],[36,5582,5584,5587,5590],{"className":5583},[71],[36,5585],{"className":5586,"style":205},[75],[36,5588,15],{"className":5589},[80,81],[36,5591,168],{"className":5592},[219],[15,5594,5595,5596,5624],{},"对 ",[36,5597,5599,5612],{"className":5598,"translate":40},[39],[36,5600,5602],{"className":5601},[44],[46,5603,5604],{"xmlns":48},[50,5605,5606,5610],{},[53,5607,5608],{},[56,5609,15],{},[60,5611,15],{"encoding":62},[36,5613,5615],{"className":5614,"ariaHidden":67},[66],[36,5616,5618,5621],{"className":5617},[71],[36,5619],{"className":5620,"style":108},[75],[36,5622,15],{"className":5623},[80,81]," 求导并令其为零：",[36,5626,5628],{"className":5627,"translate":40},[314],[36,5629,5631,5703],{"className":5630,"translate":40},[39],[36,5632,5634],{"className":5633},[44],[46,5635,5636],{"xmlns":48,"display":323},[50,5637,5638,5700],{},[53,5639,5640,5654,5656,5670,5672,5696,5698],{},[4169,5641,5642,5648],{},[53,5643,5644,5646],{},[56,5645,4175],{"mathvariant":1388},[56,5647,3428],{"mathvariant":1388},[53,5649,5650,5652],{},[56,5651,4175],{"mathvariant":1388},[56,5653,15],{},[160,5655,346],{},[4169,5657,5658,5668],{},[53,5659,5660,5662],{},[160,5661,3464],{},[329,5663,5664,5666],{},[56,5665,2119],{},[56,5667,3146],{},[56,5669,15],{},[160,5671,178],{},[4169,5673,5674,5688],{},[53,5675,5676,5678,5680,5682],{},[56,5677,58],{},[160,5679,178],{},[160,5681,3464],{},[329,5683,5684,5686],{},[56,5685,2119],{},[56,5687,3146],{},[53,5689,5690,5692,5694],{},[173,5691,175],{},[160,5693,178],{},[56,5695,15],{},[160,5697,346],{},[173,5699,4198],{},[60,5701,5702],{"encoding":62},"\\frac{\\partial \\ell}{\\partial p} = \\frac{\\sum x_i}{p} - \\frac{n-\\sum x_i}{1-p} = 0",[36,5704,5706,5792,5915,6059],{"className":5705,"ariaHidden":67},[66],[36,5707,5709,5713,5783,5786,5789],{"className":5708},[71],[36,5710],{"className":5711,"style":5712},[75],"height:2.2519em;vertical-align:-0.8804em;",[36,5714,5716,5719,5780],{"className":5715},[80],[36,5717],{"className":5718},[212,4218],[36,5720,5722],{"className":5721},[4169],[36,5723,5725,5771],{"className":5724},[276,384],[36,5726,5728,5768],{"className":5727},[280],[36,5729,5732,5746,5754],{"className":5730,"style":5731},[284],"height:1.3714em;",[36,5733,5734,5737],{"style":4234},[36,5735],{"className":5736,"style":2037},[291],[36,5738,5740,5743],{"className":5739},[80],[36,5741,4175],{"className":5742,"style":920},[80],[36,5744,15],{"className":5745},[80,81],[36,5747,5748,5751],{"style":4249},[36,5749],{"className":5750,"style":2037},[291],[36,5752],{"className":5753,"style":4257},[4256],[36,5755,5756,5759],{"style":4260},[36,5757],{"className":5758,"style":2037},[291],[36,5760,5762,5765],{"className":5761},[80],[36,5763,4175],{"className":5764,"style":920},[80],[36,5766,3428],{"className":5767},[80],[36,5769,416],{"className":5770},[415],[36,5772,5774],{"className":5773},[280],[36,5775,5778],{"className":5776,"style":5777},[284],"height:0.8804em;",[36,5779],{},[36,5781],{"className":5782},[219,4218],[36,5784],{"className":5785,"style":224},[223],[36,5787,346],{"className":5788},[228],[36,5790],{"className":5791,"style":224},[223],[36,5793,5795,5799,5906,5909,5912],{"className":5794},[71],[36,5796],{"className":5797,"style":5798},[75],"height:2.3074em;vertical-align:-0.8804em;",[36,5800,5802,5805,5903],{"className":5801},[80],[36,5803],{"className":5804},[212,4218],[36,5806,5808],{"className":5807},[4169],[36,5809,5811,5895],{"className":5810},[276,384],[36,5812,5814,5892],{"className":5813},[280],[36,5815,5817,5828,5836],{"className":5816,"style":4231},[284],[36,5818,5819,5822],{"style":4234},[36,5820],{"className":5821,"style":2037},[291],[36,5823,5825],{"className":5824},[80],[36,5826,15],{"className":5827},[80,81],[36,5829,5830,5833],{"style":4249},[36,5831],{"className":5832,"style":2037},[291],[36,5834],{"className":5835,"style":4257},[4256],[36,5837,5838,5841],{"style":4260},[36,5839],{"className":5840,"style":2037},[291],[36,5842,5844,5849,5852],{"className":5843},[80],[36,5845,3464],{"className":5846,"style":5848},[2614,3303,5847],"small-op","position:relative;top:0em;",[36,5850],{"className":5851,"style":2232},[223],[36,5853,5855,5858],{"className":5854},[80],[36,5856,2119],{"className":5857},[80,81],[36,5859,5861],{"className":5860},[272],[36,5862,5864,5884],{"className":5863},[276,384],[36,5865,5867,5881],{"className":5866},[280],[36,5868,5870],{"className":5869,"style":3344},[284],[36,5871,5872,5875],{"style":2204},[36,5873],{"className":5874,"style":292},[291],[36,5876,5878],{"className":5877},[296,297,298,299],[36,5879,3146],{"className":5880},[80,81,299],[36,5882,416],{"className":5883},[415],[36,5885,5887],{"className":5886},[280],[36,5888,5890],{"className":5889,"style":423},[284],[36,5891],{},[36,5893,416],{"className":5894},[415],[36,5896,5898],{"className":5897},[280],[36,5899,5901],{"className":5900,"style":5777},[284],[36,5902],{},[36,5904],{"className":5905},[219,4218],[36,5907],{"className":5908,"style":245},[223],[36,5910,178],{"className":5911},[249],[36,5913],{"className":5914,"style":245},[223],[36,5916,5918,5921,6050,6053,6056],{"className":5917},[71],[36,5919],{"className":5920,"style":5798},[75],[36,5922,5924,5927,6047],{"className":5923},[80],[36,5925],{"className":5926},[212,4218],[36,5928,5930],{"className":5929},[4169],[36,5931,5933,6039],{"className":5932},[276,384],[36,5934,5936,6036],{"className":5935},[280],[36,5937,5939,5962,5970],{"className":5938,"style":4231},[284],[36,5940,5941,5944],{"style":4234},[36,5942],{"className":5943,"style":2037},[291],[36,5945,5947,5950,5953,5956,5959],{"className":5946},[80],[36,5948,175],{"className":5949},[80],[36,5951],{"className":5952,"style":245},[223],[36,5954,178],{"className":5955},[249],[36,5957],{"className":5958,"style":245},[223],[36,5960,15],{"className":5961},[80,81],[36,5963,5964,5967],{"style":4249},[36,5965],{"className":5966,"style":2037},[291],[36,5968],{"className":5969,"style":4257},[4256],[36,5971,5972,5975],{"style":4260},[36,5973],{"className":5974,"style":2037},[291],[36,5976,5978,5981,5984,5987,5990,5993,5996],{"className":5977},[80],[36,5979,58],{"className":5980},[80,81],[36,5982],{"className":5983,"style":245},[223],[36,5985,178],{"className":5986},[249],[36,5988],{"className":5989,"style":245},[223],[36,5991,3464],{"className":5992,"style":5848},[2614,3303,5847],[36,5994],{"className":5995,"style":2232},[223],[36,5997,5999,6002],{"className":5998},[80],[36,6000,2119],{"className":6001},[80,81],[36,6003,6005],{"className":6004},[272],[36,6006,6008,6028],{"className":6007},[276,384],[36,6009,6011,6025],{"className":6010},[280],[36,6012,6014],{"className":6013,"style":3344},[284],[36,6015,6016,6019],{"style":2204},[36,6017],{"className":6018,"style":292},[291],[36,6020,6022],{"className":6021},[296,297,298,299],[36,6023,3146],{"className":6024},[80,81,299],[36,6026,416],{"className":6027},[415],[36,6029,6031],{"className":6030},[280],[36,6032,6034],{"className":6033,"style":423},[284],[36,6035],{},[36,6037,416],{"className":6038},[415],[36,6040,6042],{"className":6041},[280],[36,6043,6045],{"className":6044,"style":5777},[284],[36,6046],{},[36,6048],{"className":6049},[219,4218],[36,6051],{"className":6052,"style":224},[223],[36,6054,346],{"className":6055},[228],[36,6057],{"className":6058,"style":224},[223],[36,6060,6062,6065],{"className":6061},[71],[36,6063],{"className":6064,"style":1356},[75],[36,6066,4198],{"className":6067},[80],[15,6069,6070],{},"解得：",[36,6072,6074],{"className":6073,"translate":40},[314],[36,6075,6077,6133],{"className":6076,"translate":40},[39],[36,6078,6080],{"className":6079},[44],[46,6081,6082],{"xmlns":48,"display":323},[50,6083,6084,6130],{},[53,6085,6086,6102,6104],{},[329,6087,6088,6094],{},[1994,6089,6090,6092],{"accent":67},[56,6091,15],{},[160,6093,2000],{},[53,6095,6096,6098,6100],{},[56,6097,4059],{"mathvariant":1388},[56,6099,2392],{"mathvariant":1388},[56,6101,4064],{"mathvariant":1388},[160,6103,346],{},[4169,6105,6106,6128],{},[53,6107,6108,6122],{},[4890,6109,6110,6112,6120],{},[160,6111,3464],{},[53,6113,6114,6116,6118],{},[56,6115,3146],{},[160,6117,346],{},[173,6119,175],{},[56,6121,58],{},[329,6123,6124,6126],{},[56,6125,2119],{},[56,6127,3146],{},[56,6129,58],{},[60,6131,6132],{"encoding":62},"\\hat{p}_{\\rm MLE} = \\frac{\\sum_{i=1}^n x_i}{n}",[36,6134,6136,6236],{"className":6135,"ariaHidden":67},[66],[36,6137,6139,6142,6227,6230,6233],{"className":6138},[71],[36,6140],{"className":6141,"style":884},[75],[36,6143,6145,6187],{"className":6144},[80],[36,6146,6148],{"className":6147},[80,2020],[36,6149,6151,6179],{"className":6150},[276,384],[36,6152,6154,6176],{"className":6153},[280],[36,6155,6157,6165],{"className":6156,"style":2030},[284],[36,6158,6159,6162],{"style":2033},[36,6160],{"className":6161,"style":2037},[291],[36,6163,15],{"className":6164},[80,81],[36,6166,6167,6170],{"style":2033},[36,6168],{"className":6169,"style":2037},[291],[36,6171,6173],{"className":6172,"style":2050},[2049],[36,6174,2000],{"className":6175},[80],[36,6177,416],{"className":6178},[415],[36,6180,6182],{"className":6181},[280],[36,6183,6185],{"className":6184,"style":2063},[284],[36,6186],{},[36,6188,6190],{"className":6189},[272],[36,6191,6193,6219],{"className":6192},[276,384],[36,6194,6196,6216],{"className":6195},[280],[36,6197,6199],{"className":6198,"style":391},[284],[36,6200,6201,6204],{"style":2204},[36,6202],{"className":6203,"style":292},[291],[36,6205,6207],{"className":6206},[296,297,298,299],[36,6208,6210],{"className":6209},[80,299],[36,6211,6213],{"className":6212},[80,299],[36,6214,3888],{"className":6215},[80,4140,299],[36,6217,416],{"className":6218},[415],[36,6220,6222],{"className":6221},[280],[36,6223,6225],{"className":6224,"style":423},[284],[36,6226],{},[36,6228],{"className":6229,"style":224},[223],[36,6231,346],{"className":6232},[228],[36,6234],{"className":6235,"style":224},[223],[36,6237,6239,6243],{"className":6238},[71],[36,6240],{"className":6241,"style":6242},[75],"height:2.18em;vertical-align:-0.686em;",[36,6244,6246,6249,6408],{"className":6245},[80],[36,6247],{"className":6248},[212,4218],[36,6250,6252],{"className":6251},[4169],[36,6253,6255,6400],{"className":6254},[276,384],[36,6256,6258,6397],{"className":6257},[280],[36,6259,6262,6273,6281],{"className":6260,"style":6261},[284],"height:1.494em;",[36,6263,6264,6267],{"style":4234},[36,6265],{"className":6266,"style":2037},[291],[36,6268,6270],{"className":6269},[80],[36,6271,58],{"className":6272},[80,81],[36,6274,6275,6278],{"style":4249},[36,6276],{"className":6277,"style":2037},[291],[36,6279],{"className":6280,"style":4257},[4256],[36,6282,6284,6287],{"style":6283},"top:-3.6897em;",[36,6285],{"className":6286,"style":2037},[291],[36,6288,6290,6354,6357],{"className":6289},[80],[36,6291,6293,6296],{"className":6292},[2614],[36,6294,3464],{"className":6295,"style":5848},[2614,3303,5847],[36,6297,6299],{"className":6298},[272],[36,6300,6302,6345],{"className":6301},[276,384],[36,6303,6305,6342],{"className":6304},[280],[36,6306,6309,6330],{"className":6307,"style":6308},[284],"height:0.8043em;",[36,6310,6312,6315],{"style":6311},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[36,6313],{"className":6314,"style":292},[291],[36,6316,6318],{"className":6317},[296,297,298,299],[36,6319,6321,6324,6327],{"className":6320},[80,299],[36,6322,3146],{"className":6323},[80,81,299],[36,6325,346],{"className":6326},[228,299],[36,6328,175],{"className":6329},[80,299],[36,6331,6333,6336],{"style":6332},"top:-3.2029em;margin-right:0.05em;",[36,6334],{"className":6335,"style":292},[291],[36,6337,6339],{"className":6338},[296,297,298,299],[36,6340,58],{"className":6341},[80,81,299],[36,6343,416],{"className":6344},[415],[36,6346,6348],{"className":6347},[280],[36,6349,6352],{"className":6350,"style":6351},[284],"height:0.2997em;",[36,6353],{},[36,6355],{"className":6356,"style":2232},[223],[36,6358,6360,6363],{"className":6359},[80],[36,6361,2119],{"className":6362},[80,81],[36,6364,6366],{"className":6365},[272],[36,6367,6369,6389],{"className":6368},[276,384],[36,6370,6372,6386],{"className":6371},[280],[36,6373,6375],{"className":6374,"style":3344},[284],[36,6376,6377,6380],{"style":2204},[36,6378],{"className":6379,"style":292},[291],[36,6381,6383],{"className":6382},[296,297,298,299],[36,6384,3146],{"className":6385},[80,81,299],[36,6387,416],{"className":6388},[415],[36,6390,6392],{"className":6391},[280],[36,6393,6395],{"className":6394,"style":423},[284],[36,6396],{},[36,6398,416],{"className":6399},[415],[36,6401,6403],{"className":6402},[280],[36,6404,6406],{"className":6405,"style":4297},[284],[36,6407],{},[36,6409],{"className":6410},[219,4218],[15,6412,6413,6414,6442],{},"在神经网络中，MLE 的参数 ",[36,6415,6417,6430],{"className":6416,"translate":40},[39],[36,6418,6420],{"className":6419},[44],[46,6421,6422],{"xmlns":48},[50,6423,6424,6428],{},[53,6425,6426],{},[56,6427,2356],{},[60,6429,2359],{"encoding":62},[36,6431,6433],{"className":6432,"ariaHidden":67},[66],[36,6434,6436,6439],{"className":6435},[71],[36,6437],{"className":6438,"style":2030},[75],[36,6440,2356],{"className":6441,"style":1433},[80,81]," 对应网络的权重和偏置，通常记为：",[36,6444,6446],{"className":6445,"translate":40},[314],[36,6447,6449,6520],{"className":6448,"translate":40},[39],[36,6450,6452],{"className":6451},[44],[46,6453,6454],{"xmlns":48,"display":323},[50,6455,6456,6517],{},[53,6457,6458,6460,6462],{},[56,6459,2356],{},[160,6461,346],{},[53,6463,6464,6471,6473,6480,6482,6488,6490,6496,6498,6501,6503,6509,6511],{},[329,6465,6466,6469],{},[56,6467,6468],{},"W",[173,6470,175],{},[160,6472,2124],{"separator":67},[329,6474,6475,6478],{},[56,6476,6477],{},"b",[173,6479,175],{},[160,6481,2124],{"separator":67},[329,6483,6484,6486],{},[56,6485,6468],{},[173,6487,1670],{},[160,6489,2124],{"separator":67},[329,6491,6492,6494],{},[56,6493,6477],{},[173,6495,1670],{},[160,6497,2124],{"separator":67},[160,6499,6500],{},"…",[160,6502,2124],{"separator":67},[329,6504,6505,6507],{},[56,6506,6468],{},[56,6508,2392],{},[160,6510,2124],{"separator":67},[329,6512,6513,6515],{},[56,6514,6477],{},[56,6516,2392],{},[60,6518,6519],{"encoding":62},"\\theta = {W_1, b_1, W_2, b_2, \\dots, W_L, b_L}",[36,6521,6523,6541],{"className":6522,"ariaHidden":67},[66],[36,6524,6526,6529,6532,6535,6538],{"className":6525},[71],[36,6527],{"className":6528,"style":2030},[75],[36,6530,2356],{"className":6531,"style":1433},[80,81],[36,6533],{"className":6534,"style":224},[223],[36,6536,346],{"className":6537},[228],[36,6539],{"className":6540,"style":224},[223],[36,6542,6544,6547],{"className":6543},[71],[36,6545],{"className":6546,"style":884},[75],[36,6548,6550,6590,6593,6596,6636,6639,6642,6682,6685,6688,6728,6731,6734,6737,6740,6743,6746,6786,6789,6792],{"className":6549},[80],[36,6551,6553,6556],{"className":6552},[80],[36,6554,6468],{"className":6555,"style":377},[80,81],[36,6557,6559],{"className":6558},[272],[36,6560,6562,6582],{"className":6561},[276,384],[36,6563,6565,6579],{"className":6564},[280],[36,6566,6568],{"className":6567,"style":2201},[284],[36,6569,6570,6573],{"style":394},[36,6571],{"className":6572,"style":292},[291],[36,6574,6576],{"className":6575},[296,297,298,299],[36,6577,175],{"className":6578},[80,299],[36,6580,416],{"className":6581},[415],[36,6583,6585],{"className":6584},[280],[36,6586,6588],{"className":6587,"style":423},[284],[36,6589],{},[36,6591,2124],{"className":6592},[2228],[36,6594],{"className":6595,"style":2232},[223],[36,6597,6599,6602],{"className":6598},[80],[36,6600,6477],{"className":6601},[80,81],[36,6603,6605],{"className":6604},[272],[36,6606,6608,6628],{"className":6607},[276,384],[36,6609,6611,6625],{"className":6610},[280],[36,6612,6614],{"className":6613,"style":2201},[284],[36,6615,6616,6619],{"style":2204},[36,6617],{"className":6618,"style":292},[291],[36,6620,6622],{"className":6621},[296,297,298,299],[36,6623,175],{"className":6624},[80,299],[36,6626,416],{"className":6627},[415],[36,6629,6631],{"className":6630},[280],[36,6632,6634],{"className":6633,"style":423},[284],[36,6635],{},[36,6637,2124],{"className":6638},[2228],[36,6640],{"className":6641,"style":2232},[223],[36,6643,6645,6648],{"className":6644},[80],[36,6646,6468],{"className":6647,"style":377},[80,81],[36,6649,6651],{"className":6650},[272],[36,6652,6654,6674],{"className":6653},[276,384],[36,6655,6657,6671],{"className":6656},[280],[36,6658,6660],{"className":6659,"style":2201},[284],[36,6661,6662,6665],{"style":394},[36,6663],{"className":6664,"style":292},[291],[36,6666,6668],{"className":6667},[296,297,298,299],[36,6669,1670],{"className":6670},[80,299],[36,6672,416],{"className":6673},[415],[36,6675,6677],{"className":6676},[280],[36,6678,6680],{"className":6679,"style":423},[284],[36,6681],{},[36,6683,2124],{"className":6684},[2228],[36,6686],{"className":6687,"style":2232},[223],[36,6689,6691,6694],{"className":6690},[80],[36,6692,6477],{"className":6693},[80,81],[36,6695,6697],{"className":6696},[272],[36,6698,6700,6720],{"className":6699},[276,384],[36,6701,6703,6717],{"className":6702},[280],[36,6704,6706],{"className":6705,"style":2201},[284],[36,6707,6708,6711],{"style":2204},[36,6709],{"className":6710,"style":292},[291],[36,6712,6714],{"className":6713},[296,297,298,299],[36,6715,1670],{"className":6716},[80,299],[36,6718,416],{"className":6719},[415],[36,6721,6723],{"className":6722},[280],[36,6724,6726],{"className":6725,"style":423},[284],[36,6727],{},[36,6729,2124],{"className":6730},[2228],[36,6732],{"className":6733,"style":2232},[223],[36,6735,6500],{"className":6736},[2282],[36,6738],{"className":6739,"style":2232},[223],[36,6741,2124],{"className":6742},[2228],[36,6744],{"className":6745,"style":2232},[223],[36,6747,6749,6752],{"className":6748},[80],[36,6750,6468],{"className":6751,"style":377},[80,81],[36,6753,6755],{"className":6754},[272],[36,6756,6758,6778],{"className":6757},[276,384],[36,6759,6761,6775],{"className":6760},[280],[36,6762,6764],{"className":6763,"style":391},[284],[36,6765,6766,6769],{"style":394},[36,6767],{"className":6768,"style":292},[291],[36,6770,6772],{"className":6771},[296,297,298,299],[36,6773,2392],{"className":6774},[80,81,299],[36,6776,416],{"className":6777},[415],[36,6779,6781],{"className":6780},[280],[36,6782,6784],{"className":6783,"style":423},[284],[36,6785],{},[36,6787,2124],{"className":6788},[2228],[36,6790],{"className":6791,"style":2232},[223],[36,6793,6795,6798],{"className":6794},[80],[36,6796,6477],{"className":6797},[80,81],[36,6799,6801],{"className":6800},[272],[36,6802,6804,6824],{"className":6803},[276,384],[36,6805,6807,6821],{"className":6806},[280],[36,6808,6810],{"className":6809,"style":391},[284],[36,6811,6812,6815],{"style":2204},[36,6813],{"className":6814,"style":292},[291],[36,6816,6818],{"className":6817},[296,297,298,299],[36,6819,2392],{"className":6820},[80,81,299],[36,6822,416],{"className":6823},[415],[36,6825,6827],{"className":6826},[280],[36,6828,6830],{"className":6829,"style":423},[284],[36,6831],{},[15,6833,6834,6835,6863,6864,6934,6935,6964,6965,7036],{},"其中 ",[36,6836,6838,6851],{"className":6837,"translate":40},[39],[36,6839,6841],{"className":6840},[44],[46,6842,6843],{"xmlns":48},[50,6844,6845,6849],{},[53,6846,6847],{},[56,6848,2392],{},[60,6850,2392],{"encoding":62},[36,6852,6854],{"className":6853,"ariaHidden":67},[66],[36,6855,6857,6860],{"className":6856},[71],[36,6858],{"className":6859,"style":139},[75],[36,6861,2392],{"className":6862},[80,81]," 是层数，",[36,6865,6867,6885],{"className":6866,"translate":40},[39],[36,6868,6870],{"className":6869},[44],[46,6871,6872],{"xmlns":48},[50,6873,6874,6882],{},[53,6875,6876],{},[329,6877,6878,6880],{},[56,6879,6468],{},[56,6881,3146],{},[60,6883,6884],{"encoding":62},"W_i",[36,6886,6888],{"className":6887,"ariaHidden":67},[66],[36,6889,6891,6894],{"className":6890},[71],[36,6892],{"className":6893,"style":552},[75],[36,6895,6897,6900],{"className":6896},[80],[36,6898,6468],{"className":6899,"style":377},[80,81],[36,6901,6903],{"className":6902},[272],[36,6904,6906,6926],{"className":6905},[276,384],[36,6907,6909,6923],{"className":6908},[280],[36,6910,6912],{"className":6911,"style":3344},[284],[36,6913,6914,6917],{"style":394},[36,6915],{"className":6916,"style":292},[291],[36,6918,6920],{"className":6919},[296,297,298,299],[36,6921,3146],{"className":6922},[80,81,299],[36,6924,416],{"className":6925},[415],[36,6927,6929],{"className":6928},[280],[36,6930,6932],{"className":6931,"style":423},[284],[36,6933],{}," 是第 ",[36,6936,6938,6951],{"className":6937,"translate":40},[39],[36,6939,6941],{"className":6940},[44],[46,6942,6943],{"xmlns":48},[50,6944,6945,6949],{},[53,6946,6947],{},[56,6948,3146],{},[60,6950,3146],{"encoding":62},[36,6952,6954],{"className":6953,"ariaHidden":67},[66],[36,6955,6957,6961],{"className":6956},[71],[36,6958],{"className":6959,"style":6960},[75],"height:0.6595em;",[36,6962,3146],{"className":6963},[80,81]," 层的权重矩阵，",[36,6966,6968,6986],{"className":6967,"translate":40},[39],[36,6969,6971],{"className":6970},[44],[46,6972,6973],{"xmlns":48},[50,6974,6975,6983],{},[53,6976,6977],{},[329,6978,6979,6981],{},[56,6980,6477],{},[56,6982,3146],{},[60,6984,6985],{"encoding":62},"b_i",[36,6987,6989],{"className":6988,"ariaHidden":67},[66],[36,6990,6992,6996],{"className":6991},[71],[36,6993],{"className":6994,"style":6995},[75],"height:0.8444em;vertical-align:-0.15em;",[36,6997,6999,7002],{"className":6998},[80],[36,7000,6477],{"className":7001},[80,81],[36,7003,7005],{"className":7004},[272],[36,7006,7008,7028],{"className":7007},[276,384],[36,7009,7011,7025],{"className":7010},[280],[36,7012,7014],{"className":7013,"style":3344},[284],[36,7015,7016,7019],{"style":2204},[36,7017],{"className":7018,"style":292},[291],[36,7020,7022],{"className":7021},[296,297,298,299],[36,7023,3146],{"className":7024},[80,81,299],[36,7026,416],{"className":7027},[415],[36,7029,7031],{"className":7030},[280],[36,7032,7034],{"className":7033,"style":423},[284],[36,7035],{}," 是偏置向量。",[15,7038,7039,7040],{},"换言之：神经网络是一个函数族 ",[36,7041,7043,7068],{"className":7042,"translate":40},[39],[36,7044,7046],{"className":7045},[44],[46,7047,7048],{"xmlns":48},[50,7049,7050,7065],{},[53,7051,7052,7059,7061,7063],{},[329,7053,7054,7057],{},[56,7055,7056],{},"f",[56,7058,2356],{},[160,7060,163],{"stretchy":162},[56,7062,2119],{},[160,7064,168],{"stretchy":162},[60,7066,7067],{"encoding":62},"f_\\theta(x)",[36,7069,7071],{"className":7070,"ariaHidden":67},[66],[36,7072,7074,7077,7120,7123,7126],{"className":7073},[71],[36,7075],{"className":7076,"style":205},[75],[36,7078,7080,7084],{"className":7079},[80],[36,7081,7056],{"className":7082,"style":7083},[80,81],"margin-right:0.1076em;",[36,7085,7087],{"className":7086},[272],[36,7088,7090,7112],{"className":7089},[276,384],[36,7091,7093,7109],{"className":7092},[280],[36,7094,7097],{"className":7095,"style":7096},[284],"height:0.3361em;",[36,7098,7100,7103],{"style":7099},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[36,7101],{"className":7102,"style":292},[291],[36,7104,7106],{"className":7105},[296,297,298,299],[36,7107,2356],{"className":7108,"style":1433},[80,81,299],[36,7110,416],{"className":7111},[415],[36,7113,7115],{"className":7114},[280],[36,7116,7118],{"className":7117,"style":423},[284],[36,7119],{},[36,7121,163],{"className":7122},[212],[36,7124,2119],{"className":7125},[80,81],[36,7127,168],{"className":7128},[219],[15,7130,7131],{},"它定义了条件概率分布：",[36,7133,7135],{"className":7134,"translate":40},[314],[36,7136,7138,7167],{"className":7137,"translate":40},[39],[36,7139,7141],{"className":7140},[44],[46,7142,7143],{"xmlns":48,"display":323},[50,7144,7145,7164],{},[53,7146,7147,7153,7155,7158,7160,7162],{},[329,7148,7149,7151],{},[56,7150,333],{},[56,7152,2356],{},[160,7154,163],{"stretchy":162},[56,7156,7157],{},"y",[56,7159,1389],{"mathvariant":1388},[56,7161,2119],{},[160,7163,168],{"stretchy":162},[60,7165,7166],{"encoding":62},"P_\\theta(y|x)",[36,7168,7170],{"className":7169,"ariaHidden":67},[66],[36,7171,7173,7176,7216,7219,7223,7226,7229],{"className":7172},[71],[36,7174],{"className":7175,"style":205},[75],[36,7177,7179,7182],{"className":7178},[80],[36,7180,333],{"className":7181,"style":377},[80,81],[36,7183,7185],{"className":7184},[272],[36,7186,7188,7208],{"className":7187},[276,384],[36,7189,7191,7205],{"className":7190},[280],[36,7192,7194],{"className":7193,"style":7096},[284],[36,7195,7196,7199],{"style":394},[36,7197],{"className":7198,"style":292},[291],[36,7200,7202],{"className":7201},[296,297,298,299],[36,7203,2356],{"className":7204,"style":1433},[80,81,299],[36,7206,416],{"className":7207},[415],[36,7209,7211],{"className":7210},[280],[36,7212,7214],{"className":7213,"style":423},[284],[36,7215],{},[36,7217,163],{"className":7218},[212],[36,7220,7157],{"className":7221,"style":7222},[80,81],"margin-right:0.0359em;",[36,7224,1389],{"className":7225},[80],[36,7227,2119],{"className":7228},[80,81],[36,7230,168],{"className":7231},[219],[15,7233,7234,7235,7263],{},"训练神经网络 = 选择使观测数据出现概率最大的参数 ",[36,7236,7238,7251],{"className":7237,"translate":40},[39],[36,7239,7241],{"className":7240},[44],[46,7242,7243],{"xmlns":48},[50,7244,7245,7249],{},[53,7246,7247],{},[56,7248,2356],{},[60,7250,2359],{"encoding":62},[36,7252,7254],{"className":7253,"ariaHidden":67},[66],[36,7255,7257,7260],{"className":7256},[71],[36,7258],{"className":7259,"style":2030},[75],[36,7261,2356],{"className":7262,"style":1433},[80,81],"：",[36,7265,7267],{"className":7266,"translate":40},[314],[36,7268,7270,7344],{"className":7269,"translate":40},[39],[36,7271,7273],{"className":7272},[44],[46,7274,7275],{"xmlns":48,"display":323},[50,7276,7277,7341],{},[53,7278,7279,7295,7297,7299,7301,7311,7317,7323,7325,7331,7333,7339],{},[329,7280,7281,7287],{},[1994,7282,7283,7285],{"accent":67},[56,7284,2356],{},[160,7286,2000],{},[53,7288,7289,7291,7293],{},[56,7290,4059],{"mathvariant":1388},[56,7292,2392],{"mathvariant":1388},[56,7294,4064],{"mathvariant":1388},[160,7296,346],{},[56,7298,2498],{},[160,7300,2501],{},[2503,7302,7303,7309],{},[53,7304,7305,7307],{},[56,7306,2509],{},[160,7308,2501],{},[56,7310,2356],{},[2503,7312,7313,7315],{},[160,7314,3137],{},[56,7316,3146],{},[329,7318,7319,7321],{},[56,7320,333],{},[56,7322,2356],{},[160,7324,163],{"stretchy":162},[329,7326,7327,7329],{},[56,7328,7157],{},[56,7330,3146],{},[56,7332,1389],{"mathvariant":1388},[329,7334,7335,7337],{},[56,7336,2119],{},[56,7338,3146],{},[160,7340,168],{"stretchy":162},[60,7342,7343],{"encoding":62},"\\hat{\\theta}_{\\rm MLE} = \\arg\\max_\\theta \\prod_i P_\\theta(y_i | x_i)",[36,7345,7347,7436],{"className":7346,"ariaHidden":67},[66],[36,7348,7350,7353,7427,7430,7433],{"className":7349},[71],[36,7351],{"className":7352,"style":3901},[75],[36,7354,7356,7387],{"className":7355},[80],[36,7357,7359],{"className":7358},[80,2020],[36,7360,7362],{"className":7361},[276],[36,7363,7365],{"className":7364},[280],[36,7366,7368,7376],{"className":7367,"style":2562},[284],[36,7369,7370,7373],{"style":2033},[36,7371],{"className":7372,"style":2037},[291],[36,7374,2356],{"className":7375,"style":1433},[80,81],[36,7377,7378,7381],{"style":2585},[36,7379],{"className":7380,"style":2037},[291],[36,7382,7384],{"className":7383,"style":2050},[2049],[36,7385,2000],{"className":7386},[80],[36,7388,7390],{"className":7389},[272],[36,7391,7393,7419],{"className":7392},[276,384],[36,7394,7396,7416],{"className":7395},[280],[36,7397,7399],{"className":7398,"style":391},[284],[36,7400,7401,7404],{"style":3950},[36,7402],{"className":7403,"style":292},[291],[36,7405,7407],{"className":7406},[296,297,298,299],[36,7408,7410],{"className":7409},[80,299],[36,7411,7413],{"className":7412},[80,299],[36,7414,3888],{"className":7415},[80,4140,299],[36,7417,416],{"className":7418},[415],[36,7420,7422],{"className":7421},[280],[36,7423,7425],{"className":7424,"style":423},[284],[36,7426],{},[36,7428],{"className":7429,"style":224},[223],[36,7431,346],{"className":7432},[228],[36,7434],{"className":7435,"style":224},[223],[36,7437,7439,7443,7448,7451,7495,7498,7542,7545,7585,7588,7629,7632,7672],{"className":7438},[71],[36,7440],{"className":7441,"style":7442},[75],"height:2.3277em;vertical-align:-1.2777em;",[36,7444,2615,7446],{"className":7445},[2614],[36,7447,2619],{"style":2618},[36,7449],{"className":7450,"style":2232},[223],[36,7452,7454],{"className":7453},[2614,2626],[36,7455,7457,7487],{"className":7456},[276,384],[36,7458,7460,7484],{"className":7459},[280],[36,7461,7463,7474],{"className":7462,"style":76},[284],[36,7464,7465,7468],{"style":2638},[36,7466],{"className":7467,"style":2037},[291],[36,7469,7471],{"className":7470},[296,297,298,299],[36,7472,2356],{"className":7473,"style":1433},[80,81,299],[36,7475,7476,7479],{"style":2033},[36,7477],{"className":7478,"style":2037},[291],[36,7480,7481],{},[36,7482,2509],{"className":7483},[2614],[36,7485,416],{"className":7486},[415],[36,7488,7490],{"className":7489},[280],[36,7491,7493],{"className":7492,"style":2667},[284],[36,7494],{},[36,7496],{"className":7497,"style":2232},[223],[36,7499,7501],{"className":7500},[2614,2626],[36,7502,7504,7534],{"className":7503},[276,384],[36,7505,7507,7531],{"className":7506},[280],[36,7508,7510,7521],{"className":7509,"style":3226},[284],[36,7511,7512,7515],{"style":4999},[36,7513],{"className":7514,"style":3233},[291],[36,7516,7518],{"className":7517},[296,297,298,299],[36,7519,3146],{"className":7520},[80,81,299],[36,7522,7523,7526],{"style":3294},[36,7524],{"className":7525,"style":3233},[291],[36,7527,7528],{},[36,7529,3137],{"className":7530},[2614,3303,3304],[36,7532,416],{"className":7533},[415],[36,7535,7537],{"className":7536},[280],[36,7538,7540],{"className":7539,"style":5049},[284],[36,7541],{},[36,7543],{"className":7544,"style":2232},[223],[36,7546,7548,7551],{"className":7547},[80],[36,7549,333],{"className":7550,"style":377},[80,81],[36,7552,7554],{"className":7553},[272],[36,7555,7557,7577],{"className":7556},[276,384],[36,7558,7560,7574],{"className":7559},[280],[36,7561,7563],{"className":7562,"style":7096},[284],[36,7564,7565,7568],{"style":394},[36,7566],{"className":7567,"style":292},[291],[36,7569,7571],{"className":7570},[296,297,298,299],[36,7572,2356],{"className":7573,"style":1433},[80,81,299],[36,7575,416],{"className":7576},[415],[36,7578,7580],{"className":7579},[280],[36,7581,7583],{"className":7582,"style":423},[284],[36,7584],{},[36,7586,163],{"className":7587},[212],[36,7589,7591,7594],{"className":7590},[80],[36,7592,7157],{"className":7593,"style":7222},[80,81],[36,7595,7597],{"className":7596},[272],[36,7598,7600,7621],{"className":7599},[276,384],[36,7601,7603,7618],{"className":7602},[280],[36,7604,7606],{"className":7605,"style":3344},[284],[36,7607,7609,7612],{"style":7608},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[36,7610],{"className":7611,"style":292},[291],[36,7613,7615],{"className":7614},[296,297,298,299],[36,7616,3146],{"className":7617},[80,81,299],[36,7619,416],{"className":7620},[415],[36,7622,7624],{"className":7623},[280],[36,7625,7627],{"className":7626,"style":423},[284],[36,7628],{},[36,7630,1389],{"className":7631},[80],[36,7633,7635,7638],{"className":7634},[80],[36,7636,2119],{"className":7637},[80,81],[36,7639,7641],{"className":7640},[272],[36,7642,7644,7664],{"className":7643},[276,384],[36,7645,7647,7661],{"className":7646},[280],[36,7648,7650],{"className":7649,"style":3344},[284],[36,7651,7652,7655],{"style":2204},[36,7653],{"className":7654,"style":292},[291],[36,7656,7658],{"className":7657},[296,297,298,299],[36,7659,3146],{"className":7660},[80,81,299],[36,7662,416],{"className":7663},[415],[36,7665,7667],{"className":7666},[280],[36,7668,7670],{"className":7669,"style":423},[284],[36,7671],{},[36,7673,168],{"className":7674},[219],[15,7676,7677],{},"以分类为例：一个二分类神经网络的输出为：",[36,7679,7681],{"className":7680,"translate":40},[314],[36,7682,7684,7723],{"className":7683,"translate":40},[39],[36,7685,7687],{"className":7686},[44],[46,7688,7689],{"xmlns":48,"display":323},[50,7690,7691,7720],{},[53,7692,7693,7699,7701,7704,7706,7712,7714,7716,7718],{},[1994,7694,7695,7697],{"accent":67},[56,7696,7157],{},[160,7698,2000],{},[160,7700,346],{},[56,7702,7703],{},"σ",[160,7705,163],{"stretchy":162},[329,7707,7708,7710],{},[56,7709,7056],{},[56,7711,2356],{},[160,7713,163],{"stretchy":162},[56,7715,2119],{},[160,7717,168],{"stretchy":162},[160,7719,168],{"stretchy":162},[60,7721,7722],{"encoding":62},"\\hat{y} = \\sigma(f_\\theta(x))",[36,7724,7726,7784],{"className":7725,"ariaHidden":67},[66],[36,7727,7729,7732,7775,7778,7781],{"className":7728},[71],[36,7730],{"className":7731,"style":884},[75],[36,7733,7735],{"className":7734},[80,2020],[36,7736,7738,7767],{"className":7737},[276,384],[36,7739,7741,7764],{"className":7740},[280],[36,7742,7744,7752],{"className":7743,"style":2030},[284],[36,7745,7746,7749],{"style":2033},[36,7747],{"className":7748,"style":2037},[291],[36,7750,7157],{"className":7751,"style":7222},[80,81],[36,7753,7754,7757],{"style":2033},[36,7755],{"className":7756,"style":2037},[291],[36,7758,7761],{"className":7759,"style":7760},[2049],"left:-0.1944em;",[36,7762,2000],{"className":7763},[80],[36,7765,416],{"className":7766},[415],[36,7768,7770],{"className":7769},[280],[36,7771,7773],{"className":7772,"style":2063},[284],[36,7774],{},[36,7776],{"className":7777,"style":224},[223],[36,7779,346],{"className":7780},[228],[36,7782],{"className":7783,"style":224},[223],[36,7785,7787,7790,7793,7796,7836,7839,7842],{"className":7786},[71],[36,7788],{"className":7789,"style":205},[75],[36,7791,7703],{"className":7792,"style":7222},[80,81],[36,7794,163],{"className":7795},[212],[36,7797,7799,7802],{"className":7798},[80],[36,7800,7056],{"className":7801,"style":7083},[80,81],[36,7803,7805],{"className":7804},[272],[36,7806,7808,7828],{"className":7807},[276,384],[36,7809,7811,7825],{"className":7810},[280],[36,7812,7814],{"className":7813,"style":7096},[284],[36,7815,7816,7819],{"style":7099},[36,7817],{"className":7818,"style":292},[291],[36,7820,7822],{"className":7821},[296,297,298,299],[36,7823,2356],{"className":7824,"style":1433},[80,81,299],[36,7826,416],{"className":7827},[415],[36,7829,7831],{"className":7830},[280],[36,7832,7834],{"className":7833,"style":423},[284],[36,7835],{},[36,7837,163],{"className":7838},[212],[36,7840,2119],{"className":7841},[80,81],[36,7843,7845],{"className":7844},[219],"))",[15,7847,7848],{},"我们看到，最大化似然可以写成最小化交叉熵损失函数作为代价函数：",[36,7850,7852],{"className":7851,"translate":40},[314],[36,7853,7855,7954],{"className":7854,"translate":40},[39],[36,7856,7858],{"className":7857},[44],[46,7859,7860],{"xmlns":48,"display":323},[50,7861,7862,7951],{},[53,7863,7864,7880,7882,7884,7886,7896,7902,7919,7921,7923,7925,7935],{},[329,7865,7866,7872],{},[1994,7867,7868,7870],{"accent":67},[56,7869,2356],{},[160,7871,2000],{},[53,7873,7874,7876,7878],{},[56,7875,4059],{"mathvariant":1388},[56,7877,2392],{"mathvariant":1388},[56,7879,4064],{"mathvariant":1388},[160,7881,346],{},[56,7883,2498],{},[160,7885,2501],{},[2503,7887,7888,7894],{},[53,7889,7890,7892],{},[56,7891,2509],{},[160,7893,2501],{},[56,7895,2356],{},[2503,7897,7898,7900],{},[160,7899,3137],{},[56,7901,3146],{},[7903,7904,7905,7911,7913],"msubsup",{},[1994,7906,7907,7909],{"accent":67},[56,7908,7157],{},[160,7910,2000],{},[56,7912,3146],{},[329,7914,7915,7917],{},[56,7916,7157],{},[56,7918,3146],{},[160,7920,163],{"stretchy":162},[173,7922,175],{},[160,7924,178],{},[329,7926,7927,7933],{},[1994,7928,7929,7931],{"accent":67},[56,7930,7157],{},[160,7932,2000],{},[56,7934,3146],{},[182,7936,7937,7939],{},[160,7938,168],{"stretchy":162},[53,7940,7941,7943,7945],{},[173,7942,175],{},[160,7944,178],{},[329,7946,7947,7949],{},[56,7948,7157],{},[56,7950,3146],{},[60,7952,7953],{"encoding":62},"\\hat{\\theta}_{\\rm MLE} = \\arg\\max_\\theta \\prod_i \\hat{y}_i^{y_i} (1-\\hat{y}_i)^{1-y_i}",[36,7955,7957,8046,8304],{"className":7956,"ariaHidden":67},[66],[36,7958,7960,7963,8037,8040,8043],{"className":7959},[71],[36,7961],{"className":7962,"style":3901},[75],[36,7964,7966,7997],{"className":7965},[80],[36,7967,7969],{"className":7968},[80,2020],[36,7970,7972],{"className":7971},[276],[36,7973,7975],{"className":7974},[280],[36,7976,7978,7986],{"className":7977,"style":2562},[284],[36,7979,7980,7983],{"style":2033},[36,7981],{"className":7982,"style":2037},[291],[36,7984,2356],{"className":7985,"style":1433},[80,81],[36,7987,7988,7991],{"style":2585},[36,7989],{"className":7990,"style":2037},[291],[36,7992,7994],{"className":7993,"style":2050},[2049],[36,7995,2000],{"className":7996},[80],[36,7998,8000],{"className":7999},[272],[36,8001,8003,8029],{"className":8002},[276,384],[36,8004,8006,8026],{"className":8005},[280],[36,8007,8009],{"className":8008,"style":391},[284],[36,8010,8011,8014],{"style":3950},[36,8012],{"className":8013,"style":292},[291],[36,8015,8017],{"className":8016},[296,297,298,299],[36,8018,8020],{"className":8019},[80,299],[36,8021,8023],{"className":8022},[80,299],[36,8024,3888],{"className":8025},[80,4140,299],[36,8027,416],{"className":8028},[415],[36,8030,8032],{"className":8031},[280],[36,8033,8035],{"className":8034,"style":423},[284],[36,8036],{},[36,8038],{"className":8039,"style":224},[223],[36,8041,346],{"className":8042},[228],[36,8044],{"className":8045,"style":224},[223],[36,8047,8049,8052,8057,8060,8104,8107,8151,8154,8289,8292,8295,8298,8301],{"className":8048},[71],[36,8050],{"className":8051,"style":7442},[75],[36,8053,2615,8055],{"className":8054},[2614],[36,8056,2619],{"style":2618},[36,8058],{"className":8059,"style":2232},[223],[36,8061,8063],{"className":8062},[2614,2626],[36,8064,8066,8096],{"className":8065},[276,384],[36,8067,8069,8093],{"className":8068},[280],[36,8070,8072,8083],{"className":8071,"style":76},[284],[36,8073,8074,8077],{"style":2638},[36,8075],{"className":8076,"style":2037},[291],[36,8078,8080],{"className":8079},[296,297,298,299],[36,8081,2356],{"className":8082,"style":1433},[80,81,299],[36,8084,8085,8088],{"style":2033},[36,8086],{"className":8087,"style":2037},[291],[36,8089,8090],{},[36,8091,2509],{"className":8092},[2614],[36,8094,416],{"className":8095},[415],[36,8097,8099],{"className":8098},[280],[36,8100,8102],{"className":8101,"style":2667},[284],[36,8103],{},[36,8105],{"className":8106,"style":2232},[223],[36,8108,8110],{"className":8109},[2614,2626],[36,8111,8113,8143],{"className":8112},[276,384],[36,8114,8116,8140],{"className":8115},[280],[36,8117,8119,8130],{"className":8118,"style":3226},[284],[36,8120,8121,8124],{"style":4999},[36,8122],{"className":8123,"style":3233},[291],[36,8125,8127],{"className":8126},[296,297,298,299],[36,8128,3146],{"className":8129},[80,81,299],[36,8131,8132,8135],{"style":3294},[36,8133],{"className":8134,"style":3233},[291],[36,8136,8137],{},[36,8138,3137],{"className":8139},[2614,3303,3304],[36,8141,416],{"className":8142},[415],[36,8144,8146],{"className":8145},[280],[36,8147,8149],{"className":8148,"style":5049},[284],[36,8150],{},[36,8152],{"className":8153,"style":2232},[223],[36,8155,8157,8199],{"className":8156},[80],[36,8158,8160],{"className":8159},[80,2020],[36,8161,8163,8191],{"className":8162},[276,384],[36,8164,8166,8188],{"className":8165},[280],[36,8167,8169,8177],{"className":8168,"style":2030},[284],[36,8170,8171,8174],{"style":2033},[36,8172],{"className":8173,"style":2037},[291],[36,8175,7157],{"className":8176,"style":7222},[80,81],[36,8178,8179,8182],{"style":2033},[36,8180],{"className":8181,"style":2037},[291],[36,8183,8185],{"className":8184,"style":7760},[2049],[36,8186,2000],{"className":8187},[80],[36,8189,416],{"className":8190},[415],[36,8192,8194],{"className":8193},[280],[36,8195,8197],{"className":8196,"style":2063},[284],[36,8198],{},[36,8200,8202],{"className":8201},[272],[36,8203,8205,8280],{"className":8204},[276,384],[36,8206,8208,8277],{"className":8207},[280],[36,8209,8212,8224],{"className":8210,"style":8211},[284],"height:0.7823em;",[36,8213,8215,8218],{"style":8214},"top:-2.4231em;margin-left:-0.0359em;margin-right:0.05em;",[36,8216],{"className":8217,"style":292},[291],[36,8219,8221],{"className":8220},[296,297,298,299],[36,8222,3146],{"className":8223},[80,81,299],[36,8225,8227,8230],{"style":8226},"top:-3.1809em;margin-right:0.05em;",[36,8228],{"className":8229,"style":292},[291],[36,8231,8233],{"className":8232},[296,297,298,299],[36,8234,8236],{"className":8235},[80,299],[36,8237,8239,8242],{"className":8238},[80,299],[36,8240,7157],{"className":8241,"style":7222},[80,81,299],[36,8243,8245],{"className":8244},[272],[36,8246,8248,8269],{"className":8247},[276,384],[36,8249,8251,8266],{"className":8250},[280],[36,8252,8254],{"className":8253,"style":3258},[284],[36,8255,8257,8260],{"style":8256},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[36,8258],{"className":8259,"style":3265},[291],[36,8261,8263],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= - \\sum_i \\Big[ y_i \\log \\hat{y}_i + (1-y_i) \\log (1-\\hat{y}_i) 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885},[212],[36,8887,175],{"className":8888},[80],[36,8890],{"className":8891,"style":245},[223],[36,8893,178],{"className":8894},[249],[36,8896],{"className":8897,"style":245},[223],[36,8899,8901,8905,8984,8987],{"className":8900},[71],[36,8902],{"className":8903,"style":8904},[75],"height:1.8em;vertical-align:-0.65em;",[36,8906,8908,8950],{"className":8907},[80],[36,8909,8911],{"className":8910},[80,2020],[36,8912,8914,8942],{"className":8913},[276,384],[36,8915,8917,8939],{"className":8916},[280],[36,8918,8920,8928],{"className":8919,"style":2030},[284],[36,8921,8922,8925],{"style":2033},[36,8923],{"className":8924,"style":2037},[291],[36,8926,7157],{"className":8927,"style":7222},[80,81],[36,8929,8930,8933],{"style":2033},[36,8931],{"className":8932,"style":2037},[291],[36,8934,8936],{"className":8935,"style":7760},[2049],[36,8937,2000],{"className":8938},[80],[36,8940,416],{"className":8941},[415],[36,8943,8945],{"className":8944},[280],[36,8946,8948],{"className":8947,"style":2063},[284],[36,8949],{},[36,8951,8953],{"className":8952},[272],[36,8954,8956,8976],{"className":8955},[276,384],[36,8957,8959,8973],{"className":8958},[280],[36,8960,8962],{"className":8961,"style":3344},[284],[36,8963,8964,8967],{"style":7608},[36,8965],{"className":8966,"style":292},[291],[36,8968,8970],{"className":8969},[296,297,298,299],[36,8971,3146],{"className":8972},[80,81,299],[36,8974,416],{"className":8975},[415],[36,8977,8979],{"className":8978},[280],[36,8980,8982],{"className":8981,"style":423},[284],[36,8983],{},[36,8985,168],{"className":8986},[219],[36,8988,8990],{"className":8989},[80],[36,8991,8564],{"className":8992},[8664,8665],[15,8994,8995,8996,9024],{},"这里的 ",[36,8997,8999,9012],{"className":8998,"translate":40},[39],[36,9000,9002],{"className":9001},[44],[46,9003,9004],{"xmlns":48},[50,9005,9006,9010],{},[53,9007,9008],{},[56,9009,2356],{},[60,9011,2359],{"encoding":62},[36,9013,9015],{"className":9014,"ariaHidden":67},[66],[36,9016,9018,9021],{"className":9017},[71],[36,9019],{"className":9020,"style":2030},[75],[36,9022,2356],{"className":9023,"style":1433},[80,81]," 表示所有权重和偏置。",[15,9026,9027],{},"再以回归为例：假设",[36,9029,9031],{"className":9030,"translate":40},[314],[36,9032,9034,9109],{"className":9033,"translate":40},[39],[36,9035,9037],{"className":9036},[44],[46,9038,9039],{"xmlns":48,"display":323},[50,9040,9041,9106],{},[53,9042,9043,9049,9051,9057,9059,9065,9067,9069,9076,9078,9081,9087,9090,9092,9094,9096,9098,9104],{},[329,9044,9045,9047],{},[56,9046,7157],{},[56,9048,3146],{},[160,9050,346],{},[329,9052,9053,9055],{},[56,9054,7056],{},[56,9056,2356],{},[160,9058,163],{"stretchy":162},[329,9060,9061,9063],{},[56,9062,2119],{},[56,9064,3146],{},[160,9066,168],{"stretchy":162},[160,9068,5284],{},[329,9070,9071,9074],{},[56,9072,9073],{},"ϵ",[56,9075,3146],{},[160,9077,2124],{"separator":67},[223,9079],{"width":9080},"1em",[329,9082,9083,9085],{},[56,9084,9073],{},[56,9086,3146],{},[160,9088,9089],{},"∼",[56,9091,127],{},[160,9093,163],{"stretchy":162},[173,9095,4198],{},[160,9097,2124],{"separator":67},[182,9099,9100,9102],{},[56,9101,7703],{},[173,9103,1670],{},[160,9105,168],{"stretchy":162},[60,9107,9108],{"encoding":62},"y_i = f_\\theta(x_i) + \\epsilon_i, \\quad \\epsilon_i \\sim N(0, \\sigma^2)",[36,9110,9112,9167,9268,9373],{"className":9111,"ariaHidden":67},[66],[36,9113,9115,9118,9158,9161,9164],{"className":9114},[71],[36,9116],{"className":9117,"style":108},[75],[36,9119,9121,9124],{"className":9120},[80],[36,9122,7157],{"className":9123,"style":7222},[80,81],[36,9125,9127],{"className":9126},[272],[36,9128,9130,9150],{"className":9129},[276,384],[36,9131,9133,9147],{"className":9132},[280],[36,9134,9136],{"className":9135,"style":3344},[284],[36,9137,9138,9141],{"style":7608},[36,9139],{"className":9140,"style":292},[291],[36,9142,9144],{"className":9143},[296,297,298,299],[36,9145,3146],{"className":9146},[80,81,299],[36,9148,416],{"className":9149},[415],[36,9151,9153],{"className":9152},[280],[36,9154,9156],{"className":9155,"style":423},[284],[36,9157],{},[36,9159],{"className":9160,"style":224},[223],[36,9162,346],{"className":9163},[228],[36,9165],{"className":9166,"style":224},[223],[36,9168,9170,9173,9213,9216,9256,9259,9262,9265],{"className":9169},[71],[36,9171],{"className":9172,"style":205},[75],[36,9174,9176,9179],{"className":9175},[80],[36,9177,7056],{"className":9178,"style":7083},[80,81],[36,9180,9182],{"className":9181},[272],[36,9183,9185,9205],{"className":9184},[276,384],[36,9186,9188,9202],{"className":9187},[280],[36,9189,9191],{"className":9190,"style":7096},[284],[36,9192,9193,9196],{"style":7099},[36,9194],{"className":9195,"style":292},[291],[36,9197,9199],{"className":9198},[296,297,298,299],[36,9200,2356],{"className":9201,"style":1433},[80,81,299],[36,9203,416],{"className":9204},[415],[36,9206,9208],{"className":9207},[280],[36,9209,9211],{"className":9210,"style":423},[284],[36,9212],{},[36,9214,163],{"className":9215},[212],[36,9217,9219,9222],{"className":9218},[80],[36,9220,2119],{"className":9221},[80,81],[36,9223,9225],{"className":9224},[272],[36,9226,9228,9248],{"className":9227},[276,384],[36,9229,9231,9245],{"className":9230},[280],[36,9232,9234],{"className":9233,"style":3344},[284],[36,9235,9236,9239],{"style":2204},[36,9237],{"className":9238,"style":292},[291],[36,9240,9242],{"className":9241},[296,297,298,299],[36,9243,3146],{"className":9244},[80,81,299],[36,9246,416],{"className":9247},[415],[36,9249,9251],{"className":9250},[280],[36,9252,9254],{"className":9253,"style":423},[284],[36,9255],{},[36,9257,168],{"className":9258},[219],[36,9260],{"className":9261,"style":245},[223],[36,9263,5284],{"className":9264},[249],[36,9266],{"className":9267,"style":245},[223],[36,9269,9271,9274,9314,9317,9321,9324,9364,9367,9370],{"className":9270},[71],[36,9272],{"className":9273,"style":108},[75],[36,9275,9277,9280],{"className":9276},[80],[36,9278,9073],{"className":9279},[80,81],[36,9281,9283],{"className":9282},[272],[36,9284,9286,9306],{"className":9285},[276,384],[36,9287,9289,9303],{"className":9288},[280],[36,9290,9292],{"className":9291,"style":3344},[284],[36,9293,9294,9297],{"style":2204},[36,9295],{"className":9296,"style":292},[291],[36,9298,9300],{"className":9299},[296,297,298,299],[36,9301,3146],{"className":9302},[80,81,299],[36,9304,416],{"className":9305},[415],[36,9307,9309],{"className":9308},[280],[36,9310,9312],{"className":9311,"style":423},[284],[36,9313],{},[36,9315,2124],{"className":9316},[2228],[36,9318],{"className":9319,"style":9320},[223],"margin-right:1em;",[36,9322],{"className":9323,"style":2232},[223],[36,9325,9327,9330],{"className":9326},[80],[36,9328,9073],{"className":9329},[80,81],[36,9331,9333],{"className":9332},[272],[36,9334,9336,9356],{"className":9335},[276,384],[36,9337,9339,9353],{"className":9338},[280],[36,9340,9342],{"className":9341,"style":3344},[284],[36,9343,9344,9347],{"style":2204},[36,9345],{"className":9346,"style":292},[291],[36,9348,9350],{"className":9349},[296,297,298,299],[36,9351,3146],{"className":9352},[80,81,299],[36,9354,416],{"className":9355},[415],[36,9357,9359],{"className":9358},[280],[36,9360,9362],{"className":9361,"style":423},[284],[36,9363],{},[36,9365],{"className":9366,"style":224},[223],[36,9368,9089],{"className":9369},[228],[36,9371],{"className":9372,"style":224},[223],[36,9374,9376,9379,9382,9385,9388,9391,9394,9423],{"className":9375},[71],[36,9377],{"className":9378,"style":5145},[75],[36,9380,127],{"className":9381,"style":143},[80,81],[36,9383,163],{"className":9384},[212],[36,9386,4198],{"className":9387},[80],[36,9389,2124],{"className":9390},[2228],[36,9392],{"className":9393,"style":2232},[223],[36,9395,9397,9400],{"className":9396},[80],[36,9398,7703],{"className":9399,"style":7222},[80,81],[36,9401,9403],{"className":9402},[272],[36,9404,9406],{"className":9405},[276],[36,9407,9409],{"className":9408},[280],[36,9410,9412],{"className":9411,"style":1476},[284],[36,9413,9414,9417],{"style":480},[36,9415],{"className":9416,"style":292},[291],[36,9418,9420],{"className":9419},[296,297,298,299],[36,9421,1670],{"className":9422},[80,299],[36,9424,168],{"className":9425},[219],[15,9427,9428],{},"对数似然：",[36,9430,9432],{"className":9431,"translate":40},[314],[36,9433,9435,9516],{"className":9434,"translate":40},[39],[36,9436,9438],{"className":9437},[44],[46,9439,9440],{"xmlns":48,"display":323},[50,9441,9442,9513],{},[53,9443,9444,9446,9448,9450,9452,9454,9456,9470,9476,9478,9484,9486,9492,9494,9500,9502,9508,9510],{},[56,9445,3428],{"mathvariant":1388},[160,9447,163],{"stretchy":162},[56,9449,2356],{},[160,9451,168],{"stretchy":162},[160,9453,346],{},[160,9455,178],{},[4169,9457,9458,9460],{},[173,9459,175],{},[53,9461,9462,9464],{},[173,9463,1670],{},[182,9465,9466,9468],{},[56,9467,7703],{},[173,9469,1670],{},[2503,9471,9472,9474],{},[160,9473,3464],{},[56,9475,3146],{},[160,9477,163],{"stretchy":162},[329,9479,9480,9482],{},[56,9481,7157],{},[56,9483,3146],{},[160,9485,178],{},[329,9487,9488,9490],{},[56,9489,7056],{},[56,9491,2356],{},[160,9493,163],{"stretchy":162},[329,9495,9496,9498],{},[56,9497,2119],{},[56,9499,3146],{},[160,9501,168],{"stretchy":162},[182,9503,9504,9506],{},[160,9505,168],{"stretchy":162},[173,9507,1670],{},[160,9509,5284],{},[56,9511,9512],{},"C",[60,9514,9515],{"encoding":62},"\\ell(\\theta) = -\\frac{1}{2\\sigma^2} \\sum_i (y_i - f_\\theta(x_i))^2 + C",[36,9517,9519,9546,9747,9877],{"className":9518,"ariaHidden":67},[66],[36,9520,9522,9525,9528,9531,9534,9537,9540,9543],{"className":9521},[71],[36,9523],{"className":9524,"style":205},[75],[36,9526,3428],{"className":9527},[80],[36,9529,163],{"className":9530},[212],[36,9532,2356],{"className":9533,"style":1433},[80,81],[36,9535,168],{"className":9536},[219],[36,9538],{"className":9539,"style":224},[223],[36,9541,346],{"className":9542},[228],[36,9544],{"className":9545,"style":224},[223],[36,9547,9549,9553,9556,9648,9651,9695,9698,9738,9741,9744],{"className":9548},[71],[36,9550],{"className":9551,"style":9552},[75],"height:2.5991em;vertical-align:-1.2777em;",[36,9554,178],{"className":9555},[80],[36,9557,9559,9562,9645],{"className":9558},[80],[36,9560],{"className":9561},[212,4218],[36,9563,9565],{"className":9564},[4169],[36,9566,9568,9637],{"className":9567},[276,384],[36,9569,9571,9634],{"className":9570},[280],[36,9572,9575,9615,9623],{"className":9573,"style":9574},[284],"height:1.3214em;",[36,9576,9577,9580],{"style":4234},[36,9578],{"className":9579,"style":2037},[291],[36,9581,9583,9586],{"className":9582},[80],[36,9584,1670],{"className":9585},[80],[36,9587,9589,9592],{"className":9588},[80],[36,9590,7703],{"className":9591,"style":7222},[80,81],[36,9593,9595],{"className":9594},[272],[36,9596,9598],{"className":9597},[276],[36,9599,9601],{"className":9600},[280],[36,9602,9604],{"className":9603,"style":4437},[284],[36,9605,9606,9609],{"style":4440},[36,9607],{"className":9608,"style":292},[291],[36,9610,9612],{"className":9611},[296,297,298,299],[36,9613,1670],{"className":9614},[80,299],[36,9616,9617,9620],{"style":4249},[36,9618],{"className":9619,"style":2037},[291],[36,9621],{"className":9622,"style":4257},[4256],[36,9624,9625,9628],{"style":4260},[36,9626],{"className":9627,"style":2037},[291],[36,9629,9631],{"className":9630},[80],[36,9632,175],{"className":9633},[80],[36,9635,416],{"className":9636},[415],[36,9638,9640],{"className":9639},[280],[36,9641,9643],{"className":9642,"style":4297},[284],[36,9644],{},[36,9646],{"className":9647},[219,4218],[36,9649],{"className":9650,"style":2232},[223],[36,9652,9654],{"className":9653},[2614,2626],[36,9655,9657,9687],{"className":9656},[276,384],[36,9658,9660,9684],{"className":9659},[280],[36,9661,9663,9674],{"className":9662,"style":3226},[284],[36,9664,9665,9668],{"style":4999},[36,9666],{"className":9667,"style":3233},[291],[36,9669,9671],{"className":9670},[296,297,298,299],[36,9672,3146],{"className":9673},[80,81,299],[36,9675,9676,9679],{"style":3294},[36,9677],{"className":9678,"style":3233},[291],[36,9680,9681],{},[36,9682,3464],{"className":9683},[2614,3303,3304],[36,9685,416],{"className":9686},[415],[36,9688,9690],{"className":9689},[280],[36,9691,9693],{"className":9692,"style":5049},[284],[36,9694],{},[36,9696,163],{"className":9697},[212],[36,9699,9701,9704],{"className":9700},[80],[36,9702,7157],{"className":9703,"style":7222},[80,81],[36,9705,9707],{"className":9706},[272],[36,9708,9710,9730],{"className":9709},[276,384],[36,9711,9713,9727],{"className":9712},[280],[36,9714,9716],{"className":9715,"style":3344},[284],[36,9717,9718,9721],{"style":7608},[36,9719],{"className":9720,"style":292},[291],[36,9722,9724],{"className":9723},[296,297,298,299],[36,9725,3146],{"className":9726},[80,81,299],[36,9728,416],{"className":9729},[415],[36,9731,9733],{"className":9732},[280],[36,9734,9736],{"className":9735,"style":423},[284],[36,9737],{},[36,9739],{"className":9740,"style":245},[223],[36,9742,178],{"className":9743},[249],[36,9745],{"className":9746,"style":245},[223],[36,9748,9750,9753,9793,9796,9836,9839,9868,9871,9874],{"className":9749},[71],[36,9751],{"className":9752,"style":5145},[75],[36,9754,9756,9759],{"className":9755},[80],[36,9757,7056],{"className":9758,"style":7083},[80,81],[36,9760,9762],{"className":9761},[272],[36,9763,9765,9785],{"className":9764},[276,384],[36,9766,9768,9782],{"className":9767},[280],[36,9769,9771],{"className":9770,"style":7096},[284],[36,9772,9773,9776],{"style":7099},[36,9774],{"className":9775,"style":292},[291],[36,9777,9779],{"className":9778},[296,297,298,299],[36,9780,2356],{"className":9781,"style":1433},[80,81,299],[36,9783,416],{"className":9784},[415],[36,9786,9788],{"className":9787},[280],[36,9789,9791],{"className":9790,"style":423},[284],[36,9792],{},[36,9794,163],{"className":9795},[212],[36,9797,9799,9802],{"className":9798},[80],[36,9800,2119],{"className":9801},[80,81],[36,9803,9805],{"className":9804},[272],[36,9806,9808,9828],{"className":9807},[276,384],[36,9809,9811,9825],{"className":9810},[280],[36,9812,9814],{"className":9813,"style":3344},[284],[36,9815,9816,9819],{"style":2204},[36,9817],{"className":9818,"style":292},[291],[36,9820,9822],{"className":9821},[296,297,298,299],[36,9823,3146],{"className":9824},[80,81,299],[36,9826,416],{"className":9827},[415],[36,9829,9831],{"className":9830},[280],[36,9832,9834],{"className":9833,"style":423},[284],[36,9835],{},[36,9837,168],{"className":9838},[219],[36,9840,9842,9845],{"className":9841},[219],[36,9843,168],{"className":9844},[219],[36,9846,9848],{"className":9847},[272],[36,9849,9851],{"className":9850},[276],[36,9852,9854],{"className":9853},[280],[36,9855,9857],{"className":9856,"style":1476},[284],[36,9858,9859,9862],{"style":480},[36,9860],{"className":9861,"style":292},[291],[36,9863,9865],{"className":9864},[296,297,298,299],[36,9866,1670],{"className":9867},[80,299],[36,9869],{"className":9870,"style":245},[223],[36,9872,5284],{"className":9873},[249],[36,9875],{"className":9876,"style":245},[223],[36,9878,9880,9883],{"className":9879},[71],[36,9881],{"className":9882,"style":139},[75],[36,9884,9512],{"className":9885,"style":9886},[80,81],"margin-right:0.0715em;",[15,9888,9889,9890,9918],{},"我们看到，最大似然目标等价于最小化均方误差，其中 ",[36,9891,9893,9906],{"className":9892,"translate":40},[39],[36,9894,9896],{"className":9895},[44],[46,9897,9898],{"xmlns":48},[50,9899,9900,9904],{},[53,9901,9902],{},[56,9903,2356],{},[60,9905,2359],{"encoding":62},[36,9907,9909],{"className":9908,"ariaHidden":67},[66],[36,9910,9912,9915],{"className":9911},[71],[36,9913],{"className":9914,"style":2030},[75],[36,9916,2356],{"className":9917,"style":1433},[80,81]," = 神经网络的权重和偏置。",[15,9920,9921],{},"这里隐含的假设是误差服从高斯分布。如果我们改为假设拉普拉斯分布，MLE 就会导出 MAE 损失。这就解释了为什么不同任务选择不同的损失函数——它们对应着关于数据分布的不同假设。",[1090,9923,9924],{},[15,9925,9926],{},"在神经网络中，最大似然估计就是在所有权重和偏置的组合中，找出最可能让训练数据出现的那一组。",[15,9928,9929],{},"因此，每次我们用梯度下降更新权重时，本质上都在做 MLE 优化。",{"title":1041,"searchDepth":1042,"depth":1042,"links":9931},[9932,9933,9934],{"id":1257,"depth":1042,"text":1257},{"id":2089,"depth":1042,"text":2089},{"id":3976,"depth":1042,"text":3976},{},"\u002Fblog\u002F2026\u002F2026-06-08-explanation-of-neural-network-from-maximum-likelihood-estimation",{"title":1252,"description":1041},"blog\u002F2026\u002F2026-06-08-explanation-of-neural-network-from-maximum-likelihood-estimation","从统计学习的视角看，现代神经网络确实可以理解为一个大规模的最大似然估计（MLE）过程。具体而言，神经网络是一个参数化函数，而最常见的训练方式正是在数据上做最大似然估计。",[9941,1071,9942],"mathematics","statistics","2026-06-08T00:00:00+08:00","KqJWdWWflfGEXMq50rYAbKlpB_dzorqoPRucn8TA3Xw",{"left":9946,"top":9946,"width":9947,"height":9947,"rotate":9946,"vFlip":1060,"hFlip":1060,"body":9948},0,24,"\u003Cg fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-linejoin=\"round\" stroke-width=\"2\">\u003Cpath d=\"m4.748 7.645l-.331 3.644c-.05.54-.074.811-.03 1.07a2 2 0 0 0 .238.655c.131.228.325.422.711.808l5.176 5.176c.787.787 1.18 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