[{"data":1,"prerenderedAt":121808},["ShallowReactive",2],{"post-\u002Fblog\u002F2025\u002F2025-09-11-representer-theorem":3,"i-ci:tag":33443,"i-mingcute:time-line":33443},{"post":4,"nextPost":33453,"prevPost":61578},{"id":5,"title":6,"body":7,"description":33439,"draft":33440,"enableCodeHighlight":33440,"enableCodeHightlight":33440,"enableComment":33441,"enableD2":33441,"enableKatex":33441,"extension":33442,"image":33407,"important":33440,"location":33443,"meta":33444,"navigation":33441,"ogImage":33443,"path":33445,"seo":33446,"stem":33447,"summary":33448,"tags":33449,"time":33451,"__hash__":33452},"blog\u002Fblog\u002F2025\u002F2025-09-11-representer-theorem.md","核方法基石：Representer Theorem",{"type":8,"value":9,"toc":33406},"minimark",[10,91,94,99,291,423,601,604,812,845,954,1125,1128,1724,1899,1903,1908,2404,2834,3053,3770,3864,4139,4198,4270,4523,4526,5232,5305,5309,5312,5477,5608,5900,6142,6145,6149,7109,7241,7245,7249,8717,8720,9081,9316,9352,9356,9359,9675,9730,10159,10396,11034,11111,11114,11610,11642,11919,11922,12597,12962,13527,13539,13600,13604,13858,14293,14603,15461,15546,15854,16115,16119,16130,17039,17381,17684,17688,17692,18065,18069,18101,18646,18860,18863,19009,19065,19068,19352,20136,20143,20147,20150,20289,20389,20392,20546,20771,20775,20779,20903,21363,21611,21787,21819,22057,22272,22407,22468,22977,23177,23181,23184,23772,23878,24183,24186,24963,24966,25188,25441,25444,26308,26408,26746,27010,27014,27017,27547,27795,28328,28360,28364,28367,28469,28472,28833,28836,29073,29076,29408,29568,29572,29706,29939,29942,30428,30570,30573,30577,31414,31418,31879,31911,32692,32724,32728,32735,32738,33042,33210,33213,33217,33224,33342],[11,12,13,14,90],"p",{},"核方法最反直觉的一点是：你明明在一个可能是无限维的函数空间里做优化，最后却只需要解一个 ",[15,16,20,49],"span",{"className":17,"translate":19},[18],"katex","no",[15,21,24],{"className":22},[23],"katex-mathml",[25,26,28],"math",{"xmlns":27},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[29,30,31,44],"semantics",{},[32,33,34,38,42],"mrow",{},[35,36,37],"mi",{},"n",[39,40,41],"mo",{},"×",[35,43,37],{},[45,46,48],"annotation",{"encoding":47},"application\u002Fx-tex","n\\times n",[15,50,54,80],{"className":51,"ariaHidden":53},[52],"katex-html","true",[15,55,58,63,68,73,77],{"className":56},[57],"base",[15,59],{"className":60,"style":62},[61],"strut","height:0.6667em;vertical-align:-0.0833em;",[15,64,37],{"className":65},[66,67],"mord","mathnormal",[15,69],{"className":70,"style":72},[71],"mspace","margin-right:0.2222em;",[15,74,41],{"className":75},[76],"mbin",[15,78],{"className":79,"style":72},[71],[15,81,83,87],{"className":82},[57],[15,84],{"className":85,"style":86},[61],"height:0.4306em;",[15,88,37],{"className":89},[66,67]," 的线性系统或二次规划。把这件事说清楚的，就是表示定理。",[11,92,93],{},"本文会把这个结论拆成三件事：RKHS 为什么能定义「范数正则化」；正交分解为什么迫使最优解落在训练点张成的子空间里；以及这个结论的边界——什么样的正则项会让它失效。",[95,96,98],"h2",{"id":97},"一无穷维优化","一、无穷维优化",[11,100,101,102,232,233,261,262,290],{},"线性模型 ",[15,103,105,145],{"className":104,"translate":19},[18],[15,106,108],{"className":107},[23],[25,109,110],{"xmlns":27},[29,111,112,142],{},[32,113,114,117,121,124,127,130,140],{},[35,115,116],{},"f",[39,118,120],{"stretchy":119},"false","(",[35,122,123],{},"x",[39,125,126],{"stretchy":119},")",[39,128,129],{},"=",[131,132,133,136],"msup",{},[35,134,135],{},"w",[35,137,139],{"mathvariant":138},"normal","⊤",[35,141,123],{},[45,143,144],{"encoding":47},"f(x)=w^\\top x",[15,146,148,181],{"className":147,"ariaHidden":53},[52],[15,149,151,155,159,163,166,170,174,178],{"className":150},[57],[15,152],{"className":153,"style":154},[61],"height:1em;vertical-align:-0.25em;",[15,156,116],{"className":157,"style":158},[66,67],"margin-right:0.1076em;",[15,160,120],{"className":161},[162],"mopen",[15,164,123],{"className":165},[66,67],[15,167,126],{"className":168},[169],"mclose",[15,171],{"className":172,"style":173},[71],"margin-right:0.2778em;",[15,175,129],{"className":176},[177],"mrel",[15,179],{"className":180,"style":173},[71],[15,182,184,188,229],{"className":183},[57],[15,185],{"className":186,"style":187},[61],"height:0.8491em;",[15,189,191,195],{"className":190},[66],[15,192,135],{"className":193,"style":194},[66,67],"margin-right:0.0269em;",[15,196,199],{"className":197},[198],"msupsub",[15,200,203],{"className":201},[202],"vlist-t",[15,204,207],{"className":205},[206],"vlist-r",[15,208,211],{"className":209,"style":187},[210],"vlist",[15,212,214,219],{"style":213},"top:-3.063em;margin-right:0.05em;",[15,215],{"className":216,"style":218},[217],"pstrut","height:2.7em;",[15,220,226],{"className":221},[222,223,224,225],"sizing","reset-size6","size3","mtight",[15,227,139],{"className":228},[66,225],[15,230,123],{"className":231},[66,67]," 的表达能力受限于特征 ",[15,234,236,249],{"className":235,"translate":19},[18],[15,237,239],{"className":238},[23],[25,240,241],{"xmlns":27},[29,242,243,247],{},[32,244,245],{},[35,246,123],{},[45,248,123],{"encoding":47},[15,250,252],{"className":251,"ariaHidden":53},[52],[15,253,255,258],{"className":254},[57],[15,256],{"className":257,"style":86},[61],[15,259,123],{"className":260},[66,67]," 的坐标。想拟合非线性关系，一个标准做法是先把 ",[15,263,265,278],{"className":264,"translate":19},[18],[15,266,268],{"className":267},[23],[25,269,270],{"xmlns":27},[29,271,272,276],{},[32,273,274],{},[35,275,123],{},[45,277,123],{"encoding":47},[15,279,281],{"className":280,"ariaHidden":53},[52],[15,282,284,287],{"className":283},[57],[15,285],{"className":286,"style":86},[61],[15,288,123],{"className":289},[66,67]," 映射到一个更大的特征空间：",[15,292,295],{"className":293,"translate":19},[294],"katex-display",[15,296,298,339],{"className":297,"translate":19},[18],[15,299,301],{"className":300},[23],[25,302,304],{"xmlns":27,"display":303},"block",[29,305,306,336],{},[32,307,308,310,312,314,316,318,324,327,329,331,333],{},[35,309,116],{},[39,311,120],{"stretchy":119},[35,313,123],{},[39,315,126],{"stretchy":119},[39,317,129],{},[131,319,320,322],{},[35,321,135],{},[35,323,139],{"mathvariant":138},[35,325,326],{},"φ",[39,328,120],{"stretchy":119},[35,330,123],{},[39,332,126],{"stretchy":119},[39,334,335],{"separator":53},",",[45,337,338],{"encoding":47},"f(x)=w^\\top \\varphi(x),",[15,340,342,369],{"className":341,"ariaHidden":53},[52],[15,343,345,348,351,354,357,360,363,366],{"className":344},[57],[15,346],{"className":347,"style":154},[61],[15,349,116],{"className":350,"style":158},[66,67],[15,352,120],{"className":353},[162],[15,355,123],{"className":356},[66,67],[15,358,126],{"className":359},[169],[15,361],{"className":362,"style":173},[71],[15,364,129],{"className":365},[177],[15,367],{"className":368,"style":173},[71],[15,370,372,376,407,410,413,416,419],{"className":371},[57],[15,373],{"className":374,"style":375},[61],"height:1.1491em;vertical-align:-0.25em;",[15,377,379,382],{"className":378},[66],[15,380,135],{"className":381,"style":194},[66,67],[15,383,385],{"className":384},[198],[15,386,388],{"className":387},[202],[15,389,391],{"className":390},[206],[15,392,395],{"className":393,"style":394},[210],"height:0.8991em;",[15,396,398,401],{"style":397},"top:-3.113em;margin-right:0.05em;",[15,399],{"className":400,"style":218},[217],[15,402,404],{"className":403},[222,223,224,225],[15,405,139],{"className":406},[66,225],[15,408,326],{"className":409},[66,67],[15,411,120],{"className":412},[162],[15,414,123],{"className":415},[66,67],[15,417,126],{"className":418},[169],[15,420,335],{"className":421},[422],"mpunct",[11,424,425,426,541,542,571,572,600],{},"其中 ",[15,427,429,462],{"className":428,"translate":19},[18],[15,430,432],{"className":431},[23],[25,433,434],{"xmlns":27},[29,435,436,459],{},[32,437,438,440,443,447,450],{},[35,439,326],{},[39,441,442],{},":",[35,444,446],{"mathvariant":445},"script","X",[39,448,449],{},"→",[131,451,452,456],{},[35,453,455],{"mathvariant":454},"double-struck","R",[35,457,458],{},"d",[45,460,461],{"encoding":47},"\\varphi:\\mathcal X\\to\\mathbb R^d",[15,463,465,484,505],{"className":464,"ariaHidden":53},[52],[15,466,468,472,475,478,481],{"className":467},[57],[15,469],{"className":470,"style":471},[61],"height:0.625em;vertical-align:-0.1944em;",[15,473,326],{"className":474},[66,67],[15,476],{"className":477,"style":173},[71],[15,479,442],{"className":480},[177],[15,482],{"className":483,"style":173},[71],[15,485,487,491,496,499,502],{"className":486},[57],[15,488],{"className":489,"style":490},[61],"height:0.6833em;",[15,492,446],{"className":493,"style":495},[66,494],"mathcal","margin-right:0.1464em;",[15,497],{"className":498,"style":173},[71],[15,500,449],{"className":501},[177],[15,503],{"className":504,"style":173},[71],[15,506,508,511],{"className":507},[57],[15,509],{"className":510,"style":187},[61],[15,512,514,518],{"className":513},[66],[15,515,455],{"className":516},[66,517],"mathbb",[15,519,521],{"className":520},[198],[15,522,524],{"className":523},[202],[15,525,527],{"className":526},[206],[15,528,530],{"className":529,"style":187},[210],[15,531,532,535],{"style":213},[15,533],{"className":534,"style":218},[217],[15,536,538],{"className":537},[222,223,224,225],[15,539,458],{"className":540},[66,67,225]," 是特征映射。多项式回归、样条基、径向基函数都属于这个套路。问题在于，",[15,543,545,559],{"className":544,"translate":19},[18],[15,546,548],{"className":547},[23],[25,549,550],{"xmlns":27},[29,551,552,556],{},[32,553,554],{},[35,555,326],{},[45,557,558],{"encoding":47},"\\varphi",[15,560,562],{"className":561,"ariaHidden":53},[52],[15,563,565,568],{"className":564},[57],[15,566],{"className":567,"style":471},[61],[15,569,326],{"className":570},[66,67]," 的维数可以很大，甚至是无穷维。例如高斯核对应的是一个无穷维的特征空间；如果显式地写出 ",[15,573,575,588],{"className":574,"translate":19},[18],[15,576,578],{"className":577},[23],[25,579,580],{"xmlns":27},[29,581,582,586],{},[32,583,584],{},[35,585,326],{},[45,587,558],{"encoding":47},[15,589,591],{"className":590,"ariaHidden":53},[52],[15,592,594,597],{"className":593},[57],[15,595],{"className":596,"style":471},[61],[15,598,326],{"className":599},[66,67],"，优化根本无从下手。",[11,602,603],{},"核方法绕开了显式特征：它只通过内积",[15,605,607],{"className":606,"translate":19},[294],[15,608,610,671],{"className":609,"translate":19},[18],[15,611,613],{"className":612},[23],[25,614,615],{"xmlns":27,"display":303},[29,616,617,668],{},[32,618,619,622,624,626,628,636,638,640,643,645,647,649,651,653,655,657,663,665],{},[35,620,621],{},"k",[39,623,120],{"stretchy":119},[35,625,123],{},[39,627,335],{"separator":53},[131,629,630,632],{},[35,631,123],{},[39,633,635],{"mathvariant":138,"lspace":634,"rspace":634},"0em","′",[39,637,126],{"stretchy":119},[39,639,129],{},[39,641,642],{"stretchy":119},"⟨",[35,644,326],{},[39,646,120],{"stretchy":119},[35,648,123],{},[39,650,126],{"stretchy":119},[39,652,335],{"separator":53},[35,654,326],{},[39,656,120],{"stretchy":119},[131,658,659,661],{},[35,660,123],{},[39,662,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,664,126],{"stretchy":119},[39,666,667],{"stretchy":119},"⟩",[45,669,670],{"encoding":47},"k(x,x')=\\langle \\varphi(x),\\varphi(x')\\rangle",[15,672,674,743],{"className":673,"ariaHidden":53},[52],[15,675,677,681,685,688,691,694,698,731,734,737,740],{"className":676},[57],[15,678],{"className":679,"style":680},[61],"height:1.0519em;vertical-align:-0.25em;",[15,682,621],{"className":683,"style":684},[66,67],"margin-right:0.0315em;",[15,686,120],{"className":687},[162],[15,689,123],{"className":690},[66,67],[15,692,335],{"className":693},[422],[15,695],{"className":696,"style":697},[71],"margin-right:0.1667em;",[15,699,701,704],{"className":700},[66],[15,702,123],{"className":703},[66,67],[15,705,707],{"className":706},[198],[15,708,710],{"className":709},[202],[15,711,713],{"className":712},[206],[15,714,717],{"className":715,"style":716},[210],"height:0.8019em;",[15,718,719,722],{"style":397},[15,720],{"className":721,"style":218},[217],[15,723,725],{"className":724},[222,223,224,225],[15,726,728],{"className":727},[66,225],[15,729,635],{"className":730},[66,225],[15,732,126],{"className":733},[169],[15,735],{"className":736,"style":173},[71],[15,738,129],{"className":739},[177],[15,741],{"className":742,"style":173},[71],[15,744,746,749,752,755,758,761,764,767,770,773,776,808],{"className":745},[57],[15,747],{"className":748,"style":680},[61],[15,750,642],{"className":751},[162],[15,753,326],{"className":754},[66,67],[15,756,120],{"className":757},[162],[15,759,123],{"className":760},[66,67],[15,762,126],{"className":763},[169],[15,765,335],{"className":766},[422],[15,768],{"className":769,"style":697},[71],[15,771,326],{"className":772},[66,67],[15,774,120],{"className":775},[162],[15,777,779,782],{"className":778},[66],[15,780,123],{"className":781},[66,67],[15,783,785],{"className":784},[198],[15,786,788],{"className":787},[202],[15,789,791],{"className":790},[206],[15,792,794],{"className":793,"style":716},[210],[15,795,796,799],{"style":397},[15,797],{"className":798,"style":218},[217],[15,800,802],{"className":801},[222,223,224,225],[15,803,805],{"className":804},[66,225],[15,806,635],{"className":807},[66,225],[15,809,811],{"className":810},[169],")⟩",[11,813,814,815,844],{},"来访问特征空间。这个 ",[15,816,818,831],{"className":817,"translate":19},[18],[15,819,821],{"className":820},[23],[25,822,823],{"xmlns":27},[29,824,825,829],{},[32,826,827],{},[35,828,621],{},[45,830,621],{"encoding":47},[15,832,834],{"className":833,"ariaHidden":53},[52],[15,835,837,841],{"className":836},[57],[15,838],{"className":839,"style":840},[61],"height:0.6944em;",[15,842,621],{"className":843,"style":684},[66,67]," 就是核函数。于是，一个在特征空间中的线性模型，可以等价地写成",[15,846,848],{"className":847,"translate":19},[294],[15,849,851,891],{"className":850,"translate":19},[18],[15,852,854],{"className":853},[23],[25,855,856],{"xmlns":27,"display":303},[29,857,858,888],{},[32,859,860,862,864,866,868,870,872,874,876,878,880,882,884,886],{},[35,861,116],{},[39,863,120],{"stretchy":119},[35,865,123],{},[39,867,126],{"stretchy":119},[39,869,129],{},[39,871,642],{"stretchy":119},[35,873,135],{},[39,875,335],{"separator":53},[35,877,326],{},[39,879,120],{"stretchy":119},[35,881,123],{},[39,883,126],{"stretchy":119},[39,885,667],{"stretchy":119},[39,887,335],{"separator":53},[45,889,890],{"encoding":47},"f(x)=\\langle w,\\varphi(x)\\rangle,",[15,892,894,921],{"className":893,"ariaHidden":53},[52],[15,895,897,900,903,906,909,912,915,918],{"className":896},[57],[15,898],{"className":899,"style":154},[61],[15,901,116],{"className":902,"style":158},[66,67],[15,904,120],{"className":905},[162],[15,907,123],{"className":908},[66,67],[15,910,126],{"className":911},[169],[15,913],{"className":914,"style":173},[71],[15,916,129],{"className":917},[177],[15,919],{"className":920,"style":173},[71],[15,922,924,927,930,933,936,939,942,945,948,951],{"className":923},[57],[15,925],{"className":926,"style":154},[61],[15,928,642],{"className":929},[162],[15,931,135],{"className":932,"style":194},[66,67],[15,934,335],{"className":935},[422],[15,937],{"className":938,"style":697},[71],[15,940,326],{"className":941},[66,67],[15,943,120],{"className":944},[162],[15,946,123],{"className":947},[66,67],[15,949,811],{"className":950},[169],[15,952,335],{"className":953},[422],[11,955,956,957,985,986,1014,1015,1043,1044,1124],{},"而 ",[15,958,960,973],{"className":959,"translate":19},[18],[15,961,963],{"className":962},[23],[25,964,965],{"xmlns":27},[29,966,967,971],{},[32,968,969],{},[35,970,135],{},[45,972,135],{"encoding":47},[15,974,976],{"className":975,"ariaHidden":53},[52],[15,977,979,982],{"className":978},[57],[15,980],{"className":981,"style":86},[61],[15,983,135],{"className":984,"style":194},[66,67]," 所在的希尔伯特空间（更准确地说，是 ",[15,987,989,1002],{"className":988,"translate":19},[18],[15,990,992],{"className":991},[23],[25,993,994],{"xmlns":27},[29,995,996,1000],{},[32,997,998],{},[35,999,326],{},[45,1001,558],{"encoding":47},[15,1003,1005],{"className":1004,"ariaHidden":53},[52],[15,1006,1008,1011],{"className":1007},[57],[15,1009],{"className":1010,"style":471},[61],[15,1012,326],{"className":1013},[66,67]," 张成的闭包）就是核 ",[15,1016,1018,1031],{"className":1017,"translate":19},[18],[15,1019,1021],{"className":1020},[23],[25,1022,1023],{"xmlns":27},[29,1024,1025,1029],{},[32,1026,1027],{},[35,1028,621],{},[45,1030,621],{"encoding":47},[15,1032,1034],{"className":1033,"ariaHidden":53},[52],[15,1035,1037,1040],{"className":1036},[57],[15,1038],{"className":1039,"style":840},[61],[15,1041,621],{"className":1042,"style":684},[66,67]," 对应的再生核希尔伯特空间 ",[15,1045,1047,1067],{"className":1046,"translate":19},[18],[15,1048,1050],{"className":1049},[23],[25,1051,1052],{"xmlns":27},[29,1053,1054,1064],{},[32,1055,1056],{},[1057,1058,1059,1062],"msub",{},[35,1060,1061],{"mathvariant":445},"H",[35,1063,621],{},[45,1065,1066],{"encoding":47},"\\mathcal 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empirical risk 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是无穷维的，这看起来是一个无穷维优化问题。表示定理说：它其实是一个 ",[15,1871,1873,1886],{"className":1872,"translate":19},[18],[15,1874,1876],{"className":1875},[23],[25,1877,1878],{"xmlns":27},[29,1879,1880,1884],{},[32,1881,1882],{},[35,1883,37],{},[45,1885,37],{"encoding":47},[15,1887,1889],{"className":1888,"ariaHidden":53},[52],[15,1890,1892,1895],{"className":1891},[57],[15,1893],{"className":1894,"style":86},[61],[15,1896,37],{"className":1897},[66,67]," 维问题。",[95,1900,1902],{"id":1901},"二rkhs-速览","二、RKHS 速览",[1904,1905,1907],"h3",{"id":1906},"_21-正定核与再生性","2.1 正定核与再生性",[11,1909,1910,1911,1940,1941,2038,2039,2043,2044,2072,2073,2239,2240,2403],{},"设 ",[15,1912,1914,1928],{"className":1913,"translate":19},[18],[15,1915,1917],{"className":1916},[23],[25,1918,1919],{"xmlns":27},[29,1920,1921,1925],{},[32,1922,1923],{},[35,1924,446],{"mathvariant":445},[45,1926,1927],{"encoding":47},"\\mathcal 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R",[15,1971,1973,1991,2010,2028],{"className":1972,"ariaHidden":53},[52],[15,1974,1976,1979,1982,1985,1988],{"className":1975},[57],[15,1977],{"className":1978,"style":840},[61],[15,1980,621],{"className":1981,"style":684},[66,67],[15,1983],{"className":1984,"style":173},[71],[15,1986,442],{"className":1987},[177],[15,1989],{"className":1990,"style":173},[71],[15,1992,1994,1998,2001,2004,2007],{"className":1993},[57],[15,1995],{"className":1996,"style":1997},[61],"height:0.7667em;vertical-align:-0.0833em;",[15,1999,446],{"className":2000,"style":495},[66,494],[15,2002],{"className":2003,"style":72},[71],[15,2005,41],{"className":2006},[76],[15,2008],{"className":2009,"style":72},[71],[15,2011,2013,2016,2019,2022,2025],{"className":2012},[57],[15,2014],{"className":2015,"style":490},[61],[15,2017,446],{"className":2018,"style":495},[66,494],[15,2020],{"className":2021,"style":173},[71],[15,2023,449],{"className":2024},[177],[15,2026],{"className":2027,"style":173},[71],[15,2029,2031,2035],{"className":2030},[57],[15,2032],{"className":2033,"style":2034},[61],"height:0.6889em;",[15,2036,455],{"className":2037},[66,517]," 称为",[2040,2041,2042],"strong",{},"正定核","（positive definite kernel），如果对任意 ",[15,2045,2047,2060],{"className":2046,"translate":19},[18],[15,2048,2050],{"className":2049},[23],[25,2051,2052],{"xmlns":27},[29,2053,2054,2058],{},[32,2055,2056],{},[35,2057,37],{},[45,2059,37],{"encoding":47},[15,2061,2063],{"className":2062,"ariaHidden":53},[52],[15,2064,2066,2069],{"className":2065},[57],[15,2067],{"className":2068,"style":86},[61],[15,2070,37],{"className":2071},[66,67],"、任意 ",[15,2074,2076,2111],{"className":2075,"translate":19},[18],[15,2077,2079],{"className":2078},[23],[25,2080,2081],{"xmlns":27},[29,2082,2083,2108],{},[32,2084,2085,2091,2093,2096,2098,2104,2106],{},[1057,2086,2087,2089],{},[35,2088,123],{},[1209,2090,1211],{},[39,2092,335],{"separator":53},[39,2094,2095],{},"…",[39,2097,335],{"separator":53},[1057,2099,2100,2102],{},[35,2101,123],{},[35,2103,37],{},[39,2105,1187],{},[35,2107,446],{"mathvariant":445},[45,2109,2110],{"encoding":47},"x_1,\\dots,x_n\\in\\mathcal X",[15,2112,2114,2230],{"className":2113,"ariaHidden":53},[52],[15,2115,2117,2121,2162,2165,2168,2171,2174,2177,2180,2221,2224,2227],{"className":2116},[57],[15,2118],{"className":2119,"style":2120},[61],"height:0.7335em;vertical-align:-0.1944em;",[15,2122,2124,2127],{"className":2123},[66],[15,2125,123],{"className":2126},[66,67],[15,2128,2130],{"className":2129},[198],[15,2131,2133,2154],{"className":2132},[202,1090],[15,2134,2136,2151],{"className":2135},[206],[15,2137,2140],{"className":2138,"style":2139},[210],"height:0.3011em;",[15,2141,2142,2145],{"style":1584},[15,2143],{"className":2144,"style":218},[217],[15,2146,2148],{"className":2147},[222,223,224,225],[15,2149,1211],{"className":2150},[66,225],[15,2152,1114],{"className":2153},[1113],[15,2155,2157],{"className":2156},[206],[15,2158,2160],{"className":2159,"style":1121},[210],[15,2161],{},[15,2163,335],{"className":2164},[422],[15,2166],{"className":2167,"style":697},[71],[15,2169,2095],{"className":2170},[1413],[15,2172],{"className":2173,"style":697},[71],[15,2175,335],{"className":2176},[422],[15,2178],{"className":2179,"style":697},[71],[15,2181,2183,2186],{"className":2182},[66],[15,2184,123],{"className":2185},[66,67],[15,2187,2189],{"className":2188},[198],[15,2190,2192,2213],{"className":2191},[202,1090],[15,2193,2195,2210],{"className":2194},[206],[15,2196,2199],{"className":2197,"style":2198},[210],"height:0.1514em;",[15,2200,2201,2204],{"style":1584},[15,2202],{"className":2203,"style":218},[217],[15,2205,2207],{"className":2206},[222,223,224,225],[15,2208,37],{"className":2209},[66,67,225],[15,2211,1114],{"className":2212},[1113],[15,2214,2216],{"className":2215},[206],[15,2217,2219],{"className":2218,"style":1121},[210],[15,2220],{},[15,2222],{"className":2223,"style":173},[71],[15,2225,1187],{"className":2226},[177],[15,2228],{"className":2229,"style":173},[71],[15,2231,2233,2236],{"className":2232},[57],[15,2234],{"className":2235,"style":490},[61],[15,2237,446],{"className":2238,"style":495},[66,494]," 和任意 ",[15,2241,2243,2278],{"className":2242,"translate":19},[18],[15,2244,2246],{"className":2245},[23],[25,2247,2248],{"xmlns":27},[29,2249,2250,2275],{},[32,2251,2252,2259,2261,2263,2265,2271,2273],{},[1057,2253,2254,2257],{},[35,2255,2256],{},"c",[1209,2258,1211],{},[39,2260,335],{"separator":53},[39,2262,2095],{},[39,2264,335],{"separator":53},[1057,2266,2267,2269],{},[35,2268,2256],{},[35,2270,37],{},[39,2272,1187],{},[35,2274,455],{"mathvariant":454},[45,2276,2277],{"encoding":47},"c_1,\\dots,c_n\\in\\mathbb R",[15,2279,2281,2394],{"className":2280,"ariaHidden":53},[52],[15,2282,2284,2287,2327,2330,2333,2336,2339,2342,2345,2385,2388,2391],{"className":2283},[57],[15,2285],{"className":2286,"style":2120},[61],[15,2288,2290,2293],{"className":2289},[66],[15,2291,2256],{"className":2292},[66,67],[15,2294,2296],{"className":2295},[198],[15,2297,2299,2319],{"className":2298},[202,1090],[15,2300,2302,2316],{"className":2301},[206],[15,2303,2305],{"className":2304,"style":2139},[210],[15,2306,2307,2310],{"style":1584},[15,2308],{"className":2309,"style":218},[217],[15,2311,2313],{"className":2312},[222,223,224,225],[15,2314,1211],{"className":2315},[66,225],[15,2317,1114],{"className":2318},[1113],[15,2320,2322],{"className":2321},[206],[15,2323,2325],{"className":2324,"style":1121},[210],[15,2326],{},[15,2328,335],{"className":2329},[422],[15,2331],{"className":2332,"style":697},[71],[15,2334,2095],{"className":2335},[1413],[15,2337],{"className":2338,"style":697},[71],[15,2340,335],{"className":2341},[422],[15,2343],{"className":2344,"style":697},[71],[15,2346,2348,2351],{"className":2347},[66],[15,2349,2256],{"className":2350},[66,67],[15,2352,2354],{"className":2353},[198],[15,2355,2357,2377],{"className":2356},[202,1090],[15,2358,2360,2374],{"className":2359},[206],[15,2361,2363],{"className":2362,"style":2198},[210],[15,2364,2365,2368],{"style":1584},[15,2366],{"className":2367,"style":218},[217],[15,2369,2371],{"className":2370},[222,223,224,225],[15,2372,37],{"className":2373},[66,67,225],[15,2375,1114],{"className":2376},[1113],[15,2378,2380],{"className":2379},[206],[15,2381,2383],{"className":2382,"style":1121},[210],[15,2384],{},[15,2386],{"className":2387,"style":173},[71],[15,2389,1187],{"className":2390},[177],[15,2392],{"className":2393,"style":173},[71],[15,2395,2397,2400],{"className":2396},[57],[15,2398],{"className":2399,"style":2034},[61],[15,2401,455],{"className":2402},[66,517],"，都有",[15,2405,2407],{"className":2406,"translate":19},[294],[15,2408,2410,2491],{"className":2409,"translate":19},[18],[15,2411,2413],{"className":2412},[23],[25,2414,2415],{"xmlns":27,"display":303},[29,2416,2417,2488],{},[32,2418,2419,2433,2448,2454,2460,2463,2465,2467,2473,2475,2481,2483,2485],{},[1196,2420,2421,2423,2431],{},[39,2422,1200],{},[32,2424,2425,2427,2429],{},[35,2426,1205],{},[39,2428,129],{},[1209,2430,1211],{},[35,2432,37],{},[1196,2434,2435,2437,2446],{},[39,2436,1200],{},[32,2438,2439,2442,2444],{},[35,2440,2441],{},"j",[39,2443,129],{},[1209,2445,1211],{},[35,2447,37],{},[1057,2449,2450,2452],{},[35,2451,2256],{},[35,2453,1205],{},[1057,2455,2456,2458],{},[35,2457,2256],{},[35,2459,2441],{},[1274,2461,2462],{}," ",[35,2464,621],{},[39,2466,120],{"stretchy":119},[1057,2468,2469,2471],{},[35,2470,123],{},[35,2472,1205],{},[39,2474,335],{"separator":53},[1057,2476,2477,2479],{},[35,2478,123],{},[35,2480,2441],{},[39,2482,126],{"stretchy":119},[39,2484,1801],{},[1209,2486,2487],{},"0.",[45,2489,2490],{"encoding":47},"\\sum_{i=1}^n\\sum_{j=1}^n c_i c_j\\,k(x_i,x_j)\\ge 0.",[15,2492,2494,2825],{"className":2493,"ariaHidden":53},[52],[15,2495,2497,2501,2565,2568,2634,2637,2677,2718,2721,2724,2727,2767,2770,2773,2813,2816,2819,2822],{"className":2496},[57],[15,2498],{"className":2499,"style":2500},[61],"height:3.0652em;vertical-align:-1.4138em;",[15,2502,2504],{"className":2503},[1339,1351],[15,2505,2507,2557],{"className":2506},[202,1090],[15,2508,2510,2554],{"className":2509},[206],[15,2511,2513,2533,2543],{"className":2512,"style":1436},[210],[15,2514,2515,2518],{"style":1439},[15,2516],{"className":2517,"style":1443},[217],[15,2519,2521],{"className":2520},[222,223,224,225],[15,2522,2524,2527,2530],{"className":2523},[66,225],[15,2525,1205],{"className":2526},[66,67,225],[15,2528,129],{"className":2529},[177,225],[15,2531,1211],{"className":2532},[66,225],[15,2534,2535,2538],{"style":1461},[15,2536],{"className":2537,"style":1443},[217],[15,2539,2540],{},[15,2541,1200],{"className":2542},[1339,1470,1471],[15,2544,2545,2548],{"style":1474},[15,2546],{"className":2547,"style":1443},[217],[15,2549,2551],{"className":2550},[222,223,224,225],[15,2552,37],{"className":2553},[66,67,225],[15,2555,1114],{"className":2556},[1113],[15,2558,2560],{"className":2559},[206],[15,2561,2563],{"className":2562,"style":1493},[210],[15,2564],{},[15,2566],{"className":2567,"style":697},[71],[15,2569,2571],{"className":2570},[1339,1351],[15,2572,2574,2625],{"className":2573},[202,1090],[15,2575,2577,2622],{"className":2576},[206],[15,2578,2580,2601,2611],{"className":2579,"style":1436},[210],[15,2581,2582,2585],{"style":1439},[15,2583],{"className":2584,"style":1443},[217],[15,2586,2588],{"className":2587},[222,223,224,225],[15,2589,2591,2595,2598],{"className":2590},[66,225],[15,2592,2441],{"className":2593,"style":2594},[66,67,225],"margin-right:0.0572em;",[15,2596,129],{"className":2597},[177,225],[15,2599,1211],{"className":2600},[66,225],[15,2602,2603,2606],{"style":1461},[15,2604],{"className":2605,"style":1443},[217],[15,2607,2608],{},[15,2609,1200],{"className":2610},[1339,1470,1471],[15,2612,2613,2616],{"style":1474},[15,2614],{"className":2615,"style":1443},[217],[15,2617,2619],{"className":2618},[222,223,224,225],[15,2620,37],{"className":2621},[66,67,225],[15,2623,1114],{"className":2624},[1113],[15,2626,2628],{"className":2627},[206],[15,2629,2632],{"className":2630,"style":2631},[210],"height:1.4138em;",[15,2633],{},[15,2635],{"className":2636,"style":697},[71],[15,2638,2640,2643],{"className":2639},[66],[15,2641,2256],{"className":2642},[66,67],[15,2644,2646],{"className":2645},[198],[15,2647,2649,2669],{"className":2648},[202,1090],[15,2650,2652,2666],{"className":2651},[206],[15,2653,2655],{"className":2654,"style":1528},[210],[15,2656,2657,2660],{"style":1584},[15,2658],{"className":2659,"style":218},[217],[15,2661,2663],{"className":2662},[222,223,224,225],[15,2664,1205],{"className":2665},[66,67,225],[15,2667,1114],{"className":2668},[1113],[15,2670,2672],{"className":2671},[206],[15,2673,2675],{"className":2674,"style":1121},[210],[15,2676],{},[15,2678,2680,2683],{"className":2679},[66],[15,2681,2256],{"className":2682},[66,67],[15,2684,2686],{"className":2685},[198],[15,2687,2689,2709],{"className":2688},[202,1090],[15,2690,2692,2706],{"className":2691},[206],[15,2693,2695],{"className":2694,"style":1528},[210],[15,2696,2697,2700],{"style":1584},[15,2698],{"className":2699,"style":218},[217],[15,2701,2703],{"className":2702},[222,223,224,225],[15,2704,2441],{"className":2705,"style":2594},[66,67,225],[15,2707,1114],{"className":2708},[1113],[15,2710,2712],{"className":2711},[206],[15,2713,2716],{"className":2714,"style":2715},[210],"height:0.2861em;",[15,2717],{},[15,2719],{"className":2720,"style":697},[71],[15,2722,621],{"className":2723,"style":684},[66,67],[15,2725,120],{"className":2726},[162],[15,2728,2730,2733],{"className":2729},[66],[15,2731,123],{"className":2732},[66,67],[15,2734,2736],{"className":2735},[198],[15,2737,2739,2759],{"className":2738},[202,1090],[15,2740,2742,2756],{"className":2741},[206],[15,2743,2745],{"className":2744,"style":1528},[210],[15,2746,2747,2750],{"style":1584},[15,2748],{"className":2749,"style":218},[217],[15,2751,2753],{"className":2752},[222,223,224,225],[15,2754,1205],{"className":2755},[66,67,225],[15,2757,1114],{"className":2758},[1113],[15,2760,2762],{"className":2761},[206],[15,2763,2765],{"className":2764,"style":1121},[210],[15,2766],{},[15,2768,335],{"className":2769},[422],[15,2771],{"className":2772,"style":697},[71],[15,2774,2776,2779],{"className":2775},[66],[15,2777,123],{"className":2778},[66,67],[15,2780,2782],{"className":2781},[198],[15,2783,2785,2805],{"className":2784},[202,1090],[15,2786,2788,2802],{"className":2787},[206],[15,2789,2791],{"className":2790,"style":1528},[210],[15,2792,2793,2796],{"style":1584},[15,2794],{"className":2795,"style":218},[217],[15,2797,2799],{"className":2798},[222,223,224,225],[15,2800,2441],{"className":2801,"style":2594},[66,67,225],[15,2803,1114],{"className":2804},[1113],[15,2806,2808],{"className":2807},[206],[15,2809,2811],{"className":2810,"style":2715},[210],[15,2812],{},[15,2814,126],{"className":2815},[169],[15,2817],{"className":2818,"style":173},[71],[15,2820,1801],{"className":2821},[177],[15,2823],{"className":2824,"style":173},[71],[15,2826,2828,2831],{"className":2827},[57],[15,2829],{"className":2830,"style":1836},[61],[15,2832,2487],{"className":2833},[66],[11,2835,2836,2837,3052],{},"也就是说，核矩阵 ",[15,2838,2840,2885],{"className":2839,"translate":19},[18],[15,2841,2843],{"className":2842},[23],[25,2844,2845],{"xmlns":27},[29,2846,2847,2882],{},[32,2848,2849,2860,2862,2864,2866,2872,2874,2880],{},[1057,2850,2851,2854],{},[35,2852,2853],{},"K",[32,2855,2856,2858],{},[35,2857,1205],{},[35,2859,2441],{},[39,2861,129],{},[35,2863,621],{},[39,2865,120],{"stretchy":119},[1057,2867,2868,2870],{},[35,2869,123],{},[35,2871,1205],{},[39,2873,335],{"separator":53},[1057,2875,2876,2878],{},[35,2877,123],{},[35,2879,2441],{},[39,2881,126],{"stretchy":119},[45,2883,2884],{"encoding":47},"K_{ij}=k(x_i,x_j)",[15,2886,2888,2950],{"className":2887,"ariaHidden":53},[52],[15,2889,2891,2895,2941,2944,2947],{"className":2890},[57],[15,2892],{"className":2893,"style":2894},[61],"height:0.9694em;vertical-align:-0.2861em;",[15,2896,2898,2902],{"className":2897},[66],[15,2899,2853],{"className":2900,"style":2901},[66,67],"margin-right:0.0715em;",[15,2903,2905],{"className":2904},[198],[15,2906,2908,2933],{"className":2907},[202,1090],[15,2909,2911,2930],{"className":2910},[206],[15,2912,2914],{"className":2913,"style":1528},[210],[15,2915,2917,2920],{"style":2916},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[15,2918],{"className":2919,"style":218},[217],[15,2921,2923],{"className":2922},[222,223,224,225],[15,2924,2926],{"className":2925},[66,225],[15,2927,2929],{"className":2928,"style":2594},[66,67,225],"ij",[15,2931,1114],{"className":2932},[1113],[15,2934,2936],{"className":2935},[206],[15,2937,2939],{"className":2938,"style":2715},[210],[15,2940],{},[15,2942],{"className":2943,"style":173},[71],[15,2945,129],{"className":2946},[177],[15,2948],{"className":2949,"style":173},[71],[15,2951,2953,2957,2960,2963,3003,3006,3009,3049],{"className":2952},[57],[15,2954],{"className":2955,"style":2956},[61],"height:1.0361em;vertical-align:-0.2861em;",[15,2958,621],{"className":2959,"style":684},[66,67],[15,2961,120],{"className":2962},[162],[15,2964,2966,2969],{"className":2965},[66],[15,2967,123],{"className":2968},[66,67],[15,2970,2972],{"className":2971},[198],[15,2973,2975,2995],{"className":2974},[202,1090],[15,2976,2978,2992],{"className":2977},[206],[15,2979,2981],{"className":2980,"style":1528},[210],[15,2982,2983,2986],{"style":1584},[15,2984],{"className":2985,"style":218},[217],[15,2987,2989],{"className":2988},[222,223,224,225],[15,2990,1205],{"className":2991},[66,67,225],[15,2993,1114],{"className":2994},[1113],[15,2996,2998],{"className":2997},[206],[15,2999,3001],{"className":3000,"style":1121},[210],[15,3002],{},[15,3004,335],{"className":3005},[422],[15,3007],{"className":3008,"style":697},[71],[15,3010,3012,3015],{"className":3011},[66],[15,3013,123],{"className":3014},[66,67],[15,3016,3018],{"className":3017},[198],[15,3019,3021,3041],{"className":3020},[202,1090],[15,3022,3024,3038],{"className":3023},[206],[15,3025,3027],{"className":3026,"style":1528},[210],[15,3028,3029,3032],{"style":1584},[15,3030],{"className":3031,"style":218},[217],[15,3033,3035],{"className":3034},[222,223,224,225],[15,3036,2441],{"className":3037,"style":2594},[66,67,225],[15,3039,1114],{"className":3040},[1113],[15,3042,3044],{"className":3043},[206],[15,3045,3047],{"className":3046,"style":2715},[210],[15,3048],{},[15,3050,126],{"className":3051},[169]," 总是半正定的。",[3054,3055,3057,3188,3516,3735],"theorem",{"title":3056},"定理 2.1（Moore–Aronszajn）",[11,3058,3059,3060,3088,3089,3158,3159,3187],{},"对任意正定核 ",[15,3061,3063,3076],{"className":3062,"translate":19},[18],[15,3064,3066],{"className":3065},[23],[25,3067,3068],{"xmlns":27},[29,3069,3070,3074],{},[32,3071,3072],{},[35,3073,621],{},[45,3075,621],{"encoding":47},[15,3077,3079],{"className":3078,"ariaHidden":53},[52],[15,3080,3082,3085],{"className":3081},[57],[15,3083],{"className":3084,"style":840},[61],[15,3086,621],{"className":3087,"style":684},[66,67],"，存在唯一的希尔伯特空间 ",[15,3090,3092,3109],{"className":3091,"translate":19},[18],[15,3093,3095],{"className":3094},[23],[25,3096,3097],{"xmlns":27},[29,3098,3099,3107],{},[32,3100,3101],{},[1057,3102,3103,3105],{},[35,3104,1061],{"mathvariant":445},[35,3106,621],{},[45,3108,1066],{"encoding":47},[15,3110,3112],{"className":3111,"ariaHidden":53},[52],[15,3113,3115,3118],{"className":3114},[57],[15,3116],{"className":3117,"style":1076},[61],[15,3119,3121,3124],{"className":3120},[66],[15,3122,1061],{"className":3123,"style":1083},[66,494],[15,3125,3127],{"className":3126},[198],[15,3128,3130,3150],{"className":3129},[202,1090],[15,3131,3133,3147],{"className":3132},[206],[15,3134,3136],{"className":3135,"style":1097},[210],[15,3137,3138,3141],{"style":1100},[15,3139],{"className":3140,"style":218},[217],[15,3142,3144],{"className":3143},[222,223,224,225],[15,3145,621],{"className":3146,"style":684},[66,67,225],[15,3148,1114],{"className":3149},[1113],[15,3151,3153],{"className":3152},[206],[15,3154,3156],{"className":3155,"style":1121},[210],[15,3157],{},"，由 ",[15,3160,3162,3175],{"className":3161,"translate":19},[18],[15,3163,3165],{"className":3164},[23],[25,3166,3167],{"xmlns":27},[29,3168,3169,3173],{},[32,3170,3171],{},[35,3172,446],{"mathvariant":445},[45,3174,1927],{"encoding":47},[15,3176,3178],{"className":3177,"ariaHidden":53},[52],[15,3179,3181,3184],{"className":3180},[57],[15,3182],{"className":3183,"style":490},[61],[15,3185,446],{"className":3186,"style":495},[66,494]," 上的函数组成，使得：",[3189,3190,3191,3370],"ol",{},[3192,3193,3194,3195,3247,3248,3369],"li",{},"对每个 ",[15,3196,3198,3216],{"className":3197,"translate":19},[18],[15,3199,3201],{"className":3200},[23],[25,3202,3203],{"xmlns":27},[29,3204,3205,3213],{},[32,3206,3207,3209,3211],{},[35,3208,123],{},[39,3210,1187],{},[35,3212,446],{"mathvariant":445},[45,3214,3215],{"encoding":47},"x\\in\\mathcal X",[15,3217,3219,3238],{"className":3218,"ariaHidden":53},[52],[15,3220,3222,3226,3229,3232,3235],{"className":3221},[57],[15,3223],{"className":3224,"style":3225},[61],"height:0.5782em;vertical-align:-0.0391em;",[15,3227,123],{"className":3228},[66,67],[15,3230],{"className":3231,"style":173},[71],[15,3233,1187],{"className":3234},[177],[15,3236],{"className":3237,"style":173},[71],[15,3239,3241,3244],{"className":3240},[57],[15,3242],{"className":3243,"style":490},[61],[15,3245,446],{"className":3246,"style":495},[66,494],"，",[15,3249,3251,3284],{"className":3250,"translate":19},[18],[15,3252,3254],{"className":3253},[23],[25,3255,3256],{"xmlns":27},[29,3257,3258,3281],{},[32,3259,3260,3262,3264,3267,3269,3271,3273,3275],{},[35,3261,621],{},[39,3263,120],{"stretchy":119},[39,3265,3266],{},"⋅",[39,3268,335],{"separator":53},[35,3270,123],{},[39,3272,126],{"stretchy":119},[39,3274,1187],{},[1057,3276,3277,3279],{},[35,3278,1061],{"mathvariant":445},[35,3280,621],{},[45,3282,3283],{"encoding":47},"k(\\cdot,x)\\in\\mathcal H_k",[15,3285,3287,3323],{"className":3286,"ariaHidden":53},[52],[15,3288,3290,3293,3296,3299,3302,3305,3308,3311,3314,3317,3320],{"className":3289},[57],[15,3291],{"className":3292,"style":154},[61],[15,3294,621],{"className":3295,"style":684},[66,67],[15,3297,120],{"className":3298},[162],[15,3300,3266],{"className":3301},[66],[15,3303,335],{"className":3304},[422],[15,3306],{"className":3307,"style":697},[71],[15,3309,123],{"className":3310},[66,67],[15,3312,126],{"className":3313},[169],[15,3315],{"className":3316,"style":173},[71],[15,3318,1187],{"className":3319},[177],[15,3321],{"className":3322,"style":173},[71],[15,3324,3326,3329],{"className":3325},[57],[15,3327],{"className":3328,"style":1076},[61],[15,3330,3332,3335],{"className":3331},[66],[15,3333,1061],{"className":3334,"style":1083},[66,494],[15,3336,3338],{"className":3337},[198],[15,3339,3341,3361],{"className":3340},[202,1090],[15,3342,3344,3358],{"className":3343},[206],[15,3345,3347],{"className":3346,"style":1097},[210],[15,3348,3349,3352],{"style":1100},[15,3350],{"className":3351,"style":218},[217],[15,3353,3355],{"className":3354},[222,223,224,225],[15,3356,621],{"className":3357,"style":684},[66,67,225],[15,3359,1114],{"className":3360},[1113],[15,3362,3364],{"className":3363},[206],[15,3365,3367],{"className":3366,"style":1121},[210],[15,3368],{},"；",[3192,3371,3194,3372,3465,3466,3247],{},[15,3373,3375,3397],{"className":3374,"translate":19},[18],[15,3376,3378],{"className":3377},[23],[25,3379,3380],{"xmlns":27},[29,3381,3382,3394],{},[32,3383,3384,3386,3388],{},[35,3385,116],{},[39,3387,1187],{},[1057,3389,3390,3392],{},[35,3391,1061],{"mathvariant":445},[35,3393,621],{},[45,3395,3396],{"encoding":47},"f\\in\\mathcal H_k",[15,3398,3400,3419],{"className":3399,"ariaHidden":53},[52],[15,3401,3403,3407,3410,3413,3416],{"className":3402},[57],[15,3404],{"className":3405,"style":3406},[61],"height:0.8889em;vertical-align:-0.1944em;",[15,3408,116],{"className":3409,"style":158},[66,67],[15,3411],{"className":3412,"style":173},[71],[15,3414,1187],{"className":3415},[177],[15,3417],{"className":3418,"style":173},[71],[15,3420,3422,3425],{"className":3421},[57],[15,3423],{"className":3424,"style":1076},[61],[15,3426,3428,3431],{"className":3427},[66],[15,3429,1061],{"className":3430,"style":1083},[66,494],[15,3432,3434],{"className":3433},[198],[15,3435,3437,3457],{"className":3436},[202,1090],[15,3438,3440,3454],{"className":3439},[206],[15,3441,3443],{"className":3442,"style":1097},[210],[15,3444,3445,3448],{"style":1100},[15,3446],{"className":3447,"style":218},[217],[15,3449,3451],{"className":3450},[222,223,224,225],[15,3452,621],{"className":3453,"style":684},[66,67,225],[15,3455,1114],{"className":3456},[1113],[15,3458,3460],{"className":3459},[206],[15,3461,3463],{"className":3462,"style":1121},[210],[15,3464],{}," 和 ",[15,3467,3469,3486],{"className":3468,"translate":19},[18],[15,3470,3472],{"className":3471},[23],[25,3473,3474],{"xmlns":27},[29,3475,3476,3484],{},[32,3477,3478,3480,3482],{},[35,3479,123],{},[39,3481,1187],{},[35,3483,446],{"mathvariant":445},[45,3485,3215],{"encoding":47},[15,3487,3489,3507],{"className":3488,"ariaHidden":53},[52],[15,3490,3492,3495,3498,3501,3504],{"className":3491},[57],[15,3493],{"className":3494,"style":3225},[61],[15,3496,123],{"className":3497},[66,67],[15,3499],{"className":3500,"style":173},[71],[15,3502,1187],{"className":3503},[177],[15,3505],{"className":3506,"style":173},[71],[15,3508,3510,3513],{"className":3509},[57],[15,3511],{"className":3512,"style":490},[61],[15,3514,446],{"className":3515,"style":495},[66,494],[15,3517,3519],{"className":3518,"translate":19},[294],[15,3520,3522,3575],{"className":3521,"translate":19},[18],[15,3523,3525],{"className":3524},[23],[25,3526,3527],{"xmlns":27,"display":303},[29,3528,3529,3572],{},[32,3530,3531,3533,3535,3537,3539,3541,3543,3545,3547,3549,3559,3561,3563,3565,3567,3569],{},[39,3532,642],{"stretchy":119},[35,3534,116],{},[39,3536,335],{"separator":53},[35,3538,621],{},[39,3540,120],{"stretchy":119},[39,3542,3266],{},[39,3544,335],{"separator":53},[35,3546,123],{},[39,3548,126],{"stretchy":119},[1057,3550,3551,3553],{},[39,3552,667],{"stretchy":119},[1057,3554,3555,3557],{},[35,3556,1061],{"mathvariant":445},[35,3558,621],{},[39,3560,129],{},[35,3562,116],{},[39,3564,120],{"stretchy":119},[35,3566,123],{},[39,3568,126],{"stretchy":119},[35,3570,3571],{"mathvariant":138},".",[45,3573,3574],{"encoding":47},"\\langle f,k(\\cdot,x)\\rangle_{\\mathcal H_k}=f(x).",[15,3576,3578,3714],{"className":3577,"ariaHidden":53},[52],[15,3579,3581,3585,3588,3591,3594,3597,3600,3603,3606,3609,3612,3615,3618,3705,3708,3711],{"className":3580},[57],[15,3582],{"className":3583,"style":3584},[61],"height:1.0059em;vertical-align:-0.2559em;",[15,3586,642],{"className":3587},[162],[15,3589,116],{"className":3590,"style":158},[66,67],[15,3592,335],{"className":3593},[422],[15,3595],{"className":3596,"style":697},[71],[15,3598,621],{"className":3599,"style":684},[66,67],[15,3601,120],{"className":3602},[162],[15,3604,3266],{"className":3605},[66],[15,3607,335],{"className":3608},[422],[15,3610],{"className":3611,"style":697},[71],[15,3613,123],{"className":3614},[66,67],[15,3616,126],{"className":3617},[169],[15,3619,3621,3624],{"className":3620},[169],[15,3622,667],{"className":3623},[169],[15,3625,3627],{"className":3626},[198],[15,3628,3630,3696],{"className":3629},[202,1090],[15,3631,3633,3693],{"className":3632},[206],[15,3634,3637],{"className":3635,"style":3636},[210],"height:0.3283em;",[15,3638,3639,3642],{"style":1584},[15,3640],{"className":3641,"style":218},[217],[15,3643,3645],{"className":3644},[222,223,224,225],[15,3646,3648],{"className":3647},[66,225],[15,3649,3651,3654],{"className":3650},[66,225],[15,3652,1061],{"className":3653,"style":1083},[66,494,225],[15,3655,3657],{"className":3656},[198],[15,3658,3660,3684],{"className":3659},[202,1090],[15,3661,3663,3681],{"className":3662},[206],[15,3664,3667],{"className":3665,"style":3666},[210],"height:0.3448em;",[15,3668,3670,3674],{"style":3669},"top:-2.3488em;margin-left:-0.0097em;margin-right:0.0714em;",[15,3671],{"className":3672,"style":3673},[217],"height:2.5em;",[15,3675,3678],{"className":3676},[222,3677,1508,225],"reset-size3",[15,3679,621],{"className":3680,"style":684},[66,67,225],[15,3682,1114],{"className":3683},[1113],[15,3685,3687],{"className":3686},[206],[15,3688,3691],{"className":3689,"style":3690},[210],"height:0.1512em;",[15,3692],{},[15,3694,1114],{"className":3695},[1113],[15,3697,3699],{"className":3698},[206],[15,3700,3703],{"className":3701,"style":3702},[210],"height:0.2559em;",[15,3704],{},[15,3706],{"className":3707,"style":173},[71],[15,3709,129],{"className":3710},[177],[15,3712],{"className":3713,"style":173},[71],[15,3715,3717,3720,3723,3726,3729,3732],{"className":3716},[57],[15,3718],{"className":3719,"style":154},[61],[15,3721,116],{"className":3722,"style":158},[66,67],[15,3724,120],{"className":3725},[162],[15,3727,123],{"className":3728},[66,67],[15,3730,126],{"className":3731},[169],[15,3733,3571],{"className":3734},[66],[11,3736,3737,3738,3766,3767,1124],{},"这个空间称为 ",[15,3739,3741,3754],{"className":3740,"translate":19},[18],[15,3742,3744],{"className":3743},[23],[25,3745,3746],{"xmlns":27},[29,3747,3748,3752],{},[32,3749,3750],{},[35,3751,621],{},[45,3753,621],{"encoding":47},[15,3755,3757],{"className":3756,"ariaHidden":53},[52],[15,3758,3760,3763],{"className":3759},[57],[15,3761],{"className":3762,"style":840},[61],[15,3764,621],{"className":3765,"style":684},[66,67]," 的",[2040,3768,3769],{},"再生核希尔伯特空间",[11,3771,3772,3773,3776,3777,3805,3806,3863],{},"第二条就是",[2040,3774,3775],{},"再生性","（reproducing property）。它把「函数在 ",[15,3778,3780,3793],{"className":3779,"translate":19},[18],[15,3781,3783],{"className":3782},[23],[25,3784,3785],{"xmlns":27},[29,3786,3787,3791],{},[32,3788,3789],{},[35,3790,123],{},[45,3792,123],{"encoding":47},[15,3794,3796],{"className":3795,"ariaHidden":53},[52],[15,3797,3799,3802],{"className":3798},[57],[15,3800],{"className":3801,"style":86},[61],[15,3803,123],{"className":3804},[66,67]," 点的取值」变成了「与 ",[15,3807,3809,3833],{"className":3808,"translate":19},[18],[15,3810,3812],{"className":3811},[23],[25,3813,3814],{"xmlns":27},[29,3815,3816,3830],{},[32,3817,3818,3820,3822,3824,3826,3828],{},[35,3819,621],{},[39,3821,120],{"stretchy":119},[39,3823,3266],{},[39,3825,335],{"separator":53},[35,3827,123],{},[39,3829,126],{"stretchy":119},[45,3831,3832],{"encoding":47},"k(\\cdot,x)",[15,3834,3836],{"className":3835,"ariaHidden":53},[52],[15,3837,3839,3842,3845,3848,3851,3854,3857,3860],{"className":3838},[57],[15,3840],{"className":3841,"style":154},[61],[15,3843,621],{"className":3844,"style":684},[66,67],[15,3846,120],{"className":3847},[162],[15,3849,3266],{"className":3850},[66],[15,3852,335],{"className":3853},[422],[15,3855],{"className":3856,"style":697},[71],[15,3858,123],{"className":3859},[66,67],[15,3861,126],{"className":3862},[169]," 的内积」。等价地说，取值泛函",[15,3865,3867],{"className":3866,"translate":19},[294],[15,3868,3870,3928],{"className":3869,"translate":19},[18],[15,3871,3873],{"className":3872},[23],[25,3874,3875],{"xmlns":27,"display":303},[29,3876,3877,3925],{},[32,3878,3879,3886,3888,3894,3896,3898,3900,3903,3909,3911,3913,3915,3917,3919,3921,3923],{},[1057,3880,3881,3884],{},[35,3882,3883],{},"δ",[35,3885,123],{},[39,3887,442],{},[1057,3889,3890,3892],{},[35,3891,1061],{"mathvariant":445},[35,3893,621],{},[39,3895,449],{},[35,3897,455],{"mathvariant":454},[39,3899,335],{"separator":53},[71,3901],{"width":3902},"2em",[1057,3904,3905,3907],{},[35,3906,3883],{},[35,3908,123],{},[39,3910,120],{"stretchy":119},[35,3912,116],{},[39,3914,126],{"stretchy":119},[39,3916,129],{},[35,3918,116],{},[39,3920,120],{"stretchy":119},[35,3922,123],{},[39,3924,126],{"stretchy":119},[45,3926,3927],{"encoding":47},"\\delta_x:\\mathcal H_k\\to\\mathbb R,\\qquad \\delta_x(f)=f(x)",[15,3929,3931,3989,4044,4121],{"className":3930,"ariaHidden":53},[52],[15,3932,3934,3938,3980,3983,3986],{"className":3933},[57],[15,3935],{"className":3936,"style":3937},[61],"height:0.8444em;vertical-align:-0.15em;",[15,3939,3941,3945],{"className":3940},[66],[15,3942,3883],{"className":3943,"style":3944},[66,67],"margin-right:0.0379em;",[15,3946,3948],{"className":3947},[198],[15,3949,3951,3972],{"className":3950},[202,1090],[15,3952,3954,3969],{"className":3953},[206],[15,3955,3957],{"className":3956,"style":2198},[210],[15,3958,3960,3963],{"style":3959},"top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;",[15,3961],{"className":3962,"style":218},[217],[15,3964,3966],{"className":3965},[222,223,224,225],[15,3967,123],{"className":3968},[66,67,225],[15,3970,1114],{"className":3971},[1113],[15,3973,3975],{"className":3974},[206],[15,3976,3978],{"className":3977,"style":1121},[210],[15,3979],{},[15,3981],{"className":3982,"style":173},[71],[15,3984,442],{"className":3985},[177],[15,3987],{"className":3988,"style":173},[71],[15,3990,3992,3995,4035,4038,4041],{"className":3991},[57],[15,3993],{"className":3994,"style":1076},[61],[15,3996,3998,4001],{"className":3997},[66],[15,3999,1061],{"className":4000,"style":1083},[66,494],[15,4002,4004],{"className":4003},[198],[15,4005,4007,4027],{"className":4006},[202,1090],[15,4008,4010,4024],{"className":4009},[206],[15,4011,4013],{"className":4012,"style":1097},[210],[15,4014,4015,4018],{"style":1100},[15,4016],{"className":4017,"style":218},[217],[15,4019,4021],{"className":4020},[222,223,224,225],[15,4022,621],{"className":4023,"style":684},[66,67,225],[15,4025,1114],{"className":4026},[1113],[15,4028,4030],{"className":4029},[206],[15,4031,4033],{"className":4032,"style":1121},[210],[15,4034],{},[15,4036],{"className":4037,"style":173},[71],[15,4039,449],{"className":4040},[177],[15,4042],{"className":4043,"style":173},[71],[15,4045,4047,4050,4053,4056,4060,4063,4103,4106,4109,4112,4115,4118],{"className":4046},[57],[15,4048],{"className":4049,"style":154},[61],[15,4051,455],{"className":4052},[66,517],[15,4054,335],{"className":4055},[422],[15,4057],{"className":4058,"style":4059},[71],"margin-right:2em;",[15,4061],{"className":4062,"style":697},[71],[15,4064,4066,4069],{"className":4065},[66],[15,4067,3883],{"className":4068,"style":3944},[66,67],[15,4070,4072],{"className":4071},[198],[15,4073,4075,4095],{"className":4074},[202,1090],[15,4076,4078,4092],{"className":4077},[206],[15,4079,4081],{"className":4080,"style":2198},[210],[15,4082,4083,4086],{"style":3959},[15,4084],{"className":4085,"style":218},[217],[15,4087,4089],{"className":4088},[222,223,224,225],[15,4090,123],{"className":4091},[66,67,225],[15,4093,1114],{"className":4094},[1113],[15,4096,4098],{"className":4097},[206],[15,4099,4101],{"className":4100,"style":1121},[210],[15,4102],{},[15,4104,120],{"className":4105},[162],[15,4107,116],{"className":4108,"style":158},[66,67],[15,4110,126],{"className":4111},[169],[15,4113],{"className":4114,"style":173},[71],[15,4116,129],{"className":4117},[177],[15,4119],{"className":4120,"style":173},[71],[15,4122,4124,4127,4130,4133,4136],{"className":4123},[57],[15,4125],{"className":4126,"style":154},[61],[15,4128,116],{"className":4129,"style":158},[66,67],[15,4131,120],{"className":4132},[162],[15,4134,123],{"className":4135},[66,67],[15,4137,126],{"className":4138},[169],[11,4140,4141,4142,1124],{},"是有界线性泛函，且由 Riesz 表示定理，它对应的表示元正是 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\\sum_i a_i k(\\cdot,x_i),\\sum_j b_j k(\\cdot,x_j)\\Bigr\\rangle\n=\\sum_{i,j}a_i b_j\\,k(x_i,x_j),",[15,4660,4662,4988],{"className":4661,"ariaHidden":53},[52],[15,4663,4665,4669,4676,4721,4724,4764,4767,4770,4773,4776,4779,4819,4822,4825,4828,4872,4875,4915,4918,4921,4924,4927,4930,4970,4973,4979,4982,4985],{"className":4664},[57],[15,4666],{"className":4667,"style":4668},[61],"height:2.5638em;vertical-align:-1.4138em;",[15,4670,4672],{"className":4671},[162],[15,4673,642],{"className":4674},[1422,4675],"size2",[15,4677,4679],{"className":4678},[1339,1351],[15,4680,4682,4713],{"className":4681},[202,1090],[15,4683,4685,4710],{"className":4684},[206],[15,4686,4689,4700],{"className":4687,"style":4688},[210],"height:1.05em;",[15,4690,4691,4694],{"style":1439},[15,4692],{"className":4693,"style":1443},[217],[15,4695,4697],{"className":4696},[222,223,224,225],[15,4698,1205],{"className":4699},[66,67,225],[15,4701,4702,4705],{"style":1461},[15,4703],{"className":4704,"style":1443},[217],[15,4706,4707],{},[15,4708,1200],{"className":4709},[1339,1470,1471],[15,4711,1114],{"className":4712},[1113],[15,4714,4716],{"className":4715},[206],[15,4717,4719],{"className":4718,"style":1493},[210],[15,4720],{},[15,4722],{"className":4723,"style":697},[71],[15,4725,4727,4730],{"className":4726},[66],[15,4728,4554],{"className":4729},[66,67],[15,4731,4733],{"className":4732},[198],[15,4734,4736,4756],{"className":4735},[202,1090],[15,4737,4739,4753],{"className":4738},[206],[15,4740,4742],{"className":4741,"style":1528},[210],[15,4743,4744,4747],{"style":1584},[15,4745],{"className":4746,"style":218},[217],[15,4748,4750],{"className":4749},[222,223,224,225],[15,4751,1205],{"className":4752},[66,67,225],[15,4754,1114],{"className":4755},[1113],[15,4757,4759],{"className":4758},[206],[15,4760,4762],{"className":4761,"style":1121},[210],[15,4763],{},[15,4765,621],{"className":4766,"style":684},[66,67],[15,4768,120],{"className":4769},[162],[15,4771,3266],{"className":4772},[66],[15,4774,335],{"className":4775},[422],[15,4777],{"className":4778,"style":697},[71],[15,4780,4782,4785],{"className":4781},[66],[15,4783,123],{"className":4784},[66,67],[15,4786,4788],{"className":4787},[198],[15,4789,4791,4811],{"className":4790},[202,1090],[15,4792,4794,4808],{"className":4793},[206],[15,4795,4797],{"className":4796,"style":1528},[210],[15,4798,4799,4802],{"style":1584},[15,4800],{"className":4801,"style":218},[217],[15,4803,4805],{"className":4804},[222,223,224,225],[15,4806,1205],{"className":4807},[66,67,225],[15,4809,1114],{"className":4810},[1113],[15,4812,4814],{"className":4813},[206],[15,4815,4817],{"className":4816,"style":1121},[210],[15,4818],{},[15,4820,126],{"className":4821},[169],[15,4823,335],{"className":4824},[422],[15,4826],{"className":4827,"style":697},[71],[15,4829,4831],{"className":4830},[1339,1351],[15,4832,4834,4864],{"className":4833},[202,1090],[15,4835,4837,4861],{"className":4836},[206],[15,4838,4840,4851],{"className":4839,"style":4688},[210],[15,4841,4842,4845],{"style":1439},[15,4843],{"className":4844,"style":1443},[217],[15,4846,4848],{"className":4847},[222,223,224,225],[15,4849,2441],{"className":4850,"style":2594},[66,67,225],[15,4852,4853,4856],{"style":1461},[15,4854],{"className":4855,"style":1443},[217],[15,4857,4858],{},[15,4859,1200],{"className":4860},[1339,1470,1471],[15,4862,1114],{"className":4863},[1113],[15,4865,4867],{"className":4866},[206],[15,4868,4870],{"className":4869,"style":2631},[210],[15,4871],{},[15,4873],{"className":4874,"style":697},[71],[15,4876,4878,4881],{"className":4877},[66],[15,4879,4585],{"className":4880},[66,67],[15,4882,4884],{"className":4883},[198],[15,4885,4887,4907],{"className":4886},[202,1090],[15,4888,4890,4904],{"className":4889},[206],[15,4891,4893],{"className":4892,"style":1528},[210],[15,4894,4895,4898],{"style":1584},[15,4896],{"className":4897,"style":218},[217],[15,4899,4901],{"className":4900},[222,223,224,225],[15,4902,2441],{"className":4903,"style":2594},[66,67,225],[15,4905,1114],{"className":4906},[1113],[15,4908,4910],{"className":4909},[206],[15,4911,4913],{"className":4912,"style":2715},[210],[15,4914],{},[15,4916,621],{"className":4917,"style":684},[66,67],[15,4919,120],{"className":4920},[162],[15,4922,3266],{"className":4923},[66],[15,4925,335],{"className":4926},[422],[15,4928],{"className":4929,"style":697},[71],[15,4931,4933,4936],{"className":4932},[66],[15,4934,123],{"className":4935},[66,67],[15,4937,4939],{"className":4938},[198],[15,4940,4942,4962],{"className":4941},[202,1090],[15,4943,4945,4959],{"className":4944},[206],[15,4946,4948],{"className":4947,"style":1528},[210],[15,4949,4950,4953],{"style":1584},[15,4951],{"className":4952,"style":218},[217],[15,4954,4956],{"className":4955},[222,223,224,225],[15,4957,2441],{"className":4958,"style":2594},[66,67,225],[15,4960,1114],{"className":4961},[1113],[15,4963,4965],{"className":4964},[206],[15,4966,4968],{"className":4967,"style":2715},[210],[15,4969],{},[15,4971,126],{"className":4972},[169],[15,4974,4976],{"className":4975},[169],[15,4977,667],{"className":4978},[1422,4675],[15,4980],{"className":4981,"style":173},[71],[15,4983,129],{"className":4984},[177],[15,4986],{"className":4987,"style":173},[71],[15,4989,4991,4995,5048,5051,5091,5131,5134,5137,5140,5180,5183,5186,5226,5229],{"className":4990},[57],[15,4992],{"className":4993,"style":4994},[61],"height:2.4638em;vertical-align:-1.4138em;",[15,4996,4998],{"className":4997},[1339,1351],[15,4999,5001,5040],{"className":5000},[202,1090],[15,5002,5004,5037],{"className":5003},[206],[15,5005,5007,5027],{"className":5006,"style":4688},[210],[15,5008,5009,5012],{"style":1439},[15,5010],{"className":5011,"style":1443},[217],[15,5013,5015],{"className":5014},[222,223,224,225],[15,5016,5018,5021,5024],{"className":5017},[66,225],[15,5019,1205],{"className":5020},[66,67,225],[15,5022,335],{"className":5023},[422,225],[15,5025,2441],{"className":5026,"style":2594},[66,67,225],[15,5028,5029,5032],{"style":1461},[15,5030],{"className":5031,"style":1443},[217],[15,5033,5034],{},[15,5035,1200],{"className":5036},[1339,1470,1471],[15,5038,1114],{"className":5039},[1113],[15,5041,5043],{"className":5042},[206],[15,5044,5046],{"className":5045,"style":2631},[210],[15,5047],{},[15,5049],{"className":5050,"style":697},[71],[15,5052,5054,5057],{"className":5053},[66],[15,5055,4554],{"className":5056},[66,67],[15,5058,5060],{"className":5059},[198],[15,5061,5063,5083],{"className":5062},[202,1090],[15,5064,5066,5080],{"className":5065},[206],[15,5067,5069],{"className":5068,"style":1528},[210],[15,5070,5071,5074],{"style":1584},[15,5072],{"className":5073,"style":218},[217],[15,5075,5077],{"className":5076},[222,223,224,225],[15,5078,1205],{"className":5079},[66,67,225],[15,5081,1114],{"className":5082},[1113],[15,5084,5086],{"className":5085},[206],[15,5087,5089],{"className":5088,"style":1121},[210],[15,5090],{},[15,5092,5094,5097],{"className":5093},[66],[15,5095,4585],{"className":5096},[66,67],[15,5098,5100],{"className":5099},[198],[15,5101,5103,5123],{"className":5102},[202,1090],[15,5104,5106,5120],{"className":5105},[206],[15,5107,5109],{"className":5108,"style":1528},[210],[15,5110,5111,5114],{"style":1584},[15,5112],{"className":5113,"style":218},[217],[15,5115,5117],{"className":5116},[222,223,224,225],[15,5118,2441],{"className":5119,"style":2594},[66,67,225],[15,5121,1114],{"className":5122},[1113],[15,5124,5126],{"className":5125},[206],[15,5127,5129],{"className":5128,"style":2715},[210],[15,5130],{},[15,5132],{"className":5133,"style":697},[71],[15,5135,621],{"className":5136,"style":684},[66,67],[15,5138,120],{"className":5139},[162],[15,5141,5143,5146],{"className":5142},[66],[15,5144,123],{"className":5145},[66,67],[15,5147,5149],{"className":5148},[198],[15,5150,5152,5172],{"className":5151},[202,1090],[15,5153,5155,5169],{"className":5154},[206],[15,5156,5158],{"className":5157,"style":1528},[210],[15,5159,5160,5163],{"style":1584},[15,5161],{"className":5162,"style":218},[217],[15,5164,5166],{"className":5165},[222,223,224,225],[15,5167,1205],{"className":5168},[66,67,225],[15,5170,1114],{"className":5171},[1113],[15,5173,5175],{"className":5174},[206],[15,5176,5178],{"className":5177,"style":1121},[210],[15,5179],{},[15,5181,335],{"className":5182},[422],[15,5184],{"className":5185,"style":697},[71],[15,5187,5189,5192],{"className":5188},[66],[15,5190,123],{"className":5191},[66,67],[15,5193,5195],{"className":5194},[198],[15,5196,5198,5218],{"className":5197},[202,1090],[15,5199,5201,5215],{"className":5200},[206],[15,5202,5204],{"className":5203,"style":1528},[210],[15,5205,5206,5209],{"style":1584},[15,5207],{"className":5208,"style":218},[217],[15,5210,5212],{"className":5211},[222,223,224,225],[15,5213,2441],{"className":5214,"style":2594},[66,67,225],[15,5216,1114],{"className":5217},[1113],[15,5219,5221],{"className":5220},[206],[15,5222,5224],{"className":5223,"style":2715},[210],[15,5225],{},[15,5227,126],{"className":5228},[169],[15,5230,335],{"className":5231},[422],[11,5233,5234,5235,5304],{},"然后取完备化。这个构造也说明了为什么 ",[15,5236,5238,5255],{"className":5237,"translate":19},[18],[15,5239,5241],{"className":5240},[23],[25,5242,5243],{"xmlns":27},[29,5244,5245,5253],{},[32,5246,5247],{},[1057,5248,5249,5251],{},[35,5250,1061],{"mathvariant":445},[35,5252,621],{},[45,5254,1066],{"encoding":47},[15,5256,5258],{"className":5257,"ariaHidden":53},[52],[15,5259,5261,5264],{"className":5260},[57],[15,5262],{"className":5263,"style":1076},[61],[15,5265,5267,5270],{"className":5266},[66],[15,5268,1061],{"className":5269,"style":1083},[66,494],[15,5271,5273],{"className":5272},[198],[15,5274,5276,5296],{"className":5275},[202,1090],[15,5277,5279,5293],{"className":5278},[206],[15,5280,5282],{"className":5281,"style":1097},[210],[15,5283,5284,5287],{"style":1100},[15,5285],{"className":5286,"style":218},[217],[15,5288,5290],{"className":5289},[222,223,224,225],[15,5291,621],{"className":5292,"style":684},[66,67,225],[15,5294,1114],{"className":5295},[1113],[15,5297,5299],{"className":5298},[206],[15,5300,5302],{"className":5301,"style":1121},[210],[15,5303],{}," 里的函数都「足够好」：它们是由核函数线性组合逼近出来的，而核的选择决定了这个空间的光滑性。",[1904,5306,5308],{"id":5307},"_22-特征映射与核技巧","2.2 特征映射与核技巧",[11,5310,5311],{},"令",[15,5313,5315],{"className":5314,"translate":19},[294],[15,5316,5318,5362],{"className":5317,"translate":19},[18],[15,5319,5321],{"className":5320},[23],[25,5322,5323],{"xmlns":27,"display":303},[29,5324,5325,5359],{},[32,5326,5327,5329,5331,5333,5335,5337,5339,5341,5343,5345,5347,5349,5351,5357],{},[35,5328,326],{},[39,5330,120],{"stretchy":119},[35,5332,123],{},[39,5334,126],{"stretchy":119},[39,5336,129],{},[35,5338,621],{},[39,5340,120],{"stretchy":119},[39,5342,3266],{},[39,5344,335],{"separator":53},[35,5346,123],{},[39,5348,126],{"stretchy":119},[39,5350,1187],{},[1057,5352,5353,5355],{},[35,5354,1061],{"mathvariant":445},[35,5356,621],{},[35,5358,3571],{"mathvariant":138},[45,5360,5361],{"encoding":47},"\\varphi(x)=k(\\cdot,x)\\in\\mathcal H_k.",[15,5363,5365,5392,5428],{"className":5364,"ariaHidden":53},[52],[15,5366,5368,5371,5374,5377,5380,5383,5386,5389],{"className":5367},[57],[15,5369],{"className":5370,"style":154},[61],[15,5372,326],{"className":5373},[66,67],[15,5375,120],{"className":5376},[162],[15,5378,123],{"className":5379},[66,67],[15,5381,126],{"className":5382},[169],[15,5384],{"className":5385,"style":173},[71],[15,5387,129],{"className":5388},[177],[15,5390],{"className":5391,"style":173},[71],[15,5393,5395,5398,5401,5404,5407,5410,5413,5416,5419,5422,5425],{"className":5394},[57],[15,5396],{"className":5397,"style":154},[61],[15,5399,621],{"className":5400,"style":684},[66,67],[15,5402,120],{"className":5403},[162],[15,5405,3266],{"className":5406},[66],[15,5408,335],{"className":5409},[422],[15,5411],{"className":5412,"style":697},[71],[15,5414,123],{"className":5415},[66,67],[15,5417,126],{"className":5418},[169],[15,5420],{"className":5421,"style":173},[71],[15,5423,1187],{"className":5424},[177],[15,5426],{"className":5427,"style":173},[71],[15,5429,5431,5434,5474],{"className":5430},[57],[15,5432],{"className":5433,"style":1076},[61],[15,5435,5437,5440],{"className":5436},[66],[15,5438,1061],{"className":5439,"style":1083},[66,494],[15,5441,5443],{"className":5442},[198],[15,5444,5446,5466],{"className":5445},[202,1090],[15,5447,5449,5463],{"className":5448},[206],[15,5450,5452],{"className":5451,"style":1097},[210],[15,5453,5454,5457],{"style":1100},[15,5455],{"className":5456,"style":218},[217],[15,5458,5460],{"className":5459},[222,223,224,225],[15,5461,621],{"className":5462,"style":684},[66,67,225],[15,5464,1114],{"className":5465},[1113],[15,5467,5469],{"className":5468},[206],[15,5470,5472],{"className":5471,"style":1121},[210],[15,5473],{},[15,5475,3571],{"className":5476},[66],[11,5478,5479,5480,5508,5509,5537,5538,5607],{},"则 ",[15,5481,5483,5496],{"className":5482,"translate":19},[18],[15,5484,5486],{"className":5485},[23],[25,5487,5488],{"xmlns":27},[29,5489,5490,5494],{},[32,5491,5492],{},[35,5493,326],{},[45,5495,558],{"encoding":47},[15,5497,5499],{"className":5498,"ariaHidden":53},[52],[15,5500,5502,5505],{"className":5501},[57],[15,5503],{"className":5504,"style":471},[61],[15,5506,326],{"className":5507},[66,67]," 就是一个从 ",[15,5510,5512,5525],{"className":5511,"translate":19},[18],[15,5513,5515],{"className":5514},[23],[25,5516,5517],{"xmlns":27},[29,5518,5519,5523],{},[32,5520,5521],{},[35,5522,446],{"mathvariant":445},[45,5524,1927],{"encoding":47},[15,5526,5528],{"className":5527,"ariaHidden":53},[52],[15,5529,5531,5534],{"className":5530},[57],[15,5532],{"className":5533,"style":490},[61],[15,5535,446],{"className":5536,"style":495},[66,494]," 到 ",[15,5539,5541,5558],{"className":5540,"translate":19},[18],[15,5542,5544],{"className":5543},[23],[25,5545,5546],{"xmlns":27},[29,5547,5548,5556],{},[32,5549,5550],{},[1057,5551,5552,5554],{},[35,5553,1061],{"mathvariant":445},[35,5555,621],{},[45,5557,1066],{"encoding":47},[15,5559,5561],{"className":5560,"ariaHidden":53},[52],[15,5562,5564,5567],{"className":5563},[57],[15,5565],{"className":5566,"style":1076},[61],[15,5568,5570,5573],{"className":5569},[66],[15,5571,1061],{"className":5572,"style":1083},[66,494],[15,5574,5576],{"className":5575},[198],[15,5577,5579,5599],{"className":5578},[202,1090],[15,5580,5582,5596],{"className":5581},[206],[15,5583,5585],{"className":5584,"style":1097},[210],[15,5586,5587,5590],{"style":1100},[15,5588],{"className":5589,"style":218},[217],[15,5591,5593],{"className":5592},[222,223,224,225],[15,5594,621],{"className":5595,"style":684},[66,67,225],[15,5597,1114],{"className":5598},[1113],[15,5600,5602],{"className":5601},[206],[15,5603,5605],{"className":5604,"style":1121},[210],[15,5606],{}," 的特征映射，并且",[15,5609,5611],{"className":5610,"translate":19},[294],[15,5612,5614,5680],{"className":5613,"translate":19},[18],[15,5615,5617],{"className":5616},[23],[25,5618,5619],{"xmlns":27,"display":303},[29,5620,5621,5677],{},[32,5622,5623,5625,5627,5629,5631,5637,5639,5641,5643,5645,5647,5649,5651,5653,5655,5657,5663,5665,5675],{},[35,5624,621],{},[39,5626,120],{"stretchy":119},[35,5628,123],{},[39,5630,335],{"separator":53},[131,5632,5633,5635],{},[35,5634,123],{},[39,5636,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,5638,126],{"stretchy":119},[39,5640,129],{},[39,5642,642],{"stretchy":119},[35,5644,326],{},[39,5646,120],{"stretchy":119},[35,5648,123],{},[39,5650,126],{"stretchy":119},[39,5652,335],{"separator":53},[35,5654,326],{},[39,5656,120],{"stretchy":119},[131,5658,5659,5661],{},[35,5660,123],{},[39,5662,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,5664,126],{"stretchy":119},[1057,5666,5667,5669],{},[39,5668,667],{"stretchy":119},[1057,5670,5671,5673],{},[35,5672,1061],{"mathvariant":445},[35,5674,621],{},[35,5676,3571],{"mathvariant":138},[45,5678,5679],{"encoding":47},"k(x,x')=\\langle \\varphi(x),\\varphi(x')\\rangle_{\\mathcal 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trick）的来源：所有只涉及特征内积的算法，都可以把 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替换成 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[66],[15,6668,1251],{"className":6669},[66],[15,6671,123],{"className":6672},[66,67],[15,6674],{"className":6675,"style":72},[71],[15,6677,6600],{"className":6678},[76],[15,6680],{"className":6681,"style":72},[71],[15,6683,6685,6688,6720,6774,6777,6780,6783,6812,6815],{"className":6684},[57],[15,6686],{"className":6687,"style":4398},[61],[15,6689,6691,6694],{"className":6690},[66],[15,6692,123],{"className":6693},[66,67],[15,6695,6697],{"className":6696},[198],[15,6698,6700],{"className":6699},[202],[15,6701,6703],{"className":6702},[206],[15,6704,6706],{"className":6705,"style":6003},[210],[15,6707,6708,6711],{"style":213},[15,6709],{"className":6710,"style":218},[217],[15,6712,6714],{"className":6713},[222,223,224,225],[15,6715,6717],{"className":6716},[66,225],[15,6718,635],{"className":6719},[66,225],[15,6721,6723,6726],{"className":6722},[66],[15,6724,1251],{"className":6725},[66],[15,6727,6729],{"className":6728},[198],[15,6730,6732,6765],{"className":6731},[202,1090],[15,6733,6735,6762],{"className":6734},[206],[15,6736,6739,6751],{"className":6737,"style":6738},[210],"height:0.8141em;",[15,6740,6742,6745],{"style":6741},"top:-2.4519em;margin-left:0em;margin-right:0.05em;",[15,6743],{"className":6744,"style":218},[217],[15,6746,6748],{"className":6747},[222,223,224,225],[15,6749,1263],{"className":6750},[66,225],[15,6752,6753,6756],{"style":213},[15,6754],{"className":6755,"style":218},[217],[15,6757,6759],{"className":6758},[222,223,224,225],[15,6760,1263],{"className":6761},[66,225],[15,6763,1114],{"className":6764},[1113],[15,6766,6768],{"className":6767},[206],[15,6769,6772],{"className":6770,"style":6771},[210],"height:0.2481em;",[15,6773],{},[15,6775,6623],{"className":6776},[66],[15,6778,120],{"className":6779},[162],[15,6781,1263],{"className":6782},[66],[15,6784,6786,6789],{"className":6785},[66],[15,6787,6632],{"className":6788,"style":1515},[66,67],[15,6790,6792],{"className":6791},[198],[15,6793,6795],{"className":6794},[202],[15,6796,6798],{"className":6797},[206],[15,6799,6801],{"className":6800,"style":6738},[210],[15,6802,6803,6806],{"style":213},[15,6804],{"className":6805,"style":218},[217],[15,6807,6809],{"className":6808},[222,223,224,225],[15,6810,1263],{"className":6811},[66,225],[15,6813,126],{"className":6814},[169],[15,6816,6818],{"className":6817},[169],[15,6819,126],{"className":6820},[1422,1508],[6174,6822,6823],{},"无穷维，光滑函数",[6156,6825,6826,6829,7009],{},[6174,6827,6828],{},"Laplacian 核",[6174,6830,6831],{},[15,6832,6834,6880],{"className":6833,"translate":19},[18],[15,6835,6837],{"className":6836},[23],[25,6838,6839],{"xmlns":27},[29,6840,6841,6877],{},[32,6842,6843,6845,6847,6849,6851,6853,6855,6857,6859,6865,6871,6873,6875],{},[35,6844,6590],{},[39,6846,1170],{},[1274,6848,6595],{},[39,6850,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,6852,6600],{},[35,6854,1251],{"mathvariant":138},[35,6856,123],{},[39,6858,6600],{},[131,6860,6861,6863],{},[35,6862,123],{},[39,6864,635],{"mathvariant":138,"lspace":634,"rspace":634},[1057,6866,6867,6869],{},[35,6868,1251],{"mathvariant":138},[1209,6870,1211],{},[35,6872,6623],{"mathvariant":138},[35,6874,6632],{},[39,6876,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[45,6878,6879],{"encoding":47},"\\exp\\!\\bigl(-\\|x-x'\\|_1\u002F\\sigma\\bigr)",[15,6881,6883,6919],{"className":6882,"ariaHidden":53},[52],[15,6884,6886,6889,6892,6895,6901,6904,6907,6910,6913,6916],{"className":6885},[57],[15,6887],{"className":6888,"style":4398},[61],[15,6890,6590],{"className":6891},[1339],[15,6893],{"className":6894,"style":6657},[71],[15,6896,6898],{"className":6897},[162],[15,6899,120],{"className":6900},[1422,1508],[15,6902,6600],{"className":6903},[66],[15,6905,1251],{"className":6906},[66],[15,6908,123],{"className":6909},[66,67],[15,6911],{"className":6912,"style":72},[71],[15,6914,6600],{"className":6915},[76],[15,6917],{"className":6918,"style":72},[71],[15,6920,6922,6925,6957,6997,7000,7003],{"className":6921},[57],[15,6923],{"className":6924,"style":4398},[61],[15,6926,6928,6931],{"className":6927},[66],[15,6929,123],{"className":6930},[66,67],[15,6932,6934],{"className":6933},[198],[15,6935,6937],{"className":6936},[202],[15,6938,6940],{"className":6939},[206],[15,6941,6943],{"className":6942,"style":6003},[210],[15,6944,6945,6948],{"style":213},[15,6946],{"className":6947,"style":218},[217],[15,6949,6951],{"className":6950},[222,223,224,225],[15,6952,6954],{"className":6953},[66,225],[15,6955,635],{"className":6956},[66,225],[15,6958,6960,6963],{"className":6959},[66],[15,6961,1251],{"className":6962},[66],[15,6964,6966],{"className":6965},[198],[15,6967,6969,6989],{"className":6968},[202,1090],[15,6970,6972,6986],{"className":6971},[206],[15,6973,6975],{"className":6974,"style":2139},[210],[15,6976,6977,6980],{"style":1584},[15,6978],{"className":6979,"style":218},[217],[15,6981,6983],{"className":6982},[222,223,224,225],[15,6984,1211],{"className":6985},[66,225],[15,6987,1114],{"className":6988},[1113],[15,6990,6992],{"className":6991},[206],[15,6993,6995],{"className":6994,"style":1121},[210],[15,6996],{},[15,6998,6623],{"className":6999},[66],[15,7001,6632],{"className":7002,"style":1515},[66,67],[15,7004,7006],{"className":7005},[169],[15,7007,126],{"className":7008},[1422,1508],[6174,7010,7011],{},"无穷维，Sobolev 型",[6156,7013,7014,7017,7020],{},[6174,7015,7016],{},"Matérn 核",[6174,7018,7019],{},"由修正 Bessel 函数给出",[6174,7021,7022,7023],{},"Sobolev 空间 ",[15,7024,7026,7055],{"className":7025,"translate":19},[18],[15,7027,7029],{"className":7028},[23],[25,7030,7031],{"xmlns":27},[29,7032,7033,7052],{},[32,7034,7035],{},[131,7036,7037,7039],{},[35,7038,1061],{},[32,7040,7041,7044,7046,7048,7050],{},[35,7042,7043],{},"ν",[39,7045,1245],{},[35,7047,458],{},[35,7049,6623],{"mathvariant":138},[1209,7051,1263],{},[45,7053,7054],{"encoding":47},"H^{\\nu+d\u002F2}",[15,7056,7058],{"className":7057,"ariaHidden":53},[52],[15,7059,7061,7065],{"className":7060},[57],[15,7062],{"className":7063,"style":7064},[61],"height:0.888em;",[15,7066,7068,7072],{"className":7067},[66],[15,7069,1061],{"className":7070,"style":7071},[66,67],"margin-right:0.0813em;",[15,7073,7075],{"className":7074},[198],[15,7076,7078],{"className":7077},[202],[15,7079,7081],{"className":7080},[206],[15,7082,7084],{"className":7083,"style":7064},[210],[15,7085,7086,7089],{"style":213},[15,7087],{"className":7088,"style":218},[217],[15,7090,7092],{"className":7091},[222,223,224,225],[15,7093,7095,7099,7102,7105],{"className":7094},[66,225],[15,7096,7043],{"className":7097,"style":7098},[66,67,225],"margin-right:0.0637em;",[15,7100,1245],{"className":7101},[76,225],[15,7103,458],{"className":7104},[66,67,225],[15,7106,7108],{"className":7107},[66,225],"\u002F2",[11,7110,7111,7112,7181,7182,1124],{},"核的选择等价于选择函数空间的先验：高斯核偏好非常光滑的函数，Laplacian 核允许更尖锐的变化，线性核则把模型限制在线性函数里。表示定理本身对核的选择没有偏好；它只负责把优化问题从 ",[15,7113,7115,7132],{"className":7114,"translate":19},[18],[15,7116,7118],{"className":7117},[23],[25,7119,7120],{"xmlns":27},[29,7121,7122,7130],{},[32,7123,7124],{},[1057,7125,7126,7128],{},[35,7127,1061],{"mathvariant":445},[35,7129,621],{},[45,7131,1066],{"encoding":47},[15,7133,7135],{"className":7134,"ariaHidden":53},[52],[15,7136,7138,7141],{"className":7137},[57],[15,7139],{"className":7140,"style":1076},[61],[15,7142,7144,7147],{"className":7143},[66],[15,7145,1061],{"className":7146,"style":1083},[66,494],[15,7148,7150],{"className":7149},[198],[15,7151,7153,7173],{"className":7152},[202,1090],[15,7154,7156,7170],{"className":7155},[206],[15,7157,7159],{"className":7158,"style":1097},[210],[15,7160,7161,7164],{"style":1100},[15,7162],{"className":7163,"style":218},[217],[15,7165,7167],{"className":7166},[222,223,224,225],[15,7168,621],{"className":7169,"style":684},[66,67,225],[15,7171,1114],{"className":7172},[1113],[15,7174,7176],{"className":7175},[206],[15,7177,7179],{"className":7178,"style":1121},[210],[15,7180],{}," 降到 ",[15,7183,7185,7203],{"className":7184,"translate":19},[18],[15,7186,7188],{"className":7187},[23],[25,7189,7190],{"xmlns":27},[29,7191,7192,7200],{},[32,7193,7194],{},[131,7195,7196,7198],{},[35,7197,455],{"mathvariant":454},[35,7199,37],{},[45,7201,7202],{"encoding":47},"\\mathbb R^n",[15,7204,7206],{"className":7205,"ariaHidden":53},[52],[15,7207,7209,7212],{"className":7208},[57],[15,7210],{"className":7211,"style":2034},[61],[15,7213,7215,7218],{"className":7214},[66],[15,7216,455],{"className":7217},[66,517],[15,7219,7221],{"className":7220},[198],[15,7222,7224],{"className":7223},[202],[15,7225,7227],{"className":7226},[206],[15,7228,7230],{"className":7229,"style":6436},[210],[15,7231,7232,7235],{"style":213},[15,7233],{"className":7234,"style":218},[217],[15,7236,7238],{"className":7237},[222,223,224,225],[15,7239,37],{"className":7240},[66,67,225],[95,7242,7244],{"id":7243},"三表示定理","三、表示定理",[1904,7246,7248],{"id":7247},"_31-定理","3.1 定理",[3054,7250,7252,7821,8321,8324,8330,8677],{"title":7251},"定理 3.1（表示定理）",[11,7253,1910,7254,1755,7282,7310,7311,7579,7580,7722,7723,7820],{},[15,7255,7257,7270],{"className":7256,"translate":19},[18],[15,7258,7260],{"className":7259},[23],[25,7261,7262],{"xmlns":27},[29,7263,7264,7268],{},[32,7265,7266],{},[35,7267,1061],{"mathvariant":445},[45,7269,1742],{"encoding":47},[15,7271,7273],{"className":7272,"ariaHidden":53},[52],[15,7274,7276,7279],{"className":7275},[57],[15,7277],{"className":7278,"style":490},[61],[15,7280,1061],{"className":7281,"style":1083},[66,494],[15,7283,7285,7298],{"className":7284,"translate":19},[18],[15,7286,7288],{"className":7287},[23],[25,7289,7290],{"xmlns":27},[29,7291,7292,7296],{},[32,7293,7294],{},[35,7295,621],{},[45,7297,621],{"encoding":47},[15,7299,7301],{"className":7300,"ariaHidden":53},[52],[15,7302,7304,7307],{"className":7303},[57],[15,7305],{"className":7306,"style":840},[61],[15,7308,621],{"className":7309,"style":684},[66,67]," 对应的 RKHS，给定训练数据 ",[15,7312,7314,7368],{"className":7313,"translate":19},[18],[15,7315,7317],{"className":7316},[23],[25,7318,7319],{"xmlns":27},[29,7320,7321,7365],{},[32,7322,7323,7325,7331,7333,7339,7341,7343,7345,7347,7349,7355,7357,7363],{},[39,7324,120],{"stretchy":119},[1057,7326,7327,7329],{},[35,7328,123],{},[1209,7330,1211],{},[39,7332,335],{"separator":53},[1057,7334,7335,7337],{},[35,7336,1224],{},[1209,7338,1211],{},[39,7340,126],{"stretchy":119},[39,7342,335],{"separator":53},[39,7344,2095],{},[39,7346,335],{"separator":53},[39,7348,120],{"stretchy":119},[1057,7350,7351,7353],{},[35,7352,123],{},[35,7354,37],{},[39,7356,335],{"separator":53},[1057,7358,7359,7361],{},[35,7360,1224],{},[35,7362,37],{},[39,7364,126],{"stretchy":119},[45,7366,7367],{"encoding":47},"(x_1,y_1),\\dots,(x_n,y_n)",[15,7369,7371],{"className":7370,"ariaHidden":53},[52],[15,7372,7374,7377,7380,7420,7423,7426,7466,7469,7472,7475,7478,7481,7484,7487,7490,7530,7533,7536,7576],{"className":7373},[57],[15,7375],{"className":7376,"style":154},[61],[15,7378,120],{"className":7379},[162],[15,7381,7383,7386],{"className":7382},[66],[15,7384,123],{"className":7385},[66,67],[15,7387,7389],{"className":7388},[198],[15,7390,7392,7412],{"className":7391},[202,1090],[15,7393,7395,7409],{"className":7394},[206],[15,7396,7398],{"className":7397,"style":2139},[210],[15,7399,7400,7403],{"style":1584},[15,7401],{"className":7402,"style":218},[217],[15,7404,7406],{"className":7405},[222,223,224,225],[15,7407,1211],{"className":7408},[66,225],[15,7410,1114],{"className":7411},[1113],[15,7413,7415],{"className":7414},[206],[15,7416,7418],{"className":7417,"style":1121},[210],[15,7419],{},[15,7421,335],{"className":7422},[422],[15,7424],{"className":7425,"style":697},[71],[15,7427,7429,7432],{"className":7428},[66],[15,7430,1224],{"className":7431,"style":1515},[66,67],[15,7433,7435],{"className":7434},[198],[15,7436,7438,7458],{"className":7437},[202,1090],[15,7439,7441,7455],{"className":7440},[206],[15,7442,7444],{"className":7443,"style":2139},[210],[15,7445,7446,7449],{"style":1531},[15,7447],{"className":7448,"style":218},[217],[15,7450,7452],{"className":7451},[222,223,224,225],[15,7453,1211],{"className":7454},[66,225],[15,7456,1114],{"className":7457},[1113],[15,7459,7461],{"className":7460},[206],[15,7462,7464],{"className":7463,"style":1121},[210],[15,7465],{},[15,7467,126],{"className":7468},[169],[15,7470,335],{"className":7471},[422],[15,7473],{"className":7474,"style":697},[71],[15,7476,2095],{"className":7477},[1413],[15,7479],{"className":7480,"style":697},[71],[15,7482,335],{"className":7483},[422],[15,7485],{"className":7486,"style":697},[71],[15,7488,120],{"className":7489},[162],[15,7491,7493,7496],{"className":7492},[66],[15,7494,123],{"className":7495},[66,67],[15,7497,7499],{"className":7498},[198],[15,7500,7502,7522],{"className":7501},[202,1090],[15,7503,7505,7519],{"className":7504},[206],[15,7506,7508],{"className":7507,"style":2198},[210],[15,7509,7510,7513],{"style":1584},[15,7511],{"className":7512,"style":218},[217],[15,7514,7516],{"className":7515},[222,223,224,225],[15,7517,37],{"className":7518},[66,67,225],[15,7520,1114],{"className":7521},[1113],[15,7523,7525],{"className":7524},[206],[15,7526,7528],{"className":7527,"style":1121},[210],[15,7529],{},[15,7531,335],{"className":7532},[422],[15,7534],{"className":7535,"style":697},[71],[15,7537,7539,7542],{"className":7538},[66],[15,7540,1224],{"className":7541,"style":1515},[66,67],[15,7543,7545],{"className":7544},[198],[15,7546,7548,7568],{"className":7547},[202,1090],[15,7549,7551,7565],{"className":7550},[206],[15,7552,7554],{"className":7553,"style":2198},[210],[15,7555,7556,7559],{"style":1531},[15,7557],{"className":7558,"style":218},[217],[15,7560,7562],{"className":7561},[222,223,224,225],[15,7563,37],{"className":7564},[66,67,225],[15,7566,1114],{"className":7567},[1113],[15,7569,7571],{"className":7570},[206],[15,7572,7574],{"className":7573,"style":1121},[210],[15,7575],{},[15,7577,126],{"className":7578},[169],"。设 ",[15,7581,7583,7621],{"className":7582,"translate":19},[18],[15,7584,7586],{"className":7585},[23],[25,7587,7588],{"xmlns":27},[29,7589,7590,7618],{},[32,7591,7592,7594,7596,7602,7604,7606,7609,7611,7613,7616],{},[35,7593,1216],{},[39,7595,442],{},[131,7597,7598,7600],{},[35,7599,455],{"mathvariant":454},[35,7601,37],{},[39,7603,449],{},[35,7605,455],{"mathvariant":454},[39,7607,7608],{},"∪",[39,7610,4305],{"stretchy":119},[39,7612,1245],{},[35,7614,7615],{"mathvariant":138},"∞",[39,7617,4328],{"stretchy":119},[45,7619,7620],{"encoding":47},"L:\\mathbb R^n\\to\\mathbb R\\cup\\{+\\infty\\}",[15,7622,7624,7642,7686,7704],{"className":7623,"ariaHidden":53},[52],[15,7625,7627,7630,7633,7636,7639],{"className":7626},[57],[15,7628],{"className":7629,"style":490},[61],[15,7631,1216],{"className":7632},[66,67],[15,7634],{"className":7635,"style":173},[71],[15,7637,442],{"className":7638},[177],[15,7640],{"className":7641,"style":173},[71],[15,7643,7645,7648,7677,7680,7683],{"className":7644},[57],[15,7646],{"className":7647,"style":2034},[61],[15,7649,7651,7654],{"className":7650},[66],[15,7652,455],{"className":7653},[66,517],[15,7655,7657],{"className":7656},[198],[15,7658,7660],{"className":7659},[202],[15,7661,7663],{"className":7662},[206],[15,7664,7666],{"className":7665,"style":6436},[210],[15,7667,7668,7671],{"style":213},[15,7669],{"className":7670,"style":218},[217],[15,7672,7674],{"className":7673},[222,223,224,225],[15,7675,37],{"className":7676},[66,67,225],[15,7678],{"className":7679,"style":173},[71],[15,7681,449],{"className":7682},[177],[15,7684],{"className":7685,"style":173},[71],[15,7687,7689,7692,7695,7698,7701],{"className":7688},[57],[15,7690],{"className":7691,"style":2034},[61],[15,7693,455],{"className":7694},[66,517],[15,7696],{"className":7697,"style":72},[71],[15,7699,7608],{"className":7700},[76],[15,7702],{"className":7703,"style":72},[71],[15,7705,7707,7710,7713,7716,7719],{"className":7706},[57],[15,7708],{"className":7709,"style":154},[61],[15,7711,4305],{"className":7712},[162],[15,7714,1245],{"className":7715},[66],[15,7717,7615],{"className":7718},[66],[15,7720,4328],{"className":7721},[169]," 是任意损失函数，",[15,7724,7726,7757],{"className":7725,"translate":19},[18],[15,7727,7729],{"className":7728},[23],[25,7730,7731],{"xmlns":27},[29,7732,7733,7754],{},[32,7734,7735,7738,7740,7742,7744,7746,7748,7750,7752],{},[35,7736,7737],{"mathvariant":138},"Ω",[39,7739,442],{},[39,7741,1194],{"stretchy":119},[1209,7743,1804],{},[39,7745,335],{"separator":53},[35,7747,7615],{"mathvariant":138},[39,7749,126],{"stretchy":119},[39,7751,449],{},[35,7753,455],{"mathvariant":454},[45,7755,7756],{"encoding":47},"\\Omega:[0,\\infty)\\to\\mathbb R",[15,7758,7760,7778,7811],{"className":7759,"ariaHidden":53},[52],[15,7761,7763,7766,7769,7772,7775],{"className":7762},[57],[15,7764],{"className":7765,"style":490},[61],[15,7767,7737],{"className":7768},[66],[15,7770],{"className":7771,"style":173},[71],[15,7773,442],{"className":7774},[177],[15,7776],{"className":7777,"style":173},[71],[15,7779,7781,7784,7787,7790,7793,7796,7799,7802,7805,7808],{"className":7780},[57],[15,7782],{"className":7783,"style":154},[61],[15,7785,1194],{"className":7786},[162],[15,7788,1804],{"className":7789},[66],[15,7791,335],{"className":7792},[422],[15,7794],{"className":7795,"style":697},[71],[15,7797,7615],{"className":7798},[66],[15,7800,126],{"className":7801},[169],[15,7803],{"className":7804,"style":173},[71],[15,7806,449],{"className":7807},[177],[15,7809],{"className":7810,"style":173},[71],[15,7812,7814,7817],{"className":7813},[57],[15,7815],{"className":7816,"style":2034},[61],[15,7818,455],{"className":7819},[66,517]," 是非减函数。考虑",[15,7822,7824],{"className":7823,"translate":19},[294],[15,7825,7827,7946],{"className":7826,"translate":19},[18],[15,7828,7830],{"className":7829},[23],[25,7831,7832],{"xmlns":27,"display":303},[29,7833,7834,7943],{},[1142,7835,7836],{"width":1144},[1146,7837,7838,7840,7936,7938],{},[1149,7839],{"width":1151},[1149,7841,7842],{},[32,7843,7844,7850,7852,7854,7856,7872,7934],{},[131,7845,7846,7848],{},[35,7847,116],{},[39,7849,1162],{},[39,7851,1187],{},[35,7853,1167],{},[39,7855,1170],{},[1172,7857,7858,7864],{},[32,7859,7860,7862],{},[35,7861,1178],{},[39,7863,1170],{},[32,7865,7866,7868,7870],{},[35,7867,116],{},[39,7869,1187],{},[35,7871,1061],{"mathvariant":445},[32,7873,7874,7876,7878,7880,7882,7884,7890,7892,7894,7896,7898,7900,7902,7908,7910,7912,7914,7916,7918,7920,7922,7930,7932],{},[39,7875,4305],{"fence":53},[35,7877,1216],{},[39,7879,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,7881,116],{},[39,7883,120],{"stretchy":119},[1057,7885,7886,7888],{},[35,7887,123],{},[1209,7889,1211],{},[39,7891,126],{"stretchy":119},[39,7893,335],{"separator":53},[39,7895,2095],{},[39,7897,335],{"separator":53},[35,7899,116],{},[39,7901,120],{"stretchy":119},[1057,7903,7904,7906],{},[35,7905,123],{},[35,7907,37],{},[39,7909,126],{"stretchy":119},[39,7911,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,7913,1245],{},[35,7915,7737],{"mathvariant":138},[39,7917,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,7919,1251],{"mathvariant":138},[35,7921,116],{},[1255,7923,7924,7926,7928],{},[35,7925,1251],{"mathvariant":138},[35,7927,1061],{"mathvariant":445},[1209,7929,1263],{},[39,7931,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,7933,4328],{"fence":53},[35,7935,3571],{"mathvariant":138},[1149,7937],{"width":1151},[1149,7939,7940],{},[1274,7941,7942],{},"(2)",[45,7944,7945],{"encoding":47},"f^*\\in\\arg\\min_{f\\in\\mathcal 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存在一个最优解具有形式",[15,8331,8333],{"className":8332,"translate":19},[294],[15,8334,8336,8419],{"className":8335,"translate":19},[18],[15,8337,8339],{"className":8338},[23],[25,8340,8341],{"xmlns":27,"display":303},[29,8342,8343,8416],{},[1142,8344,8345],{"width":1144},[1146,8346,8347,8349,8409,8411],{},[1149,8348],{"width":1151},[1149,8350,8351],{},[32,8352,8353,8359,8361,8363,8365,8367,8381,8388,8390,8392,8394,8396,8398,8404,8406],{},[131,8354,8355,8357],{},[35,8356,116],{},[39,8358,1162],{},[39,8360,120],{"stretchy":119},[35,8362,123],{},[39,8364,126],{"stretchy":119},[39,8366,129],{},[1196,8368,8369,8371,8379],{},[39,8370,1200],{},[32,8372,8373,8375,8377],{},[35,8374,1205],{},[39,8376,129],{},[1209,8378,1211],{},[35,8380,37],{},[1057,8382,8383,8386],{},[35,8384,8385],{},"α",[35,8387,1205],{},[1274,8389,2462],{},[35,8391,621],{},[39,8393,120],{"stretchy":119},[35,8395,123],{},[39,8397,335],{"separator":53},[1057,8399,8400,8402],{},[35,8401,123],{},[35,8403,1205],{},[39,8405,126],{"stretchy":119},[39,8407,8408],{"separator":53},";",[1149,8410],{"width":1151},[1149,8412,8413],{},[1274,8414,8415],{},"(3)",[45,8417,8418],{"encoding":47},"f^*(x)=\\sum_{i=1}^n\\alpha_i\\,k(x,x_i);\n\\tag{3}",[15,8420,8422,8475,8655],{"className":8421,"ariaHidden":53},[52],[15,8423,8425,8428,8457,8460,8463,8466,8469,8472],{"className":8424},[57],[15,8426],{"className":8427,"style":154},[61],[15,8429,8431,8434],{"className":8430},[66],[15,8432,116],{"className":8433,"style":158},[66,67],[15,8435,8437],{"className":8436},[198],[15,8438,8440],{"className":8439},[202],[15,8441,8443],{"className":8442},[206],[15,8444,8446],{"className":8445,"style":1308},[210],[15,8447,8448,8451],{"style":397},[15,8449],{"className":8450,"style":218},[217],[15,8452,8454],{"className":8453},[222,223,224,225],[15,8455,1162],{"className":8456},[76,225],[15,8458,120],{"className":8459},[162],[15,8461,123],{"className":8462},[66,67],[15,8464,126],{"className":8465},[169],[15,8467],{"className":8468,"style":173},[71],[15,8470,129],{"className":8471},[177],[15,8473],{"className":8474,"style":173},[71],[15,8476,8478,8482,8546,8549,8591,8594,8597,8600,8603,8606,8609,8649,8652],{"className":8477},[57],[15,8479],{"className":8480,"style":8481},[61],"height:2.9291em;vertical-align:-1.2777em;",[15,8483,8485],{"className":8484},[1339,1351],[15,8486,8488,8538],{"className":8487},[202,1090],[15,8489,8491,8535],{"className":8490},[206],[15,8492,8494,8514,8524],{"className":8493,"style":1436},[210],[15,8495,8496,8499],{"style":1439},[15,8497],{"className":8498,"style":1443},[217],[15,8500,8502],{"className":8501},[222,223,224,225],[15,8503,8505,8508,8511],{"className":8504},[66,225],[15,8506,1205],{"className":8507},[66,67,225],[15,8509,129],{"className":8510},[177,225],[15,8512,1211],{"className":8513},[66,225],[15,8515,8516,8519],{"style":1461},[15,8517],{"className":8518,"style":1443},[217],[15,8520,8521],{},[15,8522,1200],{"className":8523},[1339,1470,1471],[15,8525,8526,8529],{"style":1474},[15,8527],{"className":8528,"style":1443},[217],[15,8530,8532],{"className":8531},[222,223,224,225],[15,8533,37],{"className":8534},[66,67,225],[15,8536,1114],{"className":8537},[1113],[15,8539,8541],{"className":8540},[206],[15,8542,8544],{"className":8543,"style":1493},[210],[15,8545],{},[15,8547],{"className":8548,"style":697},[71],[15,8550,8552,8556],{"className":8551},[66],[15,8553,8385],{"className":8554,"style":8555},[66,67],"margin-right:0.0037em;",[15,8557,8559],{"className":8558},[198],[15,8560,8562,8583],{"className":8561},[202,1090],[15,8563,8565,8580],{"className":8564},[206],[15,8566,8568],{"className":8567,"style":1528},[210],[15,8569,8571,8574],{"style":8570},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[15,8572],{"className":8573,"style":218},[217],[15,8575,8577],{"className":8576},[222,223,224,225],[15,8578,1205],{"className":8579},[66,67,225],[15,8581,1114],{"className":8582},[1113],[15,8584,8586],{"className":8585},[206],[15,8587,8589],{"className":8588,"style":1121},[210],[15,8590],{},[15,8592],{"className":8593,"style":697},[71],[15,8595,621],{"className":8596,"style":684},[66,67],[15,8598,120],{"className":8599},[162],[15,8601,123],{"className":8602},[66,67],[15,8604,335],{"className":8605},[422],[15,8607],{"className":8608,"style":697},[71],[15,8610,8612,8615],{"className":8611},[66],[15,8613,123],{"className":8614},[66,67],[15,8616,8618],{"className":8617},[198],[15,8619,8621,8641],{"className":8620},[202,1090],[15,8622,8624,8638],{"className":8623},[206],[15,8625,8627],{"className":8626,"style":1528},[210],[15,8628,8629,8632],{"style":1584},[15,8630],{"className":8631,"style":218},[217],[15,8633,8635],{"className":8634},[222,223,224,225],[15,8636,1205],{"className":8637},[66,67,225],[15,8639,1114],{"className":8640},[1113],[15,8642,8644],{"className":8643},[206],[15,8645,8647],{"className":8646,"style":1121},[210],[15,8648],{},[15,8650,126],{"className":8651},[169],[15,8653,8408],{"className":8654},[422],[15,8656,8658,8661],{"className":8657},[1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若 ",[15,8684,8686,8700],{"className":8685,"translate":19},[18],[15,8687,8689],{"className":8688},[23],[25,8690,8691],{"xmlns":27},[29,8692,8693,8697],{},[32,8694,8695],{},[35,8696,7737],{"mathvariant":138},[45,8698,8699],{"encoding":47},"\\Omega",[15,8701,8703],{"className":8702,"ariaHidden":53},[52],[15,8704,8706,8709],{"className":8705},[57],[15,8707],{"className":8708,"style":490},[61],[15,8710,7737],{"className":8711},[66]," 严格递增，则",[2040,8714,8715],{},"每个","最优解都具有这个形式。",[11,8718,8719],{},"这里有三个细节值得强调。",[11,8721,8722,8723,8751,8752,8922,8923,8951,8952,8980,8981,9051,9052,9080],{},"第一，",[15,8724,8726,8739],{"className":8725,"translate":19},[18],[15,8727,8729],{"className":8728},[23],[25,8730,8731],{"xmlns":27},[29,8732,8733,8737],{},[32,8734,8735],{},[35,8736,1216],{},[45,8738,1216],{"encoding":47},[15,8740,8742],{"className":8741,"ariaHidden":53},[52],[15,8743,8745,8748],{"className":8744},[57],[15,8746],{"className":8747,"style":490},[61],[15,8749,1216],{"className":8750},[66,67]," 只通过 ",[15,8753,8755,8797],{"className":8754,"translate":19},[18],[15,8756,8758],{"className":8757},[23],[25,8759,8760],{"xmlns":27},[29,8761,8762,8794],{},[32,8763,8764,8766,8768,8774,8776,8778,8780,8782,8784,8786,8792],{},[35,8765,116],{},[39,8767,120],{"stretchy":119},[1057,8769,8770,8772],{},[35,8771,123],{},[1209,8773,1211],{},[39,8775,126],{"stretchy":119},[39,8777,335],{"separator":53},[39,8779,2095],{},[39,8781,335],{"separator":53},[35,8783,116],{},[39,8785,120],{"stretchy":119},[1057,8787,8788,8790],{},[35,8789,123],{},[35,8791,37],{},[39,8793,126],{"stretchy":119},[45,8795,8796],{"encoding":47},"f(x_1),\\dots,f(x_n)",[15,8798,8800],{"className":8799,"ariaHidden":53},[52],[15,8801,8803,8806,8809,8812,8852,8855,8858,8861,8864,8867,8870,8873,8876,8879,8919],{"className":8802},[57],[15,8804],{"className":8805,"style":154},[61],[15,8807,116],{"className":8808,"style":158},[66,67],[15,8810,120],{"className":8811},[162],[15,8813,8815,8818],{"className":8814},[66],[15,8816,123],{"className":8817},[66,67],[15,8819,8821],{"className":8820},[198],[15,8822,8824,8844],{"className":8823},[202,1090],[15,8825,8827,8841],{"className":8826},[206],[15,8828,8830],{"className":8829,"style":2139},[210],[15,8831,8832,8835],{"style":1584},[15,8833],{"className":8834,"style":218},[217],[15,8836,8838],{"className":8837},[222,223,224,225],[15,8839,1211],{"className":8840},[66,225],[15,8842,1114],{"className":8843},[1113],[15,8845,8847],{"className":8846},[206],[15,8848,8850],{"className":8849,"style":1121},[210],[15,8851],{},[15,8853,126],{"className":8854},[169],[15,8856,335],{"className":8857},[422],[15,8859],{"className":8860,"style":697},[71],[15,8862,2095],{"className":8863},[1413],[15,8865],{"className":8866,"style":697},[71],[15,8868,335],{"className":8869},[422],[15,8871],{"className":8872,"style":697},[71],[15,8874,116],{"className":8875,"style":158},[66,67],[15,8877,120],{"className":8878},[162],[15,8880,8882,8885],{"className":8881},[66],[15,8883,123],{"className":8884},[66,67],[15,8886,8888],{"className":8887},[198],[15,8889,8891,8911],{"className":8890},[202,1090],[15,8892,8894,8908],{"className":8893},[206],[15,8895,8897],{"className":8896,"style":2198},[210],[15,8898,8899,8902],{"style":1584},[15,8900],{"className":8901,"style":218},[217],[15,8903,8905],{"className":8904},[222,223,224,225],[15,8906,37],{"className":8907},[66,67,225],[15,8909,1114],{"className":8910},[1113],[15,8912,8914],{"className":8913},[206],[15,8915,8917],{"className":8916,"style":1121},[210],[15,8918],{},[15,8920,126],{"className":8921},[169]," 这 ",[15,8924,8926,8939],{"className":8925,"translate":19},[18],[15,8927,8929],{"className":8928},[23],[25,8930,8931],{"xmlns":27},[29,8932,8933,8937],{},[32,8934,8935],{},[35,8936,37],{},[45,8938,37],{"encoding":47},[15,8940,8942],{"className":8941,"ariaHidden":53},[52],[15,8943,8945,8948],{"className":8944},[57],[15,8946],{"className":8947,"style":86},[61],[15,8949,37],{"className":8950},[66,67]," 个值依赖 ",[15,8953,8955,8968],{"className":8954,"translate":19},[18],[15,8956,8958],{"className":8957},[23],[25,8959,8960],{"xmlns":27},[29,8961,8962,8966],{},[32,8963,8964],{},[35,8965,116],{},[45,8967,116],{"encoding":47},[15,8969,8971],{"className":8970,"ariaHidden":53},[52],[15,8972,8974,8977],{"className":8973},[57],[15,8975],{"className":8976,"style":3406},[61],[15,8978,116],{"className":8979,"style":158},[66,67],"（标签 ",[15,8982,8984,9002],{"className":8983,"translate":19},[18],[15,8985,8987],{"className":8986},[23],[25,8988,8989],{"xmlns":27},[29,8990,8991,8999],{},[32,8992,8993],{},[1057,8994,8995,8997],{},[35,8996,1224],{},[35,8998,1205],{},[45,9000,9001],{"encoding":47},"y_i",[15,9003,9005],{"className":9004,"ariaHidden":53},[52],[15,9006,9008,9011],{"className":9007},[57],[15,9009],{"className":9010,"style":471},[61],[15,9012,9014,9017],{"className":9013},[66],[15,9015,1224],{"className":9016,"style":1515},[66,67],[15,9018,9020],{"className":9019},[198],[15,9021,9023,9043],{"className":9022},[202,1090],[15,9024,9026,9040],{"className":9025},[206],[15,9027,9029],{"className":9028,"style":1528},[210],[15,9030,9031,9034],{"style":1531},[15,9032],{"className":9033,"style":218},[217],[15,9035,9037],{"className":9036},[222,223,224,225],[15,9038,1205],{"className":9039},[66,67,225],[15,9041,1114],{"className":9042},[1113],[15,9044,9046],{"className":9045},[206],[15,9047,9049],{"className":9048,"style":1121},[210],[15,9050],{}," 已吸收进 ",[15,9053,9055,9068],{"className":9054,"translate":19},[18],[15,9056,9058],{"className":9057},[23],[25,9059,9060],{"xmlns":27},[29,9061,9062,9066],{},[32,9063,9064],{},[35,9065,1216],{},[45,9067,1216],{"encoding":47},[15,9069,9071],{"className":9070,"ariaHidden":53},[52],[15,9072,9074,9077],{"className":9073},[57],[15,9075],{"className":9076,"style":490},[61],[15,9078,1216],{"className":9079},[66,67],"）。如果损失还依赖导数、积分或在其他点上的取值，定理的形式会改变。表示定理的标准版本针对的正是这种「点值型」经验风险。",[11,9082,9083,9084,9112,9113,9188,9189,9261,9262,9315],{},"第二，",[15,9085,9087,9100],{"className":9086,"translate":19},[18],[15,9088,9090],{"className":9089},[23],[25,9091,9092],{"xmlns":27},[29,9093,9094,9098],{},[32,9095,9096],{},[35,9097,7737],{"mathvariant":138},[45,9099,8699],{"encoding":47},[15,9101,9103],{"className":9102,"ariaHidden":53},[52],[15,9104,9106,9109],{"className":9105},[57],[15,9107],{"className":9108,"style":490},[61],[15,9110,7737],{"className":9111},[66]," 只要求非减，不要求凸、不要求可微。式 (1) 中的 ",[15,9114,9116,9140],{"className":9115,"translate":19},[18],[15,9117,9119],{"className":9118},[23],[25,9120,9121],{"xmlns":27},[29,9122,9123,9137],{},[32,9124,9125,9127,9129,9131],{},[35,9126,1248],{},[35,9128,1251],{"mathvariant":138},[35,9130,116],{},[131,9132,9133,9135],{},[35,9134,1251],{"mathvariant":138},[1209,9136,1263],{},[45,9138,9139],{"encoding":47},"\\lambda\\|f\\|^2",[15,9141,9143],{"className":9142,"ariaHidden":53},[52],[15,9144,9146,9150,9153,9156,9159],{"className":9145},[57],[15,9147],{"className":9148,"style":9149},[61],"height:1.0641em;vertical-align:-0.25em;",[15,9151,1248],{"className":9152},[66,67],[15,9154,1251],{"className":9155},[66],[15,9157,116],{"className":9158,"style":158},[66,67],[15,9160,9162,9165],{"className":9161},[66],[15,9163,1251],{"className":9164},[66],[15,9166,9168],{"className":9167},[198],[15,9169,9171],{"className":9170},[202],[15,9172,9174],{"className":9173},[206],[15,9175,9177],{"className":9176,"style":6738},[210],[15,9178,9179,9182],{"style":213},[15,9180],{"className":9181,"style":218},[217],[15,9183,9185],{"className":9184},[222,223,224,225],[15,9186,1263],{"className":9187},[66,225]," 对应 ",[15,9190,9192,9219],{"className":9191,"translate":19},[18],[15,9193,9195],{"className":9194},[23],[25,9196,9197],{"xmlns":27},[29,9198,9199,9216],{},[32,9200,9201,9203,9205,9208,9210,9212,9214],{},[35,9202,7737],{"mathvariant":138},[39,9204,120],{"stretchy":119},[35,9206,9207],{},"t",[39,9209,126],{"stretchy":119},[39,9211,129],{},[35,9213,1248],{},[35,9215,9207],{},[45,9217,9218],{"encoding":47},"\\Omega(t)=\\lambda t",[15,9220,9222,9249],{"className":9221,"ariaHidden":53},[52],[15,9223,9225,9228,9231,9234,9237,9240,9243,9246],{"className":9224},[57],[15,9226],{"className":9227,"style":154},[61],[15,9229,7737],{"className":9230},[66],[15,9232,120],{"className":9233},[162],[15,9235,9207],{"className":9236},[66,67],[15,9238,126],{"className":9239},[169],[15,9241],{"className":9242,"style":173},[71],[15,9244,129],{"className":9245},[177],[15,9247],{"className":9248,"style":173},[71],[15,9250,9252,9255,9258],{"className":9251},[57],[15,9253],{"className":9254,"style":840},[61],[15,9256,1248],{"className":9257},[66,67],[15,9259,9207],{"className":9260},[66,67],"（",[15,9263,9265,9284],{"className":9264,"translate":19},[18],[15,9266,9268],{"className":9267},[23],[25,9269,9270],{"xmlns":27},[29,9271,9272,9281],{},[32,9273,9274,9276,9279],{},[35,9275,1248],{},[39,9277,9278],{},">",[1209,9280,1804],{},[45,9282,9283],{"encoding":47},"\\lambda>0",[15,9285,9287,9306],{"className":9286,"ariaHidden":53},[52],[15,9288,9290,9294,9297,9300,9303],{"className":9289},[57],[15,9291],{"className":9292,"style":9293},[61],"height:0.7335em;vertical-align:-0.0391em;",[15,9295,1248],{"className":9296},[66,67],[15,9298],{"className":9299,"style":173},[71],[15,9301,9278],{"className":9302},[177],[15,9304],{"className":9305,"style":173},[71],[15,9307,9309,9312],{"className":9308},[57],[15,9310],{"className":9311,"style":1836},[61],[15,9313,1804],{"className":9314},[66]," 时严格递增），是最常见的特例。",[11,9317,9318,9319,9322,9323,9351],{},"第三，第 1 条是",[2040,9320,9321],{},"存在性","，第 2 条才是「任意最优解」。如果 ",[15,9324,9326,9339],{"className":9325,"translate":19},[18],[15,9327,9329],{"className":9328},[23],[25,9330,9331],{"xmlns":27},[29,9332,9333,9337],{},[32,9334,9335],{},[35,9336,7737],{"mathvariant":138},[45,9338,8699],{"encoding":47},[15,9340,9342],{"className":9341,"ariaHidden":53},[52],[15,9343,9345,9348],{"className":9344},[57],[15,9346],{"className":9347,"style":490},[61],[15,9349,7737],{"className":9350},[66]," 只是非减而非严格递增，可能存在落在子空间之外的最优解；我们会在第四节看到这一点。",[1904,9353,9355],{"id":9354},"_32-证明正交分解","3.2 证明：正交分解",[11,9357,9358],{},"证明的核心是一个正交分解。令",[15,9360,9362],{"className":9361,"translate":19},[294],[15,9363,9365,9438],{"className":9364,"translate":19},[18],[15,9366,9368],{"className":9367},[23],[25,9369,9370],{"xmlns":27,"display":303},[29,9371,9372,9435],{},[32,9373,9374,9380,9382,9384,9386,9388,9390,9392,9394,9396,9402,9404,9406,9408,9410,9412,9414,9416,9418,9424,9426,9428,9431,9433],{},[1057,9375,9376,9378],{},[35,9377,1061],{"mathvariant":445},[1209,9379,1804],{},[39,9381,129],{},[35,9383,15],{"mathvariant":138},[39,9385,1170],{},[39,9387,4305],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,9389,621],{},[39,9391,120],{"stretchy":119},[39,9393,3266],{},[39,9395,335],{"separator":53},[1057,9397,9398,9400],{},[35,9399,123],{},[1209,9401,1211],{},[39,9403,126],{"stretchy":119},[39,9405,335],{"separator":53},[39,9407,2095],{},[39,9409,335],{"separator":53},[35,9411,621],{},[39,9413,120],{"stretchy":119},[39,9415,3266],{},[39,9417,335],{"separator":53},[1057,9419,9420,9422],{},[35,9421,123],{},[35,9423,37],{},[39,9425,126],{"stretchy":119},[39,9427,4328],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,9429,9430],{},"⊆",[35,9432,1061],{"mathvariant":445},[35,9434,3571],{"mathvariant":138},[45,9436,9437],{"encoding":47},"\\mathcal H_0=\\operatorname{span}\\bigl\\{k(\\cdot,x_1),\\dots,k(\\cdot,x_n)\\bigr\\}\\subseteq\\mathcal H.",[15,9439,9441,9496,9663],{"className":9440,"ariaHidden":53},[52],[15,9442,9444,9447,9487,9490,9493],{"className":9443},[57],[15,9445],{"className":9446,"style":1076},[61],[15,9448,9450,9453],{"className":9449},[66],[15,9451,1061],{"className":9452,"style":1083},[66,494],[15,9454,9456],{"className":9455},[198],[15,9457,9459,9479],{"className":9458},[202,1090],[15,9460,9462,9476],{"className":9461},[206],[15,9463,9465],{"className":9464,"style":2139},[210],[15,9466,9467,9470],{"style":1100},[15,9468],{"className":9469,"style":218},[217],[15,9471,9473],{"className":9472},[222,223,224,225],[15,9474,1804],{"className":9475},[66,225],[15,9477,1114],{"className":9478},[1113],[15,9480,9482],{"className":9481},[206],[15,9483,9485],{"className":9484,"style":1121},[210],[15,9486],{},[15,9488],{"className":9489,"style":173},[71],[15,9491,129],{"className":9492},[177],[15,9494],{"className":9495,"style":173},[71],[15,9497,9499,9502,9508,9514,9517,9520,9523,9526,9529,9569,9572,9575,9578,9581,9584,9587,9590,9593,9596,9599,9602,9605,9645,9648,9654,9657,9660],{"className":9498},[57],[15,9500],{"className":9501,"style":4398},[61],[15,9503,9505],{"className":9504},[1339],[15,9506,15],{"className":9507},[66,4427],[15,9509,9511],{"className":9510},[162],[15,9512,4305],{"className":9513},[1422,1508],[15,9515,621],{"className":9516,"style":684},[66,67],[15,9518,120],{"className":9519},[162],[15,9521,3266],{"className":9522},[66],[15,9524,335],{"className":9525},[422],[15,9527],{"className":9528,"style":697},[71],[15,9530,9532,9535],{"className":9531},[66],[15,9533,123],{"className":9534},[66,67],[15,9536,9538],{"className":9537},[198],[15,9539,9541,9561],{"className":9540},[202,1090],[15,9542,9544,9558],{"className":9543},[206],[15,9545,9547],{"className":9546,"style":2139},[210],[15,9548,9549,9552],{"style":1584},[15,9550],{"className":9551,"style":218},[217],[15,9553,9555],{"className":9554},[222,223,224,225],[15,9556,1211],{"className":9557},[66,225],[15,9559,1114],{"className":9560},[1113],[15,9562,9564],{"className":9563},[206],[15,9565,9567],{"className":9566,"style":1121},[210],[15,9568],{},[15,9570,126],{"className":9571},[169],[15,9573,335],{"className":9574},[422],[15,9576],{"className":9577,"style":697},[71],[15,9579,2095],{"className":9580},[1413],[15,9582],{"className":9583,"style":697},[71],[15,9585,335],{"className":9586},[422],[15,9588],{"className":9589,"style":697},[71],[15,9591,621],{"className":9592,"style":684},[66,67],[15,9594,120],{"className":9595},[162],[15,9597,3266],{"className":9598},[66],[15,9600,335],{"className":9601},[422],[15,9603],{"className":9604,"style":697},[71],[15,9606,9608,9611],{"className":9607},[66],[15,9609,123],{"className":9610},[66,67],[15,9612,9614],{"className":9613},[198],[15,9615,9617,9637],{"className":9616},[202,1090],[15,9618,9620,9634],{"className":9619},[206],[15,9621,9623],{"className":9622,"style":2198},[210],[15,9624,9625,9628],{"style":1584},[15,9626],{"className":9627,"style":218},[217],[15,9629,9631],{"className":9630},[222,223,224,225],[15,9632,37],{"className":9633},[66,67,225],[15,9635,1114],{"className":9636},[1113],[15,9638,9640],{"className":9639},[206],[15,9641,9643],{"className":9642,"style":1121},[210],[15,9644],{},[15,9646,126],{"className":9647},[169],[15,9649,9651],{"className":9650},[169],[15,9652,4328],{"className":9653},[1422,1508],[15,9655],{"className":9656,"style":173},[71],[15,9658,9430],{"className":9659},[177],[15,9661],{"className":9662,"style":173},[71],[15,9664,9666,9669,9672],{"className":9665},[57],[15,9667],{"className":9668,"style":490},[61],[15,9670,1061],{"className":9671,"style":1083},[66,494],[15,9673,3571],{"className":9674},[66],[11,9676,9677,9678,9729],{},"对任意 ",[15,9679,9681,9699],{"className":9680,"translate":19},[18],[15,9682,9684],{"className":9683},[23],[25,9685,9686],{"xmlns":27},[29,9687,9688,9696],{},[32,9689,9690,9692,9694],{},[35,9691,116],{},[39,9693,1187],{},[35,9695,1061],{"mathvariant":445},[45,9697,9698],{"encoding":47},"f\\in\\mathcal H",[15,9700,9702,9720],{"className":9701,"ariaHidden":53},[52],[15,9703,9705,9708,9711,9714,9717],{"className":9704},[57],[15,9706],{"className":9707,"style":3406},[61],[15,9709,116],{"className":9710,"style":158},[66,67],[15,9712],{"className":9713,"style":173},[71],[15,9715,1187],{"className":9716},[177],[15,9718],{"className":9719,"style":173},[71],[15,9721,9723,9726],{"className":9722},[57],[15,9724],{"className":9725,"style":490},[61],[15,9727,1061],{"className":9728,"style":1083},[66,494],"，把它分解为",[15,9731,9733],{"className":9732,"translate":19},[294],[15,9734,9736,9808],{"className":9735,"translate":19},[18],[15,9737,9739],{"className":9738},[23],[25,9740,9741],{"xmlns":27,"display":303},[29,9742,9743,9805],{},[32,9744,9745,9747,9749,9755,9757,9764,9766,9768,9774,9776,9782,9784,9787,9793,9795,9803],{},[35,9746,116],{},[39,9748,129],{},[1057,9750,9751,9753],{},[35,9752,116],{},[39,9754,1251],{},[39,9756,1245],{},[1057,9758,9759,9761],{},[35,9760,116],{},[39,9762,9763],{},"⊥",[39,9765,335],{"separator":53},[71,9767],{"width":3902},[1057,9769,9770,9772],{},[35,9771,116],{},[39,9773,1251],{},[39,9775,1187],{},[1057,9777,9778,9780],{},[35,9779,1061],{"mathvariant":445},[1209,9781,1804],{},[39,9783,335],{"separator":53},[71,9785],{"width":9786},"1em",[1057,9788,9789,9791],{},[35,9790,116],{},[39,9792,9763],{},[39,9794,1187],{},[1255,9796,9797,9799,9801],{},[35,9798,1061],{"mathvariant":445},[1209,9800,1804],{},[39,9802,9763],{},[35,9804,3571],{"mathvariant":138},[45,9806,9807],{"encoding":47},"f=f_\\parallel+f_\\perp,\\qquad f_\\parallel\\in\\mathcal H_0,\\quad f_\\perp\\in\\mathcal H_0^\\perp.",[15,9809,9811,9829,9887,9992,10097],{"className":9810,"ariaHidden":53},[52],[15,9812,9814,9817,9820,9823,9826],{"className":9813},[57],[15,9815],{"className":9816,"style":3406},[61],[15,9818,116],{"className":9819,"style":158},[66,67],[15,9821],{"className":9822,"style":173},[71],[15,9824,129],{"className":9825},[177],[15,9827],{"className":9828,"style":173},[71],[15,9830,9832,9836,9878,9881,9884],{"className":9831},[57],[15,9833],{"className":9834,"style":9835},[61],"height:1.0496em;vertical-align:-0.3552em;",[15,9837,9839,9842],{"className":9838},[66],[15,9840,116],{"className":9841,"style":158},[66,67],[15,9843,9845],{"className":9844},[198],[15,9846,9848,9869],{"className":9847},[202,1090],[15,9849,9851,9866],{"className":9850},[206],[15,9852,9854],{"className":9853,"style":3666},[210],[15,9855,9857,9860],{"style":9856},"top:-2.5198em;margin-left:-0.1076em;margin-right:0.05em;",[15,9858],{"className":9859,"style":218},[217],[15,9861,9863],{"className":9862},[222,223,224,225],[15,9864,1251],{"className":9865},[177,225],[15,9867,1114],{"className":9868},[1113],[15,9870,9872],{"className":9871},[206],[15,9873,9876],{"className":9874,"style":9875},[210],"height:0.3552em;",[15,9877],{},[15,9879],{"className":9880,"style":72},[71],[15,9882,1245],{"className":9883},[76],[15,9885],{"className":9886,"style":72},[71],[15,9888,9890,9893,9934,9937,9940,9943,9983,9986,9989],{"className":9889},[57],[15,9891],{"className":9892,"style":9835},[61],[15,9894,9896,9899],{"className":9895},[66],[15,9897,116],{"className":9898,"style":158},[66,67],[15,9900,9902],{"className":9901},[198],[15,9903,9905,9926],{"className":9904},[202,1090],[15,9906,9908,9923],{"className":9907},[206],[15,9909,9911],{"className":9910,"style":1097},[210],[15,9912,9914,9917],{"style":9913},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[15,9915],{"className":9916,"style":218},[217],[15,9918,9920],{"className":9919},[222,223,224,225],[15,9921,9763],{"className":9922},[177,225],[15,9924,1114],{"className":9925},[1113],[15,9927,9929],{"className":9928},[206],[15,9930,9932],{"className":9931,"style":1121},[210],[15,9933],{},[15,9935,335],{"className":9936},[422],[15,9938],{"className":9939,"style":4059},[71],[15,9941],{"className":9942,"style":697},[71],[15,9944,9946,9949],{"className":9945},[66],[15,9947,116],{"className":9948,"style":158},[66,67],[15,9950,9952],{"className":9951},[198],[15,9953,9955,9975],{"className":9954},[202,1090],[15,9956,9958,9972],{"className":9957},[206],[15,9959,9961],{"className":9960,"style":3666},[210],[15,9962,9963,9966],{"style":9856},[15,9964],{"className":9965,"style":218},[217],[15,9967,9969],{"className":9968},[222,223,224,225],[15,9970,1251],{"className":9971},[177,225],[15,9973,1114],{"className":9974},[1113],[15,9976,9978],{"className":9977},[206],[15,9979,9981],{"className":9980,"style":9875},[210],[15,9982],{},[15,9984],{"className":9985,"style":173},[71],[15,9987,1187],{"className":9988},[177],[15,9990],{"className":9991,"style":173},[71],[15,9993,9995,9998,10038,10041,10045,10048,10088,10091,10094],{"className":9994},[57],[15,9996],{"className":9997,"style":3406},[61],[15,9999,10001,10004],{"className":10000},[66],[15,10002,1061],{"className":10003,"style":1083},[66,494],[15,10005,10007],{"className":10006},[198],[15,10008,10010,10030],{"className":10009},[202,1090],[15,10011,10013,10027],{"className":10012},[206],[15,10014,10016],{"className":10015,"style":2139},[210],[15,10017,10018,10021],{"style":1100},[15,10019],{"className":10020,"style":218},[217],[15,10022,10024],{"className":10023},[222,223,224,225],[15,10025,1804],{"className":10026},[66,225],[15,10028,1114],{"className":10029},[1113],[15,10031,10033],{"className":10032},[206],[15,10034,10036],{"className":10035,"style":1121},[210],[15,10037],{},[15,10039,335],{"className":10040},[422],[15,10042],{"className":10043,"style":10044},[71],"margin-right:1em;",[15,10046],{"className":10047,"style":697},[71],[15,10049,10051,10054],{"className":10050},[66],[15,10052,116],{"className":10053,"style":158},[66,67],[15,10055,10057],{"className":10056},[198],[15,10058,10060,10080],{"className":10059},[202,1090],[15,10061,10063,10077],{"className":10062},[206],[15,10064,10066],{"className":10065,"style":1097},[210],[15,10067,10068,10071],{"style":9913},[15,10069],{"className":10070,"style":218},[217],[15,10072,10074],{"className":10073},[222,223,224,225],[15,10075,9763],{"className":10076},[177,225],[15,10078,1114],{"className":10079},[1113],[15,10081,10083],{"className":10082},[206],[15,10084,10086],{"className":10085,"style":1121},[210],[15,10087],{},[15,10089],{"className":10090,"style":173},[71],[15,10092,1187],{"className":10093},[177],[15,10095],{"className":10096,"style":173},[71],[15,10098,10100,10104,10156],{"className":10099},[57],[15,10101],{"className":10102,"style":10103},[61],"height:1.1461em;vertical-align:-0.247em;",[15,10105,10107,10110],{"className":10106},[66],[15,10108,1061],{"className":10109,"style":1083},[66,494],[15,1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",[15,10163,10165,10183],{"className":10164,"translate":19},[18],[15,10166,10168],{"className":10167},[23],[25,10169,10170],{"xmlns":27},[29,10171,10172,10180],{},[32,10173,10174],{},[1057,10175,10176,10178],{},[35,10177,123],{},[35,10179,1205],{},[45,10181,10182],{"encoding":47},"x_i",[15,10184,10186],{"className":10185,"ariaHidden":53},[52],[15,10187,10189,10193],{"className":10188},[57],[15,10190],{"className":10191,"style":10192},[61],"height:0.5806em;vertical-align:-0.15em;",[15,10194,10196,10199],{"className":10195},[66],[15,10197,123],{"className":10198},[66,67],[15,10200,10202],{"className":10201},[198],[15,10203,10205,10225],{"className":10204},[202,1090],[15,10206,10208,10222],{"className":10207},[206],[15,10209,10211],{"className":10210,"style":1528},[210],[15,10212,10213,10216],{"style":1584},[15,10214],{"className":10215,"style":218},[217],[15,10217,10219],{"className":10218},[222,223,224,225],[15,10220,1205],{"className":10221},[66,67,225],[15,10223,1114],{"className":10224},[1113],[15,10226,10228],{"className":10227},[206],[15,10229,10231],{"className":10230,"style":1121},[210],[15,10232],{},"，因为 ",[15,10235,10237,10273],{"className":10236,"translate":19},[18],[15,10238,10240],{"className":10239},[23],[25,10241,10242],{"xmlns":27},[29,10243,10244,10270],{},[32,10245,10246,10248,10250,10252,10254,10260,10262,10264],{},[35,10247,621],{},[39,10249,120],{"stretchy":119},[39,10251,3266],{},[39,10253,335],{"separator":53},[1057,10255,10256,10258],{},[35,10257,123],{},[35,10259,1205],{},[39,10261,126],{"stretchy":119},[39,10263,1187],{},[1057,10265,10266,10268],{},[35,10267,1061],{"mathvariant":445},[1209,10269,1804],{},[45,10271,10272],{"encoding":47},"k(\\cdot,x_i)\\in\\mathcal H_0",[15,10274,10276,10349],{"className":10275,"ariaHidden":53},[52],[15,10277,10279,10282,10285,10288,10291,10294,10297,10337,10340,10343,10346],{"className":10278},[57],[15,10280],{"className":10281,"style":154},[61],[15,10283,621],{"className":10284,"style":684},[66,67],[15,10286,120],{"className":10287},[162],[15,10289,3266],{"className":10290},[66],[15,10292,335],{"className":10293},[422],[15,10295],{"className":10296,"style":697},[71],[15,10298,10300,10303],{"className":10299},[66],[15,10301,123],{"className":10302},[66,67],[15,10304,10306],{"className":10305},[198],[15,10307,10309,10329],{"className":10308},[202,1090],[15,10310,10312,10326],{"className":10311},[206],[15,10313,10315],{"className":10314,"style":1528},[210],[15,10316,10317,10320],{"style":1584},[15,10318],{"className":10319,"style":218},[217],[15,10321,10323],{"className":10322},[222,223,224,225],[15,10324,1205],{"className":10325},[66,67,225],[15,10327,1114],{"className":10328},[1113],[15,10330,10332],{"className":10331},[206],[15,10333,10335],{"className":10334,"style":1121},[210],[15,10336],{},[15,10338,126],{"className":10339},[169],[15,10341],{"className":10342,"style":173},[71],[15,10344,1187],{"className":10345},[177],[15,10347],{"className":10348,"style":173},[71],[15,10350,10352,10355],{"className":10351},[57],[15,10353],{"className":10354,"style":1076},[61],[15,10356,10358,10361],{"className":10357},[66],[15,10359,1061],{"className":10360,"style":1083},[66,494],[15,10362,10364],{"className":10363},[198],[15,10365,10367,10387],{"className":10366},[202,1090],[15,10368,10370,10384],{"className":10369},[206],[15,10371,10373],{"className":10372,"style":2139},[210],[15,10374,10375,10378],{"style":1100},[15,10376],{"className":10377,"style":218},[217],[15,10379,10381],{"className":10380},[222,223,224,225],[15,10382,1804],{"className":10383},[66,225],[15,10385,1114],{"className":10386},[1113],[15,10388,10390],{"className":10389},[206],[15,10391,10393],{"className":10392,"style":1121},[210],[15,10394],{},"，所以由再生性",[15,10397,10399],{"className":10398,"translate":19},[294],[15,10400,10402,10535],{"className":10401,"translate":19},[18],[15,10403,10405],{"className":10404},[23],[25,10406,10407],{"xmlns":27,"display":303},[29,10408,10409,10532],{},[32,10410,10411,10413,10415,10421,10423,10425,10427,10429,10431,10433,10435,10437,10439,10445,10447,10449,10452,10454,10456,10462,10464,10466,10468,10470,10472,10478,10480,10482,10484,10486,10492,10494,10496,10498,10500,10502,10508,10510,10512,10514,10520,10522,10528,10530],{},[35,10412,116],{},[39,10414,120],{"stretchy":119},[1057,10416,10417,10419],{},[35,10418,123],{},[35,10420,1205],{},[39,10422,126],{"stretchy":119},[39,10424,129],{},[39,10426,642],{"stretchy":119},[35,10428,116],{},[39,10430,335],{"separator":53},[35,10432,621],{},[39,10434,120],{"stretchy":119},[39,10436,3266],{},[39,10438,335],{"separator":53},[1057,10440,10441,10443],{},[35,10442,123],{},[35,10444,1205],{},[39,10446,126],{"stretchy":119},[39,10448,667],{"stretchy":119},[71,10450],{"linebreak":10451},"newline",[39,10453,129],{},[39,10455,642],{"stretchy":119},[1057,10457,10458,10460],{},[35,10459,116],{},[39,10461,1251],{},[39,10463,335],{"separator":53},[35,10465,621],{},[39,10467,120],{"stretchy":119},[39,10469,3266],{},[39,10471,335],{"separator":53},[1057,10473,10474,10476],{},[35,10475,123],{},[35,10477,1205],{},[39,10479,126],{"stretchy":119},[39,10481,667],{"stretchy":119},[39,10483,1245],{},[39,10485,642],{"stretchy":119},[1057,10487,10488,10490],{},[35,10489,116],{},[39,10491,9763],{},[39,10493,335],{"separator":53},[35,10495,621],{},[39,10497,120],{"stretchy":119},[39,10499,3266],{},[39,10501,335],{"separator":53},[1057,10503,10504,10506],{},[35,10505,123],{},[35,10507,1205],{},[39,10509,126],{"stretchy":119},[39,10511,667],{"stretchy":119},[39,10513,129],{},[1057,10515,10516,10518],{},[35,10517,116],{},[39,10519,1251],{},[39,10521,120],{"stretchy":119},[1057,10523,10524,10526],{},[35,10525,123],{},[35,10527,1205],{},[39,10529,126],{"stretchy":119},[35,10531,3571],{"mathvariant":138},[45,10533,10534],{"encoding":47},"f(x_i)=\\langle f,k(\\cdot,x_i)\\rangle \\\\\n=\\langle f_\\parallel,k(\\cdot,x_i)\\rangle+\\langle f_\\perp,k(\\cdot,x_i)\\rangle\n=f_\\parallel(x_i).",[15,10536,10538,10602,10678,10681,10694,10817,10939],{"className":10537,"ariaHidden":53},[52],[15,10539,10541,10544,10547,10550,10590,10593,10596,10599],{"className":10540},[57],[15,10542],{"className":10543,"style":154},[61],[15,10545,116],{"className":10546,"style":158},[66,67],[15,10548,120],{"className":10549},[162],[15,10551,10553,10556],{"className":10552},[66],[15,10554,123],{"className":10555},[66,67],[15,10557,10559],{"className":10558},[198],[15,10560,10562,10582],{"className":10561},[202,1090],[15,10563,10565,10579],{"className":10564},[206],[15,10566,10568],{"className":10567,"style":1528},[210],[15,10569,10570,10573],{"style":1584},[15,10571],{"className":10572,"style":218},[217],[15,10574,10576],{"className":10575},[222,223,224,225],[15,10577,1205],{"className":10578},[66,67,225],[15,10580,1114],{"className":10581},[1113],[15,10583,10585],{"className":10584},[206],[15,10586,10588],{"className":10587,"style":1121},[210],[15,10589],{},[15,10591,126],{"className":10592},[169],[15,10594],{"className":10595,"style":173},[71],[15,10597,129],{"className":10598},[177],[15,10600],{"className":10601,"style":173},[71],[15,10603,10605,10608,10611,10614,10617,10620,10623,10626,10629,10632,10635,10675],{"className":10604},[57],[15,10606],{"className":10607,"style":154},[61],[15,10609,642],{"className":10610},[162],[15,10612,116],{"className":10613,"style":158},[66,67],[15,10615,335],{"className":10616},[422],[15,10618],{"className":10619,"style":697},[71],[15,10621,621],{"className":10622,"style":684},[66,67],[15,10624,120],{"className":10625},[162],[15,10627,3266],{"className":10628},[66],[15,10630,335],{"className":10631},[422],[15,10633],{"className":10634,"style":697},[71],[15,10636,10638,10641],{"className":10637},[66],[15,10639,123],{"className":10640},[66,67],[15,10642,10644],{"className":10643},[198],[15,10645,10647,10667],{"className":10646},[202,1090],[15,10648,10650,10664],{"className":10649},[206],[15,10651,10653],{"className":10652,"style":1528},[210],[15,10654,10655,10658],{"style":1584},[15,10656],{"className":10657,"style":218},[217],[15,10659,10661],{"className":10660},[222,223,224,225],[15,10662,1205],{"className":10663},[66,67,225],[15,10665,1114],{"className":10666},[1113],[15,10668,10670],{"className":10669},[206],[15,10671,10673],{"className":10672,"style":1121},[210],[15,10674],{},[15,10676,811],{"className":10677},[169],[15,10679],{"className":10680},[71,10451],[15,10682,10684,10688,10691],{"className":10683},[57],[15,10685],{"className":10686,"style":10687},[61],"height:0.3669em;",[15,10689,129],{"className":10690},[177],[15,10692],{"className":10693,"style":173},[71],[15,10695,10697,10701,10704,10744,10747,10750,10753,10756,10759,10762,10765,10805,10808,10811,10814],{"className":10696},[57],[15,10698],{"className":10699,"style":10700},[61],"height:1.1052em;vertical-align:-0.3552em;",[15,10702,642],{"className":10703},[162],[15,10705,10707,10710],{"className":10706},[66],[15,10708,116],{"className":10709,"style":158},[66,67],[15,10711,10713],{"className":10712},[198],[15,10714,10716,10736],{"className":10715},[202,1090],[15,10717,10719,10733],{"className":10718},[206],[15,10720,10722],{"className":10721,"style":3666},[210],[15,10723,10724,10727],{"style":9856},[15,10725],{"className":10726,"style":218},[217],[15,10728,10730],{"className":10729},[222,223,224,225],[15,10731,1251],{"className":10732},[177,225],[15,10734,1114],{"className":10735},[1113],[15,10737,10739],{"className":10738},[206],[15,10740,10742],{"className":10741,"style":9875},[210],[15,10743],{},[15,10745,335],{"className":10746},[422],[15,10748],{"className":10749,"style":697},[71],[15,10751,621],{"className":10752,"style":684},[66,67],[15,10754,120],{"className":10755},[162],[15,10757,3266],{"className":10758},[66],[15,10760,335],{"className":10761},[422],[15,10763],{"className":10764,"style":697},[71],[15,10766,10768,10771],{"className":10767},[66],[15,10769,123],{"className":10770},[66,67],[15,10772,10774],{"className":10773},[198],[15,10775,10777,10797],{"className":10776},[202,1090],[15,10778,10780,10794],{"className":10779},[206],[15,10781,10783],{"className":10782,"style":1528},[210],[15,10784,10785,10788],{"style":1584},[15,10786],{"className":10787,"style":218},[217],[15,10789,10791],{"className":10790},[222,223,224,225],[15,10792,1205],{"className":10793},[66,67,225],[15,10795,1114],{"className":10796},[1113],[15,10798,10800],{"className":10799},[206],[15,10801,10803],{"className":10802,"style":1121},[210],[15,10804],{},[15,10806,811],{"className":10807},[169],[15,10809],{"className":10810,"style":72},[71],[15,10812,1245],{"className":10813},[76],[15,10815],{"className":10816,"style":72},[71],[15,10818,10820,10823,10826,10866,10869,10872,10875,10878,10881,10884,10887,10927,10930,10933,10936],{"className":10819},[57],[15,10821],{"className":10822,"style":154},[61],[15,10824,642],{"className":10825},[162],[15,10827,10829,10832],{"className":10828},[66],[15,10830,116],{"className":10831,"style":158},[66,67],[15,10833,10835],{"className":10834},[198],[15,10836,10838,10858],{"className":10837},[202,1090],[15,10839,10841,10855],{"className":10840},[206],[15,10842,10844],{"className":10843,"style":1097},[210],[15,10845,10846,10849],{"style":9913},[15,10847],{"className":10848,"style":218},[217],[15,10850,10852],{"className":10851},[222,223,224,225],[15,10853,9763],{"className":10854},[177,225],[15,10856,1114],{"className":10857},[1113],[15,10859,10861],{"className":10860},[206],[15,10862,10864],{"className":10863,"style":1121},[210],[15,10865],{},[15,10867,335],{"className":10868},[422],[15,10870],{"className":10871,"style":697},[71],[15,10873,621],{"className":10874,"style":684},[66,67],[15,10876,120],{"className":10877},[162],[15,10879,3266],{"className":10880},[66],[15,10882,335],{"className":10883},[422],[15,10885],{"className":10886,"style":697},[71],[15,10888,10890,10893],{"className":10889},[66],[15,10891,123],{"className":10892},[66,67],[15,10894,10896],{"className":10895},[198],[15,10897,10899,10919],{"className":10898},[202,1090],[15,10900,10902,10916],{"className":10901},[206],[15,10903,10905],{"className":10904,"style":1528},[210],[15,10906,10907,10910],{"style":1584},[15,10908],{"className":10909,"style":218},[217],[15,10911,10913],{"className":10912},[222,223,224,225],[15,10914,1205],{"className":10915},[66,67,225],[15,10917,1114],{"className":10918},[1113],[15,10920,10922],{"className":10921},[206],[15,10923,10925],{"className":10924,"style":1121},[210],[15,10926],{},[15,10928,811],{"className":10929},[169],[15,10931],{"className":10932,"style":173},[71],[15,10934,129],{"className":10935},[177],[15,10937],{"className":10938,"style":173},[71],[15,10940,10942,10945,10985,10988,11028,11031],{"className":10941},[57],[15,10943],{"className":10944,"style":10700},[61],[15,10946,10948,10951],{"className":10947},[66],[15,10949,116],{"className":10950,"style":158},[66,67],[15,10952,10954],{"className":10953},[198],[15,10955,10957,10977],{"className":10956},[202,1090],[15,10958,10960,10974],{"className":10959},[206],[15,10961,10963],{"className":10962,"style":3666},[210],[15,10964,10965,10968],{"style":9856},[15,10966],{"className":10967,"style":218},[217],[15,10969,10971],{"className":10970},[222,223,224,225],[15,10972,1251],{"className":10973},[177,225],[15,10975,1114],{"className":10976},[1113],[15,10978,10980],{"className":10979},[206],[15,10981,10983],{"className":10982,"style":9875},[210],[15,10984],{},[15,10986,120],{"className":10987},[162],[15,10989,10991,10994],{"className":10990},[66],[15,10992,123],{"className":10993},[66,67],[15,10995,10997],{"className":10996},[198],[15,10998,11000,11020],{"className":10999},[202,1090],[15,11001,11003,11017],{"className":11002},[206],[15,11004,11006],{"className":11005,"style":1528},[210],[15,11007,11008,11011],{"style":1584},[15,11009],{"className":11010,"style":218},[217],[15,11012,11014],{"className":11013},[222,223,224,225],[15,11015,1205],{"className":11016},[66,67,225],[15,11018,1114],{"className":11019},[1113],[15,11021,11023],{"className":11022},[206],[15,11024,11026],{"className":11025,"style":1121},[210],[15,11027],{},[15,11029,126],{"className":11030},[169],[15,11032,3571],{"className":11033},[66],[11,11035,11036,11037,11110],{},"也就是说，",[2040,11038,11039,11109],{},[15,11040,11042,11060],{"className":11041,"translate":19},[18],[15,11043,11045],{"className":11044},[23],[25,11046,11047],{"xmlns":27},[29,11048,11049,11057],{},[32,11050,11051],{},[1057,11052,11053,11055],{},[35,11054,116],{},[39,11056,9763],{},[45,11058,11059],{"encoding":47},"f_\\perp",[15,11061,11063],{"className":11062,"ariaHidden":53},[52],[15,11064,11066,11069],{"className":11065},[57],[15,11067],{"className":11068,"style":3406},[61],[15,11070,11072,11075],{"className":11071},[66],[15,11073,116],{"className":11074,"style":158},[66,67],[15,11076,11078],{"className":11077},[198],[15,11079,11081,11101],{"className":11080},[202,1090],[15,11082,11084,11098],{"className":11083},[206],[15,11085,11087],{"className":11086,"style":1097},[210],[15,11088,11089,11092],{"style":9913},[15,11090],{"className":11091,"style":218},[217],[15,11093,11095],{"className":11094},[222,223,224,225],[15,11096,9763],{"className":11097},[177,225],[15,11099,1114],{"className":11100},[1113],[15,11102,11104],{"className":11103},[206],[15,11105,11107],{"className":11106,"style":1121},[210],[15,11108],{}," 在所有训练点上的取值都是零","。它对损失函数没有任何贡献。",[11,11112,11113],{},"再看正则项。由勾股定理，",[15,11115,11117],{"className":11116,"translate":19},[294],[15,11118,11120,11200],{"className":11119,"translate":19},[18],[15,11121,11123],{"className":11122},[23],[25,11124,11125],{"xmlns":27,"display":303},[29,11126,11127,11197],{},[32,11128,11129,11131,11133,11141,11143,11145,11151,11159,11161,11163,11169,11177,11179,11181,11187,11195],{},[35,11130,1251],{"mathvariant":138},[35,11132,116],{},[1255,11134,11135,11137,11139],{},[35,11136,1251],{"mathvariant":138},[35,11138,1061],{"mathvariant":445},[1209,11140,1263],{},[39,11142,129],{},[35,11144,1251],{"mathvariant":138},[1057,11146,11147,11149],{},[35,11148,116],{},[39,11150,1251],{},[1255,11152,11153,11155,11157],{},[35,11154,1251],{"mathvariant":138},[35,11156,1061],{"mathvariant":445},[1209,11158,1263],{},[39,11160,1245],{},[35,11162,1251],{"mathvariant":138},[1057,11164,11165,11167],{},[35,11166,116],{},[39,11168,9763],{},[1255,11170,11171,11173,11175],{},[35,11172,1251],{"mathvariant":138},[35,11174,1061],{"mathvariant":445},[1209,11176,1263],{},[39,11178,1801],{},[35,11180,1251],{"mathvariant":138},[1057,11182,11183,11185],{},[35,11184,116],{},[39,11186,1251],{},[1255,11188,11189,11191,11193],{},[35,11190,1251],{"mathvariant":138},[35,11192,1061],{"mathvariant":445},[1209,11194,1263],{},[35,11196,3571],{"mathvariant":138},[45,11198,11199],{"encoding":47},"\\|f\\|_{\\mathcal H}^2=\\|f_\\parallel\\|_{\\mathcal H}^2+\\|f_\\perp\\|_{\\mathcal H}^2\n\\ge \\|f_\\parallel\\|_{\\mathcal H}^2.",[15,11201,11203,11279,11392,11504],{"className":11202,"ariaHidden":53},[52],[15,11204,11206,11210,11213,11216,11270,11273,11276],{"className":11205},[57],[15,11207],{"className":11208,"style":11209},[61],"height:1.1141em;vertical-align:-0.25em;",[15,11211,1251],{"className":11212},[66],[15,11214,116],{"className":11215,"style":158},[66,67],[15,11217,11219,11222],{"className":11218},[66],[15,11220,1251],{"className":11221},[66],[15,11223,11225],{"className":11224},[198],[15,11226,11228,11262],{"className":11227},[202,1090],[15,11229,11231,11259],{"className":11230},[206],[15,11232,11234,11248],{"className":11233,"style":1650},[210],[15,11235,11236,11239],{"style":1653},[15,11237],{"className":11238,"style":218},[217],[15,11240,11242],{"className":11241},[222,223,224,225],[15,11243,11245],{"className":11244},[66,225],[15,11246,1061],{"className":11247,"style":1083},[66,494,225],[15,11249,11250,11253],{"style":397},[15,11251],{"className":11252,"style":218},[217],[15,11254,11256],{"className":11255},[222,223,224,225],[15,11257,1263],{"className":11258},[66,225],[15,11260,1114],{"className":11261},[1113],[15,11263,11265],{"className":11264},[206],[15,11266,11268],{"className":11267,"style":1686},[210],[15,11269],{},[15,11271],{"className":11272,"style":173},[71],[15,11274,129],{"className":11275},[177],[15,11277],{"className":11278,"style":173},[71],[15,11280,11282,11286,11289,11329,11383,11386,11389],{"className":11281},[57],[15,11283],{"className":11284,"style":11285},[61],"height:1.2193em;vertical-align:-0.3552em;",[15,11287,1251],{"className":11288},[66],[15,11290,11292,11295],{"className":11291},[66],[15,11293,116],{"className":11294,"style":158},[66,67],[15,11296,11298],{"className":11297},[198],[15,11299,11301,11321],{"className":11300},[202,1090],[15,11302,11304,11318],{"className":11303},[206],[15,11305,11307],{"className":11306,"style":3666},[210],[15,11308,11309,11312],{"style":9856},[15,11310],{"className":11311,"style":218},[217],[15,11313,11315],{"className":11314},[222,223,224,225],[15,11316,1251],{"className":11317},[177,225],[15,11319,1114],{"className":11320},[1113],[15,11322,11324],{"className":11323},[206],[15,11325,11327],{"className":11326,"style":9875},[210],[15,11328],{},[15,11330,11332,11335],{"className":11331},[66],[15,11333,1251],{"className":11334},[66],[15,11336,11338],{"className":11337},[198],[15,11339,11341,11375],{"className":11340},[202,1090],[15,11342,11344,11372],{"className":11343},[206],[15,11345,11347,11361],{"className":11346,"style":1650},[210],[15,11348,11349,11352],{"style":1653},[15,11350],{"className":11351,"style":218},[217],[15,11353,11355],{"className":11354},[222,223,224,225],[15,11356,11358],{"className":11357},[66,225],[15,11359,1061],{"className":11360,"style":1083},[66,494,225],[15,11362,11363,11366],{"style":397},[15,11364],{"className":11365,"style":218},[217],[15,11367,11369],{"className":11368},[222,223,224,225],[15,11370,1263],{"className":11371},[66,225],[15,11373,1114],{"className":11374},[1113],[15,11376,11378],{"className":11377},[206],[15,11379,11381],{"className":11380,"style":1686},[210],[15,11382],{},[15,11384],{"className":11385,"style":72},[71],[15,11387,1245],{"className":11388},[76],[15,11390],{"className":11391,"style":72},[71],[15,11393,11395,11398,11401,11441,11495,11498,11501],{"className":11394},[57],[15,11396],{"className":11397,"style":11209},[61],[15,11399,1251],{"className":11400},[66],[15,11402,11404,11407],{"className":11403},[66],[15,11405,116],{"className":11406,"style":158},[66,67],[15,11408,11410],{"className":11409},[198],[15,11411,11413,11433],{"className":11412},[202,1090],[15,11414,11416,11430],{"className":11415},[206],[15,11417,11419],{"className":11418,"style":1097},[210],[15,11420,11421,11424],{"style":9913},[15,11422],{"className":11423,"style":218},[217],[15,11425,11427],{"className":11426},[222,223,224,225],[15,11428,9763],{"className":11429},[177,225],[15,11431,1114],{"className":11432},[1113],[15,11434,11436],{"className":11435},[206],[15,11437,11439],{"className":11438,"style":1121},[210],[15,11440],{},[15,11442,11444,11447],{"className":11443},[66],[15,11445,1251],{"className":11446},[66],[15,11448,11450],{"className":11449},[198],[15,11451,11453,11487],{"className":11452},[202,1090],[15,11454,11456,11484],{"className":11455},[206],[15,11457,11459,11473],{"className":11458,"style":1650},[210],[15,11460,11461,11464],{"style":1653},[15,11462],{"className":11463,"style":218},[217],[15,11465,11467],{"className":11466},[222,223,224,225],[15,11468,11470],{"className":11469},[66,225],[15,11471,1061],{"className":11472,"style":1083},[66,494,225],[15,11474,11475,11478],{"style":397},[15,11476],{"className":11477,"style":218},[217],[15,11479,11481],{"className":11480},[222,223,224,225],[15,11482,1263],{"className":11483},[66,225],[15,11485,1114],{"className":11486},[1113],[15,11488,11490],{"className":11489},[206],[15,11491,11493],{"className":11492,"style":1686},[210],[15,11494],{},[15,11496],{"className":11497,"style":173},[71],[15,11499,1801],{"className":11500},[177],[15,11502],{"className":11503,"style":173},[71],[15,11505,11507,11510,11513,11553,11607],{"className":11506},[57],[15,11508],{"className":11509,"style":11285},[61],[15,11511,1251],{"className":11512},[66],[15,11514,11516,11519],{"className":11515},[66],[15,11517,116],{"className":11518,"style":158},[66,67],[15,11520,11522],{"className":11521},[198],[15,11523,11525,11545],{"className":11524},[202,1090],[15,11526,11528,11542],{"className":11527},[206],[15,11529,11531],{"className":11530,"style":3666},[210],[15,11532,11533,11536],{"style":9856},[15,11534],{"className":11535,"style":218},[217],[15,11537,11539],{"className":11538},[222,223,224,225],[15,11540,1251],{"className":11541},[177,225],[15,11543,1114],{"className":11544},[1113],[15,11546,11548],{"className":11547},[206],[15,11549,11551],{"className":11550,"style":9875},[210],[15,11552],{},[15,11554,11556,11559],{"className":11555},[66],[15,11557,1251],{"className":11558},[66],[15,11560,11562],{"className":11561},[198],[15,11563,11565,11599],{"className":11564},[202,1090],[15,11566,11568,11596],{"className":11567},[206],[15,11569,11571,11585],{"className":11570,"style":1650},[210],[15,11572,11573,11576],{"style":1653},[15,11574],{"className":11575,"style":218},[217],[15,11577,11579],{"className":11578},[222,223,224,225],[15,11580,11582],{"className":11581},[66,225],[15,11583,1061],{"className":11584,"style":1083},[66,494,225],[15,11586,11587,11590],{"style":397},[15,11588],{"className":11589,"style":218},[217],[15,11591,11593],{"className":11592},[222,223,224,225],[15,11594,1263],{"className":11595},[66,225],[15,11597,1114],{"className":11598},[1113],[15,11600,11602],{"className":11601},[206],[15,11603,11605],{"className":11604,"style":1686},[210],[15,11606],{},[15,11608,3571],{"className":11609},[66],[11,11611,11612,11613,11641],{},"因为 ",[15,11614,11616,11629],{"className":11615,"translate":19},[18],[15,11617,11619],{"className":11618},[23],[25,11620,11621],{"xmlns":27},[29,11622,11623,11627],{},[32,11624,11625],{},[35,11626,7737],{"mathvariant":138},[45,11628,8699],{"encoding":47},[15,11630,11632],{"className":11631,"ariaHidden":53},[52],[15,11633,11635,11638],{"className":11634},[57],[15,11636],{"className":11637,"style":490},[61],[15,11639,7737],{"className":11640},[66]," 非减，",[15,11643,11645],{"className":11644,"translate":19},[294],[15,11646,11648,11704],{"className":11647,"translate":19},[18],[15,11649,11651],{"className":11650},[23],[25,11652,11653],{"xmlns":27,"display":303},[29,11654,11655,11701],{},[32,11656,11657,11659,11661,11663,11665,11673,11675,11677,11679,11681,11683,11689,11697,11699],{},[35,11658,7737],{"mathvariant":138},[39,11660,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,11662,1251],{"mathvariant":138},[35,11664,116],{},[1255,11666,11667,11669,11671],{},[35,11668,1251],{"mathvariant":138},[35,11670,1061],{"mathvariant":445},[1209,11672,1263],{},[39,11674,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,11676,1801],{},[35,11678,7737],{"mathvariant":138},[39,11680,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,11682,1251],{"mathvariant":138},[1057,11684,11685,11687],{},[35,11686,116],{},[39,11688,1251],{},[1255,11690,11691,11693,11695],{},[35,11692,1251],{"mathvariant":138},[35,11694,1061],{"mathvariant":445},[1209,11696,1263],{},[39,11698,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,11700,3571],{"mathvariant":138},[45,11702,11703],{"encoding":47},"\\Omega\\bigl(\\|f\\|_{\\mathcal H}^2\\bigr)\\ge  \\Omega\\bigl(\\|f_\\parallel\\|_{\\mathcal H}^2\\bigr).",[15,11705,11707,11798],{"className":11706,"ariaHidden":53},[52],[15,11708,11710,11714,11717,11723,11726,11729,11783,11789,11792,11795],{"className":11709},[57],[15,11711],{"className":11712,"style":11713},[61],"height:1.2141em;vertical-align:-0.35em;",[15,11715,7737],{"className":11716},[66],[15,11718,11720],{"className":11719},[162],[15,11721,120],{"className":11722},[1422,1508],[15,11724,1251],{"className":11725},[66],[15,11727,116],{"className":11728,"style":158},[66,67],[15,11730,11732,11735],{"className":11731},[66],[15,11733,1251],{"className":11734},[66],[15,11736,11738],{"className":11737},[198],[15,11739,11741,11775],{"className":11740},[202,1090],[15,11742,11744,11772],{"className":11743},[206],[15,11745,11747,11761],{"className":11746,"style":1650},[210],[15,11748,11749,11752],{"style":1653},[15,11750],{"className":11751,"style":218},[217],[15,11753,11755],{"className":11754},[222,223,224,225],[15,11756,11758],{"className":11757},[66,225],[15,11759,1061],{"className":11760,"style":1083},[66,494,225],[15,11762,11763,11766],{"style":397},[15,11764],{"className":11765,"style":218},[217],[15,11767,11769],{"className":11768},[222,223,224,225],[15,11770,1263],{"className":11771},[66,225],[15,11773,1114],{"className":11774},[1113],[15,11776,11778],{"className":11777},[206],[15,11779,11781],{"className":11780,"style":1686},[210],[15,11782],{},[15,11784,11786],{"className":11785},[169],[15,11787,126],{"className":11788},[1422,1508],[15,11790],{"className":11791,"style":173},[71],[15,11793,1801],{"className":11794},[177],[15,11796],{"className":11797,"style":173},[71],[15,11799,11801,11804,11807,11813,11816,11856,11910,11916],{"className":11800},[57],[15,11802],{"className":11803,"style":11285},[61],[15,11805,7737],{"className":11806},[66],[15,11808,11810],{"className":11809},[162],[15,11811,120],{"className":11812},[1422,1508],[15,11814,1251],{"className":11815},[66],[15,11817,11819,11822],{"className":11818},[66],[15,11820,116],{"className":11821,"style":158},[66,67],[15,11823,11825],{"className":11824},[198],[15,11826,11828,11848],{"className":11827},[202,1090],[15,11829,11831,11845],{"className":11830},[206],[15,11832,11834],{"className":11833,"style":3666},[210],[15,11835,11836,11839],{"style":9856},[15,11837],{"className":11838,"style":218},[217],[15,11840,11842],{"className":11841},[222,223,224,225],[15,11843,1251],{"className":11844},[177,225],[15,11846,1114],{"className":11847},[1113],[15,11849,11851],{"className":11850},[206],[15,11852,11854],{"className":11853,"style":9875},[210],[15,11855],{},[15,11857,11859,11862],{"className":11858},[66],[15,11860,1251],{"className":11861},[66],[15,11863,11865],{"className":11864},[198],[15,11866,11868,11902],{"className":11867},[202,1090],[15,11869,11871,11899],{"className":11870},[206],[15,11872,11874,11888],{"className":11873,"style":1650},[210],[15,11875,11876,11879],{"style":1653},[15,11877],{"className":11878,"style":218},[217],[15,11880,11882],{"className":11881},[222,223,224,225],[15,11883,11885],{"className":11884},[66,225],[15,11886,1061],{"className":11887,"style":1083},[66,494,225],[15,11889,11890,11893],{"style":397},[15,11891],{"className":11892,"style":218},[217],[15,11894,11896],{"className":11895},[222,223,224,225],[15,11897,1263],{"className":11898},[66,225],[15,11900,1114],{"className":11901},[1113],[15,11903,11905],{"className":11904},[206],[15,11906,11908],{"className":11907,"style":1686},[210],[15,11909],{},[15,11911,11913],{"className":11912},[169],[15,11914,126],{"className":11915},[1422,1508],[15,11917,3571],{"className":11918},[66],[11,11920,11921],{},"把两件事合起来：",[15,11923,11925],{"className":11924,"translate":19},[294],[15,11926,11928,12066],{"className":11927,"translate":19},[18],[15,11929,11931],{"className":11930},[23],[25,11932,11933],{"xmlns":27,"display":303},[29,11934,11935,12063],{},[32,11936,11937,11939,11941,11943,11945,11951,11953,11955,11957,11959,11961,11963,11969,11971,11973,11975,11977,11979,11981,11983,11989,11991,11993,11995,11997,11999,12005,12007,12013,12015,12017,12019,12021,12027,12029,12035,12037,12039,12041,12043,12045,12047,12053,12059,12061],{},[35,11938,1216],{},[39,11940,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,11942,116],{},[39,11944,120],{"stretchy":119},[1057,11946,11947,11949],{},[35,11948,123],{},[1209,11950,1211],{},[39,11952,126],{"stretchy":119},[39,11954,335],{"separator":53},[39,11956,2095],{},[39,11958,335],{"separator":53},[35,11960,116],{},[39,11962,120],{"stretchy":119},[1057,11964,11965,11967],{},[35,11966,123],{},[35,11968,37],{},[39,11970,126],{"stretchy":119},[39,11972,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,11974,1245],{},[35,11976,7737],{"mathvariant":138},[39,11978,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,11980,1251],{"mathvariant":138},[35,11982,116],{},[131,11984,11985,11987],{},[35,11986,1251],{"mathvariant":138},[1209,11988,1263],{},[39,11990,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,11992,1801],{},[71,11994],{"linebreak":10451},[35,11996,1216],{},[39,11998,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,12000,12001,12003],{},[35,12002,116],{},[39,12004,1251],{},[39,12006,120],{"stretchy":119},[1057,12008,12009,12011],{},[35,12010,123],{},[1209,12012,1211],{},[39,12014,126],{"stretchy":119},[39,12016,335],{"separator":53},[39,12018,2095],{},[39,12020,335],{"separator":53},[1057,12022,12023,12025],{},[35,12024,116],{},[39,12026,1251],{},[39,12028,120],{"stretchy":119},[1057,12030,12031,12033],{},[35,12032,123],{},[35,12034,37],{},[39,12036,126],{"stretchy":119},[39,12038,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,12040,1245],{},[35,12042,7737],{"mathvariant":138},[39,12044,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,12046,1251],{"mathvariant":138},[1057,12048,12049,12051],{},[35,12050,116],{},[39,12052,1251],{},[131,12054,12055,12057],{},[35,12056,1251],{"mathvariant":138},[1209,12058,1263],{},[39,12060,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,12062,3571],{"mathvariant":138},[45,12064,12065],{"encoding":47},"L\\bigl(f(x_1),\\dots,f(x_n)\\bigr)+\\Omega\\bigl(\\|f\\|^2\\bigr)\n\\ge \n \\\\ \nL\\bigl(f_\\parallel(x_1),\\dots,f_\\parallel(x_n)\\bigr)+\\Omega\\bigl(\\|f_\\parallel\\|^2\\bigr).",[15,12067,12069,12215,12277,12280,12501],{"className":12068,"ariaHidden":53},[52],[15,12070,12072,12075,12078,12084,12087,12090,12130,12133,12136,12139,12142,12145,12148,12151,12154,12157,12197,12200,12206,12209,12212],{"className":12071},[57],[15,12073],{"className":12074,"style":4398},[61],[15,12076,1216],{"className":12077},[66,67],[15,12079,12081],{"className":12080},[162],[15,12082,120],{"className":12083},[1422,1508],[15,12085,116],{"className":12086,"style":158},[66,67],[15,12088,120],{"className":12089},[162],[15,12091,12093,12096],{"className":12092},[66],[15,12094,123],{"className":12095},[66,67],[15,12097,12099],{"className":12098},[198],[15,12100,12102,12122],{"className":12101},[202,1090],[15,12103,12105,12119],{"className":12104},[206],[15,12106,12108],{"className":12107,"style":2139},[210],[15,12109,12110,12113],{"style":1584},[15,12111],{"className":12112,"style":218},[217],[15,12114,12116],{"className":12115},[222,223,224,225],[15,12117,1211],{"className":12118},[66,225],[15,12120,1114],{"className":12121},[1113],[15,12123,12125],{"className":12124},[206],[15,12126,12128],{"className":12127,"style":1121},[210],[15,12129],{},[15,12131,126],{"className":12132},[169],[15,12134,335],{"className":12135},[422],[15,12137],{"className":12138,"style":697},[71],[15,12140,2095],{"className":12141},[1413],[15,12143],{"className":12144,"style":697},[71],[15,12146,335],{"className":12147},[422],[15,12149],{"className":12150,"style":697},[71],[15,12152,116],{"className":12153,"style":158},[66,67],[15,12155,120],{"className":12156},[162],[15,12158,12160,12163],{"className":12159},[66],[15,12161,123],{"className":12162},[66,67],[15,12164,12166],{"className":12165},[198],[15,12167,12169,12189],{"className":12168},[202,1090],[15,12170,12172,12186],{"className":12171},[206],[15,12173,12175],{"className":12174,"style":2198},[210],[15,12176,12177,12180],{"style":1584},[15,12178],{"className":12179,"style":218},[217],[15,12181,12183],{"className":12182},[222,223,224,225],[15,12184,37],{"className":12185},[66,67,225],[15,12187,1114],{"className":12188},[1113],[15,12190,12192],{"className":12191},[206],[15,12193,12195],{"className":12194,"style":1121},[210],[15,12196],{},[15,12198,126],{"className":12199},[169],[15,12201,12203],{"className":12202},[169],[15,12204,126],{"className":12205},[1422,1508],[15,12207],{"className":12208,"style":72},[71],[15,12210,1245],{"className":12211},[76],[15,12213],{"className":12214,"style":72},[71],[15,12216,12218,12221,12224,12230,12233,12236,12265,12271,12274],{"className":12217},[57],[15,12219],{"className":12220,"style":11713},[61],[15,12222,7737],{"className":12223},[66],[15,12225,12227],{"className":12226},[162],[15,12228,120],{"className":12229},[1422,1508],[15,12231,1251],{"className":12232},[66],[15,12234,116],{"className":12235,"style":158},[66,67],[15,12237,12239,12242],{"className":12238},[66],[15,12240,1251],{"className":12241},[66],[15,12243,12245],{"className":12244},[198],[15,12246,12248],{"className":12247},[202],[15,12249,12251],{"className":12250},[206],[15,12252,12254],{"className":12253,"style":1650},[210],[15,12255,12256,12259],{"style":397},[15,12257],{"className":12258,"style":218},[217],[15,12260,12262],{"className":12261},[222,223,224,225],[15,12263,1263],{"className":12264},[66,225],[15,12266,12268],{"className":12267},[169],[15,12269,126],{"className":12270},[1422,1508],[15,12272],{"className":12273,"style":173},[71],[15,12275,1801],{"className":12276},[177],[15,12278],{"className":12279},[71,10451],[15,12281,12283,12287,12290,12296,12336,12339,12379,12382,12385,12388,12391,12394,12397,12400,12440,12443,12483,12486,12492,12495,12498],{"className":12282},[57],[15,12284],{"className":12285,"style":12286},[61],"height:1.2052em;vertical-align:-0.3552em;",[15,12288,1216],{"className":12289},[66,67],[15,12291,12293],{"className":12292},[162],[15,12294,120],{"className":12295},[1422,1508],[15,12297,12299,12302],{"className":12298},[66],[15,12300,116],{"className":12301,"style":158},[66,67],[15,12303,12305],{"className":12304},[198],[15,12306,12308,12328],{"className":12307},[202,1090],[15,12309,12311,12325],{"className":12310},[206],[15,12312,12314],{"className":12313,"style":3666},[210],[15,12315,12316,12319],{"style":9856},[15,12317],{"className":12318,"style":218},[217],[15,12320,12322],{"className":12321},[222,223,224,225],[15,12323,1251],{"className":12324},[177,225],[15,12326,1114],{"className":12327},[1113],[15,12329,12331],{"className":12330},[206],[15,12332,12334],{"className":12333,"style":9875},[210],[15,12335],{},[15,12337,120],{"className":12338},[162],[15,12340,12342,12345],{"className":12341},[66],[15,12343,123],{"className":12344},[66,67],[15,12346,12348],{"className":12347},[198],[15,12349,12351,12371],{"className":12350},[202,1090],[15,12352,12354,12368],{"className":12353},[206],[15,12355,12357],{"className":12356,"style":2139},[210],[15,12358,12359,12362],{"style":1584},[15,12360],{"className":12361,"style":218},[217],[15,12363,12365],{"className":12364},[222,223,224,225],[15,12366,1211],{"className":12367},[66,225],[15,12369,1114],{"className":12370},[1113],[15,12372,12374],{"className":12373},[206],[15,12375,12377],{"className":12376,"style":1121},[210],[15,12378],{},[15,12380,126],{"className":12381},[169],[15,12383,335],{"className":12384},[422],[15,12386],{"className":12387,"style":697},[71],[15,12389,2095],{"className":12390},[1413],[15,12392],{"className":12393,"style":697},[71],[15,12395,335],{"className":12396},[422],[15,12398],{"className":12399,"style":697},[71],[15,12401,12403,12406],{"className":12402},[66],[15,12404,116],{"className":12405,"style":158},[66,67],[15,12407,12409],{"className":12408},[198],[15,12410,12412,12432],{"className":12411},[202,1090],[15,12413,12415,12429],{"className":12414},[206],[15,12416,12418],{"className":12417,"style":3666},[210],[15,12419,12420,12423],{"style":9856},[15,12421],{"className":12422,"style":218},[217],[15,12424,12426],{"className":12425},[222,223,224,225],[15,12427,1251],{"className":12428},[177,225],[15,12430,1114],{"className":12431},[1113],[15,12433,12435],{"className":12434},[206],[15,12436,12438],{"className":12437,"style":9875},[210],[15,12439],{},[15,12441,120],{"className":12442},[162],[15,12444,12446,12449],{"className":12445},[66],[15,12447,123],{"className":12448},[66,67],[15,12450,12452],{"className":12451},[198],[15,12453,12455,12475],{"className":12454},[202,1090],[15,12456,12458,12472],{"className":12457},[206],[15,12459,12461],{"className":12460,"style":2198},[210],[15,12462,12463,12466],{"style":1584},[15,12464],{"className":12465,"style":218},[217],[15,12467,12469],{"className":12468},[222,223,224,225],[15,12470,37],{"className":12471},[66,67,225],[15,12473,1114],{"className":12474},[1113],[15,12476,12478],{"className":12477},[206],[15,12479,12481],{"className":12480,"style":1121},[210],[15,12482],{},[15,12484,126],{"className":12485},[169],[15,12487,12489],{"className":12488},[169],[15,12490,126],{"className":12491},[1422,1508],[15,12493],{"className":12494,"style":72},[71],[15,12496,1245],{"className":12497},[76],[15,12499],{"className":12500,"style":72},[71],[15,12502,12504,12507,12510,12516,12519,12559,12588,12594],{"className":12503},[57],[15,12505],{"className":12506,"style":11285},[61],[15,12508,7737],{"className":12509},[66],[15,12511,12513],{"className":12512},[162],[15,12514,120],{"className":12515},[1422,1508],[15,12517,1251],{"className":12518},[66],[15,12520,12522,12525],{"className":12521},[66],[15,12523,116],{"className":12524,"style":158},[66,67],[15,12526,12528],{"className":12527},[198],[15,12529,12531,12551],{"className":12530},[202,1090],[15,12532,12534,12548],{"className":12533},[206],[15,12535,12537],{"className":12536,"style":3666},[210],[15,12538,12539,12542],{"style":9856},[15,12540],{"className":12541,"style":218},[217],[15,12543,12545],{"className":12544},[222,223,224,225],[15,12546,1251],{"className":12547},[177,225],[15,12549,1114],{"className":12550},[1113],[15,12552,12554],{"className":12553},[206],[15,12555,12557],{"className":12556,"style":9875},[210],[15,12558],{},[15,12560,12562,12565],{"className":12561},[66],[15,12563,1251],{"className":12564},[66],[15,12566,12568],{"className":12567},[198],[15,12569,12571],{"className":12570},[202],[15,12572,12574],{"className":12573},[206],[15,12575,12577],{"className":12576,"style":1650},[210],[15,12578,12579,12582],{"style":397},[15,12580],{"className":12581,"style":218},[217],[15,12583,12585],{"className":12584},[222,223,224,225],[15,12586,1263],{"className":12587},[66,225],[15,12589,12591],{"className":12590},[169],[15,12592,126],{"className":12593},[1422,1508],[15,12595,3571],{"className":12596},[66],[11,12598,12599,12600,12628,12629,12699,12700,12728,12729,12757,12758,12827,12828,12961],{},"所以，对任意 ",[15,12601,12603,12616],{"className":12602,"translate":19},[18],[15,12604,12606],{"className":12605},[23],[25,12607,12608],{"xmlns":27},[29,12609,12610,12614],{},[32,12611,12612],{},[35,12613,116],{},[45,12615,116],{"encoding":47},[15,12617,12619],{"className":12618,"ariaHidden":53},[52],[15,12620,12622,12625],{"className":12621},[57],[15,12623],{"className":12624,"style":3406},[61],[15,12626,116],{"className":12627,"style":158},[66,67],"，它的投影 ",[15,12630,12632,12650],{"className":12631,"translate":19},[18],[15,12633,12635],{"className":12634},[23],[25,12636,12637],{"xmlns":27},[29,12638,12639,12647],{},[32,12640,12641],{},[1057,12642,12643,12645],{},[35,12644,116],{},[39,12646,1251],{},[45,12648,12649],{"encoding":47},"f_\\parallel",[15,12651,12653],{"className":12652,"ariaHidden":53},[52],[15,12654,12656,12659],{"className":12655},[57],[15,12657],{"className":12658,"style":9835},[61],[15,12660,12662,12665],{"className":12661},[66],[15,12663,116],{"className":12664,"style":158},[66,67],[15,12666,12668],{"className":12667},[198],[15,12669,12671,12691],{"className":12670},[202,1090],[15,12672,12674,12688],{"className":12673},[206],[15,12675,12677],{"className":12676,"style":3666},[210],[15,12678,12679,12682],{"style":9856},[15,12680],{"className":12681,"style":218},[217],[15,12683,12685],{"className":12684},[222,223,224,225],[15,12686,1251],{"className":12687},[177,225],[15,12689,1114],{"className":12690},[1113],[15,12692,12694],{"className":12693},[206],[15,12695,12697],{"className":12696,"style":9875},[210],[15,12698],{}," 的目标值都不比 ",[15,12701,12703,12716],{"className":12702,"translate":19},[18],[15,12704,12706],{"className":12705},[23],[25,12707,12708],{"xmlns":27},[29,12709,12710,12714],{},[32,12711,12712],{},[35,12713,116],{},[45,12715,116],{"encoding":47},[15,12717,12719],{"className":12718,"ariaHidden":53},[52],[15,12720,12722,12725],{"className":12721},[57],[15,12723],{"className":12724,"style":3406},[61],[15,12726,116],{"className":12727,"style":158},[66,67]," 差。若 ",[15,12730,12732,12745],{"className":12731,"translate":19},[18],[15,12733,12735],{"className":12734},[23],[25,12736,12737],{"xmlns":27},[29,12738,12739,12743],{},[32,12740,12741],{},[35,12742,116],{},[45,12744,116],{"encoding":47},[15,12746,12748],{"className":12747,"ariaHidden":53},[52],[15,12749,12751,12754],{"className":12750},[57],[15,12752],{"className":12753,"style":3406},[61],[15,12755,116],{"className":12756,"style":158},[66,67]," 是最优解，则 ",[15,12759,12761,12778],{"className":12760,"translate":19},[18],[15,12762,12764],{"className":12763},[23],[25,12765,12766],{"xmlns":27},[29,12767,12768,12776],{},[32,12769,12770],{},[1057,12771,12772,12774],{},[35,12773,116],{},[39,12775,1251],{},[45,12777,12649],{"encoding":47},[15,12779,12781],{"className":12780,"ariaHidden":53},[52],[15,12782,12784,12787],{"className":12783},[57],[15,12785],{"className":12786,"style":9835},[61],[15,12788,12790,12793],{"className":12789},[66],[15,12791,116],{"className":12792,"style":158},[66,67],[15,12794,12796],{"className":12795},[198],[15,12797,12799,12819],{"className":12798},[202,1090],[15,12800,12802,12816],{"className":12801},[206],[15,12803,12805],{"className":12804,"style":3666},[210],[15,12806,12807,12810],{"style":9856},[15,12808],{"className":12809,"style":218},[217],[15,12811,12813],{"className":12812},[222,223,224,225],[15,12814,1251],{"className":12815},[177,225],[15,12817,1114],{"className":12818},[1113],[15,12820,12822],{"className":12821},[206],[15,12823,12825],{"className":12824,"style":9875},[210],[15,12826],{}," 也是最优解，且 ",[15,12829,12831,12857],{"className":12830,"translate":19},[18],[15,12832,12834],{"className":12833},[23],[25,12835,12836],{"xmlns":27},[29,12837,12838,12854],{},[32,12839,12840,12846,12848],{},[1057,12841,12842,12844],{},[35,12843,116],{},[39,12845,1251],{},[39,12847,1187],{},[1057,12849,12850,12852],{},[35,12851,1061],{"mathvariant":445},[1209,12853,1804],{},[45,12855,12856],{"encoding":47},"f_\\parallel\\in\\mathcal H_0",[15,12858,12860,12915],{"className":12859,"ariaHidden":53},[52],[15,12861,12863,12866,12906,12909,12912],{"className":12862},[57],[15,12864],{"className":12865,"style":9835},[61],[15,12867,12869,12872],{"className":12868},[66],[15,12870,116],{"className":12871,"style":158},[66,67],[15,12873,12875],{"className":12874},[198],[15,12876,12878,12898],{"className":12877},[202,1090],[15,12879,12881,12895],{"className":12880},[206],[15,12882,12884],{"className":12883,"style":3666},[210],[15,12885,12886,12889],{"style":9856},[15,12887],{"className":12888,"style":218},[217],[15,12890,12892],{"className":12891},[222,223,224,225],[15,12893,1251],{"className":12894},[177,225],[15,12896,1114],{"className":12897},[1113],[15,12899,12901],{"className":12900},[206],[15,12902,12904],{"className":12903,"style":9875},[210],[15,12905],{},[15,12907],{"className":12908,"style":173},[71],[15,12910,1187],{"className":12911},[177],[15,12913],{"className":12914,"style":173},[71],[15,12916,12918,12921],{"className":12917},[57],[15,12919],{"className":12920,"style":1076},[61],[15,12922,12924,12927],{"className":12923},[66],[15,12925,1061],{"className":12926,"style":1083},[66,494],[15,12928,12930],{"className":12929},[198],[15,12931,12933,12953],{"className":12932},[202,1090],[15,12934,12936,12950],{"className":12935},[206],[15,12937,12939],{"className":12938,"style":2139},[210],[15,12940,12941,12944],{"style":1100},[15,12942],{"className":12943,"style":218},[217],[15,12945,12947],{"className":12946},[222,223,224,225],[15,12948,1804],{"className":12949},[66,225],[15,12951,1114],{"className":12952},[1113],[15,12954,12956],{"className":12955},[206],[15,12957,12959],{"className":12958,"style":1121},[210],[15,12960],{},"。这就证明了第 1 条。",[11,12963,12964,12965,12993,12994,13127,13128,13301,13302,13330,13331,13423,13424,13494,13495],{},"若 ",[15,12966,12968,12981],{"className":12967,"translate":19},[18],[15,12969,12971],{"className":12970},[23],[25,12972,12973],{"xmlns":27},[29,12974,12975,12979],{},[32,12976,12977],{},[35,12978,7737],{"mathvariant":138},[45,12980,8699],{"encoding":47},[15,12982,12984],{"className":12983,"ariaHidden":53},[52],[15,12985,12987,12990],{"className":12986},[57],[15,12988],{"className":12989,"style":490},[61],[15,12991,7737],{"className":12992},[66]," 严格递增且 ",[15,12995,12997,13020],{"className":12996,"translate":19},[18],[15,12998,13000],{"className":12999},[23],[25,13001,13002],{"xmlns":27},[29,13003,13004,13017],{},[32,13005,13006,13012,13015],{},[1057,13007,13008,13010],{},[35,13009,116],{},[39,13011,9763],{},[39,13013,13014],{"mathvariant":138},"≠",[1209,13016,1804],{},[45,13018,13019],{"encoding":47},"f_\\perp\\neq 0",[15,13021,13023,13118],{"className":13022,"ariaHidden":53},[52],[15,13024,13026,13029,13069,13072,13115],{"className":13025},[57],[15,13027],{"className":13028,"style":3406},[61],[15,13030,13032,13035],{"className":13031},[66],[15,13033,116],{"className":13034,"style":158},[66,67],[15,13036,13038],{"className":13037},[198],[15,13039,13041,13061],{"className":13040},[202,1090],[15,13042,13044,13058],{"className":13043},[206],[15,13045,13047],{"className":13046,"style":1097},[210],[15,13048,13049,13052],{"style":9913},[15,13050],{"className":13051,"style":218},[217],[15,13053,13055],{"className":13054},[222,223,224,225],[15,13056,9763],{"className":13057},[177,225],[15,13059,1114],{"className":13060},[1113],[15,13062,13064],{"className":13063},[206],[15,13065,13067],{"className":13066,"style":1121},[210],[15,13068],{},[15,13070],{"className":13071,"style":173},[71],[15,13073,13075,13108,13112],{"className":13074},[177],[15,13076,13078],{"className":13077},[177],[15,13079,13082],{"className":13080},[66,13081],"vbox",[15,13083,13086],{"className":13084},[13085],"thinbox",[15,13087,13090,13093,13104],{"className":13088},[13089],"rlap",[15,13091],{"className":13092,"style":3406},[61],[15,13094,13097],{"className":13095},[13096],"inner",[15,13098,13100],{"className":13099},[66],[15,13101,13103],{"className":13102},[177],"",[15,13105],{"className":13106},[13107],"fix",[15,13109],{"className":13110},[71,13111],"nobreak",[15,13113,129],{"className":13114},[177],[15,13116],{"className":13117,"style":173},[71],[15,13119,13121,13124],{"className":13120},[57],[15,13122],{"className":13123,"style":1836},[61],[15,13125,1804],{"className":13126},[66],"，则 ",[15,13129,13131,13169],{"className":13130,"translate":19},[18],[15,13132,13134],{"className":13133},[23],[25,13135,13136],{"xmlns":27},[29,13137,13138,13166],{},[32,13139,13140,13142,13144,13150,13152,13154,13160],{},[35,13141,1251],{"mathvariant":138},[35,13143,116],{},[131,13145,13146,13148],{},[35,13147,1251],{"mathvariant":138},[1209,13149,1263],{},[39,13151,9278],{},[35,13153,1251],{"mathvariant":138},[1057,13155,13156,13158],{},[35,13157,116],{},[39,13159,1251],{},[131,13161,13162,13164],{},[35,13163,1251],{"mathvariant":138},[1209,13165,1263],{},[45,13167,13168],{"encoding":47},"\\|f\\|^2>\\|f_\\parallel\\|^2",[15,13170,13172,13222],{"className":13171,"ariaHidden":53},[52],[15,13173,13175,13178,13181,13184,13213,13216,13219],{"className":13174},[57],[15,13176],{"className":13177,"style":9149},[61],[15,13179,1251],{"className":13180},[66],[15,13182,116],{"className":13183,"style":158},[66,67],[15,13185,13187,13190],{"className":13186},[66],[15,13188,1251],{"className":13189},[66],[15,13191,13193],{"className":13192},[198],[15,13194,13196],{"className":13195},[202],[15,13197,13199],{"className":13198},[206],[15,13200,13202],{"className":13201,"style":6738},[210],[15,13203,13204,13207],{"style":213},[15,13205],{"className":13206,"style":218},[217],[15,13208,13210],{"className":13209},[222,223,224,225],[15,13211,1263],{"className":13212},[66,225],[15,13214],{"className":13215,"style":173},[71],[15,13217,9278],{"className":13218},[177],[15,13220],{"className":13221,"style":173},[71],[15,13223,13225,13229,13232,13272],{"className":13224},[57],[15,13226],{"className":13227,"style":13228},[61],"height:1.1693em;vertical-align:-0.3552em;",[15,13230,1251],{"className":13231},[66],[15,13233,13235,13238],{"className":13234},[66],[15,13236,116],{"className":13237,"style":158},[66,67],[15,13239,13241],{"className":13240},[198],[15,13242,13244,13264],{"className":13243},[202,1090],[15,13245,13247,13261],{"className":13246},[206],[15,13248,13250],{"className":13249,"style":3666},[210],[15,13251,13252,13255],{"style":9856},[15,13253],{"className":13254,"style":218},[217],[15,13256,13258],{"className":13257},[222,223,224,225],[15,13259,1251],{"className":13260},[177,225],[15,13262,1114],{"className":13263},[1113],[15,13265,13267],{"className":13266},[206],[15,13268,13270],{"className":13269,"style":9875},[210],[15,13271],{},[15,13273,13275,13278],{"className":13274},[66],[15,13276,1251],{"className":13277},[66],[15,13279,13281],{"className":13280},[198],[15,13282,13284],{"className":13283},[202],[15,13285,13287],{"className":13286},[206],[15,13288,13290],{"className":13289,"style":6738},[210],[15,13291,13292,13295],{"style":213},[15,13293],{"className":13294,"style":218},[217],[15,13296,13298],{"className":13297},[222,223,224,225],[15,13299,1263],{"className":13300},[66,225],"，上面的不等式是严格的，",[15,13303,13305,13318],{"className":13304,"translate":19},[18],[15,13306,13308],{"className":13307},[23],[25,13309,13310],{"xmlns":27},[29,13311,13312,13316],{},[32,13313,13314],{},[35,13315,116],{},[45,13317,116],{"encoding":47},[15,13319,13321],{"className":13320,"ariaHidden":53},[52],[15,13322,13324,13327],{"className":13323},[57],[15,13325],{"className":13326,"style":3406},[61],[15,13328,116],{"className":13329,"style":158},[66,67]," 不可能是最优解。因此每个最优解都必须满足 ",[15,13332,13334,13356],{"className":13333,"translate":19},[18],[15,13335,13337],{"className":13336},[23],[25,13338,13339],{"xmlns":27},[29,13340,13341,13353],{},[32,13342,13343,13349,13351],{},[1057,13344,13345,13347],{},[35,13346,116],{},[39,13348,9763],{},[39,13350,129],{},[1209,13352,1804],{},[45,13354,13355],{"encoding":47},"f_\\perp=0",[15,13357,13359,13414],{"className":13358,"ariaHidden":53},[52],[15,13360,13362,13365,13405,13408,13411],{"className":13361},[57],[15,13363],{"className":13364,"style":3406},[61],[15,13366,13368,13371],{"className":13367},[66],[15,13369,116],{"className":13370,"style":158},[66,67],[15,13372,13374],{"className":13373},[198],[15,13375,13377,13397],{"className":13376},[202,1090],[15,13378,13380,13394],{"className":13379},[206],[15,13381,13383],{"className":13382,"style":1097},[210],[15,13384,13385,13388],{"style":9913},[15,13386],{"className":13387,"style":218},[217],[15,13389,13391],{"className":13390},[222,223,224,225],[15,13392,9763],{"className":13393},[177,225],[15,13395,1114],{"className":13396},[1113],[15,13398,13400],{"className":13399},[206],[15,13401,13403],{"className":13402,"style":1121},[210],[15,13404],{},[15,13406],{"className":13407,"style":173},[71],[15,13409,129],{"className":13410},[177],[15,13412],{"className":13413,"style":173},[71],[15,13415,13417,13420],{"className":13416},[57],[15,13418],{"className":13419,"style":1836},[61],[15,13421,1804],{"className":13422},[66],"，即落在 ",[15,13425,13427,13445],{"className":13426,"translate":19},[18],[15,13428,13430],{"className":13429},[23],[25,13431,13432],{"xmlns":27},[29,13433,13434,13442],{},[32,13435,13436],{},[1057,13437,13438,13440],{},[35,13439,1061],{"mathvariant":445},[1209,13441,1804],{},[45,13443,13444],{"encoding":47},"\\mathcal H_0",[15,13446,13448],{"className":13447,"ariaHidden":53},[52],[15,13449,13451,13454],{"className":13450},[57],[15,13452],{"className":13453,"style":1076},[61],[15,13455,13457,13460],{"className":13456},[66],[15,13458,1061],{"className":13459,"style":1083},[66,494],[15,13461,13463],{"className":13462},[198],[15,13464,13466,13486],{"className":13465},[202,1090],[15,13467,13469,13483],{"className":13468},[206],[15,13470,13472],{"className":13471,"style":2139},[210],[15,13473,13474,13477],{"style":1100},[15,13475],{"className":13476,"style":218},[217],[15,13478,13480],{"className":13479},[222,223,224,225],[15,13481,1804],{"className":13482},[66,225],[15,13484,1114],{"className":13485},[1113],[15,13487,13489],{"className":13488},[206],[15,13490,13492],{"className":13491,"style":1121},[210],[15,13493],{}," 中。第 2 条得证。",[15,13496,13498,13513],{"className":13497,"translate":19},[18],[15,13499,13501],{"className":13500},[23],[25,13502,13503],{"xmlns":27},[29,13504,13505,13510],{},[32,13506,13507],{},[35,13508,13509],{"mathvariant":138},"□",[45,13511,13512],{"encoding":47},"\\square",[15,13514,13516],{"className":13515,"ariaHidden":53},[52],[15,13517,13519,13523],{"className":13518},[57],[15,13520],{"className":13521,"style":13522},[61],"height:0.675em;",[15,13524,13509],{"className":13525},[66,13526],"amsrm",[13528,13529,13530],"div",{},[13531,13532],"img",{"height":13533,"width":13534,"style":13535,"data-d2-theme":13536,"alt":13537,"src":13538},277,760,"max-width:100%;height:auto","default","表示定理的证明结构：正交分解与有限维归约","data:image\u002Fsvg+xml;base64,<svg width="1415" height="277" xmlns="http://www.w3.org/2000/svg" data-d2-version="v0.9.0" viewBox="0 0 1415 277"><svg width="1415" height="277" class="d2-3061015132 d2-svg" viewBox="-1 -1 1415 277"><rect width="1415" height="277" x="-1" y="-1" 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stroke="#0d32b2" d="M413 209h142v66H413z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="484" y="247.5" fill="#0a0f25" class="text-bold fill-N1" style="text-anchor:middle;font-size:16px">f(xᵢ) = f∥(xᵢ)</text></g><g class="RQ=="><g class="shape"><path fill="#f7f8fe" stroke="#0d32b2" d="M655 37h197v66H655z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="753.5" y="75.5" fill="#0a0f25" class="text-bold fill-N1" style="text-anchor:middle;font-size:16px">‖f‖² = ‖f∥‖² + ‖f⊥‖²</text></g><g class="Rg=="><g class="shape"><path fill="#f7f8fe" stroke="#0d32b2" d="M682 163h144v66H682z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="754" y="201.5" fill="#0a0f25" class="text-bold fill-N1" style="text-anchor:middle;font-size:16px">损失只看 f∥</text></g><g class="Rw=="><g class="shape"><path fill="#f7f8fe" stroke="#0d32b2" d="M952 100h200v66H952z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="1052" y="138.5" fill="#0a0f25" class="text-bold 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k(\\cdot,x_j)",[15,13663,13665,13683],{"className":13664,"ariaHidden":53},[52],[15,13666,13668,13671,13674,13677,13680],{"className":13667},[57],[15,13669],{"className":13670,"style":3406},[61],[15,13672,116],{"className":13673,"style":158},[66,67],[15,13675],{"className":13676,"style":173},[71],[15,13678,129],{"className":13679},[177],[15,13681],{"className":13682,"style":173},[71],[15,13684,13686,13690,13756,13759,13799,13802,13805,13808,13811,13814,13854],{"className":13685},[57],[15,13687],{"className":13688,"style":13689},[61],"height:1.2401em;vertical-align:-0.4358em;",[15,13691,13693,13698],{"className":13692},[1339],[15,13694,1200],{"className":13695,"style":13697},[1339,1470,13696],"small-op","position:relative;top:0em;",[15,13699,13701],{"className":13700},[198],[15,13702,13704,13747],{"className":13703},[202,1090],[15,13705,13707,13744],{"className":13706},[206],[15,13708,13711,13732],{"className":13709,"style":13710},[210],"height:0.8043em;",[15,13712,13714,13717],{"style":13713},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[15,13715],{"className":13716,"style":218},[217],[15,13718,13720],{"className":13719},[222,223,224,225],[15,13721,13723,13726,13729],{"className":13722},[66,225],[15,13724,2441],{"className":13725,"style":2594},[66,67,225],[15,13727,129],{"className":13728},[177,225],[15,13730,1211],{"className":13731},[66,225],[15,13733,13735,13738],{"style":13734},"top:-3.2029em;margin-right:0.05em;",[15,13736],{"className":13737,"style":218},[217],[15,13739,13741],{"className":13740},[222,223,224,225],[15,13742,37],{"className":13743},[66,67,225],[15,13745,1114],{"className":13746},[1113],[15,13748,13750],{"className":13749},[206],[15,13751,13754],{"className":13752,"style":13753},[210],"height:0.4358em;",[15,13755],{},[15,13757],{"className":13758,"style":697},[71],[15,13760,13762,13765],{"className":13761},[66],[15,13763,8385],{"className":13764,"style":8555},[66,67],[15,13766,13768],{"className":13767},[198],[15,13769,13771,13791],{"className":13770},[202,1090],[15,13772,13774,13788],{"className":13773},[206],[15,13775,13777],{"className":13776,"style":1528},[210],[15,13778,13779,13782],{"style":8570},[15,13780],{"className":13781,"style":218},[217],[15,13783,13785],{"className":13784},[222,223,224,225],[15,13786,2441],{"className":13787,"style":2594},[66,67,225],[15,13789,1114],{"className":13790},[1113],[15,13792,13794],{"className":13793},[206],[15,13795,13797],{"className":13796,"style":2715},[210],[15,13798],{},[15,13800,621],{"className":13801,"style":684},[66,67],[15,13803,120],{"className":13804},[162],[15,13806,3266],{"className":13807},[66],[15,13809,335],{"className":13810},[422],[15,13812],{"className":13813,"style":697},[71],[15,13815,13817,13820],{"className":13816},[66],[15,13818,123],{"className":13819},[66,67],[15,13821,13823],{"className":13822},[198],[15,13824,13826,13846],{"className":13825},[202,1090],[15,13827,13829,13843],{"className":13828},[206],[15,13830,13832],{"className":13831,"style":1528},[210],[15,13833,13834,13837],{"style":1584},[15,13835],{"className":13836,"style":218},[217],[15,13838,13840],{"className":13839},[222,223,224,225],[15,13841,2441],{"className":13842,"style":2594},[66,67,225],[15,13844,1114],{"className":13845},[1113],[15,13847,13849],{"className":13848},[206],[15,13850,13852],{"className":13851,"style":2715},[210],[15,13853],{},[15,13855,126],{"className":13856},[169]," 代回目标函数。对每个训练点，",[15,13859,13861],{"className":13860,"translate":19},[294],[15,13862,13864,13948],{"className":13863,"translate":19},[18],[15,13865,13867],{"className":13866},[23],[25,13868,13869],{"xmlns":27,"display":303},[29,13870,13871,13945],{},[32,13872,13873,13875,13877,13883,13885,13887,13901,13907,13909,13911,13913,13919,13921,13927,13929,13931,13933,13935,13937,13943],{},[35,13874,116],{},[39,13876,120],{"stretchy":119},[1057,13878,13879,13881],{},[35,13880,123],{},[35,13882,1205],{},[39,13884,126],{"stretchy":119},[39,13886,129],{},[1196,13888,13889,13891,13899],{},[39,13890,1200],{},[32,13892,13893,13895,13897],{},[35,13894,2441],{},[39,13896,129],{},[1209,13898,1211],{},[35,13900,37],{},[1057,13902,13903,13905],{},[35,13904,8385],{},[35,13906,2441],{},[1274,13908,2462],{},[35,13910,621],{},[39,13912,120],{"stretchy":119},[1057,13914,13915,13917],{},[35,13916,123],{},[35,13918,1205],{},[39,13920,335],{"separator":53},[1057,13922,13923,13925],{},[35,13924,123],{},[35,13926,2441],{},[39,13928,126],{"stretchy":119},[39,13930,129],{},[39,13932,120],{"stretchy":119},[35,13934,2853],{},[35,13936,8385],{},[1057,13938,13939,13941],{},[39,13940,126],{"stretchy":119},[35,13942,1205],{},[39,13944,335],{"separator":53},[45,13946,13947],{"encoding":47},"f(x_i)=\\sum_{j=1}^n\\alpha_j\\,k(x_i,x_j)=(K\\alpha)_i,",[15,13949,13951,14015,14235],{"className":13950,"ariaHidden":53},[52],[15,13952,13954,13957,13960,13963,14003,14006,14009,14012],{"className":13953},[57],[15,13955],{"className":13956,"style":154},[61],[15,13958,116],{"className":13959,"style":158},[66,67],[15,13961,120],{"className":13962},[162],[15,13964,13966,13969],{"className":13965},[66],[15,13967,123],{"className":13968},[66,67],[15,13970,13972],{"className":13971},[198],[15,13973,13975,13995],{"className":13974},[202,1090],[15,13976,13978,13992],{"className":13977},[206],[15,13979,13981],{"className":13980,"style":1528},[210],[15,13982,13983,13986],{"style":1584},[15,13984],{"className":13985,"style":218},[217],[15,13987,13989],{"className":13988},[222,223,224,225],[15,13990,1205],{"className":13991},[66,67,225],[15,13993,1114],{"className":13994},[1113],[15,13996,13998],{"className":13997},[206],[15,13999,14001],{"className":14000,"style":1121},[210],[15,14002],{},[15,14004,126],{"className":14005},[169],[15,14007],{"className":14008,"style":173},[71],[15,14010,129],{"className":14011},[177],[15,14013],{"className":14014,"style":173},[71],[15,14016,14018,14021,14085,14088,14128,14131,14134,14137,14177,14180,14183,14223,14226,14229,14232],{"className":14017},[57],[15,14019],{"className":14020,"style":2500},[61],[15,14022,14024],{"className":14023},[1339,1351],[15,14025,14027,14077],{"className":14026},[202,1090],[15,14028,14030,14074],{"className":14029},[206],[15,14031,14033,14053,14063],{"className":14032,"style":1436},[210],[15,14034,14035,14038],{"style":1439},[15,14036],{"className":14037,"style":1443},[217],[15,14039,14041],{"className":14040},[222,223,224,225],[15,14042,14044,14047,14050],{"className":14043},[66,225],[15,14045,2441],{"className":14046,"style":2594},[66,67,225],[15,14048,129],{"className":14049},[177,225],[15,14051,1211],{"className":14052},[66,225],[15,14054,14055,14058],{"style":1461},[15,14056],{"className":14057,"style":1443},[217],[15,14059,14060],{},[15,14061,1200],{"className":14062},[1339,1470,1471],[15,14064,14065,14068],{"style":1474},[15,14066],{"className":14067,"style":1443},[217],[15,14069,14071],{"className":14070},[222,223,224,225],[15,14072,37],{"className":14073},[66,67,225],[15,14075,1114],{"className":14076},[1113],[15,14078,14080],{"className":14079},[206],[15,14081,14083],{"className":14082,"style":2631},[210],[15,14084],{},[15,14086],{"className":14087,"style":697},[71],[15,14089,14091,14094],{"className":14090},[66],[15,14092,8385],{"className":14093,"style":8555},[66,67],[15,14095,14097],{"className":14096},[198],[15,14098,14100,14120],{"className":14099},[202,1090],[15,14101,14103,14117],{"className":14102},[206],[15,14104,14106],{"className":14105,"style":1528},[210],[15,14107,14108,14111],{"style":8570},[15,14109],{"className":14110,"style":218},[217],[15,14112,14114],{"className":14113},[222,223,224,225],[15,14115,2441],{"className":14116,"style":2594},[66,67,225],[15,14118,1114],{"className":14119},[1113],[15,14121,14123],{"className":14122},[206],[15,14124,14126],{"className":14125,"style":2715},[210],[15,14127],{},[15,14129],{"className":14130,"style":697},[71],[15,14132,621],{"className":14133,"style":684},[66,67],[15,14135,120],{"className":14136},[162],[15,14138,14140,14143],{"className":14139},[66],[15,14141,123],{"className":14142},[66,67],[15,14144,14146],{"className":14145},[198],[15,14147,14149,14169],{"className":14148},[202,1090],[15,14150,14152,14166],{"className":14151},[206],[15,14153,14155],{"className":14154,"style":1528},[210],[15,14156,14157,14160],{"style":1584},[15,14158],{"className":14159,"style":218},[217],[15,14161,14163],{"className":14162},[222,223,224,225],[15,14164,1205],{"className":14165},[66,67,225],[15,14167,1114],{"className":14168},[1113],[15,14170,14172],{"className":14171},[206],[15,14173,14175],{"className":14174,"style":1121},[210],[15,14176],{},[15,14178,335],{"className":14179},[422],[15,14181],{"className":14182,"style":697},[71],[15,14184,14186,14189],{"className":14185},[66],[15,14187,123],{"className":14188},[66,67],[15,14190,14192],{"className":14191},[198],[15,14193,14195,14215],{"className":14194},[202,1090],[15,14196,14198,14212],{"className":14197},[206],[15,14199,14201],{"className":14200,"style":1528},[210],[15,14202,14203,14206],{"style":1584},[15,14204],{"className":14205,"style":218},[217],[15,14207,14209],{"className":14208},[222,223,224,225],[15,14210,2441],{"className":14211,"style":2594},[66,67,225],[15,14213,1114],{"className":14214},[1113],[15,14216,14218],{"className":14217},[206],[15,14219,14221],{"className":14220,"style":2715},[210],[15,14222],{},[15,14224,126],{"className":14225},[169],[15,14227],{"className":14228,"style":173},[71],[15,14230,129],{"className":14231},[177],[15,14233],{"className":14234,"style":173},[71],[15,14236,14238,14241,14244,14247,14250,14290],{"className":14237},[57],[15,14239],{"className":14240,"style":154},[61],[15,14242,120],{"className":14243},[162],[15,14245,2853],{"className":14246,"style":2901},[66,67],[15,14248,8385],{"className":14249,"style":8555},[66,67],[15,14251,14253,14256],{"className":14252},[169],[15,14254,126],{"className":14255},[169],[15,14257,14259],{"className":14258},[198],[15,14260,14262,14282],{"className":14261},[202,1090],[15,14263,14265,14279],{"className":14264},[206],[15,14266,14268],{"className":14267,"style":1528},[210],[15,14269,14270,14273],{"style":1584},[15,14271],{"className":14272,"style":218},[217],[15,14274,14276],{"className":14275},[222,223,224,225],[15,14277,1205],{"className":14278},[66,67,225],[15,14280,1114],{"className":14281},[1113],[15,14283,14285],{"className":14284},[206],[15,14286,14288],{"className":14287,"style":1121},[210],[15,14289],{},[15,14291,335],{"className":14292},[422],[11,14294,425,14295,14393,14394,14602],{},[15,14296,14298,14326],{"className":14297,"translate":19},[18],[15,14299,14301],{"className":14300},[23],[25,14302,14303],{"xmlns":27},[29,14304,14305,14323],{},[32,14306,14307,14309,14311],{},[35,14308,2853],{},[39,14310,1187],{},[131,14312,14313,14315],{},[35,14314,455],{"mathvariant":454},[32,14316,14317,14319,14321],{},[35,14318,37],{},[39,14320,41],{},[35,14322,37],{},[45,14324,14325],{"encoding":47},"K\\in\\mathbb R^{n\\times n}",[15,14327,14329,14348],{"className":14328,"ariaHidden":53},[52],[15,14330,14332,14336,14339,14342,14345],{"className":14331},[57],[15,14333],{"className":14334,"style":14335},[61],"height:0.7224em;vertical-align:-0.0391em;",[15,14337,2853],{"className":14338,"style":2901},[66,67],[15,14340],{"className":14341,"style":173},[71],[15,14343,1187],{"className":14344},[177],[15,14346],{"className":14347,"style":173},[71],[15,14349,14351,14355],{"className":14350},[57],[15,14352],{"className":14353,"style":14354},[61],"height:0.7713em;",[15,14356,14358,14361],{"className":14357},[66],[15,14359,455],{"className":14360},[66,517],[15,14362,14364],{"className":14363},[198],[15,14365,14367],{"className":14366},[202],[15,14368,14370],{"className":14369},[206],[15,14371,14373],{"className":14372,"style":14354},[210],[15,14374,14375,14378],{"style":213},[15,14376],{"className":14377,"style":218},[217],[15,14379,14381],{"className":14380},[222,223,224,225],[15,14382,14384,14387,14390],{"className":14383},[66,225],[15,14385,37],{"className":14386},[66,67,225],[15,14388,41],{"className":14389},[76,225],[15,14391,37],{"className":14392},[66,67,225]," 是核矩阵，",[15,14395,14397,14440],{"className":14396,"translate":19},[18],[15,14398,14400],{"className":14399},[23],[25,14401,14402],{"xmlns":27},[29,14403,14404,14438],{},[32,14405,14406,14416,14418,14420,14422,14428,14430,14436],{},[1057,14407,14408,14410],{},[35,14409,2853],{},[32,14411,14412,14414],{},[35,14413,1205],{},[35,14415,2441],{},[39,14417,129],{},[35,14419,621],{},[39,14421,120],{"stretchy":119},[1057,14423,14424,14426],{},[35,14425,123],{},[35,14427,1205],{},[39,14429,335],{"separator":53},[1057,14431,14432,14434],{},[35,14433,123],{},[35,14435,2441],{},[39,14437,126],{"stretchy":119},[45,14439,2884],{"encoding":47},[15,14441,14443,14501],{"className":14442,"ariaHidden":53},[52],[15,14444,14446,14449,14492,14495,14498],{"className":14445},[57],[15,14447],{"className":14448,"style":2894},[61],[15,14450,14452,14455],{"className":14451},[66],[15,14453,2853],{"className":14454,"style":2901},[66,67],[15,14456,14458],{"className":14457},[198],[15,14459,14461,14484],{"className":14460},[202,1090],[15,14462,14464,14481],{"className":14463},[206],[15,14465,14467],{"className":14466,"style":1528},[210],[15,14468,14469,14472],{"style":2916},[15,14470],{"className":14471,"style":218},[217],[15,14473,14475],{"className":14474},[222,223,224,225],[15,14476,14478],{"className":14477},[66,225],[15,14479,2929],{"className":14480,"style":2594},[66,67,225],[15,14482,1114],{"className":14483},[1113],[15,14485,14487],{"className":14486},[206],[15,14488,14490],{"className":14489,"style":2715},[210],[15,14491],{},[15,14493],{"className":14494,"style":173},[71],[15,14496,129],{"className":14497},[177],[15,14499],{"className":14500,"style":173},[71],[15,14502,14504,14507,14510,14513,14553,14556,14559,14599],{"className":14503},[57],[15,14505],{"className":14506,"style":2956},[61],[15,14508,621],{"className":14509,"style":684},[66,67],[15,14511,120],{"className":14512},[162],[15,14514,14516,14519],{"className":14515},[66],[15,14517,123],{"className":14518},[66,67],[15,14520,14522],{"className":14521},[198],[15,14523,14525,14545],{"className":14524},[202,1090],[15,14526,14528,14542],{"className":14527},[206],[15,14529,14531],{"className":14530,"style":1528},[210],[15,14532,14533,14536],{"style":1584},[15,14534],{"className":14535,"style":218},[217],[15,14537,14539],{"className":14538},[222,223,224,225],[15,14540,1205],{"className":14541},[66,67,225],[15,14543,1114],{"className":14544},[1113],[15,14546,14548],{"className":14547},[206],[15,14549,14551],{"className":14550,"style":1121},[210],[15,14552],{},[15,14554,335],{"className":14555},[422],[15,14557],{"className":14558,"style":697},[71],[15,14560,14562,14565],{"className":14561},[66],[15,14563,123],{"className":14564},[66,67],[15,14566,14568],{"className":14567},[198],[15,14569,14571,14591],{"className":14570},[202,1090],[15,14572,14574,14588],{"className":14573},[206],[15,14575,14577],{"className":14576,"style":1528},[210],[15,14578,14579,14582],{"style":1584},[15,14580],{"className":14581,"style":218},[217],[15,14583,14585],{"className":14584},[222,223,224,225],[15,14586,2441],{"className":14587,"style":2594},[66,67,225],[15,14589,1114],{"className":14590},[1113],[15,14592,14594],{"className":14593},[206],[15,14595,14597],{"className":14596,"style":2715},[210],[15,14598],{},[15,14600,126],{"className":14601},[169],"。正则项则是",[15,14604,14606],{"className":14605,"translate":19},[294],[15,14607,14609,14761],{"className":14608,"translate":19},[18],[15,14610,14612],{"className":14611},[23],[25,14613,14614],{"xmlns":27,"display":303},[29,14615,14616,14758],{},[32,14617,14618,14620,14622,14630,14632,14634,14640,14646,14648,14650,14652,14654,14660,14662,14664,14670,14676,14678,14680,14682,14684,14690,14692,14694,14696,14698,14710,14716,14722,14724,14726,14728,14734,14736,14742,14744,14746,14752,14754,14756],{},[35,14619,1251],{"mathvariant":138},[35,14621,116],{},[1255,14623,14624,14626,14628],{},[35,14625,1251],{"mathvariant":138},[35,14627,1061],{"mathvariant":445},[1209,14629,1263],{},[39,14631,129],{},[39,14633,642],{"fence":53,"stretchy":53,"minsize":4543,"maxsize":4543},[1172,14635,14636,14638],{},[39,14637,1200],{},[35,14639,1205],{},[1057,14641,14642,14644],{},[35,14643,8385],{},[35,14645,1205],{},[35,14647,621],{},[39,14649,120],{"stretchy":119},[39,14651,3266],{},[39,14653,335],{"separator":53},[1057,14655,14656,14658],{},[35,14657,123],{},[35,14659,1205],{},[39,14661,126],{"stretchy":119},[39,14663,335],{"separator":53},[1172,14665,14666,14668],{},[39,14667,1200],{},[35,14669,2441],{},[1057,14671,14672,14674],{},[35,14673,8385],{},[35,14675,2441],{},[35,14677,621],{},[39,14679,120],{"stretchy":119},[39,14681,3266],{},[39,14683,335],{"separator":53},[1057,14685,14686,14688],{},[35,14687,123],{},[35,14689,2441],{},[39,14691,126],{"stretchy":119},[39,14693,667],{"fence":53,"stretchy":53,"minsize":4543,"maxsize":4543},[71,14695],{"linebreak":10451},[39,14697,129],{},[1172,14699,14700,14702],{},[39,14701,1200],{},[32,14703,14704,14706,14708],{},[35,14705,1205],{},[39,14707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H}^2\n=\\Bigl\\langle\\sum_i\\alpha_i k(\\cdot,x_i),\\sum_j\\alpha_j k(\\cdot,x_j)\\Bigr\\rangle \\\\\n=\\sum_{i,j}\\alpha_i\\alpha_j\\,k(x_i,x_j)\n=\\alpha^\\top 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的有限维问题：",[15,15547,15549],{"className":15548,"translate":19},[294],[15,15550,15552,15635],{"className":15551,"translate":19},[18],[15,15553,15555],{"className":15554},[23],[25,15556,15557],{"xmlns":27,"display":303},[29,15558,15559,15632],{},[1142,15560,15561],{"width":1144},[1146,15562,15563,15565,15625,15627],{},[1149,15564],{"width":1151},[1149,15566,15567],{},[32,15568,15569,15589,15623],{},[1172,15570,15571,15577],{},[32,15572,15573,15575],{},[35,15574,1178],{},[39,15576,1170],{},[32,15578,15579,15581,15583],{},[35,15580,8385],{},[39,15582,1187],{},[131,15584,15585,15587],{},[35,15586,455],{"mathvariant":454},[35,15588,37],{},[32,15590,15591,15593,15595,15597,15599,15601,15603,15605,15607,15609,15615,15617,15619,15621],{},[39,15592,4305],{"fence":53},[35,15594,1216],{},[39,15596,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,15598,2853],{},[35,15600,8385],{},[39,15602,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,15604,1245],{},[35,15606,7737],{"mathvariant":138},[39,15608,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[131,15610,15611,15613],{},[35,15612,8385],{},[35,15614,139],{"mathvariant":138},[35,15616,2853],{},[35,15618,8385],{},[39,15620,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,15622,4328],{"fence":53},[35,15624,3571],{"mathvariant":138},[1149,15626],{"width":1151},[1149,15628,15629],{},[1274,15630,15631],{},"(4)",[45,15633,15634],{"encoding":47},"\\min_{\\alpha\\in\\mathbb R^n}\\left\\{L\\bigl(K\\alpha\\bigr)+\\Omega\\bigl(\\alpha^\\top K\\alpha\\bigr)\\right\\}.\n\\tag{4}",[15,15636,15638,15832],{"className":15637,"ariaHidden":53},[52],[15,15639,15641,15645,15728,15731,15826,15829],{"className":15640},[57],[15,15642],{"className":15643,"style":15644},[61],"height:1.6747em;vertical-align:-0.7756em;",[15,15646,15648],{"className":15647},[1339,1351],[15,15649,15651,15719],{"className":15650},[202,1090],[15,15652,15654,15716],{"className":15653},[206],[15,15655,15657,15706],{"className":15656,"style":1361},[210],[15,15658,15660,15663],{"style":15659},"top:-2.3518em;margin-left:0em;",[15,15661],{"className":15662,"style":1368},[217],[15,15664,15666],{"className":15665},[222,223,224,225],[15,15667,15669,15672,15675],{"className":15668},[66,225],[15,15670,8385],{"className":15671,"style":8555},[66,67,225],[15,15673,1187],{"className":15674},[177,225],[15,15676,15678,15681],{"className":15677},[66,225],[15,15679,455],{"className":15680},[66,517,225],[15,15682,15684],{"className":15683},[198],[15,15685,15687],{"className":15686},[202],[15,15688,15690],{"className":15689},[206],[15,15691,15694],{"className":15692,"style":15693},[210],"height:0.5935em;",[15,15695,15697,15700],{"style":15696},"top:-2.786em;margin-right:0.0714em;",[15,15698],{"className":15699,"style":3673},[217],[15,15701,15703],{"className":15702},[222,3677,1508,225],[15,15704,37],{"className":15705},[66,67,225],[15,15707,15708,15711],{"style":1386},[15,15709],{"className":15710,"style":1368},[217],[15,15712,15713],{},[15,15714,1178],{"className":15715},[1339],[15,15717,1114],{"className":15718},[1113],[15,15720,15722],{"className":15721},[206],[15,15723,15726],{"className":15724,"style":15725},[210],"height:0.7756em;",[15,15727],{},[15,15729],{"className":15730,"style":697},[71],[15,15732,15734,15740,15743,15749,15752,15755,15761,15764,15767,15770,15773,15779,15808,15811,15814,15820],{"className":15733},[1413],[15,15735,15737],{"className":15736,"style":1418},[162,1417],[15,15738,4305],{"className":15739},[1422,1508],[15,15741,1216],{"className":15742},[66,67],[15,15744,15746],{"className":15745},[162],[15,15747,120],{"className":15748},[1422,1508],[15,15750,2853],{"className":15751,"style":2901},[66,67],[15,15753,8385],{"className":15754,"style":8555},[66,67],[15,15756,15758],{"className":15757},[169],[15,15759,126],{"className":15760},[1422,1508],[15,15762],{"className":15763,"style":72},[71],[15,15765,1245],{"className":15766},[76],[15,15768],{"className":15769,"style":72},[71],[15,15771,7737],{"className":15772},[66],[15,15774,15776],{"className":15775},[162],[15,15777,120],{"className":15778},[1422,1508],[15,15780,15782,15785],{"className":15781},[66],[15,15783,8385],{"className":15784,"style":8555},[66,67],[15,15786,15788],{"className":15787},[198],[15,15789,15791],{"className":15790},[202],[15,15792,15794],{"className":15793},[206],[15,15795,15797],{"className":15796,"style":394},[210],[15,15798,15799,15802],{"style":397},[15,15800],{"className":15801,"style":218},[217],[15,15803,15805],{"className":15804},[222,223,224,225],[15,15806,139],{"className":15807},[66,225],[15,15809,2853],{"className":15810,"style":2901},[66,67],[15,15812,8385],{"className":15813,"style":8555},[66,67],[15,15815,15817],{"className":15816},[169],[15,15818,126],{"className":15819},[1422,1508],[15,15821,15823],{"className":15822,"style":1418},[169,1417],[15,15824,4328],{"className":15825},[1422,1508],[15,15827],{"className":15828,"style":697},[71],[15,15830,3571],{"className":15831},[66],[15,15833,15835,15838],{"className":15834},[1704],[15,15836],{"className":15837,"style":15644},[61],[15,15839,15841,15844,15851],{"className":15840},[66,1711],[15,15842,120],{"className":15843},[66],[15,15845,15847],{"className":15846},[66],[15,15848,15850],{"className":15849},[66],"4",[15,15852,126],{"className":15853},[66],[11,15855,15856,15857,15885,15886,15914,15915,15943,15944,15972,15973,16026,16027,16055,16056,16085,16086,16114],{},"这就是表示定理的全部实用价值：原本在无穷维空间里找 ",[15,15858,15860,15873],{"className":15859,"translate":19},[18],[15,15861,15863],{"className":15862},[23],[25,15864,15865],{"xmlns":27},[29,15866,15867,15871],{},[32,15868,15869],{},[35,15870,116],{},[45,15872,116],{"encoding":47},[15,15874,15876],{"className":15875,"ariaHidden":53},[52],[15,15877,15879,15882],{"className":15878},[57],[15,15880],{"className":15881,"style":3406},[61],[15,15883,116],{"className":15884,"style":158},[66,67],"，现在只需找 ",[15,15887,15889,15902],{"className":15888,"translate":19},[18],[15,15890,15892],{"className":15891},[23],[25,15893,15894],{"xmlns":27},[29,15895,15896,15900],{},[32,15897,15898],{},[35,15899,37],{},[45,15901,37],{"encoding":47},[15,15903,15905],{"className":15904,"ariaHidden":53},[52],[15,15906,15908,15911],{"className":15907},[57],[15,15909],{"className":15910,"style":86},[61],[15,15912,37],{"className":15913},[66,67]," 个系数。核矩阵 ",[15,15916,15918,15931],{"className":15917,"translate":19},[18],[15,15919,15921],{"className":15920},[23],[25,15922,15923],{"xmlns":27},[29,15924,15925,15929],{},[32,15926,15927],{},[35,15928,2853],{},[45,15930,2853],{"encoding":47},[15,15932,15934],{"className":15933,"ariaHidden":53},[52],[15,15935,15937,15940],{"className":15936},[57],[15,15938],{"className":15939,"style":490},[61],[15,15941,2853],{"className":15942,"style":2901},[66,67]," 是半正定的；若 ",[15,15945,15947,15960],{"className":15946,"translate":19},[18],[15,15948,15950],{"className":15949},[23],[25,15951,15952],{"xmlns":27},[29,15953,15954,15958],{},[32,15955,15956],{},[35,15957,2853],{},[45,15959,2853],{"encoding":47},[15,15961,15963],{"className":15962,"ariaHidden":53},[52],[15,15964,15966,15969],{"className":15965},[57],[15,15967],{"className":15968,"style":490},[61],[15,15970,2853],{"className":15971,"style":2901},[66,67]," 可逆，则 ",[15,15974,15976,15995],{"className":15975,"translate":19},[18],[15,15977,15979],{"className":15978},[23],[25,15980,15981],{"xmlns":27},[29,15982,15983,15992],{},[32,15984,15985,15987,15990],{},[35,15986,8385],{},[39,15988,15989],{},"↦",[35,15991,116],{},[45,15993,15994],{"encoding":47},"\\alpha\\mapsto f",[15,15996,15998,16017],{"className":15997,"ariaHidden":53},[52],[15,15999,16001,16005,16008,16011,16014],{"className":16000},[57],[15,16002],{"className":16003,"style":16004},[61],"height:0.522em;vertical-align:-0.011em;",[15,16006,8385],{"className":16007,"style":8555},[66,67],[15,16009],{"className":16010,"style":173},[71],[15,16012,15989],{"className":16013},[177],[15,16015],{"className":16016,"style":173},[71],[15,16018,16020,16023],{"className":16019},[57],[15,16021],{"className":16022,"style":3406},[61],[15,16024,116],{"className":16025,"style":158},[66,67]," 是一一对应；若 ",[15,16028,16030,16043],{"className":16029,"translate":19},[18],[15,16031,16033],{"className":16032},[23],[25,16034,16035],{"xmlns":27},[29,16036,16037,16041],{},[32,16038,16039],{},[35,16040,2853],{},[45,16042,2853],{"encoding":47},[15,16044,16046],{"className":16045,"ariaHidden":53},[52],[15,16047,16049,16052],{"className":16048},[57],[15,16050],{"className":16051,"style":490},[61],[15,16053,2853],{"className":16054,"style":2901},[66,67]," 奇异，不同的 ",[15,16057,16059,16073],{"className":16058,"translate":19},[18],[15,16060,16062],{"className":16061},[23],[25,16063,16064],{"xmlns":27},[29,16065,16066,16070],{},[32,16067,16068],{},[35,16069,8385],{},[45,16071,16072],{"encoding":47},"\\alpha",[15,16074,16076],{"className":16075,"ariaHidden":53},[52],[15,16077,16079,16082],{"className":16078},[57],[15,16080],{"className":16081,"style":86},[61],[15,16083,8385],{"className":16084,"style":8555},[66,67]," 可能对应同一个函数（相差一个 ",[15,16087,16089,16102],{"className":16088,"translate":19},[18],[15,16090,16092],{"className":16091},[23],[25,16093,16094],{"xmlns":27},[29,16095,16096,16100],{},[32,16097,16098],{},[35,16099,2853],{},[45,16101,2853],{"encoding":47},[15,16103,16105],{"className":16104,"ariaHidden":53},[52],[15,16106,16108,16111],{"className":16107},[57],[15,16109],{"className":16110,"style":490},[61],[15,16112,2853],{"className":16113,"style":2901},[66,67]," 的零空间向量，其范数为零），但函数本身仍然是良定义的。",[1904,16116,16118],{"id":16117},"_34-唯一性系数与偏置项","3.4 唯一性、系数与偏置项",[11,16120,16121,16122,16125,16126,16129],{},"表示定理给出的是",[2040,16123,16124],{},"解的形式","，不是",[2040,16127,16128],{},"解的唯一性","。以下几点容易混淆：",[16131,16132,16133,16561,16626],"ul",{},[3192,16134,16135,8682,16138,16166,16167,16195,16196,16256,16257,16307,16308,16336,16337,16395,16396,16531,16532,16560],{},[2040,16136,16137],{},"函数唯一 vs 系数唯一。",[15,16139,16141,16154],{"className":16140,"translate":19},[18],[15,16142,16144],{"className":16143},[23],[25,16145,16146],{"xmlns":27},[29,16147,16148,16152],{},[32,16149,16150],{},[35,16151,2853],{},[45,16153,2853],{"encoding":47},[15,16155,16157],{"className":16156,"ariaHidden":53},[52],[15,16158,16160,16163],{"className":16159},[57],[15,16161],{"className":16162,"style":490},[61],[15,16164,2853],{"className":16165,"style":2901},[66,67]," 奇异，",[15,16168,16170,16183],{"className":16169,"translate":19},[18],[15,16171,16173],{"className":16172},[23],[25,16174,16175],{"xmlns":27},[29,16176,16177,16181],{},[32,16178,16179],{},[35,16180,8385],{},[45,16182,16072],{"encoding":47},[15,16184,16186],{"className":16185,"ariaHidden":53},[52],[15,16187,16189,16192],{"className":16188},[57],[15,16190],{"className":16191,"style":86},[61],[15,16193,8385],{"className":16194,"style":8555},[66,67]," 不唯一，但 ",[15,16197,16199,16217],{"className":16198,"translate":19},[18],[15,16200,16202],{"className":16201},[23],[25,16203,16204],{"xmlns":27},[29,16205,16206,16214],{},[32,16207,16208],{},[131,16209,16210,16212],{},[35,16211,116],{},[39,16213,1162],{},[45,16215,16216],{"encoding":47},"f^*",[15,16218,16220],{"className":16219,"ariaHidden":53},[52],[15,16221,16223,16226],{"className":16222},[57],[15,16224],{"className":16225,"style":3406},[61],[15,16227,16229,16232],{"className":16228},[66],[15,16230,116],{"className":16231,"style":158},[66,67],[15,16233,16235],{"className":16234},[198],[15,16236,16238],{"className":16237},[202],[15,16239,16241],{"className":16240},[206],[15,16242,16245],{"className":16243,"style":16244},[210],"height:0.6887em;",[15,16246,16247,16250],{"style":213},[15,16248],{"className":16249,"style":218},[217],[15,16251,16253],{"className":16252},[222,223,224,225],[15,16254,1162],{"className":16255},[76,225]," 可能唯一。核岭回归在 ",[15,16258,16260,16277],{"className":16259,"translate":19},[18],[15,16261,16263],{"className":16262},[23],[25,16264,16265],{"xmlns":27},[29,16266,16267,16275],{},[32,16268,16269,16271,16273],{},[35,16270,1248],{},[39,16272,9278],{},[1209,16274,1804],{},[45,16276,9283],{"encoding":47},[15,16278,16280,16298],{"className":16279,"ariaHidden":53},[52],[15,16281,16283,16286,16289,16292,16295],{"className":16282},[57],[15,16284],{"className":16285,"style":9293},[61],[15,16287,1248],{"className":16288},[66,67],[15,16290],{"className":16291,"style":173},[71],[15,16293,9278],{"className":16294},[177],[15,16296],{"className":16297,"style":173},[71],[15,16299,16301,16304],{"className":16300},[57],[15,16302],{"className":16303,"style":1836},[61],[15,16305,1804],{"className":16306},[66]," 时就是如此：目标函数关于 ",[15,16309,16311,16324],{"className":16310,"translate":19},[18],[15,16312,16314],{"className":16313},[23],[25,16315,16316],{"xmlns":27},[29,16317,16318,16322],{},[32,16319,16320],{},[35,16321,116],{},[45,16323,116],{"encoding":47},[15,16325,16327],{"className":16326,"ariaHidden":53},[52],[15,16328,16330,16333],{"className":16329},[57],[15,16331],{"className":16332,"style":3406},[61],[15,16334,116],{"className":16335,"style":158},[66,67]," 严格凸，",[15,16338,16340,16357],{"className":16339,"translate":19},[18],[15,16341,16343],{"className":16342},[23],[25,16344,16345],{"xmlns":27},[29,16346,16347,16355],{},[32,16348,16349],{},[131,16350,16351,16353],{},[35,16352,116],{},[39,16354,1162],{},[45,16356,16216],{"encoding":47},[15,16358,16360],{"className":16359,"ariaHidden":53},[52],[15,16361,16363,16366],{"className":16362},[57],[15,16364],{"className":16365,"style":3406},[61],[15,16367,16369,16372],{"className":16368},[66],[15,16370,116],{"className":16371,"style":158},[66,67],[15,16373,16375],{"className":16374},[198],[15,16376,16378],{"className":16377},[202],[15,16379,16381],{"className":16380},[206],[15,16382,16384],{"className":16383,"style":16244},[210],[15,16385,16386,16389],{"style":213},[15,16387],{"className":16388,"style":218},[217],[15,16390,16392],{"className":16391},[222,223,224,225],[15,16393,1162],{"className":16394},[76,225]," 唯一；而 ",[15,16397,16399,16438],{"className":16398,"translate":19},[18],[15,16400,16402],{"className":16401},[23],[25,16403,16404],{"xmlns":27},[29,16405,16406,16435],{},[32,16407,16408,16410,16412,16414,16416,16418,16420,16423,16433],{},[35,16409,8385],{},[39,16411,129],{},[39,16413,120],{"stretchy":119},[35,16415,2853],{},[39,16417,1245],{},[35,16419,1248],{},[35,16421,16422],{},"I",[131,16424,16425,16427],{},[39,16426,126],{"stretchy":119},[32,16428,16429,16431],{},[39,16430,6600],{},[1209,16432,1211],{},[35,16434,1224],{},[45,16436,16437],{"encoding":47},"\\alpha=(K+\\lambda I)^{-1}y",[15,16439,16441,16459,16480],{"className":16440,"ariaHidden":53},[52],[15,16442,16444,16447,16450,16453,16456],{"className":16443},[57],[15,16445],{"className":16446,"style":86},[61],[15,16448,8385],{"className":16449,"style":8555},[66,67],[15,16451],{"className":16452,"style":173},[71],[15,16454,129],{"className":16455},[177],[15,16457],{"className":16458,"style":173},[71],[15,16460,16462,16465,16468,16471,16474,16477],{"className":16461},[57],[15,16463],{"className":16464,"style":154},[61],[15,16466,120],{"className":16467},[162],[15,16469,2853],{"className":16470,"style":2901},[66,67],[15,16472],{"className":16473,"style":72},[71],[15,16475,1245],{"className":16476},[76],[15,16478],{"className":16479,"style":72},[71],[15,16481,16483,16486,16489,16493,16528],{"className":16482},[57],[15,16484],{"className":16485,"style":9149},[61],[15,16487,1248],{"className":16488},[66,67],[15,16490,16422],{"className":16491,"style":16492},[66,67],"margin-right:0.0785em;",[15,16494,16496,16499],{"className":16495},[169],[15,16497,126],{"className":16498},[169],[15,16500,16502],{"className":16501},[198],[15,16503,16505],{"className":16504},[202],[15,16506,16508],{"className":16507},[206],[15,16509,16511],{"className":16510,"style":6738},[210],[15,16512,16513,16516],{"style":213},[15,16514],{"className":16515,"style":218},[217],[15,16517,16519],{"className":16518},[222,223,224,225],[15,16520,16522,16525],{"className":16521},[66,225],[15,16523,6600],{"className":16524},[66,225],[15,16526,1211],{"className":16527},[66,225],[15,16529,1224],{"className":16530,"style":1515},[66,67]," 只是其中一个系数表示，所有相差 ",[15,16533,16535,16548],{"className":16534,"translate":19},[18],[15,16536,16538],{"className":16537},[23],[25,16539,16540],{"xmlns":27},[29,16541,16542,16546],{},[32,16543,16544],{},[35,16545,2853],{},[45,16547,2853],{"encoding":47},[15,16549,16551],{"className":16550,"ariaHidden":53},[52],[15,16552,16554,16557],{"className":16553},[57],[15,16555],{"className":16556,"style":490},[61],[15,16558,2853],{"className":16559,"style":2901},[66,67]," 零空间的系数给出同一个函数。",[3192,16562,16563,16625],{},[2040,16564,16565,16566,16624],{},"损失不严格凸时，",[15,16567,16569,16586],{"className":16568,"translate":19},[18],[15,16570,16572],{"className":16571},[23],[25,16573,16574],{"xmlns":27},[29,16575,16576,16584],{},[32,16577,16578],{},[131,16579,16580,16582],{},[35,16581,116],{},[39,16583,1162],{},[45,16585,16216],{"encoding":47},[15,16587,16589],{"className":16588,"ariaHidden":53},[52],[15,16590,16592,16595],{"className":16591},[57],[15,16593],{"className":16594,"style":3406},[61],[15,16596,16598,16601],{"className":16597},[66],[15,16599,116],{"className":16600,"style":158},[66,67],[15,16602,16604],{"className":16603},[198],[15,16605,16607],{"className":16606},[202],[15,16608,16610],{"className":16609},[206],[15,16611,16613],{"className":16612,"style":16244},[210],[15,16614,16615,16618],{"style":213},[15,16616],{"className":16617,"style":218},[217],[15,16619,16621],{"className":16620},[222,223,224,225],[15,16622,1162],{"className":16623},[76,225]," 也可能不唯一。"," 例如 hinge loss 在某些数据上会有平坦区域；表示定理仍然成立，但最优解可能是一个集合。",[3192,16627,16628,16631,16632,16698,16699,16727,16728,16756,16757,17005,17006,17034,17035,17038],{},[2040,16629,16630],{},"偏置项要单独处理。"," 如果模型是 ",[15,16633,16635,16659],{"className":16634,"translate":19},[18],[15,16636,16638],{"className":16637},[23],[25,16639,16640],{"xmlns":27},[29,16641,16642,16656],{},[32,16643,16644,16646,16648,16650,16652,16654],{},[35,16645,116],{},[39,16647,120],{"stretchy":119},[35,16649,123],{},[39,16651,126],{"stretchy":119},[39,16653,1245],{},[35,16655,4585],{},[45,16657,16658],{"encoding":47},"f(x)+b",[15,16660,16662,16689],{"className":16661,"ariaHidden":53},[52],[15,16663,16665,16668,16671,16674,16677,16680,16683,16686],{"className":16664},[57],[15,16666],{"className":16667,"style":154},[61],[15,16669,116],{"className":16670,"style":158},[66,67],[15,16672,120],{"className":16673},[162],[15,16675,123],{"className":16676},[66,67],[15,16678,126],{"className":16679},[169],[15,16681],{"className":16682,"style":72},[71],[15,16684,1245],{"className":16685},[76],[15,16687],{"className":16688,"style":72},[71],[15,16690,16692,16695],{"className":16691},[57],[15,16693],{"className":16694,"style":840},[61],[15,16696,4585],{"className":16697},[66,67],"，而 ",[15,16700,16702,16715],{"className":16701,"translate":19},[18],[15,16703,16705],{"className":16704},[23],[25,16706,16707],{"xmlns":27},[29,16708,16709,16713],{},[32,16710,16711],{},[35,16712,4585],{},[45,16714,4585],{"encoding":47},[15,16716,16718],{"className":16717,"ariaHidden":53},[52],[15,16719,16721,16724],{"className":16720},[57],[15,16722],{"className":16723,"style":840},[61],[15,16725,4585],{"className":16726},[66,67]," 不受正则项约束，那么表示定理直接作用于 ",[15,16729,16731,16744],{"className":16730,"translate":19},[18],[15,16732,16734],{"className":16733},[23],[25,16735,16736],{"xmlns":27},[29,16737,16738,16742],{},[32,16739,16740],{},[35,16741,116],{},[45,16743,116],{"encoding":47},[15,16745,16747],{"className":16746,"ariaHidden":53},[52],[15,16748,16750,16753],{"className":16749},[57],[15,16751],{"className":16752,"style":3406},[61],[15,16754,116],{"className":16755,"style":158},[66,67]," 的部分仍然给出 ",[15,16758,16760,16808],{"className":16759,"translate":19},[18],[15,16761,16763],{"className":16762},[23],[25,16764,16765],{"xmlns":27},[29,16766,16767,16805],{},[32,16768,16769,16775,16777,16783,16789,16791,16793,16795,16797,16803],{},[131,16770,16771,16773],{},[35,16772,116],{},[39,16774,1162],{},[39,16776,129],{},[1057,16778,16779,16781],{},[39,16780,1200],{},[35,16782,1205],{},[1057,16784,16785,16787],{},[35,16786,8385],{},[35,16788,1205],{},[35,16790,621],{},[39,16792,120],{"stretchy":119},[39,16794,3266],{},[39,16796,335],{"separator":53},[1057,16798,16799,16801],{},[35,16800,123],{},[35,16802,1205],{},[39,16804,126],{"stretchy":119},[45,16806,16807],{"encoding":47},"f^*=\\sum_i\\alpha_i k(\\cdot,x_i)",[15,16809,16811,16855],{"className":16810,"ariaHidden":53},[52],[15,16812,16814,16817,16846,16849,16852],{"className":16813},[57],[15,16815],{"className":16816,"style":3406},[61],[15,16818,16820,16823],{"className":16819},[66],[15,16821,116],{"className":16822,"style":158},[66,67],[15,16824,16826],{"className":16825},[198],[15,16827,16829],{"className":16828},[202],[15,16830,16832],{"className":16831},[206],[15,16833,16835],{"className":16834,"style":16244},[210],[15,16836,16837,16840],{"style":213},[15,16838],{"className":16839,"style":218},[217],[15,16841,16843],{"className":16842},[222,223,224,225],[15,16844,1162],{"className":16845},[76,225],[15,16847],{"className":16848,"style":173},[71],[15,16850,129],{"className":16851},[177],[15,16853],{"className":16854,"style":173},[71],[15,16856,16858,16862,16904,16907,16947,16950,16953,16956,16959,16962,17002],{"className":16857},[57],[15,16859],{"className":16860,"style":16861},[61],"height:1.0497em;vertical-align:-0.2997em;",[15,16863,16865,16868],{"className":16864},[1339],[15,16866,1200],{"className":16867,"style":13697},[1339,1470,13696],[15,16869,16871],{"className":16870},[198],[15,16872,16874,16895],{"className":16873},[202,1090],[15,16875,16877,16892],{"className":16876},[206],[15,16878,16881],{"className":16879,"style":16880},[210],"height:0.162em;",[15,16882,16883,16886],{"style":13713},[15,16884],{"className":16885,"style":218},[217],[15,16887,16889],{"className":16888},[222,223,224,225],[15,16890,1205],{"className":16891},[66,67,225],[15,16893,1114],{"className":16894},[1113],[15,16896,16898],{"className":16897},[206],[15,16899,16902],{"className":16900,"style":16901},[210],"height:0.2997em;",[15,16903],{},[15,16905],{"className":16906,"style":697},[71],[15,16908,16910,16913],{"className":16909},[66],[15,16911,8385],{"className":16912,"style":8555},[66,67],[15,16914,16916],{"className":16915},[198],[15,16917,16919,16939],{"className":16918},[202,1090],[15,16920,16922,16936],{"className":16921},[206],[15,16923,16925],{"className":16924,"style":1528},[210],[15,16926,16927,16930],{"style":8570},[15,16928],{"className":16929,"style":218},[217],[15,16931,16933],{"className":16932},[222,223,224,225],[15,16934,1205],{"className":16935},[66,67,225],[15,16937,1114],{"className":16938},[1113],[15,16940,16942],{"className":16941},[206],[15,16943,16945],{"className":16944,"style":1121},[210],[15,16946],{},[15,16948,621],{"className":16949,"style":684},[66,67],[15,16951,120],{"className":16952},[162],[15,16954,3266],{"className":16955},[66],[15,16957,335],{"className":16958},[422],[15,16960],{"className":16961,"style":697},[71],[15,16963,16965,16968],{"className":16964},[66],[15,16966,123],{"className":16967},[66,67],[15,16969,16971],{"className":16970},[198],[15,16972,16974,16994],{"className":16973},[202,1090],[15,16975,16977,16991],{"className":16976},[206],[15,16978,16980],{"className":16979,"style":1528},[210],[15,16981,16982,16985],{"style":1584},[15,16983],{"className":16984,"style":218},[217],[15,16986,16988],{"className":16987},[222,223,224,225],[15,16989,1205],{"className":16990},[66,67,225],[15,16992,1114],{"className":16993},[1113],[15,16995,16997],{"className":16996},[206],[15,16998,17000],{"className":16999,"style":1121},[210],[15,17001],{},[15,17003,126],{"className":17004},[169],"，但 ",[15,17007,17009,17022],{"className":17008,"translate":19},[18],[15,17010,17012],{"className":17011},[23],[25,17013,17014],{"xmlns":27},[29,17015,17016,17020],{},[32,17017,17018],{},[35,17019,4585],{},[45,17021,4585],{"encoding":47},[15,17023,17025],{"className":17024,"ariaHidden":53},[52],[15,17026,17028,17031],{"className":17027},[57],[15,17029],{"className":17030,"style":840},[61],[15,17032,4585],{"className":17033},[66,67]," 需要通过有限维优化或 KKT 条件单独求出。更系统的处理是",[2040,17036,17037],{},"半参数表示定理","（semi-parametric representer theorem）：若模型为",[15,17040,17042],{"className":17041,"translate":19},[294],[15,17043,17045,17123],{"className":17044,"translate":19},[18],[15,17046,17048],{"className":17047},[23],[25,17049,17050],{"xmlns":27,"display":303},[29,17051,17052,17120],{},[32,17053,17054,17056,17058,17060,17062,17064,17078,17085,17092,17094,17096,17098,17100,17102,17104,17106,17108,17110,17112,17114,17116,17118],{},[35,17055,116],{},[39,17057,120],{"stretchy":119},[35,17059,123],{},[39,17061,126],{"stretchy":119},[39,17063,129],{},[1196,17065,17066,17068,17076],{},[39,17067,1200],{},[32,17069,17070,17072,17074],{},[35,17071,2441],{},[39,17073,129],{},[1209,17075,1211],{},[35,17077,11],{},[1057,17079,17080,17083],{},[35,17081,17082],{},"β",[35,17084,2441],{},[1057,17086,17087,17090],{},[35,17088,17089],{},"ϕ",[35,17091,2441],{},[39,17093,120],{"stretchy":119},[35,17095,123],{},[39,17097,126],{"stretchy":119},[39,17099,1245],{},[35,17101,1344],{},[39,17103,120],{"stretchy":119},[35,17105,123],{},[39,17107,126],{"stretchy":119},[39,17109,335],{"separator":53},[71,17111],{"width":3902},[35,17113,1344],{},[39,17115,1187],{},[35,17117,1061],{"mathvariant":445},[39,17119,335],{"separator":53},[45,17121,17122],{"encoding":47},"f(x)=\\sum_{j=1}^p\\beta_j\\phi_j(x)+g(x),\\qquad g\\in\\mathcal H,",[15,17124,17126,17153,17329,17368],{"className":17125,"ariaHidden":53},[52],[15,17127,17129,17132,17135,17138,17141,17144,17147,17150],{"className":17128},[57],[15,17130],{"className":17131,"style":154},[61],[15,17133,116],{"className":17134,"style":158},[66,67],[15,17136,120],{"className":17137},[162],[15,17139,123],{"className":17140},[66,67],[15,17142,126],{"className":17143},[169],[15,17145],{"className":17146,"style":173},[71],[15,17148,129],{"className":17149},[177],[15,17151],{"className":17152,"style":173},[71],[15,17154,17156,17160,17226,17229,17271,17311,17314,17317,17320,17323,17326],{"className":17155},[57],[15,17157],{"className":17158,"style":17159},[61],"height:3.1123em;vertical-align:-1.4138em;",[15,17161,17163],{"className":17162},[1339,1351],[15,17164,17166,17218],{"className":17165},[202,1090],[15,17167,17169,17215],{"className":17168},[206],[15,17170,17173,17193,17203],{"className":17171,"style":17172},[210],"height:1.6985em;",[15,17174,17175,17178],{"style":1439},[15,17176],{"className":17177,"style":1443},[217],[15,17179,17181],{"className":17180},[222,223,224,225],[15,17182,17184,17187,17190],{"className":17183},[66,225],[15,17185,2441],{"className":17186,"style":2594},[66,67,225],[15,17188,129],{"className":17189},[177,225],[15,17191,1211],{"className":17192},[66,225],[15,17194,17195,17198],{"style":1461},[15,17196],{"className":17197,"style":1443},[217],[15,17199,17200],{},[15,17201,1200],{"className":17202},[1339,1470,1471],[15,17204,17206,17209],{"style":17205},"top:-4.3471em;margin-left:0em;",[15,17207],{"className":17208,"style":1443},[217],[15,17210,17212],{"className":17211},[222,223,224,225],[15,17213,11],{"className":17214},[66,67,225],[15,17216,1114],{"className":17217},[1113],[15,17219,17221],{"className":17220},[206],[15,17222,17224],{"className":17223,"style":2631},[210],[15,17225],{},[15,17227],{"className":17228,"style":697},[71],[15,17230,17232,17236],{"className":17231},[66],[15,17233,17082],{"className":17234,"style":17235},[66,67],"margin-right:0.0528em;",[15,17237,17239],{"className":17238},[198],[15,17240,17242,17263],{"className":17241},[202,1090],[15,17243,17245,17260],{"className":17244},[206],[15,17246,17248],{"className":17247,"style":1528},[210],[15,17249,17251,17254],{"style":17250},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[15,17252],{"className":17253,"style":218},[217],[15,17255,17257],{"className":17256},[222,223,224,225],[15,17258,2441],{"className":17259,"style":2594},[66,67,225],[15,17261,1114],{"className":17262},[1113],[15,17264,17266],{"className":17265},[206],[15,17267,17269],{"className":17268,"style":2715},[210],[15,17270],{},[15,17272,17274,17277],{"className":17273},[66],[15,17275,17089],{"className":17276},[66,67],[15,17278,17280],{"className":17279},[198],[15,17281,17283,17303],{"className":17282},[202,1090],[15,17284,17286,17300],{"className":17285},[206],[15,17287,17289],{"className":17288,"style":1528},[210],[15,17290,17291,17294],{"style":1584},[15,17292],{"className":17293,"style":218},[217],[15,17295,17297],{"className":17296},[222,223,224,225],[15,17298,2441],{"className":17299,"style":2594},[66,67,225],[15,17301,1114],{"className":17302},[1113],[15,17304,17306],{"className":17305},[206],[15,17307,17309],{"className":17308,"style":2715},[210],[15,17310],{},[15,17312,120],{"className":17313},[162],[15,17315,123],{"className":17316},[66,67],[15,17318,126],{"className":17319},[169],[15,17321],{"className":17322,"style":72},[71],[15,17324,1245],{"className":17325},[76],[15,17327],{"className":17328,"style":72},[71],[15,17330,17332,17335,17338,17341,17344,17347,17350,17353,17356,17359,17362,17365],{"className":17331},[57],[15,17333],{"className":17334,"style":154},[61],[15,17336,1344],{"className":17337,"style":1515},[66,67],[15,17339,120],{"className":17340},[162],[15,17342,123],{"className":17343},[66,67],[15,17345,126],{"className":17346},[169],[15,17348,335],{"className":17349},[422],[15,17351],{"className":17352,"style":4059},[71],[15,17354],{"className":17355,"style":697},[71],[15,17357,1344],{"className":17358,"style":1515},[66,67],[15,17360],{"className":17361,"style":173},[71],[15,17363,1187],{"className":17364},[177],[15,17366],{"className":17367,"style":173},[71],[15,17369,17371,17375,17378],{"className":17370},[57],[15,17372],{"className":17373,"style":17374},[61],"height:0.8778em;vertical-align:-0.1944em;",[15,17376,1061],{"className":17377,"style":1083},[66,494],[15,17379,335],{"className":17380},[422],[11,17382,425,17383,17524,17525,17624,17625,17653,17654,17683],{},[15,17384,17386,17416],{"className":17385,"translate":19},[18],[15,17387,17389],{"className":17388},[23],[25,17390,17391],{"xmlns":27},[29,17392,17393,17413],{},[32,17394,17395,17401,17403,17405,17407],{},[1057,17396,17397,17399],{},[35,17398,17089],{},[1209,17400,1211],{},[39,17402,335],{"separator":53},[39,17404,2095],{},[39,17406,335],{"separator":53},[1057,17408,17409,17411],{},[35,17410,17089],{},[35,17412,11],{},[45,17414,17415],{"encoding":47},"\\phi_1,\\dots,\\phi_p",[15,17417,17419],{"className":17418,"ariaHidden":53},[52],[15,17420,17422,17426,17466,17469,17472,17475,17478,17481,17484],{"className":17421},[57],[15,17423],{"className":17424,"style":17425},[61],"height:0.9805em;vertical-align:-0.2861em;",[15,17427,17429,17432],{"className":17428},[66],[15,17430,17089],{"className":17431},[66,67],[15,17433,17435],{"className":17434},[198],[15,17436,17438,17458],{"className":17437},[202,1090],[15,17439,17441,17455],{"className":17440},[206],[15,17442,17444],{"className":17443,"style":2139},[210],[15,17445,17446,17449],{"style":1584},[15,17447],{"className":17448,"style":218},[217],[15,17450,17452],{"className":17451},[222,223,224,225],[15,17453,1211],{"className":17454},[66,225],[15,17456,1114],{"className":17457},[1113],[15,17459,17461],{"className":17460},[206],[15,17462,17464],{"className":17463,"style":1121},[210],[15,17465],{},[15,17467,335],{"className":17468},[422],[15,17470],{"className":17471,"style":697},[71],[15,17473,2095],{"className":17474},[1413],[15,17476],{"className":17477,"style":697},[71],[15,17479,335],{"className":17480},[422],[15,17482],{"className":17483,"style":697},[71],[15,17485,17487,17490],{"className":17486},[66],[15,17488,17089],{"className":17489},[66,67],[15,17491,17493],{"className":17492},[198],[15,17494,17496,17516],{"className":17495},[202,1090],[15,17497,17499,17513],{"className":17498},[206],[15,17500,17502],{"className":17501,"style":2198},[210],[15,17503,17504,17507],{"style":1584},[15,17505],{"className":17506,"style":218},[217],[15,17508,17510],{"className":17509},[222,223,224,225],[15,17511,11],{"className":17512},[66,67,225],[15,17514,1114],{"className":17515},[1113],[15,17517,17519],{"className":17518},[206],[15,17520,17522],{"className":17521,"style":2715},[210],[15,17523],{}," 张成一个有限维空间（例如常数、线性函数），正则项只惩罚 ",[15,17526,17528,17552],{"className":17527,"translate":19},[18],[15,17529,17531],{"className":17530},[23],[25,17532,17533],{"xmlns":27},[29,17534,17535,17549],{},[32,17536,17537,17539,17541],{},[35,17538,1251],{"mathvariant":138},[35,17540,1344],{},[1255,17542,17543,17545,17547],{},[35,17544,1251],{"mathvariant":138},[35,17546,1061],{"mathvariant":445},[1209,17548,1263],{},[45,17550,17551],{"encoding":47},"\\|g\\|_{\\mathcal H}^2",[15,17553,17555],{"className":17554,"ariaHidden":53},[52],[15,17556,17558,17562,17565,17568],{"className":17557},[57],[15,17559],{"className":17560,"style":17561},[61],"height:1.0894em;vertical-align:-0.2753em;",[15,17563,1251],{"className":17564},[66],[15,17566,1344],{"className":17567,"style":1515},[66,67],[15,17569,17571,17574],{"className":17570},[66],[15,17572,1251],{"className":17573},[66],[15,17575,17577],{"className":17576},[198],[15,17578,17580,17615],{"className":17579},[202,1090],[15,17581,17583,17612],{"className":17582},[206],[15,17584,17586,17601],{"className":17585,"style":6738},[210],[15,17587,17589,17592],{"style":17588},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[15,17590],{"className":17591,"style":218},[217],[15,17593,17595],{"className":17594},[222,223,224,225],[15,17596,17598],{"className":17597},[66,225],[15,17599,1061],{"className":17600,"style":1083},[66,494,225],[15,17602,17603,17606],{"style":213},[15,17604],{"className":17605,"style":218},[217],[15,17607,17609],{"className":17608},[222,223,224,225],[15,17610,1263],{"className":17611},[66,225],[15,17613,1114],{"className":17614},[1113],[15,17616,17618],{"className":17617},[206],[15,17619,17622],{"className":17620,"style":17621},[210],"height:0.2753em;",[15,17623],{},"，则最优的 ",[15,17626,17628,17641],{"className":17627,"translate":19},[18],[15,17629,17631],{"className":17630},[23],[25,17632,17633],{"xmlns":27},[29,17634,17635,17639],{},[32,17636,17637],{},[35,17638,1344],{},[45,17640,1344],{"encoding":47},[15,17642,17644],{"className":17643,"ariaHidden":53},[52],[15,17645,17647,17650],{"className":17646},[57],[15,17648],{"className":17649,"style":471},[61],[15,17651,1344],{"className":17652,"style":1515},[66,67]," 仍在训练点的张成空间中，而 ",[15,17655,17657,17671],{"className":17656,"translate":19},[18],[15,17658,17660],{"className":17659},[23],[25,17661,17662],{"xmlns":27},[29,17663,17664,17668],{},[32,17665,17666],{},[35,17667,17082],{},[45,17669,17670],{"encoding":47},"\\beta",[15,17672,17674],{"className":17673,"ariaHidden":53},[52],[15,17675,17677,17680],{"className":17676},[57],[15,17678],{"className":17679,"style":3406},[61],[15,17681,17082],{"className":17682,"style":17235},[66,67]," 通过一个有限维系统求解。SVM 的偏置、平滑样条的多项式部分都属于这种情况。",[95,17685,17687],{"id":17686},"四为什么是范数的非减函数","四、为什么是「范数的非减函数」",[1904,17689,17691],{"id":17690},"_41-充分性","4.1 充分性",[11,17693,17694,17695,17723,17724,17793,17794,17863,17864,3465,18036,18064],{},"第三节的证明已经说明：只要 ",[15,17696,17698,17711],{"className":17697,"translate":19},[18],[15,17699,17701],{"className":17700},[23],[25,17702,17703],{"xmlns":27},[29,17704,17705,17709],{},[32,17706,17707],{},[35,17708,7737],{"mathvariant":138},[45,17710,8699],{"encoding":47},[15,17712,17714],{"className":17713,"ariaHidden":53},[52],[15,17715,17717,17720],{"className":17716},[57],[15,17718],{"className":17719,"style":490},[61],[15,17721,7737],{"className":17722},[66]," 非减，投影 ",[15,17725,17727,17744],{"className":17726,"translate":19},[18],[15,17728,17730],{"className":17729},[23],[25,17731,17732],{"xmlns":27},[29,17733,17734,17742],{},[32,17735,17736],{},[1057,17737,17738,17740],{},[35,17739,116],{},[39,17741,1251],{},[45,17743,12649],{"encoding":47},[15,17745,17747],{"className":17746,"ariaHidden":53},[52],[15,17748,17750,17753],{"className":17749},[57],[15,17751],{"className":17752,"style":9835},[61],[15,17754,17756,17759],{"className":17755},[66],[15,17757,116],{"className":17758,"style":158},[66,67],[15,17760,17762],{"className":17761},[198],[15,17763,17765,17785],{"className":17764},[202,1090],[15,17766,17768,17782],{"className":17767},[206],[15,17769,17771],{"className":17770,"style":3666},[210],[15,17772,17773,17776],{"style":9856},[15,17774],{"className":17775,"style":218},[217],[15,17777,17779],{"className":17778},[222,223,224,225],[15,17780,1251],{"className":17781},[177,225],[15,17783,1114],{"className":17784},[1113],[15,17786,17788],{"className":17787},[206],[15,17789,17791],{"className":17790,"style":9875},[210],[15,17792],{}," 就不会让目标变差，因此总有一个最优解落在 ",[15,17795,17797,17814],{"className":17796,"translate":19},[18],[15,17798,17800],{"className":17799},[23],[25,17801,17802],{"xmlns":27},[29,17803,17804,17812],{},[32,17805,17806],{},[1057,17807,17808,17810],{},[35,17809,1061],{"mathvariant":445},[1209,17811,1804],{},[45,17813,13444],{"encoding":47},[15,17815,17817],{"className":17816,"ariaHidden":53},[52],[15,17818,17820,17823],{"className":17819},[57],[15,17821],{"className":17822,"style":1076},[61],[15,17824,17826,17829],{"className":17825},[66],[15,17827,1061],{"className":17828,"style":1083},[66,494],[15,17830,17832],{"className":17831},[198],[15,17833,17835,17855],{"className":17834},[202,1090],[15,17836,17838,17852],{"className":17837},[206],[15,17839,17841],{"className":17840,"style":2139},[210],[15,17842,17843,17846],{"style":1100},[15,17844],{"className":17845,"style":218},[217],[15,17847,17849],{"className":17848},[222,223,224,225],[15,17850,1804],{"className":17851},[66,225],[15,17853,1114],{"className":17854},[1113],[15,17856,17858],{"className":17857},[206],[15,17859,17861],{"className":17860,"style":1121},[210],[15,17862],{}," 中。这一步只用了 ",[15,17865,17867,17905],{"className":17866,"translate":19},[18],[15,17868,17870],{"className":17869},[23],[25,17871,17872],{"xmlns":27},[29,17873,17874,17902],{},[32,17875,17876,17878,17880,17886,17888,17890,17896],{},[35,17877,1251],{"mathvariant":138},[35,17879,116],{},[131,17881,17882,17884],{},[35,17883,1251],{"mathvariant":138},[1209,17885,1263],{},[39,17887,1801],{},[35,17889,1251],{"mathvariant":138},[1057,17891,17892,17894],{},[35,17893,116],{},[39,17895,1251],{},[131,17897,17898,17900],{},[35,17899,1251],{"mathvariant":138},[1209,17901,1263],{},[45,17903,17904],{"encoding":47},"\\|f\\|^2\\ge\\|f_\\parallel\\|^2",[15,17906,17908,17958],{"className":17907,"ariaHidden":53},[52],[15,17909,17911,17914,17917,17920,17949,17952,17955],{"className":17910},[57],[15,17912],{"className":17913,"style":9149},[61],[15,17915,1251],{"className":17916},[66],[15,17918,116],{"className":17919,"style":158},[66,67],[15,17921,17923,17926],{"className":17922},[66],[15,17924,1251],{"className":17925},[66],[15,17927,17929],{"className":17928},[198],[15,17930,17932],{"className":17931},[202],[15,17933,17935],{"className":17934},[206],[15,17936,17938],{"className":17937,"style":6738},[210],[15,17939,17940,17943],{"style":213},[15,17941],{"className":17942,"style":218},[217],[15,17944,17946],{"className":17945},[222,223,224,225],[15,17947,1263],{"className":17948},[66,225],[15,17950],{"className":17951,"style":173},[71],[15,17953,1801],{"className":17954},[177],[15,17956],{"className":17957,"style":173},[71],[15,17959,17961,17964,17967,18007],{"className":17960},[57],[15,17962],{"className":17963,"style":13228},[61],[15,17965,1251],{"className":17966},[66],[15,17968,17970,17973],{"className":17969},[66],[15,17971,116],{"className":17972,"style":158},[66,67],[15,17974,17976],{"className":17975},[198],[15,17977,17979,17999],{"className":17978},[202,1090],[15,17980,17982,17996],{"className":17981},[206],[15,17983,17985],{"className":17984,"style":3666},[210],[15,17986,17987,17990],{"style":9856},[15,17988],{"className":17989,"style":218},[217],[15,17991,17993],{"className":17992},[222,223,224,225],[15,17994,1251],{"className":17995},[177,225],[15,17997,1114],{"className":17998},[1113],[15,18000,18002],{"className":18001},[206],[15,18003,18005],{"className":18004,"style":9875},[210],[15,18006],{},[15,18008,18010,18013],{"className":18009},[66],[15,18011,1251],{"className":18012},[66],[15,18014,18016],{"className":18015},[198],[15,18017,18019],{"className":18018},[202],[15,18020,18022],{"className":18021},[206],[15,18023,18025],{"className":18024,"style":6738},[210],[15,18026,18027,18030],{"style":213},[15,18028],{"className":18029,"style":218},[217],[15,18031,18033],{"className":18032},[222,223,224,225],[15,18034,1263],{"className":18035},[66,225],[15,18037,18039,18052],{"className":18038,"translate":19},[18],[15,18040,18042],{"className":18041},[23],[25,18043,18044],{"xmlns":27},[29,18045,18046,18050],{},[32,18047,18048],{},[35,18049,7737],{"mathvariant":138},[45,18051,8699],{"encoding":47},[15,18053,18055],{"className":18054,"ariaHidden":53},[52],[15,18056,18058,18061],{"className":18057},[57],[15,18059],{"className":18060,"style":490},[61],[15,18062,7737],{"className":18063},[66]," 的单调性，没有别的东西。",[1904,18066,18068],{"id":18067},"_42-反例非单调正则项","4.2 反例：非单调正则项",[11,18070,18071,18072,18100],{},"反过来，如果 ",[15,18073,18075,18088],{"className":18074,"translate":19},[18],[15,18076,18078],{"className":18077},[23],[25,18079,18080],{"xmlns":27},[29,18081,18082,18086],{},[32,18083,18084],{},[35,18085,7737],{"mathvariant":138},[45,18087,8699],{"encoding":47},[15,18089,18091],{"className":18090,"ariaHidden":53},[52],[15,18092,18094,18097],{"className":18093},[57],[15,18095],{"className":18096,"style":490},[61],[15,18098,7737],{"className":18099},[66]," 不是非减的，表示定理就可能失效。考虑一个最小的例子。",[11,18102,18103,18104,18185,18186,18354,18355,18436,18437,18552,18553,18645],{},"取 ",[15,18105,18107,18129],{"className":18106,"translate":19},[18],[15,18108,18110],{"className":18109},[23],[25,18111,18112],{"xmlns":27},[29,18113,18114,18126],{},[32,18115,18116,18118,18120],{},[35,18117,446],{"mathvariant":445},[39,18119,129],{},[131,18121,18122,18124],{},[35,18123,455],{"mathvariant":454},[1209,18125,1263],{},[45,18127,18128],{"encoding":47},"\\mathcal X=\\mathbb R^2",[15,18130,18132,18150],{"className":18131,"ariaHidden":53},[52],[15,18133,18135,18138,18141,18144,18147],{"className":18134},[57],[15,18136],{"className":18137,"style":490},[61],[15,18139,446],{"className":18140,"style":495},[66,494],[15,18142],{"className":18143,"style":173},[71],[15,18145,129],{"className":18146},[177],[15,18148],{"className":18149,"style":173},[71],[15,18151,18153,18156],{"className":18152},[57],[15,18154],{"className":18155,"style":6738},[61],[15,18157,18159,18162],{"className":18158},[66],[15,18160,455],{"className":18161},[66,517],[15,18163,18165],{"className":18164},[198],[15,18166,18168],{"className":18167},[202],[15,18169,18171],{"className":18170},[206],[15,18172,18174],{"className":18173,"style":6738},[210],[15,18175,18176,18179],{"style":213},[15,18177],{"className":18178,"style":218},[217],[15,18180,18182],{"className":18181},[222,223,224,225],[15,18183,1263],{"className":18184},[66,225],"，线性核 ",[15,18187,18189,18233],{"className":18188,"translate":19},[18],[15,18190,18192],{"className":18191},[23],[25,18193,18194],{"xmlns":27},[29,18195,18196,18230],{},[32,18197,18198,18200,18202,18204,18206,18212,18214,18216,18218,18220,18222,18228],{},[35,18199,621],{},[39,18201,120],{"stretchy":119},[35,18203,123],{},[39,18205,335],{"separator":53},[131,18207,18208,18210],{},[35,18209,123],{},[39,18211,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,18213,126],{"stretchy":119},[39,18215,129],{},[39,18217,642],{"stretchy":119},[35,18219,123],{},[39,18221,335],{"separator":53},[131,18223,18224,18226],{},[35,18225,123],{},[39,18227,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,18229,667],{"stretchy":119},[45,18231,18232],{"encoding":47},"k(x,x')=\\langle x,x'\\rangle",[15,18234,18236,18301],{"className":18235,"ariaHidden":53},[52],[15,18237,18239,18242,18245,18248,18251,18254,18257,18289,18292,18295,18298],{"className":18238},[57],[15,18240],{"className":18241,"style":5957},[61],[15,18243,621],{"className":18244,"style":684},[66,67],[15,18246,120],{"className":18247},[162],[15,18249,123],{"className":18250},[66,67],[15,18252,335],{"className":18253},[422],[15,18255],{"className":18256,"style":697},[71],[15,18258,18260,18263],{"className":18259},[66],[15,18261,123],{"className":18262},[66,67],[15,18264,18266],{"className":18265},[198],[15,18267,18269],{"className":18268},[202],[15,18270,18272],{"className":18271},[206],[15,18273,18275],{"className":18274,"style":6003},[210],[15,18276,18277,18280],{"style":213},[15,18278],{"className":18279,"style":218},[217],[15,18281,18283],{"className":18282},[222,223,224,225],[15,18284,18286],{"className":18285},[66,225],[15,18287,635],{"className":18288},[66,225],[15,18290,126],{"className":18291},[169],[15,18293],{"className":18294,"style":173},[71],[15,18296,129],{"className":18297},[177],[15,18299],{"className":18300,"style":173},[71],[15,18302,18304,18307,18310,18313,18316,18319,18351],{"className":18303},[57],[15,18305],{"className":18306,"style":5957},[61],[15,18308,642],{"className":18309},[162],[15,18311,123],{"className":18312},[66,67],[15,18314,335],{"className":18315},[422],[15,18317],{"className":18318,"style":697},[71],[15,18320,18322,18325],{"className":18321},[66],[15,18323,123],{"className":18324},[66,67],[15,18326,18328],{"className":18327},[198],[15,18329,18331],{"className":18330},[202],[15,18332,18334],{"className":18333},[206],[15,18335,18337],{"className":18336,"style":6003},[210],[15,18338,18339,18342],{"style":213},[15,18340],{"className":18341,"style":218},[217],[15,18343,18345],{"className":18344},[222,223,224,225],[15,18346,18348],{"className":18347},[66,225],[15,18349,635],{"className":18350},[66,225],[15,18352,667],{"className":18353},[169],"，此时 ",[15,18356,18358,18380],{"className":18357,"translate":19},[18],[15,18359,18361],{"className":18360},[23],[25,18362,18363],{"xmlns":27},[29,18364,18365,18377],{},[32,18366,18367,18369,18371],{},[35,18368,1061],{"mathvariant":445},[39,18370,129],{},[131,18372,18373,18375],{},[35,18374,455],{"mathvariant":454},[1209,18376,1263],{},[45,18378,18379],{"encoding":47},"\\mathcal H=\\mathbb R^2",[15,18381,18383,18401],{"className":18382,"ariaHidden":53},[52],[15,18384,18386,18389,18392,18395,18398],{"className":18385},[57],[15,18387],{"className":18388,"style":490},[61],[15,18390,1061],{"className":18391,"style":1083},[66,494],[15,18393],{"className":18394,"style":173},[71],[15,18396,129],{"className":18397},[177],[15,18399],{"className":18400,"style":173},[71],[15,18402,18404,18407],{"className":18403},[57],[15,18405],{"className":18406,"style":6738},[61],[15,18408,18410,18413],{"className":18409},[66],[15,18411,455],{"className":18412},[66,517],[15,18414,18416],{"className":18415},[198],[15,18417,18419],{"className":18418},[202],[15,18420,18422],{"className":18421},[206],[15,18423,18425],{"className":18424,"style":6738},[210],[15,18426,18427,18430],{"style":213},[15,18428],{"className":18429,"style":218},[217],[15,18431,18433],{"className":18432},[222,223,224,225],[15,18434,1263],{"className":18435},[66,225],"。只有一个训练点 ",[15,18438,18440,18470],{"className":18439,"translate":19},[18],[15,18441,18443],{"className":18442},[23],[25,18444,18445],{"xmlns":27},[29,18446,18447,18467],{},[32,18448,18449,18455,18457,18459,18461,18463,18465],{},[1057,18450,18451,18453],{},[35,18452,123],{},[1209,18454,1211],{},[39,18456,129],{},[39,18458,120],{"stretchy":119},[1209,18460,1211],{},[39,18462,335],{"separator":53},[1209,18464,1804],{},[39,18466,126],{"stretchy":119},[45,18468,18469],{"encoding":47},"x_1=(1,0)",[15,18471,18473,18528],{"className":18472,"ariaHidden":53},[52],[15,18474,18476,18479,18519,18522,18525],{"className":18475},[57],[15,18477],{"className":18478,"style":10192},[61],[15,18480,18482,18485],{"className":18481},[66],[15,18483,123],{"className":18484},[66,67],[15,18486,18488],{"className":18487},[198],[15,18489,18491,18511],{"className":18490},[202,1090],[15,18492,18494,18508],{"className":18493},[206],[15,18495,18497],{"className":18496,"style":2139},[210],[15,18498,18499,18502],{"style":1584},[15,18500],{"className":18501,"style":218},[217],[15,18503,18505],{"className":18504},[222,223,224,225],[15,18506,1211],{"className":18507},[66,225],[15,18509,1114],{"className":18510},[1113],[15,18512,18514],{"className":18513},[206],[15,18515,18517],{"className":18516,"style":1121},[210],[15,18518],{},[15,18520],{"className":18521,"style":173},[71],[15,18523,129],{"className":18524},[177],[15,18526],{"className":18527,"style":173},[71],[15,18529,18531,18534,18537,18540,18543,18546,18549],{"className":18530},[57],[15,18532],{"className":18533,"style":154},[61],[15,18535,120],{"className":18536},[162],[15,18538,1211],{"className":18539},[66],[15,18541,335],{"className":18542},[422],[15,18544],{"className":18545,"style":697},[71],[15,18547,1804],{"className":18548},[66],[15,18550,126],{"className":18551},[169],"，标签 ",[15,18554,18556,18578],{"className":18555,"translate":19},[18],[15,18557,18559],{"className":18558},[23],[25,18560,18561],{"xmlns":27},[29,18562,18563,18575],{},[32,18564,18565,18571,18573],{},[1057,18566,18567,18569],{},[35,18568,1224],{},[1209,18570,1211],{},[39,18572,129],{},[1209,18574,1804],{},[45,18576,18577],{"encoding":47},"y_1=0",[15,18579,18581,18636],{"className":18580,"ariaHidden":53},[52],[15,18582,18584,18587,18627,18630,18633],{"className":18583},[57],[15,18585],{"className":18586,"style":471},[61],[15,18588,18590,18593],{"className":18589},[66],[15,18591,1224],{"className":18592,"style":1515},[66,67],[15,18594,18596],{"className":18595},[198],[15,18597,18599,18619],{"className":18598},[202,1090],[15,18600,18602,18616],{"className":18601},[206],[15,18603,18605],{"className":18604,"style":2139},[210],[15,18606,18607,18610],{"style":1531},[15,18608],{"className":18609,"style":218},[217],[15,18611,18613],{"className":18612},[222,223,224,225],[15,18614,1211],{"className":18615},[66,225],[15,18617,1114],{"className":18618},[1113],[15,18620,18622],{"className":18621},[206],[15,18623,18625],{"className":18624,"style":1121},[210],[15,18626],{},[15,18628],{"className":18629,"style":173},[71],[15,18631,129],{"className":18632},[177],[15,18634],{"className":18635,"style":173},[71],[15,18637,18639,18642],{"className":18638},[57],[15,18640],{"className":18641,"style":1836},[61],[15,18643,1804],{"className":18644},[66],"。损失取平方损失",[15,18647,18649],{"className":18648,"translate":19},[294],[15,18650,18652,18702],{"className":18651,"translate":19},[18],[15,18653,18655],{"className":18654},[23],[25,18656,18657],{"xmlns":27,"display":303},[29,18658,18659,18699],{},[32,18660,18661,18663,18665,18667,18669,18675,18677,18679,18681,18683,18685,18691,18697],{},[35,18662,1216],{},[39,18664,120],{"stretchy":119},[35,18666,116],{},[39,18668,120],{"stretchy":119},[1057,18670,18671,18673],{},[35,18672,123],{},[1209,18674,1211],{},[39,18676,126],{"stretchy":119},[39,18678,126],{"stretchy":119},[39,18680,129],{},[35,18682,116],{},[39,18684,120],{"stretchy":119},[1057,18686,18687,18689],{},[35,18688,123],{},[1209,18690,1211],{},[131,18692,18693,18695],{},[39,18694,126],{"stretchy":119},[1209,18696,1263],{},[39,18698,335],{"separator":53},[45,18700,18701],{"encoding":47},"L(f(x_1))=f(x_1)^2,",[15,18703,18705,18776],{"className":18704,"ariaHidden":53},[52],[15,18706,18708,18711,18714,18717,18720,18723,18763,18767,18770,18773],{"className":18707},[57],[15,18709],{"className":18710,"style":154},[61],[15,18712,1216],{"className":18713},[66,67],[15,18715,120],{"className":18716},[162],[15,18718,116],{"className":18719,"style":158},[66,67],[15,18721,120],{"className":18722},[162],[15,18724,18726,18729],{"className":18725},[66],[15,18727,123],{"className":18728},[66,67],[15,18730,18732],{"className":18731},[198],[15,18733,18735,18755],{"className":18734},[202,1090],[15,18736,18738,18752],{"className":18737},[206],[15,18739,18741],{"className":18740,"style":2139},[210],[15,18742,18743,18746],{"style":1584},[15,18744],{"className":18745,"style":218},[217],[15,18747,18749],{"className":18748},[222,223,224,225],[15,18750,1211],{"className":18751},[66,225],[15,18753,1114],{"className":18754},[1113],[15,18756,18758],{"className":18757},[206],[15,18759,18761],{"className":18760,"style":1121},[210],[15,18762],{},[15,18764,18766],{"className":18765},[169],"))",[15,18768],{"className":18769,"style":173},[71],[15,18771,129],{"className":18772},[177],[15,18774],{"className":18775,"style":173},[71],[15,18777,18779,18782,18785,18788,18828,18857],{"className":18778},[57],[15,18780],{"className":18781,"style":11209},[61],[15,18783,116],{"className":18784,"style":158},[66,67],[15,18786,120],{"className":18787},[162],[15,18789,18791,18794],{"className":18790},[66],[15,18792,123],{"className":18793},[66,67],[15,18795,18797],{"className":18796},[198],[15,18798,18800,18820],{"className":18799},[202,1090],[15,18801,18803,18817],{"className":18802},[206],[15,18804,18806],{"className":18805,"style":2139},[210],[15,18807,18808,18811],{"style":1584},[15,18809],{"className":18810,"style":218},[217],[15,18812,18814],{"className":18813},[222,223,224,225],[15,18815,1211],{"className":18816},[66,225],[15,18818,1114],{"className":18819},[1113],[15,18821,18823],{"className":18822},[206],[15,18824,18826],{"className":18825,"style":1121},[210],[15,18827],{},[15,18829,18831,18834],{"className":18830},[169],[15,18832,126],{"className":18833},[169],[15,18835,18837],{"className":18836},[198],[15,18838,18840],{"className":18839},[202],[15,18841,18843],{"className":18842},[206],[15,18844,18846],{"className":18845,"style":1650},[210],[15,18847,18848,18851],{"style":397},[15,18849],{"className":18850,"style":218},[217],[15,18852,18854],{"className":18853},[222,223,224,225],[15,18855,1263],{"className":18856},[66,225],[15,18858,335],{"className":18859},[422],[11,18861,18862],{},"正则项取",[15,18864,18866],{"className":18865,"translate":19},[294],[15,18867,18869,18911],{"className":18868,"translate":19},[18],[15,18870,18872],{"className":18871},[23],[25,18873,18874],{"xmlns":27,"display":303},[29,18875,18876,18908],{},[32,18877,18878,18880,18882,18884,18886,18888,18890,18892,18894,18896,18898,18900,18906],{},[35,18879,7737],{"mathvariant":138},[39,18881,120],{"stretchy":119},[35,18883,116],{},[39,18885,126],{"stretchy":119},[39,18887,129],{},[39,18889,120],{"stretchy":119},[35,18891,1251],{"mathvariant":138},[35,18893,116],{},[35,18895,1251],{"mathvariant":138},[39,18897,6600],{},[1209,18899,1211],{},[131,18901,18902,18904],{},[39,18903,126],{"stretchy":119},[1209,18905,1263],{},[35,18907,3571],{"mathvariant":138},[45,18909,18910],{"encoding":47},"\\Omega(f)=(\\|f\\|-1)^2.",[15,18912,18914,18941,18968],{"className":18913,"ariaHidden":53},[52],[15,18915,18917,18920,18923,18926,18929,18932,18935,18938],{"className":18916},[57],[15,18918],{"className":18919,"style":154},[61],[15,18921,7737],{"className":18922},[66],[15,18924,120],{"className":18925},[162],[15,18927,116],{"className":18928,"style":158},[66,67],[15,18930,126],{"className":18931},[169],[15,18933],{"className":18934,"style":173},[71],[15,18936,129],{"className":18937},[177],[15,18939],{"className":18940,"style":173},[71],[15,18942,18944,18947,18950,18953,18956,18959,18962,18965],{"className":18943},[57],[15,18945],{"className":18946,"style":154},[61],[15,18948,120],{"className":18949},[162],[15,18951,1251],{"className":18952},[66],[15,18954,116],{"className":18955,"style":158},[66,67],[15,18957,1251],{"className":18958},[66],[15,18960],{"className":18961,"style":72},[71],[15,18963,6600],{"className":18964},[76],[15,18966],{"className":18967,"style":72},[71],[15,18969,18971,18974,18977,19006],{"className":18970},[57],[15,18972],{"className":18973,"style":11209},[61],[15,18975,1211],{"className":18976},[66],[15,18978,18980,18983],{"className":18979},[169],[15,18981,126],{"className":18982},[169],[15,18984,18986],{"className":18985},[198],[15,18987,18989],{"className":18988},[202],[15,18990,18992],{"className":18991},[206],[15,18993,18995],{"className":18994,"style":1650},[210],[15,18996,18997,19000],{"style":397},[15,18998],{"className":18999,"style":218},[217],[15,19001,19003],{"className":19002},[222,223,224,225],[15,19004,1263],{"className":19005},[66,225],[15,19007,3571],{"className":19008},[66],[11,19010,19011,19012,19064],{},"这是一个非负的、只依赖范数的正则项，但它在 ",[15,19013,19015,19037],{"className":19014,"translate":19},[18],[15,19016,19018],{"className":19017},[23],[25,19019,19020],{"xmlns":27},[29,19021,19022,19034],{},[32,19023,19024,19026,19028,19030,19032],{},[39,19025,1194],{"stretchy":119},[1209,19027,1804],{},[39,19029,335],{"separator":53},[1209,19031,1211],{},[39,19033,1266],{"stretchy":119},[45,19035,19036],{"encoding":47},"[0,1]",[15,19038,19040],{"className":19039,"ariaHidden":53},[52],[15,19041,19043,19046,19049,19052,19055,19058,19061],{"className":19042},[57],[15,19044],{"className":19045,"style":154},[61],[15,19047,1194],{"className":19048},[162],[15,19050,1804],{"className":19051},[66],[15,19053,335],{"className":19054},[422],[15,19056],{"className":19057,"style":697},[71],[15,19059,1211],{"className":19060},[66],[15,19062,1266],{"className":19063},[169]," 上递减，不是非减函数。",[11,19066,19067],{},"训练点张成的子空间是",[15,19069,19071],{"className":19070,"translate":19},[294],[15,19072,19074,19144],{"className":19073,"translate":19},[18],[15,19075,19077],{"className":19076},[23],[25,19078,19079],{"xmlns":27,"display":303},[29,19080,19081,19141],{},[32,19082,19083,19089,19091,19093,19095,19097,19099,19101,19103,19105,19111,19113,19115,19117,19119,19121,19123,19125,19127,19129,19131,19133,19135,19137,19139],{},[1057,19084,19085,19087],{},[35,19086,1061],{"mathvariant":445},[1209,19088,1804],{},[39,19090,129],{},[35,19092,15],{"mathvariant":138},[39,19094,1170],{},[39,19096,4305],{"stretchy":119},[35,19098,621],{},[39,19100,120],{"stretchy":119},[39,19102,3266],{},[39,19104,335],{"separator":53},[1057,19106,19107,19109],{},[35,19108,123],{},[1209,19110,1211],{},[39,19112,126],{"stretchy":119},[39,19114,4328],{"stretchy":119},[39,19116,129],{},[39,19118,4305],{"stretchy":119},[39,19120,120],{"stretchy":119},[35,19122,4554],{},[39,19124,335],{"separator":53},[1209,19126,1804],{},[39,19128,126],{"stretchy":119},[39,19130,442],{},[35,19132,4554],{},[39,19134,1187],{},[35,19136,455],{"mathvariant":454},[39,19138,4328],{"stretchy":119},[35,19140,3571],{"mathvariant":138},[45,19142,19143],{"encoding":47},"\\mathcal H_0=\\operatorname{span}\\{k(\\cdot,x_1)\\}=\\{(a,0):a\\in\\mathbb R\\}.",[15,19145,19147,19202,19285,19319,19337],{"className":19146,"ariaHidden":53},[52],[15,19148,19150,19153,19193,19196,19199],{"className":19149},[57],[15,19151],{"className":19152,"style":1076},[61],[15,19154,19156,19159],{"className":19155},[66],[15,19157,1061],{"className":19158,"style":1083},[66,494],[15,19160,19162],{"className":19161},[198],[15,19163,19165,19185],{"className":19164},[202,1090],[15,19166,19168,19182],{"className":19167},[206],[15,19169,19171],{"className":19170,"style":2139},[210],[15,19172,19173,19176],{"style":1100},[15,19174],{"className":19175,"style":218},[217],[15,19177,19179],{"className":19178},[222,223,224,225],[15,19180,1804],{"className":19181},[66,225],[15,19183,1114],{"className":19184},[1113],[15,19186,19188],{"className":19187},[206],[15,19189,19191],{"className":19190,"style":1121},[210],[15,19192],{},[15,19194],{"className":19195,"style":173},[71],[15,19197,129],{"className":19198},[177],[15,19200],{"className":19201,"style":173},[71],[15,19203,19205,19208,19214,19217,19220,19223,19226,19229,19232,19272,19276,19279,19282],{"className":19204},[57],[15,19206],{"className":19207,"style":154},[61],[15,19209,19211],{"className":19210},[1339],[15,19212,15],{"className":19213},[66,4427],[15,19215,4305],{"className":19216},[162],[15,19218,621],{"className":19219,"style":684},[66,67],[15,19221,120],{"className":19222},[162],[15,19224,3266],{"className":19225},[66],[15,19227,335],{"className":19228},[422],[15,19230],{"className":19231,"style":697},[71],[15,19233,19235,19238],{"className":19234},[66],[15,19236,123],{"className":19237},[66,67],[15,19239,19241],{"className":19240},[198],[15,19242,19244,19264],{"className":19243},[202,1090],[15,19245,19247,19261],{"className":19246},[206],[15,19248,19250],{"className":19249,"style":2139},[210],[15,19251,19252,19255],{"style":1584},[15,19253],{"className":19254,"style":218},[217],[15,19256,19258],{"className":19257},[222,223,224,225],[15,19259,1211],{"className":19260},[66,225],[15,19262,1114],{"className":19263},[1113],[15,19265,19267],{"className":19266},[206],[15,19268,19270],{"className":19269,"style":1121},[210],[15,19271],{},[15,19273,19275],{"className":19274},[169],")}",[15,19277],{"className":19278,"style":173},[71],[15,19280,129],{"className":19281},[177],[15,19283],{"className":19284,"style":173},[71],[15,19286,19288,19291,19295,19298,19301,19304,19307,19310,19313,19316],{"className":19287},[57],[15,19289],{"className":19290,"style":154},[61],[15,19292,19294],{"className":19293},[162],"{(",[15,19296,4554],{"className":19297},[66,67],[15,19299,335],{"className":19300},[422],[15,19302],{"className":19303,"style":697},[71],[15,19305,1804],{"className":19306},[66],[15,19308,126],{"className":19309},[169],[15,19311],{"className":19312,"style":173},[71],[15,19314,442],{"className":19315},[177],[15,19317],{"className":19318,"style":173},[71],[15,19320,19322,19325,19328,19331,19334],{"className":19321},[57],[15,19323],{"className":19324,"style":3225},[61],[15,19326,4554],{"className":19327},[66,67],[15,19329],{"className":19330,"style":173},[71],[15,19332,1187],{"className":19333},[177],[15,19335],{"className":19336,"style":173},[71],[15,19338,19340,19343,19346,19349],{"className":19339},[57],[15,19341],{"className":19342,"style":154},[61],[15,19344,455],{"className":19345},[66,517],[15,19347,4328],{"className":19348},[169],[15,19350,3571],{"className":19351},[66],[11,19353,19354,19355,19429,19430,19537,19538,17005,19566,19627,19628,19656,19657,19685,19686,19429,19760,19866,19867,16698,19895,19627,19956,19656,19984,20012,20013,20065,20066,20135],{},"在 ",[15,19356,19358,19384],{"className":19357,"translate":19},[18],[15,19359,19361],{"className":19360},[23],[25,19362,19363],{"xmlns":27},[29,19364,19365,19381],{},[32,19366,19367,19369,19371,19373,19375,19377,19379],{},[35,19368,116],{},[39,19370,129],{},[39,19372,120],{"stretchy":119},[1209,19374,1804],{},[39,19376,335],{"separator":53},[1209,19378,1804],{},[39,19380,126],{"stretchy":119},[45,19382,19383],{"encoding":47},"f=(0,0)",[15,19385,19387,19405],{"className":19386,"ariaHidden":53},[52],[15,19388,19390,19393,19396,19399,19402],{"className":19389},[57],[15,19391],{"className":19392,"style":3406},[61],[15,19394,116],{"className":19395,"style":158},[66,67],[15,19397],{"className":19398,"style":173},[71],[15,19400,129],{"className":19401},[177],[15,19403],{"className":19404,"style":173},[71],[15,19406,19408,19411,19414,19417,19420,19423,19426],{"className":19407},[57],[15,19409],{"className":19410,"style":154},[61],[15,19412,120],{"className":19413},[162],[15,19415,1804],{"className":19416},[66],[15,19418,335],{"className":19419},[422],[15,19421],{"className":19422,"style":697},[71],[15,19424,1804],{"className":19425},[66],[15,19427,126],{"className":19428},[169]," 处，",[15,19431,19433,19461],{"className":19432,"translate":19},[18],[15,19434,19436],{"className":19435},[23],[25,19437,19438],{"xmlns":27},[29,19439,19440,19458],{},[32,19441,19442,19444,19446,19452,19454,19456],{},[35,19443,116],{},[39,19445,120],{"stretchy":119},[1057,19447,19448,19450],{},[35,19449,123],{},[1209,19451,1211],{},[39,19453,126],{"stretchy":119},[39,19455,129],{},[1209,19457,1804],{},[45,19459,19460],{"encoding":47},"f(x_1)=0",[15,19462,19464,19528],{"className":19463,"ariaHidden":53},[52],[15,19465,19467,19470,19473,19476,19516,19519,19522,19525],{"className":19466},[57],[15,19468],{"className":19469,"style":154},[61],[15,19471,116],{"className":19472,"style":158},[66,67],[15,19474,120],{"className":19475},[162],[15,19477,19479,19482],{"className":19478},[66],[15,19480,123],{"className":19481},[66,67],[15,19483,19485],{"className":19484},[198],[15,19486,19488,19508],{"className":19487},[202,1090],[15,19489,19491,19505],{"className":19490},[206],[15,19492,19494],{"className":19493,"style":2139},[210],[15,19495,19496,19499],{"style":1584},[15,19497],{"className":19498,"style":218},[217],[15,19500,19502],{"className":19501},[222,223,224,225],[15,19503,1211],{"className":19504},[66,225],[15,19506,1114],{"className":19507},[1113],[15,19509,19511],{"className":19510},[206],[15,19512,19514],{"className":19513,"style":1121},[210],[15,19515],{},[15,19517,126],{"className":19518},[169],[15,19520],{"className":19521,"style":173},[71],[15,19523,129],{"className":19524},[177],[15,19526],{"className":19527,"style":173},[71],[15,19529,19531,19534],{"className":19530},[57],[15,19532],{"className":19533,"style":1836},[61],[15,19535,1804],{"className":19536},[66],"，损失为 ",[15,19539,19541,19554],{"className":19540,"translate":19},[18],[15,19542,19544],{"className":19543},[23],[25,19545,19546],{"xmlns":27},[29,19547,19548,19552],{},[32,19549,19550],{},[1209,19551,1804],{},[45,19553,1804],{"encoding":47},[15,19555,19557],{"className":19556,"ariaHidden":53},[52],[15,19558,19560,19563],{"className":19559},[57],[15,19561],{"className":19562,"style":1836},[61],[15,19564,1804],{"className":19565},[66],[15,19567,19569,19591],{"className":19568,"translate":19},[18],[15,19570,19572],{"className":19571},[23],[25,19573,19574],{"xmlns":27},[29,19575,19576,19588],{},[32,19577,19578,19580,19582,19584,19586],{},[35,19579,1251],{"mathvariant":138},[35,19581,116],{},[35,19583,1251],{"mathvariant":138},[39,19585,129],{},[1209,19587,1804],{},[45,19589,19590],{"encoding":47},"\\|f\\|=0",[15,19592,19594,19618],{"className":19593,"ariaHidden":53},[52],[15,19595,19597,19600,19603,19606,19609,19612,19615],{"className":19596},[57],[15,19598],{"className":19599,"style":154},[61],[15,19601,1251],{"className":19602},[66],[15,19604,116],{"className":19605,"style":158},[66,67],[15,19607,1251],{"className":19608},[66],[15,19610],{"className":19611,"style":173},[71],[15,19613,129],{"className":19614},[177],[15,19616],{"className":19617,"style":173},[71],[15,19619,19621,19624],{"className":19620},[57],[15,19622],{"className":19623,"style":1836},[61],[15,19625,1804],{"className":19626},[66],"，正则项为 ",[15,19629,19631,19644],{"className":19630,"translate":19},[18],[15,19632,19634],{"className":19633},[23],[25,19635,19636],{"xmlns":27},[29,19637,19638,19642],{},[32,19639,19640],{},[1209,19641,1211],{},[45,19643,1211],{"encoding":47},[15,19645,19647],{"className":19646,"ariaHidden":53},[52],[15,19648,19650,19653],{"className":19649},[57],[15,19651],{"className":19652,"style":1836},[61],[15,19654,1211],{"className":19655},[66],"，目标值为 ",[15,19658,19660,19673],{"className":19659,"translate":19},[18],[15,19661,19663],{"className":19662},[23],[25,19664,19665],{"xmlns":27},[29,19666,19667,19671],{},[32,19668,19669],{},[1209,19670,1211],{},[45,19672,1211],{"encoding":47},[15,19674,19676],{"className":19675,"ariaHidden":53},[52],[15,19677,19679,19682],{"className":19678},[57],[15,19680],{"className":19681,"style":1836},[61],[15,19683,1211],{"className":19684},[66],"。在 ",[15,19687,19689,19715],{"className":19688,"translate":19},[18],[15,19690,19692],{"className":19691},[23],[25,19693,19694],{"xmlns":27},[29,19695,19696,19712],{},[32,19697,19698,19700,19702,19704,19706,19708,19710],{},[35,19699,116],{},[39,19701,129],{},[39,19703,120],{"stretchy":119},[1209,19705,1804],{},[39,19707,335],{"separator":53},[1209,19709,1211],{},[39,19711,126],{"stretchy":119},[45,19713,19714],{"encoding":47},"f=(0,1)",[15,19716,19718,19736],{"className":19717,"ariaHidden":53},[52],[15,19719,19721,19724,19727,19730,19733],{"className":19720},[57],[15,19722],{"className":19723,"style":3406},[61],[15,19725,116],{"className":19726,"style":158},[66,67],[15,19728],{"className":19729,"style":173},[71],[15,19731,129],{"className":19732},[177],[15,19734],{"className":19735,"style":173},[71],[15,19737,19739,19742,19745,19748,19751,19754,19757],{"className":19738},[57],[15,19740],{"className":19741,"style":154},[61],[15,19743,120],{"className":19744},[162],[15,19746,1804],{"className":19747},[66],[15,19749,335],{"className":19750},[422],[15,19752],{"className":19753,"style":697},[71],[15,19755,1211],{"className":19756},[66],[15,19758,126],{"className":19759},[169],[15,19761,19763,19790],{"className":19762,"translate":19},[18],[15,19764,19766],{"className":19765},[23],[25,19767,19768],{"xmlns":27},[29,19769,19770,19788],{},[32,19771,19772,19774,19776,19782,19784,19786],{},[35,19773,116],{},[39,19775,120],{"stretchy":119},[1057,19777,19778,19780],{},[35,19779,123],{},[1209,19781,1211],{},[39,19783,126],{"stretchy":119},[39,19785,129],{},[1209,19787,1804],{},[45,19789,19460],{"encoding":47},[15,19791,19793,19857],{"className":19792,"ariaHidden":53},[52],[15,19794,19796,19799,19802,19805,19845,19848,19851,19854],{"className":19795},[57],[15,19797],{"className":19798,"style":154},[61],[15,19800,116],{"className":19801,"style":158},[66,67],[15,19803,120],{"className":19804},[162],[15,19806,19808,19811],{"className":19807},[66],[15,19809,123],{"className":19810},[66,67],[15,19812,19814],{"className":19813},[198],[15,19815,19817,19837],{"className":19816},[202,1090],[15,19818,19820,19834],{"className":19819},[206],[15,19821,19823],{"className":19822,"style":2139},[210],[15,19824,19825,19828],{"style":1584},[15,19826],{"className":19827,"style":218},[217],[15,19829,19831],{"className":19830},[222,223,224,225],[15,19832,1211],{"className":19833},[66,225],[15,19835,1114],{"className":19836},[1113],[15,19838,19840],{"className":19839},[206],[15,19841,19843],{"className":19842,"style":1121},[210],[15,19844],{},[15,19846,126],{"className":19847},[169],[15,19849],{"className":19850,"style":173},[71],[15,19852,129],{"className":19853},[177],[15,19855],{"className":19856,"style":173},[71],[15,19858,19860,19863],{"className":19859},[57],[15,19861],{"className":19862,"style":1836},[61],[15,19864,1804],{"className":19865},[66],"，损失仍为 ",[15,19868,19870,19883],{"className":19869,"translate":19},[18],[15,19871,19873],{"className":19872},[23],[25,19874,19875],{"xmlns":27},[29,19876,19877,19881],{},[32,19878,19879],{},[1209,19880,1804],{},[45,19882,1804],{"encoding":47},[15,19884,19886],{"className":19885,"ariaHidden":53},[52],[15,19887,19889,19892],{"className":19888},[57],[15,19890],{"className":19891,"style":1836},[61],[15,19893,1804],{"className":19894},[66],[15,19896,19898,19920],{"className":19897,"translate":19},[18],[15,19899,19901],{"className":19900},[23],[25,19902,19903],{"xmlns":27},[29,19904,19905,19917],{},[32,19906,19907,19909,19911,19913,19915],{},[35,19908,1251],{"mathvariant":138},[35,19910,116],{},[35,19912,1251],{"mathvariant":138},[39,19914,129],{},[1209,19916,1211],{},[45,19918,19919],{"encoding":47},"\\|f\\|=1",[15,19921,19923,19947],{"className":19922,"ariaHidden":53},[52],[15,19924,19926,19929,19932,19935,19938,19941,19944],{"className":19925},[57],[15,19927],{"className":19928,"style":154},[61],[15,19930,1251],{"className":19931},[66],[15,19933,116],{"className":19934,"style":158},[66,67],[15,19936,1251],{"className":19937},[66],[15,19939],{"className":19940,"style":173},[71],[15,19942,129],{"className":19943},[177],[15,19945],{"className":19946,"style":173},[71],[15,19948,19950,19953],{"className":19949},[57],[15,19951],{"className":19952,"style":1836},[61],[15,19954,1211],{"className":19955},[66],[15,19957,19959,19972],{"className":19958,"translate":19},[18],[15,19960,19962],{"className":19961},[23],[25,19963,19964],{"xmlns":27},[29,19965,19966,19970],{},[32,19967,19968],{},[1209,19969,1804],{},[45,19971,1804],{"encoding":47},[15,19973,19975],{"className":19974,"ariaHidden":53},[52],[15,19976,19978,19981],{"className":19977},[57],[15,19979],{"className":19980,"style":1836},[61],[15,19982,1804],{"className":19983},[66],[15,19985,19987,20000],{"className":19986,"translate":19},[18],[15,19988,19990],{"className":19989},[23],[25,19991,19992],{"xmlns":27},[29,19993,19994,19998],{},[32,19995,19996],{},[1209,19997,1804],{},[45,19999,1804],{"encoding":47},[15,20001,20003],{"className":20002,"ariaHidden":53},[52],[15,20004,20006,20009],{"className":20005},[57],[15,20007],{"className":20008,"style":1836},[61],[15,20010,1804],{"className":20011},[66],"。所以最优解是 ",[15,20014,20016,20038],{"className":20015,"translate":19},[18],[15,20017,20019],{"className":20018},[23],[25,20020,20021],{"xmlns":27},[29,20022,20023,20035],{},[32,20024,20025,20027,20029,20031,20033],{},[39,20026,120],{"stretchy":119},[1209,20028,1804],{},[39,20030,335],{"separator":53},[1209,20032,1211],{},[39,20034,126],{"stretchy":119},[45,20036,20037],{"encoding":47},"(0,1)",[15,20039,20041],{"className":20040,"ariaHidden":53},[52],[15,20042,20044,20047,20050,20053,20056,20059,20062],{"className":20043},[57],[15,20045],{"className":20046,"style":154},[61],[15,20048,120],{"className":20049},[162],[15,20051,1804],{"className":20052},[66],[15,20054,335],{"className":20055},[422],[15,20057],{"className":20058,"style":697},[71],[15,20060,1211],{"className":20061},[66],[15,20063,126],{"className":20064},[169],"，它落在 ",[15,20067,20069,20086],{"className":20068,"translate":19},[18],[15,20070,20072],{"className":20071},[23],[25,20073,20074],{"xmlns":27},[29,20075,20076,20084],{},[32,20077,20078],{},[1057,20079,20080,20082],{},[35,20081,1061],{"mathvariant":445},[1209,20083,1804],{},[45,20085,13444],{"encoding":47},[15,20087,20089],{"className":20088,"ariaHidden":53},[52],[15,20090,20092,20095],{"className":20091},[57],[15,20093],{"className":20094,"style":1076},[61],[15,20096,20098,20101],{"className":20097},[66],[15,20099,1061],{"className":20100,"style":1083},[66,494],[15,20102,20104],{"className":20103},[198],[15,20105,20107,20127],{"className":20106},[202,1090],[15,20108,20110,20124],{"className":20109},[206],[15,20111,20113],{"className":20112,"style":2139},[210],[15,20114,20115,20118],{"style":1100},[15,20116],{"className":20117,"style":218},[217],[15,20119,20121],{"className":20120},[222,223,224,225],[15,20122,1804],{"className":20123},[66,225],[15,20125,1114],{"className":20126},[1113],[15,20128,20130],{"className":20129},[206],[15,20131,20133],{"className":20132,"style":1121},[210],[15,20134],{}," 的正交补里，而不是训练点的张成空间中。",[11,20137,20138,20139,20142],{},"这个反例说明：",[2040,20140,20141],{},"非减性不是证明技巧，而是定理的本质条件。"," 正则项如果「喜欢」某个非零范数，它就会把解推离训练点子空间。",[1904,20144,20146],{"id":20145},"_43-充要条件","4.3 充要条件",[11,20148,20149],{},"上述反例可以推广。Dinuzzo 与 Schölkopf 在 2012 年给出了完整的刻画：在相当一般的条件下，表示定理对所有损失函数和所有数据集成立，当且仅当正则项是范数的非减函数。也就是说，",[15,20151,20153],{"className":20152,"translate":19},[294],[15,20154,20156,20195],{"className":20155,"translate":19},[18],[15,20157,20159],{"className":20158},[23],[25,20160,20161],{"xmlns":27,"display":303},[29,20162,20163,20192],{},[32,20164,20165,20167,20169,20171,20173,20175,20178,20180,20182,20184,20190],{},[35,20166,7737],{"mathvariant":138},[39,20168,120],{"stretchy":119},[35,20170,116],{},[39,20172,126],{"stretchy":119},[39,20174,129],{},[35,20176,20177],{},"h",[39,20179,120],{"stretchy":119},[35,20181,1251],{"mathvariant":138},[35,20183,116],{},[1057,20185,20186,20188],{},[35,20187,1251],{"mathvariant":138},[35,20189,1061],{"mathvariant":445},[39,20191,126],{"stretchy":119},[45,20193,20194],{"encoding":47},"\\Omega(f)=h(\\|f\\|_{\\mathcal H})",[15,20196,20198,20225],{"className":20197,"ariaHidden":53},[52],[15,20199,20201,20204,20207,20210,20213,20216,20219,20222],{"className":20200},[57],[15,20202],{"className":20203,"style":154},[61],[15,20205,7737],{"className":20206},[66],[15,20208,120],{"className":20209},[162],[15,20211,116],{"className":20212,"style":158},[66,67],[15,20214,126],{"className":20215},[169],[15,20217],{"className":20218,"style":173},[71],[15,20220,129],{"className":20221},[177],[15,20223],{"className":20224,"style":173},[71],[15,20226,20228,20231,20234,20237,20240,20243,20286],{"className":20227},[57],[15,20229],{"className":20230,"style":154},[61],[15,20232,20177],{"className":20233},[66,67],[15,20235,120],{"className":20236},[162],[15,20238,1251],{"className":20239},[66],[15,20241,116],{"className":20242,"style":158},[66,67],[15,20244,20246,20249],{"className":20245},[66],[15,20247,1251],{"className":20248},[66],[15,20250,20252],{"className":20251},[198],[15,20253,20255,20278],{"className":20254},[202,1090],[15,20256,20258,20275],{"className":20257},[206],[15,20259,20261],{"className":20260,"style":3636},[210],[15,20262,20263,20266],{"style":1584},[15,20264],{"className":20265,"style":218},[217],[15,20267,20269],{"className":20268},[222,223,224,225],[15,20270,20272],{"className":20271},[66,225],[15,20273,1061],{"className":20274,"style":1083},[66,494,225],[15,20276,1114],{"className":20277},[1113],[15,20279,20281],{"className":20280},[206],[15,20282,20284],{"className":20283,"style":1121},[210],[15,20285],{},[15,20287,126],{"className":20288},[169],[11,20290,20291,20292,20388],{},"对某个非减的 ",[15,20293,20295,20325],{"className":20294,"translate":19},[18],[15,20296,20298],{"className":20297},[23],[25,20299,20300],{"xmlns":27},[29,20301,20302,20322],{},[32,20303,20304,20306,20308,20310,20312,20314,20316,20318,20320],{},[35,20305,20177],{},[39,20307,442],{},[39,20309,1194],{"stretchy":119},[1209,20311,1804],{},[39,20313,335],{"separator":53},[35,20315,7615],{"mathvariant":138},[39,20317,126],{"stretchy":119},[39,20319,449],{},[35,20321,455],{"mathvariant":454},[45,20323,20324],{"encoding":47},"h:[0,\\infty)\\to\\mathbb R",[15,20326,20328,20346,20379],{"className":20327,"ariaHidden":53},[52],[15,20329,20331,20334,20337,20340,20343],{"className":20330},[57],[15,20332],{"className":20333,"style":840},[61],[15,20335,20177],{"className":20336},[66,67],[15,20338],{"className":20339,"style":173},[71],[15,20341,442],{"className":20342},[177],[15,20344],{"className":20345,"style":173},[71],[15,20347,20349,20352,20355,20358,20361,20364,20367,20370,20373,20376],{"className":20348},[57],[15,20350],{"className":20351,"style":154},[61],[15,20353,1194],{"className":20354},[162],[15,20356,1804],{"className":20357},[66],[15,20359,335],{"className":20360},[422],[15,20362],{"className":20363,"style":697},[71],[15,20365,7615],{"className":20366},[66],[15,20368,126],{"className":20369},[169],[15,20371],{"className":20372,"style":173},[71],[15,20374,449],{"className":20375},[177],[15,20377],{"className":20378,"style":173},[71],[15,20380,20382,20385],{"className":20381},[57],[15,20383],{"className":20384,"style":2034},[61],[15,20386,455],{"className":20387},[66,517]," 成立，是表示定理的充要条件。",[11,20390,20391],{},"这里还要区分两个层次：",[16131,20393,20394,20470],{},[3192,20395,20396,20399,20400,20469],{},[2040,20397,20398],{},"非减","：保证存在一个落在 ",[15,20401,20403,20420],{"className":20402,"translate":19},[18],[15,20404,20406],{"className":20405},[23],[25,20407,20408],{"xmlns":27},[29,20409,20410,20418],{},[32,20411,20412],{},[1057,20413,20414,20416],{},[35,20415,1061],{"mathvariant":445},[1209,20417,1804],{},[45,20419,13444],{"encoding":47},[15,20421,20423],{"className":20422,"ariaHidden":53},[52],[15,20424,20426,20429],{"className":20425},[57],[15,20427],{"className":20428,"style":1076},[61],[15,20430,20432,20435],{"className":20431},[66],[15,20433,1061],{"className":20434,"style":1083},[66,494],[15,20436,20438],{"className":20437},[198],[15,20439,20441,20461],{"className":20440},[202,1090],[15,20442,20444,20458],{"className":20443},[206],[15,20445,20447],{"className":20446,"style":2139},[210],[15,20448,20449,20452],{"style":1100},[15,20450],{"className":20451,"style":218},[217],[15,20453,20455],{"className":20454},[222,223,224,225],[15,20456,1804],{"className":20457},[66,225],[15,20459,1114],{"className":20460},[1113],[15,20462,20464],{"className":20463},[206],[15,20465,20467],{"className":20466,"style":1121},[210],[15,20468],{}," 中的最优解；",[3192,20471,20472,20475,20476,20545],{},[2040,20473,20474],{},"严格递增","：保证每个最优解都落在 ",[15,20477,20479,20496],{"className":20478,"translate":19},[18],[15,20480,20482],{"className":20481},[23],[25,20483,20484],{"xmlns":27},[29,20485,20486,20494],{},[32,20487,20488],{},[1057,20489,20490,20492],{},[35,20491,1061],{"mathvariant":445},[1209,20493,1804],{},[45,20495,13444],{"encoding":47},[15,20497,20499],{"className":20498,"ariaHidden":53},[52],[15,20500,20502,20505],{"className":20501},[57],[15,20503],{"className":20504,"style":1076},[61],[15,20506,20508,20511],{"className":20507},[66],[15,20509,1061],{"className":20510,"style":1083},[66,494],[15,20512,20514],{"className":20513},[198],[15,20515,20517,20537],{"className":20516},[202,1090],[15,20518,20520,20534],{"className":20519},[206],[15,20521,20523],{"className":20522,"style":2139},[210],[15,20524,20525,20528],{"style":1100},[15,20526],{"className":20527,"style":218},[217],[15,20529,20531],{"className":20530},[222,223,224,225],[15,20532,1804],{"className":20533},[66,225],[15,20535,1114],{"className":20536},[1113],[15,20538,20540],{"className":20539},[206],[15,20541,20543],{"className":20542,"style":1121},[210],[15,20544],{}," 中。",[11,20547,20548,20549,20601,20602,20630,20631,20700,20701,20770],{},"如果 ",[15,20550,20552,20571],{"className":20551,"translate":19},[18],[15,20553,20555],{"className":20554},[23],[25,20556,20557],{"xmlns":27},[29,20558,20559,20568],{},[32,20560,20561,20563,20566],{},[35,20562,7737],{"mathvariant":138},[39,20564,20565],{},"≡",[1209,20567,1804],{},[45,20569,20570],{"encoding":47},"\\Omega\\equiv 0",[15,20572,20574,20592],{"className":20573,"ariaHidden":53},[52],[15,20575,20577,20580,20583,20586,20589],{"className":20576},[57],[15,20578],{"className":20579,"style":490},[61],[15,20581,7737],{"className":20582},[66],[15,20584],{"className":20585,"style":173},[71],[15,20587,20565],{"className":20588},[177],[15,20590],{"className":20591,"style":173},[71],[15,20593,20595,20598],{"className":20594},[57],[15,20596],{"className":20597,"style":1836},[61],[15,20599,1804],{"className":20600},[66],"，目标函数只依赖 ",[15,20603,20605,20618],{"className":20604,"translate":19},[18],[15,20606,20608],{"className":20607},[23],[25,20609,20610],{"xmlns":27},[29,20611,20612,20616],{},[32,20613,20614],{},[35,20615,116],{},[45,20617,116],{"encoding":47},[15,20619,20621],{"className":20620,"ariaHidden":53},[52],[15,20622,20624,20627],{"className":20623},[57],[15,20625],{"className":20626,"style":3406},[61],[15,20628,116],{"className":20629,"style":158},[66,67]," 在训练点上的取值。任何在这些点上取值相同的函数都是最优解，其中很多并不在 ",[15,20632,20634,20651],{"className":20633,"translate":19},[18],[15,20635,20637],{"className":20636},[23],[25,20638,20639],{"xmlns":27},[29,20640,20641,20649],{},[32,20642,20643],{},[1057,20644,20645,20647],{},[35,20646,1061],{"mathvariant":445},[1209,20648,1804],{},[45,20650,13444],{"encoding":47},[15,20652,20654],{"className":20653,"ariaHidden":53},[52],[15,20655,20657,20660],{"className":20656},[57],[15,20658],{"className":20659,"style":1076},[61],[15,20661,20663,20666],{"className":20662},[66],[15,20664,1061],{"className":20665,"style":1083},[66,494],[15,20667,20669],{"className":20668},[198],[15,20670,20672,20692],{"className":20671},[202,1090],[15,20673,20675,20689],{"className":20674},[206],[15,20676,20678],{"className":20677,"style":2139},[210],[15,20679,20680,20683],{"style":1100},[15,20681],{"className":20682,"style":218},[217],[15,20684,20686],{"className":20685},[222,223,224,225],[15,20687,1804],{"className":20688},[66,225],[15,20690,1114],{"className":20691},[1113],[15,20693,20695],{"className":20694},[206],[15,20696,20698],{"className":20697,"style":1121},[210],[15,20699],{}," 里。表示定理的「存在」版本仍然成立（取 ",[15,20702,20704,20721],{"className":20703,"translate":19},[18],[15,20705,20707],{"className":20706},[23],[25,20708,20709],{"xmlns":27},[29,20710,20711,20719],{},[32,20712,20713],{},[1057,20714,20715,20717],{},[35,20716,116],{},[39,20718,1251],{},[45,20720,12649],{"encoding":47},[15,20722,20724],{"className":20723,"ariaHidden":53},[52],[15,20725,20727,20730],{"className":20726},[57],[15,20728],{"className":20729,"style":9835},[61],[15,20731,20733,20736],{"className":20732},[66],[15,20734,116],{"className":20735,"style":158},[66,67],[15,20737,20739],{"className":20738},[198],[15,20740,20742,20762],{"className":20741},[202,1090],[15,20743,20745,20759],{"className":20744},[206],[15,20746,20748],{"className":20747,"style":3666},[210],[15,20749,20750,20753],{"style":9856},[15,20751],{"className":20752,"style":218},[217],[15,20754,20756],{"className":20755},[222,223,224,225],[15,20757,1251],{"className":20758},[177,225],[15,20760,1114],{"className":20761},[1113],[15,20763,20765],{"className":20764},[206],[15,20766,20768],{"className":20767,"style":9875},[210],[15,20769],{}," 即可），但「任意最优解」的版本不成立。因此，「任意最优解都满足」这一说法需要加上严格递增这个条件。",[95,20772,20774],{"id":20773},"五五个经典例子","五、五个经典例子",[1904,20776,20778],{"id":20777},"_51-核岭回归","5.1 核岭回归",[11,20780,20781,20782,9261,20852,20902],{},"取平方损失和 ",[15,20783,20785,20810],{"className":20784,"translate":19},[18],[15,20786,20788],{"className":20787},[23],[25,20789,20790],{"xmlns":27},[29,20791,20792,20808],{},[32,20793,20794,20796,20798,20800,20802,20804,20806],{},[35,20795,7737],{"mathvariant":138},[39,20797,120],{"stretchy":119},[35,20799,9207],{},[39,20801,126],{"stretchy":119},[39,20803,129],{},[35,20805,1248],{},[35,20807,9207],{},[45,20809,9218],{"encoding":47},[15,20811,20813,20840],{"className":20812,"ariaHidden":53},[52],[15,20814,20816,20819,20822,20825,20828,20831,20834,20837],{"className":20815},[57],[15,20817],{"className":20818,"style":154},[61],[15,20820,7737],{"className":20821},[66],[15,20823,120],{"className":20824},[162],[15,20826,9207],{"className":20827},[66,67],[15,20829,126],{"className":20830},[169],[15,20832],{"className":20833,"style":173},[71],[15,20835,129],{"className":20836},[177],[15,20838],{"className":20839,"style":173},[71],[15,20841,20843,20846,20849],{"className":20842},[57],[15,20844],{"className":20845,"style":840},[61],[15,20847,1248],{"className":20848},[66,67],[15,20850,9207],{"className":20851},[66,67],[15,20853,20855,20872],{"className":20854,"translate":19},[18],[15,20856,20858],{"className":20857},[23],[25,20859,20860],{"xmlns":27},[29,20861,20862,20870],{},[32,20863,20864,20866,20868],{},[35,20865,1248],{},[39,20867,9278],{},[1209,20869,1804],{},[45,20871,9283],{"encoding":47},[15,20873,20875,20893],{"className":20874,"ariaHidden":53},[52],[15,20876,20878,20881,20884,20887,20890],{"className":20877},[57],[15,20879],{"className":20880,"style":9293},[61],[15,20882,1248],{"className":20883},[66,67],[15,20885],{"className":20886,"style":173},[71],[15,20888,9278],{"className":20889},[177],[15,20891],{"className":20892,"style":173},[71],[15,20894,20896,20899],{"className":20895},[57],[15,20897],{"className":20898,"style":1836},[61],[15,20900,1804],{"className":20901},[66],"）：",[15,20904,20906],{"className":20905,"translate":19},[294],[15,20907,20909,21003],{"className":20908,"translate":19},[18],[15,20910,20912],{"className":20911},[23],[25,20913,20914],{"xmlns":27,"display":303},[29,20915,20916,21000],{},[32,20917,20918,20934,20998],{},[1172,20919,20920,20926],{},[32,20921,20922,20924],{},[35,20923,1178],{},[39,20925,1170],{},[32,20927,20928,20930,20932],{},[35,20929,116],{},[39,20931,1187],{},[35,20933,1061],{"mathvariant":445},[32,20935,20936,20938,20952,20954,20960,20962,20964,20966,20972,20974,20980,20982,20984,20986,20988,20996],{},[39,20937,4305],{"fence":53},[1196,20939,20940,20942,20950],{},[39,20941,1200],{},[32,20943,20944,20946,20948],{},[35,20945,1205],{},[39,20947,129],{},[1209,20949,1211],{},[35,20951,37],{},[39,20953,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,20955,20956,20958],{},[35,20957,1224],{},[35,20959,1205],{},[39,20961,6600],{},[35,20963,116],{},[39,20965,120],{"stretchy":119},[1057,20967,20968,20970],{},[35,20969,123],{},[35,20971,1205],{},[39,20973,126],{"stretchy":119},[131,20975,20976,20978],{},[39,20977,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,20979,1263],{},[39,20981,1245],{},[35,20983,1248],{},[35,20985,1251],{"mathvariant":138},[35,20987,116],{},[1255,20989,20990,20992,20994],{},[35,20991,1251],{"mathvariant":138},[35,20993,1061],{"mathvariant":445},[1209,20995,1263],{},[39,20997,4328],{"fence":53},[35,20999,3571],{"mathvariant":138},[45,21001,21002],{"encoding":47},"\\min_{f\\in\\mathcal H}\\left\\{\\sum_{i=1}^n\\bigl(y_i-f(x_i)\\bigr)^2+\\lambda\\|f\\|_{\\mathcal H}^2\\right\\}.",[15,21004,21006],{"className":21005,"ariaHidden":53},[52],[15,21007,21009,21012,21065,21068,21357,21360],{"className":21008},[57],[15,21010],{"className":21011,"style":1335},[61],[15,21013,21015],{"className":21014},[1339,1351],[15,21016,21018,21057],{"className":21017},[202,1090],[15,21019,21021,21054],{"className":21020},[206],[15,21022,21024,21044],{"className":21023,"style":1361},[210],[15,21025,21026,21029],{"style":1364},[15,21027],{"className":21028,"style":1368},[217],[15,21030,21032],{"className":21031},[222,223,224,225],[15,21033,21035,21038,21041],{"className":21034},[66,225],[15,21036,116],{"className":21037,"style":158},[66,67,225],[15,21039,1187],{"className":21040},[177,225],[15,21042,1061],{"className":21043,"style":1083},[66,494,225],[15,21045,21046,21049],{"style":1386},[15,21047],{"className":21048,"style":1368},[217],[15,21050,21051],{},[15,21052,1178],{"className":21053},[1339],[15,21055,1114],{"className":21056},[1113],[15,21058,21060],{"className":21059},[206],[15,21061,21063],{"className":21062,"style":1404},[210],[15,21064],{},[15,21066],{"className":21067,"style":697},[71],[15,21069,21071,21077,21141,21147,21187,21190,21193,21196,21199,21202,21242,21245,21279,21282,21285,21288,21291,21294,21297,21351],{"className":21070},[1413],[15,21072,21074],{"className":21073,"style":1418},[162,1417],[15,21075,4305],{"className":21076},[1422,1423],[15,21078,21080],{"className":21079},[1339,1351],[15,21081,21083,21133],{"className":21082},[202,1090],[15,21084,21086,21130],{"className":21085},[206],[15,21087,21089,21109,21119],{"className":21088,"style":1436},[210],[15,21090,21091,21094],{"style":1439},[15,21092],{"className":21093,"style":1443},[217],[15,21095,21097],{"className":21096},[222,223,224,225],[15,21098,21100,21103,21106],{"className":21099},[66,225],[15,21101,1205],{"className":21102},[66,67,225],[15,21104,129],{"className":21105},[177,225],[15,21107,1211],{"className":21108},[66,225],[15,21110,21111,21114],{"style":1461},[15,21112],{"className":21113,"style":1443},[217],[15,21115,21116],{},[15,21117,1200],{"className":21118},[1339,1470,1471],[15,21120,21121,21124],{"style":1474},[15,21122],{"className":21123,"style":1443},[217],[15,21125,21127],{"className":21126},[222,223,224,225],[15,21128,37],{"className":21129},[66,67,225],[15,21131,1114],{"className":21132},[1113],[15,21134,21136],{"className":21135},[206],[15,21137,21139],{"className":21138,"style":1493},[210],[15,21140],{},[15,21142,21144],{"className":21143},[162],[15,21145,120],{"className":21146},[1422,1508],[15,21148,21150,21153],{"className":21149},[66],[15,21151,1224],{"className":21152,"style":1515},[66,67],[15,21154,21156],{"className":21155},[198],[15,21157,21159,21179],{"className":21158},[202,1090],[15,21160,21162,21176],{"className":21161},[206],[15,21163,21165],{"className":21164,"style":1528},[210],[15,21166,21167,21170],{"style":1531},[15,21168],{"className":21169,"style":218},[217],[15,21171,21173],{"className":21172},[222,223,224,225],[15,21174,1205],{"className":21175},[66,67,225],[15,21177,1114],{"className":21178},[1113],[15,21180,21182],{"className":21181},[206],[15,21183,21185],{"className":21184,"style":1121},[210],[15,21186],{},[15,21188],{"className":21189,"style":72},[71],[15,21191,6600],{"className":21192},[76],[15,21194],{"className":21195,"style":72},[71],[15,21197,116],{"className":21198,"style":158},[66,67],[15,21200,120],{"className":21201},[162],[15,21203,21205,21208],{"className":21204},[66],[15,21206,123],{"className":21207},[66,67],[15,21209,21211],{"className":21210},[198],[15,21212,21214,21234],{"className":21213},[202,1090],[15,21215,21217,21231],{"className":21216},[206],[15,21218,21220],{"className":21219,"style":1528},[210],[15,21221,21222,21225],{"style":1584},[15,21223],{"className":21224,"style":218},[217],[15,21226,21228],{"className":21227},[222,223,224,225],[15,21229,1205],{"className":21230},[66,67,225],[15,21232,1114],{"className":21233},[1113],[15,21235,21237],{"className":21236},[206],[15,21238,21240],{"className":21239,"style":1121},[210],[15,21241],{},[15,21243,126],{"className":21244},[169],[15,21246,21248,21254],{"className":21247},[169],[15,21249,21251],{"className":21250},[169],[15,21252,126],{"className":21253},[1422,1508],[15,21255,21257],{"className":21256},[198],[15,21258,21260],{"className":21259},[202],[15,21261,21263],{"className":21262},[206],[15,21264,21267],{"className":21265,"style":21266},[210],"height:1.054em;",[15,21268,21270,21273],{"style":21269},"top:-3.3029em;margin-right:0.05em;",[15,21271],{"className":21272,"style":218},[217],[15,21274,21276],{"className":21275},[222,223,224,225],[15,21277,1263],{"className":21278},[66,225],[15,21280],{"className":21281,"style":72},[71],[15,21283,1245],{"className":21284},[76],[15,21286],{"className":21287,"style":72},[71],[15,21289,1248],{"className":21290},[66,67],[15,21292,1251],{"className":21293},[66],[15,21295,116],{"className":21296,"style":158},[66,67],[15,21298,21300,21303],{"className":21299},[66],[15,21301,1251],{"className":21302},[66],[15,21304,21306],{"className":21305},[198],[15,21307,21309,21343],{"className":21308},[202,1090],[15,21310,21312,21340],{"className":21311},[206],[15,21313,21315,21329],{"className":21314,"style":1650},[210],[15,21316,21317,21320],{"style":1653},[15,21318],{"className":21319,"style":218},[217],[15,21321,21323],{"className":21322},[222,223,224,225],[15,21324,21326],{"className":21325},[66,225],[15,21327,1061],{"className":21328,"style":1083},[66,494,225],[15,21330,21331,21334],{"style":397},[15,21332],{"className":21333,"style":218},[217],[15,21335,21337],{"className":21336},[222,223,224,225],[15,21338,1263],{"className":21339},[66,225],[15,21341,1114],{"className":21342},[1113],[15,21344,21346],{"className":21345},[206],[15,21347,21349],{"className":21348,"style":1686},[210],[15,21350],{},[15,21352,21354],{"className":21353,"style":1418},[169,1417],[15,21355,4328],{"className":21356},[1422,1423],[15,21358],{"className":21359,"style":697},[71],[15,21361,3571],{"className":21362},[66],[11,21364,21365,21366,21610],{},"由表示定理，",[15,21367,21369,21416],{"className":21368,"translate":19},[18],[15,21370,21372],{"className":21371},[23],[25,21373,21374],{"xmlns":27},[29,21375,21376,21414],{},[32,21377,21378,21384,21386,21392,21398,21400,21402,21404,21406,21412],{},[131,21379,21380,21382],{},[35,21381,116],{},[39,21383,1162],{},[39,21385,129],{},[1057,21387,21388,21390],{},[39,21389,1200],{},[35,21391,1205],{},[1057,21393,21394,21396],{},[35,21395,8385],{},[35,21397,1205],{},[35,21399,621],{},[39,21401,120],{"stretchy":119},[39,21403,3266],{},[39,21405,335],{"separator":53},[1057,21407,21408,21410],{},[35,21409,123],{},[35,21411,1205],{},[39,21413,126],{"stretchy":119},[45,21415,16807],{"encoding":47},[15,21417,21419,21463],{"className":21418,"ariaHidden":53},[52],[15,21420,21422,21425,21454,21457,21460],{"className":21421},[57],[15,21423],{"className":21424,"style":3406},[61],[15,21426,21428,21431],{"className":21427},[66],[15,21429,116],{"className":21430,"style":158},[66,67],[15,21432,21434],{"className":21433},[198],[15,21435,21437],{"className":21436},[202],[15,21438,21440],{"className":21439},[206],[15,21441,21443],{"className":21442,"style":16244},[210],[15,21444,21445,21448],{"style":213},[15,21446],{"className":21447,"style":218},[217],[15,21449,21451],{"className":21450},[222,223,224,225],[15,21452,1162],{"className":21453},[76,225],[15,21455],{"className":21456,"style":173},[71],[15,21458,129],{"className":21459},[177],[15,21461],{"className":21462,"style":173},[71],[15,21464,21466,21469,21509,21512,21552,21555,21558,21561,21564,21567,21607],{"className":21465},[57],[15,21467],{"className":21468,"style":16861},[61],[15,21470,21472,21475],{"className":21471},[1339],[15,21473,1200],{"className":21474,"style":13697},[1339,1470,13696],[15,21476,21478],{"className":21477},[198],[15,21479,21481,21501],{"className":21480},[202,1090],[15,21482,21484,21498],{"className":21483},[206],[15,21485,21487],{"className":21486,"style":16880},[210],[15,21488,21489,21492],{"style":13713},[15,21490],{"className":21491,"style":218},[217],[15,21493,21495],{"className":21494},[222,223,224,225],[15,21496,1205],{"className":21497},[66,67,225],[15,21499,1114],{"className":21500},[1113],[15,21502,21504],{"className":21503},[206],[15,21505,21507],{"className":21506,"style":16901},[210],[15,21508],{},[15,21510],{"className":21511,"style":697},[71],[15,21513,21515,21518],{"className":21514},[66],[15,21516,8385],{"className":21517,"style":8555},[66,67],[15,21519,21521],{"className":21520},[198],[15,21522,21524,21544],{"className":21523},[202,1090],[15,21525,21527,21541],{"className":21526},[206],[15,21528,21530],{"className":21529,"style":1528},[210],[15,21531,21532,21535],{"style":8570},[15,21533],{"className":21534,"style":218},[217],[15,21536,21538],{"className":21537},[222,223,224,225],[15,21539,1205],{"className":21540},[66,67,225],[15,21542,1114],{"className":21543},[1113],[15,21545,21547],{"className":21546},[206],[15,21548,21550],{"className":21549,"style":1121},[210],[15,21551],{},[15,21553,621],{"className":21554,"style":684},[66,67],[15,21556,120],{"className":21557},[162],[15,21559,3266],{"className":21560},[66],[15,21562,335],{"className":21563},[422],[15,21565],{"className":21566,"style":697},[71],[15,21568,21570,21573],{"className":21569},[66],[15,21571,123],{"className":21572},[66,67],[15,21574,21576],{"className":21575},[198],[15,21577,21579,21599],{"className":21578},[202,1090],[15,21580,21582,21596],{"className":21581},[206],[15,21583,21585],{"className":21584,"style":1528},[210],[15,21586,21587,21590],{"style":1584},[15,21588],{"className":21589,"style":218},[217],[15,21591,21593],{"className":21592},[222,223,224,225],[15,21594,1205],{"className":21595},[66,67,225],[15,21597,1114],{"className":21598},[1113],[15,21600,21602],{"className":21601},[206],[15,21603,21605],{"className":21604,"style":1121},[210],[15,21606],{},[15,21608,126],{"className":21609},[169],"。代入式 (4)：",[15,21612,21614],{"className":21613,"translate":19},[294],[15,21615,21617,21663],{"className":21616,"translate":19},[18],[15,21618,21620],{"className":21619},[23],[25,21621,21622],{"xmlns":27,"display":303},[29,21623,21624,21660],{},[32,21625,21626,21628,21630,21632,21634,21636,21642,21644,21646,21648,21654,21656,21658],{},[35,21627,1251],{"mathvariant":138},[35,21629,1224],{},[39,21631,6600],{},[35,21633,2853],{},[35,21635,8385],{},[131,21637,21638,21640],{},[35,21639,1251],{"mathvariant":138},[1209,21641,1263],{},[39,21643,1245],{},[35,21645,1248],{},[1274,21647,2462],{},[131,21649,21650,21652],{},[35,21651,8385],{},[35,21653,139],{"mathvariant":138},[35,21655,2853],{},[35,21657,8385],{},[35,21659,3571],{"mathvariant":138},[45,21661,21662],{"encoding":47},"\\|y-K\\alpha\\|^2+\\lambda\\,\\alpha^\\top K\\alpha.",[15,21664,21666,21687,21737],{"className":21665,"ariaHidden":53},[52],[15,21667,21669,21672,21675,21678,21681,21684],{"className":21668},[57],[15,21670],{"className":21671,"style":154},[61],[15,21673,1251],{"className":21674},[66],[15,21676,1224],{"className":21677,"style":1515},[66,67],[15,21679],{"className":21680,"style":72},[71],[15,21682,6600],{"className":21683},[76],[15,21685],{"className":21686,"style":72},[71],[15,21688,21690,21693,21696,21699,21728,21731,21734],{"className":21689},[57],[15,21691],{"className":21692,"style":11209},[61],[15,21694,2853],{"className":21695,"style":2901},[66,67],[15,21697,8385],{"className":21698,"style":8555},[66,67],[15,21700,21702,21705],{"className":21701},[66],[15,21703,1251],{"className":21704},[66],[15,21706,21708],{"className":21707},[198],[15,21709,21711],{"className":21710},[202],[15,21712,21714],{"className":21713},[206],[15,21715,21717],{"className":21716,"style":1650},[210],[15,21718,21719,21722],{"style":397},[15,21720],{"className":21721,"style":218},[217],[15,21723,21725],{"className":21724},[222,223,224,225],[15,21726,1263],{"className":21727},[66,225],[15,21729],{"className":21730,"style":72},[71],[15,21732,1245],{"className":21733},[76],[15,21735],{"className":21736,"style":72},[71],[15,21738,21740,21743,21746,21749,21778,21781,21784],{"className":21739},[57],[15,21741],{"className":21742,"style":394},[61],[15,21744,1248],{"className":21745},[66,67],[15,21747],{"className":21748,"style":697},[71],[15,21750,21752,21755],{"className":21751},[66],[15,21753,8385],{"className":21754,"style":8555},[66,67],[15,21756,21758],{"className":21757},[198],[15,21759,21761],{"className":21760},[202],[15,21762,21764],{"className":21763},[206],[15,21765,21767],{"className":21766,"style":394},[210],[15,21768,21769,21772],{"style":397},[15,21770],{"className":21771,"style":218},[217],[15,21773,21775],{"className":21774},[222,223,224,225],[15,21776,139],{"className":21777},[66,225],[15,21779,2853],{"className":21780,"style":2901},[66,67],[15,21782,8385],{"className":21783,"style":8555},[66,67],[15,21785,3571],{"className":21786},[66],[11,21788,21789,21790,21818],{},"对 ",[15,21791,21793,21806],{"className":21792,"translate":19},[18],[15,21794,21796],{"className":21795},[23],[25,21797,21798],{"xmlns":27},[29,21799,21800,21804],{},[32,21801,21802],{},[35,21803,8385],{},[45,21805,16072],{"encoding":47},[15,21807,21809],{"className":21808,"ariaHidden":53},[52],[15,21810,21812,21815],{"className":21811},[57],[15,21813],{"className":21814,"style":86},[61],[15,21816,8385],{"className":21817,"style":8555},[66,67]," 求梯度：",[15,21820,21822],{"className":21821,"translate":19},[294],[15,21823,21825,21895],{"className":21824,"translate":19},[18],[15,21826,21828],{"className":21827},[23],[25,21829,21830],{"xmlns":27,"display":303},[29,21831,21832,21892],{},[32,21833,21834,21836,21838,21840,21842,21844,21846,21848,21850,21852,21854,21856,21858,21860,21862,21864,21866,21868,21870,21872,21874,21876,21878,21880,21882,21884,21886,21888,21890],{},[39,21835,6600],{},[1209,21837,1263],{},[35,21839,2853],{},[39,21841,120],{"stretchy":119},[35,21843,1224],{},[39,21845,6600],{},[35,21847,2853],{},[35,21849,8385],{},[39,21851,126],{"stretchy":119},[39,21853,1245],{},[1209,21855,1263],{},[35,21857,1248],{},[35,21859,2853],{},[35,21861,8385],{},[39,21863,129],{},[1209,21865,1263],{},[35,21867,2853],{},[39,21869,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,21871,120],{"stretchy":119},[35,21873,2853],{},[39,21875,1245],{},[35,21877,1248],{},[35,21879,16422],{},[39,21881,126],{"stretchy":119},[35,21883,8385],{},[39,21885,6600],{},[35,21887,1224],{},[39,21889,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,21891,3571],{"mathvariant":138},[45,21893,21894],{"encoding":47},"-2K(y-K\\alpha)+2\\lambda K\\alpha\n=2K\\bigl((K+\\lambda I)\\alpha-y\\bigr).",[15,21896,21898,21928,21952,21979,22012,22039],{"className":21897,"ariaHidden":53},[52],[15,21899,21901,21904,21907,21910,21913,21916,21919,21922,21925],{"className":21900},[57],[15,21902],{"className":21903,"style":154},[61],[15,21905,6600],{"className":21906},[66],[15,21908,1263],{"className":21909},[66],[15,21911,2853],{"className":21912,"style":2901},[66,67],[15,21914,120],{"className":21915},[162],[15,21917,1224],{"className":21918,"style":1515},[66,67],[15,21920],{"className":21921,"style":72},[71],[15,21923,6600],{"className":21924},[76],[15,21926],{"className":21927,"style":72},[71],[15,21929,21931,21934,21937,21940,21943,21946,21949],{"className":21930},[57],[15,21932],{"className":21933,"style":154},[61],[15,21935,2853],{"className":21936,"style":2901},[66,67],[15,21938,8385],{"className":21939,"style":8555},[66,67],[15,21941,126],{"className":21942},[169],[15,21944],{"className":21945,"style":72},[71],[15,21947,1245],{"className":21948},[76],[15,21950],{"className":21951,"style":72},[71],[15,21953,21955,21958,21961,21964,21967,21970,21973,21976],{"className":21954},[57],[15,21956],{"className":21957,"style":840},[61],[15,21959,1263],{"className":21960},[66],[15,21962,1248],{"className":21963},[66,67],[15,21965,2853],{"className":21966,"style":2901},[66,67],[15,21968,8385],{"className":21969,"style":8555},[66,67],[15,21971],{"className":21972,"style":173},[71],[15,21974,129],{"className":21975},[177],[15,21977],{"className":21978,"style":173},[71],[15,21980,21982,21985,21988,21991,21997,22000,22003,22006,22009],{"className":21981},[57],[15,21983],{"className":21984,"style":4398},[61],[15,21986,1263],{"className":21987},[66],[15,21989,2853],{"className":21990,"style":2901},[66,67],[15,21992,21994],{"className":21993},[162],[15,21995,120],{"className":21996},[1422,1508],[15,21998,120],{"className":21999},[162],[15,22001,2853],{"className":22002,"style":2901},[66,67],[15,22004],{"className":22005,"style":72},[71],[15,22007,1245],{"className":22008},[76],[15,22010],{"className":22011,"style":72},[71],[15,22013,22015,22018,22021,22024,22027,22030,22033,22036],{"className":22014},[57],[15,22016],{"className":22017,"style":154},[61],[15,22019,1248],{"className":22020},[66,67],[15,22022,16422],{"className":22023,"style":16492},[66,67],[15,22025,126],{"className":22026},[169],[15,22028,8385],{"className":22029,"style":8555},[66,67],[15,22031],{"className":22032,"style":72},[71],[15,22034,6600],{"className":22035},[76],[15,22037],{"className":22038,"style":72},[71],[15,22040,22042,22045,22048,22054],{"className":22041},[57],[15,22043],{"className":22044,"style":4398},[61],[15,22046,1224],{"className":22047,"style":1515},[66,67],[15,22049,22051],{"className":22050},[169],[15,22052,126],{"className":22053},[1422,1508],[15,22055,3571],{"className":22056},[66],[11,22058,22059,22060,22163,22164,22214,22215,22271],{},"令梯度为零，得到正规方程 ",[15,22061,22063,22097],{"className":22062,"translate":19},[18],[15,22064,22066],{"className":22065},[23],[25,22067,22068],{"xmlns":27},[29,22069,22070,22094],{},[32,22071,22072,22074,22076,22078,22080,22082,22084,22086,22088,22090,22092],{},[35,22073,2853],{},[39,22075,120],{"stretchy":119},[35,22077,2853],{},[39,22079,1245],{},[35,22081,1248],{},[35,22083,16422],{},[39,22085,126],{"stretchy":119},[35,22087,8385],{},[39,22089,129],{},[35,22091,2853],{},[35,22093,1224],{},[45,22095,22096],{"encoding":47},"K(K+\\lambda I)\\alpha=Ky",[15,22098,22100,22124,22151],{"className":22099,"ariaHidden":53},[52],[15,22101,22103,22106,22109,22112,22115,22118,22121],{"className":22102},[57],[15,22104],{"className":22105,"style":154},[61],[15,22107,2853],{"className":22108,"style":2901},[66,67],[15,22110,120],{"className":22111},[162],[15,22113,2853],{"className":22114,"style":2901},[66,67],[15,22116],{"className":22117,"style":72},[71],[15,22119,1245],{"className":22120},[76],[15,22122],{"className":22123,"style":72},[71],[15,22125,22127,22130,22133,22136,22139,22142,22145,22148],{"className":22126},[57],[15,22128],{"className":22129,"style":154},[61],[15,22131,1248],{"className":22132},[66,67],[15,22134,16422],{"className":22135,"style":16492},[66,67],[15,22137,126],{"className":22138},[169],[15,22140,8385],{"className":22141,"style":8555},[66,67],[15,22143],{"className":22144,"style":173},[71],[15,22146,129],{"className":22147},[177],[15,22149],{"className":22150,"style":173},[71],[15,22152,22154,22157,22160],{"className":22153},[57],[15,22155],{"className":22156,"style":17374},[61],[15,22158,2853],{"className":22159,"style":2901},[66,67],[15,22161,1224],{"className":22162,"style":1515},[66,67],"。由于 ",[15,22165,22167,22184],{"className":22166,"translate":19},[18],[15,22168,22170],{"className":22169},[23],[25,22171,22172],{"xmlns":27},[29,22173,22174,22182],{},[32,22175,22176,22178,22180],{},[35,22177,1248],{},[39,22179,9278],{},[1209,22181,1804],{},[45,22183,9283],{"encoding":47},[15,22185,22187,22205],{"className":22186,"ariaHidden":53},[52],[15,22188,22190,22193,22196,22199,22202],{"className":22189},[57],[15,22191],{"className":22192,"style":9293},[61],[15,22194,1248],{"className":22195},[66,67],[15,22197],{"className":22198,"style":173},[71],[15,22200,9278],{"className":22201},[177],[15,22203],{"className":22204,"style":173},[71],[15,22206,22208,22211],{"className":22207},[57],[15,22209],{"className":22210,"style":1836},[61],[15,22212,1804],{"className":22213},[66]," 时 ",[15,22216,22218,22238],{"className":22217,"translate":19},[18],[15,22219,22221],{"className":22220},[23],[25,22222,22223],{"xmlns":27},[29,22224,22225,22235],{},[32,22226,22227,22229,22231,22233],{},[35,22228,2853],{},[39,22230,1245],{},[35,22232,1248],{},[35,22234,16422],{},[45,22236,22237],{"encoding":47},"K+\\lambda I",[15,22239,22241,22259],{"className":22240,"ariaHidden":53},[52],[15,22242,22244,22247,22250,22253,22256],{"className":22243},[57],[15,22245],{"className":22246,"style":1997},[61],[15,22248,2853],{"className":22249,"style":2901},[66,67],[15,22251],{"className":22252,"style":72},[71],[15,22254,1245],{"className":22255},[76],[15,22257],{"className":22258,"style":72},[71],[15,22260,22262,22265,22268],{"className":22261},[57],[15,22263],{"className":22264,"style":840},[61],[15,22266,1248],{"className":22267},[66,67],[15,22269,16422],{"className":22270,"style":16492},[66,67]," 可逆，",[15,22273,22275],{"className":22274,"translate":19},[294],[15,22276,22278,22315],{"className":22277,"translate":19},[18],[15,22279,22281],{"className":22280},[23],[25,22282,22283],{"xmlns":27,"display":303},[29,22284,22285,22313],{},[32,22286,22287,22289,22291,22293,22295,22297,22299,22301,22311],{},[35,22288,8385],{},[39,22290,129],{},[39,22292,120],{"stretchy":119},[35,22294,2853],{},[39,22296,1245],{},[35,22298,1248],{},[35,22300,16422],{},[131,22302,22303,22305],{},[39,22304,126],{"stretchy":119},[32,22306,22307,22309],{},[39,22308,6600],{},[1209,22310,1211],{},[35,22312,1224],{},[45,22314,16437],{"encoding":47},[15,22316,22318,22336,22357],{"className":22317,"ariaHidden":53},[52],[15,22319,22321,22324,22327,22330,22333],{"className":22320},[57],[15,22322],{"className":22323,"style":86},[61],[15,22325,8385],{"className":22326,"style":8555},[66,67],[15,22328],{"className":22329,"style":173},[71],[15,22331,129],{"className":22332},[177],[15,22334],{"className":22335,"style":173},[71],[15,22337,22339,22342,22345,22348,22351,22354],{"className":22338},[57],[15,22340],{"className":22341,"style":154},[61],[15,22343,120],{"className":22344},[162],[15,22346,2853],{"className":22347,"style":2901},[66,67],[15,22349],{"className":22350,"style":72},[71],[15,22352,1245],{"className":22353},[76],[15,22355],{"className":22356,"style":72},[71],[15,22358,22360,22363,22366,22369,22404],{"className":22359},[57],[15,22361],{"className":22362,"style":11209},[61],[15,22364,1248],{"className":22365},[66,67],[15,22367,16422],{"className":22368,"style":16492},[66,67],[15,22370,22372,22375],{"className":22371},[169],[15,22373,126],{"className":22374},[169],[15,22376,22378],{"className":22377},[198],[15,22379,22381],{"className":22380},[202],[15,22382,22384],{"className":22383},[206],[15,22385,22387],{"className":22386,"style":1650},[210],[15,22388,22389,22392],{"style":397},[15,22390],{"className":22391,"style":218},[217],[15,22393,22395],{"className":22394},[222,223,224,225],[15,22396,22398,22401],{"className":22397},[66,225],[15,22399,6600],{"className":22400},[66,225],[15,22402,1211],{"className":22403},[66,225],[15,22405,1224],{"className":22406,"style":1515},[66,67],[11,22408,22409,22410,22438,22439,22467],{},"是一个解（即使 ",[15,22411,22413,22426],{"className":22412,"translate":19},[18],[15,22414,22416],{"className":22415},[23],[25,22417,22418],{"xmlns":27},[29,22419,22420,22424],{},[32,22421,22422],{},[35,22423,2853],{},[45,22425,2853],{"encoding":47},[15,22427,22429],{"className":22428,"ariaHidden":53},[52],[15,22430,22432,22435],{"className":22431},[57],[15,22433],{"className":22434,"style":490},[61],[15,22436,2853],{"className":22437,"style":2901},[66,67]," 奇异，这个 ",[15,22440,22442,22455],{"className":22441,"translate":19},[18],[15,22443,22445],{"className":22444},[23],[25,22446,22447],{"xmlns":27},[29,22448,22449,22453],{},[32,22450,22451],{},[35,22452,8385],{},[45,22454,16072],{"encoding":47},[15,22456,22458],{"className":22457,"ariaHidden":53},[52],[15,22459,22461,22464],{"className":22460},[57],[15,22462],{"className":22463,"style":86},[61],[15,22465,8385],{"className":22466,"style":8555},[66,67]," 也满足正规方程；不同的解给出同一个函数）。预测为",[15,22469,22471],{"className":22470,"translate":19},[294],[15,22472,22474,22596],{"className":22473,"translate":19},[18],[15,22475,22477],{"className":22476},[23],[25,22478,22479],{"xmlns":27,"display":303},[29,22480,22481,22593],{},[32,22482,22483,22489,22491,22493,22495,22497,22499,22501,22503,22509,22511,22513,22515,22517,22519,22529,22531,22533,22535,22537,22539,22541,22543,22545,22547,22549,22551,22553,22555,22561,22563,22565,22567,22569,22571,22573,22575,22577,22583,22585,22591],{},[131,22484,22485,22487],{},[35,22486,116],{},[39,22488,1162],{},[39,22490,120],{"stretchy":119},[35,22492,123],{},[39,22494,126],{"stretchy":119},[39,22496,129],{},[35,22498,621],{},[39,22500,120],{"stretchy":119},[35,22502,123],{},[131,22504,22505,22507],{},[39,22506,126],{"stretchy":119},[35,22508,139],{"mathvariant":138},[39,22510,120],{"stretchy":119},[35,22512,2853],{},[39,22514,1245],{},[35,22516,1248],{},[35,22518,16422],{},[131,22520,22521,22523],{},[39,22522,126],{"stretchy":119},[32,22524,22525,22527],{},[39,22526,6600],{},[1209,22528,1211],{},[35,22530,1224],{},[39,22532,335],{"separator":53},[71,22534],{"linebreak":10451},[35,22536,621],{},[39,22538,120],{"stretchy":119},[35,22540,123],{},[39,22542,126],{"stretchy":119},[39,22544,129],{},[39,22546,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,22548,621],{},[39,22550,120],{"stretchy":119},[35,22552,123],{},[39,22554,335],{"separator":53},[1057,22556,22557,22559],{},[35,22558,123],{},[1209,22560,1211],{},[39,22562,126],{"stretchy":119},[39,22564,335],{"separator":53},[39,22566,2095],{},[39,22568,335],{"separator":53},[35,22570,621],{},[39,22572,120],{"stretchy":119},[35,22574,123],{},[39,22576,335],{"separator":53},[1057,22578,22579,22581],{},[35,22580,123],{},[35,22582,37],{},[39,22584,126],{"stretchy":119},[131,22586,22587,22589],{},[39,22588,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,22590,139],{"mathvariant":138},[35,22592,3571],{"mathvariant":138},[45,22594,22595],{"encoding":47},"f^*(x)=k(x)^\\top(K+\\lambda I)^{-1}y,\n\\\\\nk(x)=\\bigl(k(x,x_1),\\dots,k(x,x_n)\\bigr)^\\top.",[15,22597,22599,22652,22711,22764,22767,22794],{"className":22598,"ariaHidden":53},[52],[15,22600,22602,22605,22634,22637,22640,22643,22646,22649],{"className":22601},[57],[15,22603],{"className":22604,"style":154},[61],[15,22606,22608,22611],{"className":22607},[66],[15,22609,116],{"className":22610,"style":158},[66,67],[15,22612,22614],{"className":22613},[198],[15,22615,22617],{"className":22616},[202],[15,22618,22620],{"className":22619},[206],[15,22621,22623],{"className":22622,"style":1308},[210],[15,22624,22625,22628],{"style":397},[15,22626],{"className":22627,"style":218},[217],[15,22629,22631],{"className":22630},[222,223,224,225],[15,22632,1162],{"className":22633},[76,225],[15,22635,120],{"className":22636},[162],[15,22638,123],{"className":22639},[66,67],[15,22641,126],{"className":22642},[169],[15,22644],{"className":22645,"style":173},[71],[15,22647,129],{"className":22648},[177],[15,22650],{"className":22651,"style":173},[71],[15,22653,22655,22658,22661,22664,22667,22696,22699,22702,22705,22708],{"className":22654},[57],[15,22656],{"className":22657,"style":375},[61],[15,22659,621],{"className":22660,"style":684},[66,67],[15,22662,120],{"className":22663},[162],[15,22665,123],{"className":22666},[66,67],[15,22668,22670,22673],{"className":22669},[169],[15,22671,126],{"className":22672},[169],[15,22674,22676],{"className":22675},[198],[15,22677,22679],{"className":22678},[202],[15,22680,22682],{"className":22681},[206],[15,22683,22685],{"className":22684,"style":394},[210],[15,22686,22687,22690],{"style":397},[15,22688],{"className":22689,"style":218},[217],[15,22691,22693],{"className":22692},[222,223,224,225],[15,22694,139],{"className":22695},[66,225],[15,22697,120],{"className":22698},[162],[15,22700,2853],{"className":22701,"style":2901},[66,67],[15,22703],{"className":22704,"style":72},[71],[15,22706,1245],{"className":22707},[76],[15,22709],{"className":22710,"style":72},[71],[15,22712,22714,22717,22720,22723,22758,22761],{"className":22713},[57],[15,22715],{"className":22716,"style":11209},[61],[15,22718,1248],{"className":22719},[66,67],[15,22721,16422],{"className":22722,"style":16492},[66,67],[15,22724,22726,22729],{"className":22725},[169],[15,22727,126],{"className":22728},[169],[15,22730,22732],{"className":22731},[198],[15,22733,22735],{"className":22734},[202],[15,22736,22738],{"className":22737},[206],[15,22739,22741],{"className":22740,"style":1650},[210],[15,22742,22743,22746],{"style":397},[15,22744],{"className":22745,"style":218},[217],[15,22747,22749],{"className":22748},[222,223,224,225],[15,22750,22752,22755],{"className":22751},[66,225],[15,22753,6600],{"className":22754},[66,225],[15,22756,1211],{"className":22757},[66,225],[15,22759,1224],{"className":22760,"style":1515},[66,67],[15,22762,335],{"className":22763},[422],[15,22765],{"className":22766},[71,10451],[15,22768,22770,22773,22776,22779,22782,22785,22788,22791],{"className":22769},[57],[15,22771],{"className":22772,"style":154},[61],[15,22774,621],{"className":22775,"style":684},[66,67],[15,22777,120],{"className":22778},[162],[15,22780,123],{"className":22781},[66,67],[15,22783,126],{"className":22784},[169],[15,22786],{"className":22787,"style":173},[71],[15,22789,129],{"className":22790},[177],[15,22792],{"className":22793,"style":173},[71],[15,22795,22797,22801,22807,22810,22813,22816,22819,22822,22862,22865,22868,22871,22874,22877,22880,22883,22886,22889,22892,22895,22898,22938,22941,22974],{"className":22796},[57],[15,22798],{"className":22799,"style":22800},[61],"height:1.439em;vertical-align:-0.35em;",[15,22802,22804],{"className":22803},[162],[15,22805,120],{"className":22806},[1422,1508],[15,22808,621],{"className":22809,"style":684},[66,67],[15,22811,120],{"className":22812},[162],[15,22814,123],{"className":22815},[66,67],[15,22817,335],{"className":22818},[422],[15,22820],{"className":22821,"style":697},[71],[15,22823,22825,22828],{"className":22824},[66],[15,22826,123],{"className":22827},[66,67],[15,22829,22831],{"className":22830},[198],[15,22832,22834,22854],{"className":22833},[202,1090],[15,22835,22837,22851],{"className":22836},[206],[15,22838,22840],{"className":22839,"style":2139},[210],[15,22841,22842,22845],{"style":1584},[15,22843],{"className":22844,"style":218},[217],[15,22846,22848],{"className":22847},[222,223,224,225],[15,22849,1211],{"className":22850},[66,225],[15,22852,1114],{"className":22853},[1113],[15,22855,22857],{"className":22856},[206],[15,22858,22860],{"className":22859,"style":1121},[210],[15,22861],{},[15,22863,126],{"className":22864},[169],[15,22866,335],{"className":22867},[422],[15,22869],{"className":22870,"style":697},[71],[15,22872,2095],{"className":22873},[1413],[15,22875],{"className":22876,"style":697},[71],[15,22878,335],{"className":22879},[422],[15,22881],{"className":22882,"style":697},[71],[15,22884,621],{"className":22885,"style":684},[66,67],[15,22887,120],{"className":22888},[16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",[15,22981,22983,23008],{"className":22982,"translate":19},[18],[15,22984,22986],{"className":22985},[23],[25,22987,22988],{"xmlns":27},[29,22989,22990,23005],{},[32,22991,22992,22995,22997,23003],{},[35,22993,22994],{},"O",[39,22996,120],{"stretchy":119},[131,22998,22999,23001],{},[35,23000,37],{},[1209,23002,8673],{},[39,23004,126],{"stretchy":119},[45,23006,23007],{"encoding":47},"O(n^3)",[15,23009,23011],{"className":23010,"ariaHidden":53},[52],[15,23012,23014,23017,23021,23024,23053],{"className":23013},[57],[15,23015],{"className":23016,"style":9149},[61],[15,23018,22994],{"className":23019,"style":23020},[66,67],"margin-right:0.0278em;",[15,23022,120],{"className":23023},[162],[15,23025,23027,23030],{"className":23026},[66],[15,23028,37],{"className":23029},[66,67],[15,23031,23033],{"className":23032},[198],[15,23034,23036],{"className":23035},[202],[15,23037,23039],{"className":23038},[206],[15,23040,23042],{"className":23041,"style":6738},[210],[15,23043,23044,23047],{"style":213},[15,23045],{"className":23046,"style":218},[217],[15,23048,23050],{"className":23049},[222,223,224,225],[15,23051,8673],{"className":23052},[66,225],[15,23054,126],{"className":23055},[169]," 时间、",[15,23058,23060,23084],{"className":23059,"translate":19},[18],[15,23061,23063],{"className":23062},[23],[25,23064,23065],{"xmlns":27},[29,23066,23067,23081],{},[32,23068,23069,23071,23073,23079],{},[35,23070,22994],{},[39,23072,120],{"stretchy":119},[131,23074,23075,23077],{},[35,23076,37],{},[1209,23078,1263],{},[39,23080,126],{"stretchy":119},[45,23082,23083],{"encoding":47},"O(n^2)",[15,23085,23087],{"className":23086,"ariaHidden":53},[52],[15,23088,23090,23093,23096,23099,23128],{"className":23089},[57],[15,23091],{"className":23092,"style":9149},[61],[15,23094,22994],{"className":23095,"style":23020},[66,67],[15,23097,120],{"className":23098},[162],[15,23100,23102,23105],{"className":23101},[66],[15,23103,37],{"className":23104},[66,67],[15,23106,23108],{"className":23107},[198],[15,23109,23111],{"className":23110},[202],[15,23112,23114],{"className":23113},[206],[15,23115,23117],{"className":23116,"style":6738},[210],[15,23118,23119,23122],{"style":213},[15,23120],{"className":23121,"style":218},[217],[15,23123,23125],{"className":23124},[222,223,224,225],[15,23126,1263],{"className":23127},[66,225],[15,23129,126],{"className":23130},[169]," 内存；预测一个点需要 ",[15,23133,23135,23155],{"className":23134,"translate":19},[18],[15,23136,23138],{"className":23137},[23],[25,23139,23140],{"xmlns":27},[29,23141,23142,23152],{},[32,23143,23144,23146,23148,23150],{},[35,23145,22994],{},[39,23147,120],{"stretchy":119},[35,23149,37],{},[39,23151,126],{"stretchy":119},[45,23153,23154],{"encoding":47},"O(n)",[15,23156,23158],{"className":23157,"ariaHidden":53},[52],[15,23159,23161,23164,23167,23170,23173],{"className":23160},[57],[15,23162],{"className":23163,"style":154},[61],[15,23165,22994],{"className":23166,"style":23020},[66,67],[15,23168,120],{"className":23169},[162],[15,23171,37],{"className":23172},[66,67],[15,23174,126],{"className":23175},[169]," 次核函数求值。",[1904,23178,23180],{"id":23179},"_52-支持向量机","5.2 支持向量机",[11,23182,23183],{},"软间隔 SVM 在 RKHS 中的 primal 问题是",[15,23185,23187],{"className":23186,"translate":19},[294],[15,23188,23190,23316],{"className":23189,"translate":19},[18],[15,23191,23193],{"className":23192},[23],[25,23194,23195],{"xmlns":27,"display":303},[29,23196,23197,23313],{},[32,23198,23199,23225,23311],{},[1172,23200,23201,23207],{},[32,23202,23203,23205],{},[35,23204,1178],{},[39,23206,1170],{},[32,23208,23209,23211,23213,23215,23217,23219,23221,23223],{},[35,23210,116],{},[39,23212,1187],{},[35,23214,1061],{"mathvariant":445},[39,23216,335],{"separator":53},[1274,23218,2462],{},[35,23220,4585],{},[39,23222,1187],{},[35,23224,455],{"mathvariant":454},[32,23226,23227,23229,23235,23237,23239,23247,23249,23252,23266,23269,23271,23273,23275,23277,23279,23281,23287,23289,23291,23293,23299,23301,23303,23305,23307,23309],{},[39,23228,4305],{"fence":53},[6465,23230,23231,23233],{},[1209,23232,1211],{},[1209,23234,1263],{},[35,23236,1251],{"mathvariant":138},[35,23238,116],{},[1255,23240,23241,23243,23245],{},[35,23242,1251],{"mathvariant":138},[35,23244,1061],{"mathvariant":445},[1209,23246,1263],{},[39,23248,1245],{},[35,23250,23251],{},"C",[1196,23253,23254,23256,23264],{},[39,23255,1200],{},[32,23257,23258,23260,23262],{},[35,23259,1205],{},[39,23261,129],{},[1209,23263,1211],{},[35,23265,37],{},[35,23267,23268],{},"max",[39,23270,1170],{},[39,23272,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,23274,1804],{},[39,23276,335],{"separator":53},[1209,23278,1211],{},[39,23280,6600],{},[1057,23282,23283,23285],{},[35,23284,1224],{},[35,23286,1205],{},[39,23288,120],{"stretchy":119},[35,23290,116],{},[39,23292,120],{"stretchy":119},[1057,23294,23295,23297],{},[35,23296,123],{},[35,23298,1205],{},[39,23300,126],{"stretchy":119},[39,23302,1245],{},[35,23304,4585],{},[39,23306,126],{"stretchy":119},[39,23308,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,23310,4328],{"fence":53},[35,23312,3571],{"mathvariant":138},[45,23314,23315],{"encoding":47},"\\min_{f\\in\\mathcal H,\\,b\\in\\mathbb R}\n\\left\\{\n\\frac12\\|f\\|_{\\mathcal H}^2\n+C\\sum_{i=1}^n\\max\\bigl(0,1-y_i(f(x_i)+b)\\bigr)\n\\right\\}.",[15,23317,23319],{"className":23318,"ariaHidden":53},[52],[15,23320,23322,23325,23394,23397,23766,23769],{"className":23321},[57],[15,23323],{"className":23324,"style":1335},[61],[15,23326,23328],{"className":23327},[1339,1351],[15,23329,23331,23386],{"className":23330},[202,1090],[15,23332,23334,23383],{"className":23333},[206],[15,23335,23337,23373],{"className":23336,"style":1361},[210],[15,23338,23339,23342],{"style":1364},[15,23340],{"className":23341,"style":1368},[217],[15,23343,23345],{"className":23344},[222,223,224,225],[15,23346,23348,23351,23354,23357,23360,23364,23367,23370],{"className":23347},[66,225],[15,23349,116],{"className":23350,"style":158},[66,67,225],[15,23352,1187],{"className":23353},[177,225],[15,23355,1061],{"className":23356,"style":1083},[66,494,225],[15,23358,335],{"className":23359},[422,225],[15,23361],{"className":23362,"style":23363},[71,225],"margin-right:0.1952em;",[15,23365,4585],{"className":23366},[66,67,225],[15,23368,1187],{"className":23369},[177,225],[15,23371,455],{"className":23372},[66,517,225],[15,23374,23375,23378],{"style":1386},[15,23376],{"className":23377,"style":1368},[217],[15,23379,23380],{},[15,23381,1178],{"className":23382},[1339],[15,23384,1114],{"className":23385},[1113],[15,23387,23389],{"className":23388},[206],[15,23390,23392],{"className":23391,"style":1404},[210],[15,23393],{},[15,23395],{"className":23396,"style":697},[71],[15,23398,23400,23406,23475,23478,23481,23535,23538,23541,23544,23547,23550,23614,23617,23620,23626,23629,23632,23635,23638,23641,23644,23647,23687,23690,23693,23696,23736,23739,23742,23745,23748,23751,23754,23760],{"className":23399},[1413],[15,23401,23403],{"className":23402,"style":1418},[162,1417],[15,23404,4305],{"className":23405},[1422,1423],[15,23407,23409,23413,23472],{"className":23408},[66],[15,23410],{"className":23411},[162,23412],"nulldelimiter",[15,23414,23416],{"className":23415},[6465],[15,23417,23419,23463],{"className":23418},[202,1090],[15,23420,23422,23460],{"className":23421},[206],[15,23423,23426,23438,23448],{"className":23424,"style":23425},[210],"height:1.3214em;",[15,23427,23429,23432],{"style":23428},"top:-2.314em;",[15,23430],{"className":23431,"style":1368},[217],[15,23433,23435],{"className":23434},[66],[15,23436,1263],{"className":23437},[66],[15,23439,23441,23444],{"style":23440},"top:-3.23em;",[15,23442],{"className":23443,"style":1368},[217],[15,23445],{"className":23446,"style":4438},[23447],"frac-line",[15,23449,23451,23454],{"style":23450},"top:-3.677em;",[15,23452],{"className":23453,"style":1368},[217],[15,23455,23457],{"className":23456},[66],[15,23458,1211],{"className":23459},[66],[15,23461,1114],{"className":23462},[1113],[15,23464,23466],{"className":23465},[206],[15,23467,23470],{"className":23468,"style":23469},[210],"height:0.686em;",[15,23471],{},[15,23473],{"className":23474},[169,23412],[15,23476,1251],{"className":23477},[66],[15,23479,116],{"className":23480,"style":158},[66,67],[15,23482,23484,23487],{"className":23483},[66],[15,23485,1251],{"className":23486},[66],[15,23488,23490],{"className":23489},[198],[15,23491,23493,23527],{"className":23492},[202,1090],[15,23494,23496,23524],{"className":23495},[206],[15,23497,23499,23513],{"className":23498,"style":1650},[210],[15,23500,23501,23504],{"style":1653},[15,23502],{"className":23503,"style":218},[217],[15,23505,23507],{"className":23506},[222,223,224,225],[15,23508,23510],{"className":23509},[66,225],[15,23511,1061],{"className":23512,"style":1083},[66,494,225],[15,23514,23515,23518],{"style":397},[15,23516],{"className":23517,"style":218},[217],[15,23519,23521],{"className":23520},[222,223,224,225],[15,23522,1263],{"className":23523},[66,225],[15,23525,1114],{"className":23526},[1113],[15,23528,23530],{"className":23529},[206],[15,23531,23533],{"className":23532,"style":1686},[210],[15,23534],{},[15,23536],{"className":23537,"style":72},[71],[15,23539,1245],{"className":23540},[76],[15,23542],{"className":23543,"style":72},[71],[15,23545,23251],{"className":23546,"style":2901},[66,67],[15,23548],{"className":23549,"style":697},[71],[15,23551,23553],{"className":23552},[1339,1351],[15,23554,23556,23606],{"className":23555},[202,1090],[15,23557,23559,23603],{"className":23558},[206],[15,23560,23562,23582,23592],{"className":23561,"style":1436},[210],[15,23563,23564,23567],{"style":1439},[15,23565],{"className":23566,"style":1443},[217],[15,23568,23570],{"className":23569},[222,223,224,225],[15,23571,23573,23576,23579],{"className":23572},[66,225],[15,23574,1205],{"className":23575},[66,67,225],[15,23577,129],{"className":23578},[177,225],[15,23580,1211],{"className":23581},[66,225],[15,23583,23584,23587],{"style":1461},[15,23585],{"className":23586,"style":1443},[217],[15,23588,23589],{},[15,23590,1200],{"className":23591},[1339,1470,1471],[15,23593,23594,23597],{"style":1474},[15,23595],{"className":23596,"style":1443},[217],[15,23598,23600],{"className":23599},[222,223,224,225],[15,23601,37],{"className":23602},[66,67,225],[15,23604,1114],{"className":23605},[1113],[15,23607,23609],{"className":23608},[206],[15,23610,23612],{"className":23611,"style":1493},[210],[15,23613],{},[15,23615],{"className":23616,"style":697},[71],[15,23618,23268],{"className":23619},[1339],[15,23621,23623],{"className":23622},[162],[15,23624,120],{"className":23625},[1422,1508],[15,23627,1804],{"className":23628},[66],[15,23630,335],{"className":23631},[422],[15,23633],{"className":23634,"style":697},[71],[15,23636,1211],{"className":23637},[66],[15,23639],{"className":23640,"style":72},[71],[15,23642,6600],{"className":23643},[76],[15,23645],{"className":23646,"style":72},[71],[15,23648,23650,23653],{"className":23649},[66],[15,23651,1224],{"className":23652,"style":1515},[66,67],[15,23654,23656],{"className":23655},[198],[15,23657,23659,23679],{"className":23658},[202,1090],[15,23660,23662,23676],{"className":23661},[206],[15,23663,23665],{"className":23664,"style":1528},[210],[15,23666,23667,23670],{"style":1531},[15,23668],{"className":23669,"style":218},[217],[15,23671,23673],{"className":23672},[222,223,224,225],[15,23674,1205],{"className":23675},[66,67,225],[15,23677,1114],{"className":23678},[1113],[15,23680,23682],{"className":23681},[206],[15,23683,23685],{"className":23684,"style":1121},[210],[15,23686],{},[15,23688,120],{"className":23689},[162],[15,23691,116],{"className":23692,"style":158},[66,67],[15,23694,120],{"className":23695},[162],[15,23697,23699,23702],{"className":23698},[66],[15,23700,123],{"className":23701},[66,67],[15,23703,23705],{"className":23704},[198],[15,23706,23708,23728],{"className":23707},[202,1090],[15,23709,23711,23725],{"className":23710},[206],[15,23712,23714],{"className":23713,"style":1528},[210],[15,23715,23716,23719],{"style":1584},[15,23717],{"className":23718,"style":218},[217],[15,23720,23722],{"className":23721},[222,223,224,225],[15,23723,1205],{"className":23724},[66,67,225],[15,23726,1114],{"className":23727},[1113],[15,23729,23731],{"className":23730},[206],[15,23732,23734],{"className":23733,"style":1121},[210],[15,23735],{},[15,23737,126],{"className":23738},[169],[15,23740],{"className":23741,"style":72},[71],[15,23743,1245],{"className":23744},[76],[15,23746],{"className":23747,"style":72},[71],[15,23749,4585],{"className":23750},[66,67],[15,23752,126],{"className":23753},[169],[15,23755,23757],{"className":23756},[169],[15,23758,126],{"className":23759},[1422,1508],[15,23761,23763],{"className":23762,"style":1418},[169,1417],[15,23764,4328],{"className":23765},[1422,1423],[15,23767],{"className":23768,"style":697},[71],[15,23770,3571],{"className":23771},[66],[11,23773,23774,23775,23803,23804,23877],{},"固定 ",[15,23776,23778,23791],{"className":23777,"translate":19},[18],[15,23779,23781],{"className":23780},[23],[25,23782,23783],{"xmlns":27},[29,23784,23785,23789],{},[32,23786,23787],{},[35,23788,4585],{},[45,23790,4585],{"encoding":47},[15,23792,23794],{"className":23793,"ariaHidden":53},[52],[15,23795,23797,23800],{"className":23796},[57],[15,23798],{"className":23799,"style":840},[61],[15,23801,4585],{"className":23802},[66,67],"，正则项 ",[15,23805,23807,23835],{"className":23806,"translate":19},[18],[15,23808,23810],{"className":23809},[23],[25,23811,23812],{"xmlns":27},[29,23813,23814,23832],{},[32,23815,23816,23818,23820,23822,23824,23826,23828,23830],{},[35,23817,7737],{"mathvariant":138},[39,23819,120],{"stretchy":119},[35,23821,9207],{},[39,23823,126],{"stretchy":119},[39,23825,129],{},[35,23827,9207],{},[35,23829,6623],{"mathvariant":138},[1209,23831,1263],{},[45,23833,23834],{"encoding":47},"\\Omega(t)=t\u002F2",[15,23836,23838,23865],{"className":23837,"ariaHidden":53},[52],[15,23839,23841,23844,23847,23850,23853,23856,23859,23862],{"className":23840},[57],[15,23842],{"className":23843,"style":154},[61],[15,23845,7737],{"className":23846},[66],[15,23848,120],{"className":23849},[162],[15,23851,9207],{"className":23852},[66,67],[15,23854,126],{"className":23855},[169],[15,23857],{"className":23858,"style":173},[71],[15,23860,129],{"className":23861},[177],[15,23863],{"className":23864,"style":173},[71],[15,23866,23868,23871,23874],{"className":23867},[57],[15,23869],{"className":23870,"style":154},[61],[15,23872,9207],{"className":23873},[66,67],[15,23875,7108],{"className":23876},[66]," 严格递增，损失是 hinge loss。表示定理给出",[15,23879,23881],{"className":23880,"translate":19},[294],[15,23882,23884,23950],{"className":23883,"translate":19},[18],[15,23885,23887],{"className":23886},[23],[25,23888,23889],{"xmlns":27,"display":303},[29,23890,23891,23947],{},[32,23892,23893,23899,23901,23903,23905,23907,23921,23927,23929,23931,23933,23935,23937,23943,23945],{},[131,23894,23895,23897],{},[35,23896,116],{},[39,23898,1162],{},[39,23900,120],{"stretchy":119},[35,23902,123],{},[39,23904,126],{"stretchy":119},[39,23906,129],{},[1196,23908,23909,23911,23919],{},[39,23910,1200],{},[32,23912,23913,23915,23917],{},[35,23914,1205],{},[39,23916,129],{},[1209,23918,1211],{},[35,23920,37],{},[1057,23922,23923,23925],{},[35,23924,17082],{},[35,23926,1205],{},[1274,23928,2462],{},[35,23930,621],{},[39,23932,120],{"stretchy":119},[35,23934,123],{},[39,23936,335],{"separator":53},[1057,23938,23939,23941],{},[35,23940,123],{},[35,23942,1205],{},[39,23944,126],{"stretchy":119},[35,23946,3571],{"mathvariant":138},[45,23948,23949],{"encoding":47},"f^*(x)=\\sum_{i=1}^n\\beta_i\\,k(x,x_i).",[15,23951,23953,24006],{"className":23952,"ariaHidden":53},[52],[15,23954,23956,23959,23988,23991,23994,23997,24000,24003],{"className":23955},[57],[15,23957],{"className":23958,"style":154},[61],[15,23960,23962,23965],{"className":23961},[66],[15,23963,116],{"className":23964,"style":158},[66,67],[15,23966,23968],{"className":23967},[198],[15,23969,23971],{"className":23970},[202],[15,23972,23974],{"className":23973},[206],[15,23975,23977],{"className":23976,"style":1308},[210],[15,23978,23979,23982],{"style":397},[15,23980],{"className":23981,"style":218},[217],[15,23983,23985],{"className":23984},[222,223,224,225],[15,23986,1162],{"className":23987},[76,225],[15,23989,120],{"className":23990},[162],[15,23992,123],{"className":23993},[66,67],[15,23995,126],{"className":23996},[169],[15,23998],{"className":23999,"style":173},[71],[15,24001,129],{"className":24002},[177],[15,24004],{"className":24005,"style":173},[71],[15,24007,24009,24012,24076,24079,24119,24122,24125,24128,24131,24134,24137,24177,24180],{"className":24008},[57],[15,24010],{"className":24011,"style":8481},[61],[15,24013,24015],{"className":24014},[1339,1351],[15,24016,24018,24068],{"className":24017},[202,1090],[15,24019,24021,24065],{"className":24020},[206],[15,24022,24024,24044,24054],{"className":24023,"style":1436},[210],[15,24025,24026,24029],{"style":1439},[15,24027],{"className":24028,"style":1443},[217],[15,24030,24032],{"className":24031},[222,223,224,225],[15,24033,24035,24038,24041],{"className":24034},[66,225],[15,24036,1205],{"className":24037},[66,67,225],[15,24039,129],{"className":24040},[177,225],[15,24042,1211],{"className":24043},[66,225],[15,24045,24046,24049],{"style":1461},[15,24047],{"className":24048,"style":1443},[217],[15,24050,24051],{},[15,24052,1200],{"className":24053},[1339,1470,1471],[15,24055,24056,24059],{"style":1474},[15,24057],{"className":24058,"style":1443},[217],[15,24060,24062],{"className":24061},[222,223,224,225],[15,24063,37],{"className":24064},[66,67,225],[15,24066,1114],{"className":24067},[1113],[15,24069,24071],{"className":24070},[206],[15,24072,24074],{"className":24073,"style":1493},[210],[15,24075],{},[15,24077],{"className":24078,"style":697},[71],[15,24080,24082,24085],{"className":24081},[66],[15,24083,17082],{"className":24084,"style":17235},[66,67],[15,24086,24088],{"className":24087},[198],[15,24089,24091,24111],{"className":24090},[202,1090],[15,24092,24094,24108],{"className":24093},[206],[15,24095,24097],{"className":24096,"style":1528},[210],[15,24098,24099,24102],{"style":17250},[15,24100],{"className":24101,"style":218},[217],[15,24103,24105],{"className":24104},[222,223,224,225],[15,24106,1205],{"className":24107},[66,67,225],[15,24109,1114],{"className":24110},[1113],[15,24112,24114],{"className":24113},[206],[15,24115,24117],{"className":24116,"style":1121},[210],[15,24118],{},[15,24120],{"className":24121,"style":697},[71],[15,24123,621],{"className":24124,"style":684},[66,67],[15,24126,120],{"className":24127},[162],[15,24129,123],{"className":24130},[66,67],[15,24132,335],{"className":24133},[422],[15,24135],{"className":24136,"style":697},[71],[15,24138,24140,24143],{"className":24139},[66],[15,24141,123],{"className":24142},[66,67],[15,24144,24146],{"className":24145},[198],[15,24147,24149,24169],{"className":24148},[202,1090],[15,24150,24152,24166],{"className":24151},[206],[15,24153,24155],{"className":24154,"style":1528},[210],[15,24156,24157,24160],{"style":1584},[15,24158],{"className":24159,"style":218},[217],[15,24161,24163],{"className":24162},[222,223,224,225],[15,24164,1205],{"className":24165},[66,67,225],[15,24167,1114],{"className":24168},[1113],[15,24170,24172],{"className":24171},[206],[15,24173,24175],{"className":24174,"style":1121},[210],[15,24176],{},[15,24178,126],{"className":24179},[169],[15,24181,3571],{"className":24182},[66],[11,24184,24185],{},"代入后得到对偶问题",[15,24187,24189],{"className":24188,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y_i",[15,25298,25300,25355],{"className":25299,"ariaHidden":53},[52],[15,25301,25303,25306,25346,25349,25352],{"className":25302},[57],[15,25304],{"className":25305,"style":3406},[61],[15,25307,25309,25312],{"className":25308},[66],[15,25310,17082],{"className":25311,"style":17235},[66,67],[15,25313,25315],{"className":25314},[198],[15,25316,25318,25338],{"className":25317},[202,1090],[15,25319,25321,25335],{"className":25320},[206],[15,25322,25324],{"className":25323,"style":1528},[210],[15,25325,25326,25329],{"style":17250},[15,25327],{"className":25328,"style":218},[217],[15,25330,25332],{"className":25331},[222,223,224,225],[15,25333,1205],{"className":25334},[66,67,225],[15,25336,1114],{"className":25337},[1113],[15,25339,25341],{"className":25340},[206],[15,25342,25344],{"className":25343,"style":1121},[210],[15,25345],{},[15,25347],{"className":25348,"style":173},[71],[15,25350,129],{"className":25351},[177],[15,25353],{"className":25354,"style":173},[71],[15,25356,25358,25361,25401],{"className":25357},[57],[15,25359],{"className":25360,"style":471},[61],[15,25362,25364,25367],{"className":25363},[66],[15,25365,8385],{"className":25366,"style":8555},[66,67],[15,25368,25370],{"className":25369},[198],[15,25371,25373,25393],{"className":25372},[202,1090],[15,25374,25376,25390],{"className":25375},[206],[15,25377,25379],{"className":25378,"style":1528},[210],[15,25380,25381,25384],{"style":8570},[15,25382],{"className":25383,"style":218},[217],[15,25385,25387],{"className":25386},[222,223,224,225],[15,25388,1205],{"className":25389},[66,67,225],[15,25391,1114],{"className":25392},[1113],[15,25394,25396],{"className":25395},[206],[15,25397,25399],{"className":25398,"style":1121},[210],[15,25400],{},[15,25402,25404,25407],{"className":25403},[66],[15,25405,1224],{"className":25406,"style":1515},[66,67],[15,25408,25410],{"className":25409},[198],[15,25411,25413,25433],{"className":25412},[202,1090],[15,25414,25416,25430],{"className":25415},[206],[15,25417,25419],{"className":25418,"style":1528},[210],[15,25420,25421,25424],{"style":1531},[15,25422],{"className":25423,"style":218},[217],[15,25425,25427],{"className":25426},[222,223,224,225],[15,25428,1205],{"className":25429},[66,67,225],[15,25431,1114],{"className":25432},[1113],[15,25434,25436],{"className":25435},[206],[15,25437,25439],{"className":25438,"style":1121},[210],[15,25440],{},[11,25442,25443],{},"KKT 条件给出：",[16131,25445,25446,25725,26005],{},[3192,25447,12964,25448,13127,25633,3369],{},[15,25449,25451,25493],{"className":25450,"translate":19},[18],[15,25452,25454],{"className":25453},[23],[25,25455,25456],{"xmlns":27},[29,25457,25458,25490],{},[32,25459,25460,25466,25468,25470,25472,25478,25480,25482,25484,25486,25488],{},[1057,25461,25462,25464],{},[35,25463,1224],{},[35,25465,1205],{},[39,25467,120],{"stretchy":119},[35,25469,116],{},[39,25471,120],{"stretchy":119},[1057,25473,25474,25476],{},[35,25475,123],{},[35,25477,1205],{},[39,25479,126],{"stretchy":119},[39,25481,1245],{},[35,25483,4585],{},[39,25485,126],{"stretchy":119},[39,25487,9278],{},[1209,25489,1211],{},[45,25491,25492],{"encoding":47},"y_i(f(x_i)+b)>1",[15,25494,25496,25603,25624],{"className":25495,"ariaHidden":53},[52],[15,25497,25499,25502,25542,25545,25548,25551,25591,25594,25597,25600],{"className":25498},[57],[15,25500],{"className":25501,"style":154},[61],[15,25503,25505,25508],{"className":25504},[66],[15,25506,1224],{"className":25507,"style":1515},[66,67],[15,25509,25511],{"className":25510},[198],[15,25512,25514,25534],{"className":25513},[202,1090],[15,25515,25517,25531],{"className":25516},[206],[15,25518,25520],{"className":25519,"style":1528},[210],[15,25521,25522,25525],{"style":1531},[15,25523],{"className":25524,"style":218},[217],[15,25526,25528],{"className":25527},[222,223,224,225],[15,25529,1205],{"className":25530},[66,67,225],[15,25532,1114],{"className":25533},[1113],[15,25535,25537],{"className":25536},[206],[15,25538,25540],{"className":25539,"style":1121},[210],[15,25541],{},[15,25543,120],{"className":25544},[162],[15,25546,116],{"className":25547,"style":158},[66,67],[15,25549,120],{"className":25550},[162],[15,25552,25554,25557],{"className":25553},[66],[15,25555,123],{"className":25556},[66,67],[15,25558,25560],{"className":25559},[198],[15,25561,25563,25583],{"className":25562},[202,1090],[15,25564,25566,25580],{"className":25565},[206],[15,25567,25569],{"className":25568,"style":1528},[210],[15,25570,25571,25574],{"style":1584},[15,25572],{"className":25573,"style":218},[217],[15,25575,25577],{"className":25576},[222,223,224,225],[15,25578,1205],{"className":25579},[66,67,225],[15,25581,1114],{"className":25582},[1113],[15,25584,25586],{"className":25585},[206],[15,25587,25589],{"className":25588,"style":1121},[210],[15,25590],{},[15,25592,126],{"className":25593},[169],[15,25595],{"className":25596,"style":72},[71],[15,25598,1245],{"className":25599},[76],[15,25601],{"className":25602,"style":72},[71],[15,25604,25606,25609,25612,25615,25618,25621],{"className":25605},[57],[15,25607],{"className":25608,"style":154},[61],[15,25610,4585],{"className":25611},[66,67],[15,25613,126],{"className":25614},[169],[15,25616],{"className":25617,"style":173},[71],[15,25619,9278],{"className":25620},[177],[15,25622],{"className":25623,"style":173},[71],[15,25625,25627,25630],{"className":25626},[57],[15,25628],{"className":25629,"style":1836},[61],[15,25631,1211],{"className":25632},[66],[15,25634,25636,25658],{"className":25635,"translate":19},[18],[15,25637,25639],{"className":25638},[23],[25,25640,25641],{"xmlns":27},[29,25642,25643,25655],{},[32,25644,25645,25651,25653],{},[1057,25646,25647,25649],{},[35,25648,8385],{},[35,25650,1205],{},[39,25652,129],{},[1209,25654,1804],{},[45,25656,25657],{"encoding":47},"\\alpha_i=0",[15,25659,25661,25716],{"className":25660,"ariaHidden":53},[52],[15,25662,25664,25667,25707,25710,25713],{"className":25663},[57],[15,25665],{"className":25666,"style":10192},[61],[15,25668,25670,25673],{"className":25669},[66],[15,25671,8385],{"className":25672,"style":8555},[66,67],[15,25674,25676],{"className":25675},[198],[15,25677,25679,25699],{"className":25678},[202,1090],[15,25680,25682,25696],{"className":25681},[206],[15,25683,25685],{"className":25684,"style":1528},[210],[15,25686,25687,25690],{"style":8570},[15,25688],{"className":25689,"style":218},[217],[15,25691,25693],{"className":25692},[222,223,224,225],[15,25694,1205],{"className":25695},[66,67,225],[15,25697,1114],{"className":25698},[1113],[15,25700,25702],{"className":25701},[206],[15,25703,25705],{"className":25704,"style":1121},[210],[15,25706],{},[15,25708],{"className":25709,"style":173},[71],[15,25711,129],{"className":25712},[177],[15,25714],{"className":25715,"style":173},[71],[15,25717,25719,25722],{"className":25718},[57],[15,25720],{"className":25721,"style":1836},[61],[15,25723,1804],{"className":25724},[66],[3192,25726,12964,25727,13127,25913,3369],{},[15,25728,25730,25773],{"className":25729,"translate":19},[18],[15,25731,25733],{"className":25732},[23],[25,25734,25735],{"xmlns":27},[29,25736,25737,25770],{},[32,25738,25739,25745,25747,25749,25751,25757,25759,25761,25763,25765,25768],{},[1057,25740,25741,25743],{},[35,25742,1224],{},[35,25744,1205],{},[39,25746,120],{"stretchy":119},[35,25748,116],{},[39,25750,120],{"stretchy":119},[1057,25752,25753,25755],{},[35,25754,123],{},[35,25756,1205],{},[39,25758,126]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C",[15,26222,26224,26243,26299],{"className":26223,"ariaHidden":53},[52],[15,26225,26227,26231,26234,26237,26240],{"className":26226},[57],[15,26228],{"className":26229,"style":26230},[61],"height:0.7804em;vertical-align:-0.136em;",[15,26232,1804],{"className":26233},[66],[15,26235],{"className":26236,"style":173},[71],[15,26238,24215],{"className":26239},[177],[15,26241],{"className":26242,"style":173},[71],[15,26244,26246,26250,26290,26293,26296],{"className":26245},[57],[15,26247],{"className":26248,"style":26249},[61],"height:0.786em;vertical-align:-0.15em;",[15,26251,26253,26256],{"className":26252},[66],[15,26254,8385],{"className":26255,"style":8555},[66,67],[15,26257,26259],{"className":26258},[198],[15,26260,26262,26282],{"className":26261},[202,1090],[15,26263,26265,26279],{"className":26264},[206],[15,26266,26268],{"className":26267,"style":1528},[210],[15,26269,26270,26273],{"style":8570},[15,26271],{"className":26272,"style":218},[217],[15,26274,26276],{"className":26275},[222,223,224,225],[15,26277,1205],{"className":26278},[66,67,225],[15,26280,1114],{"className":26281},[1113],[15,26283,26285],{"className":26284},[206],[15,26286,26288],{"className":26287,"style":1121},[210],[15,26289],{},[15,26291],{"className":26292,"style":173},[71],[15,26294,24215],{"className":26295},[177],[15,26297],{"className":26298,"style":173},[71],[15,26300,26302,26305],{"className":26301},[57],[15,26303],{"className":26304,"style":490},[61],[15,26306,23251],{"className":26307,"style":2901},[66,67],[11,26309,26310,26403,26404,26407],{},[15,26311,26313,26335],{"className":26312,"translate":19},[18],[15,26314,26316],{"className":26315},[23],[25,26317,26318],{"xmlns":27},[29,26319,26320,26332],{},[32,26321,26322,26328,26330],{},[1057,26323,26324,26326],{},[35,26325,8385],{},[35,26327,1205],{},[39,26329,9278],{},[1209,26331,1804],{},[45,26333,26334],{"encoding":47},"\\alpha_i>0",[15,26336,26338,26394],{"className":26337,"ariaHidden":53},[52],[15,26339,26341,26345,26385,26388,26391],{"className":26340},[57],[15,26342],{"className":26343,"style":26344},[61],"height:0.6891em;vertical-align:-0.15em;",[15,26346,26348,26351],{"className":26347},[66],[15,26349,8385],{"className":26350,"style":8555},[66,67],[15,26352,26354],{"className":26353},[198],[15,26355,26357,26377],{"className":26356},[202,1090],[15,26358,26360,26374],{"className":26359},[206],[15,26361,26363],{"className":26362,"style":1528},[210],[15,26364,26365,26368],{"style":8570},[15,26366],{"className":26367,"style":218},[217],[15,26369,26371],{"className":26370},[222,223,224,225],[15,26372,1205],{"className":26373},[66,67,225],[15,26375,1114],{"className":26376},[1113],[15,26378,26380],{"className":26379},[206],[15,26381,26383],{"className":26382,"style":1121},[210],[15,26384],{},[15,26386],{"className":26387,"style":173},[71],[15,26389,9278],{"className":26390},[177],[15,26392],{"className":26393,"style":173},[71],[15,26395,26397,26400],{"className":26396},[57],[15,26398],{"className":26399,"style":1836},[61],[15,26401,1804],{"className":26402},[66]," 的点就是",[2040,26405,26406],{},"支持向量","。决策函数为",[15,26409,26411],{"className":26410,"translate":19},[294],[15,26412,26414,26487],{"className":26413,"translate":19},[18],[15,26415,26417],{"className":26416},[23],[25,26418,26419],{"xmlns":27,"display":303},[29,26420,26421,26484],{},[32,26422,26423,26426,26428,26482],{},[35,26424,26425],{"mathvariant":138},"sign",[39,26427,1170],{},[32,26429,26430,26432,26446,26452,26458,26460,26462,26464,26466,26468,26474,26476,26478,26480],{},[39,26431,120],{"fence":53},[1196,26433,26434,26436,26444],{},[39,26435,1200],{},[32,26437,26438,26440,26442],{},[35,26439,1205],{},[39,26441,129],{},[1209,26443,1211],{},[35,26445,37],{},[1057,26447,26448,26450],{},[35,26449,8385],{},[35,26451,1205],{},[1057,26453,26454,26456],{},[35,26455,1224],{},[35,26457,1205],{},[1274,26459,2462],{},[35,26461,621],{},[39,26463,120],{"stretchy":119},[35,26465,123],{},[39,26467,335],{"separator":53},[1057,26469,26470,26472],{},[35,26471,123],{},[35,26473,1205],{},[39,26475,126],{"stretchy":119},[39,26477,1245],{},[35,26479,4585],{},[39,26481,126],{"fence":53},[35,26483,3571],{"mathvariant":138},[45,26485,26486],{"encoding":47},"\\operatorname{sign}\\left(\\sum_{i=1}^n\\alpha_i y_i\\,k(x,x_i)+b\\right).",[15,26488,26490],{"className":26489,"ariaHidden":53},[52],[15,26491,26493,26496,26502,26505,26740,26743],{"className":26492},[57],[15,26494],{"className":26495,"style":1335},[61],[15,26497,26499],{"className":26498},[1339],[15,26500,26425],{"className":26501},[66,4427],[15,26503],{"className":26504,"style":697},[71],[15,26506,26508,26514,26578,26581,26621,26661,26664,26667,26670,26673,26676,26679,26719,26722,26725,26728,26731,26734],{"className":26507},[1413],[15,26509,26511],{"className":26510,"style":1418},[162,1417],[15,26512,120],{"className":26513},[1422,1423],[15,26515,26517],{"className":26516},[1339,1351],[15,26518,26520,26570],{"className":26519},[202,1090],[15,26521,26523,26567],{"className":26522},[206],[15,26524,26526,26546,26556],{"className":26525,"style":1436},[210],[15,26527,26528,26531],{"style":1439},[15,26529],{"className":26530,"style":1443},[217],[15,26532,26534],{"className":26533},[222,223,224,225],[15,26535,26537,26540,26543],{"className":26536},[66,225],[15,26538,1205],{"className":26539},[66,67,225],[15,26541,129],{"className":26542},[177,225],[15,26544,1211],{"className":26545},[66,225],[15,26547,26548,26551],{"style":1461},[15,26549],{"className":26550,"style":1443},[217],[15,26552,26553],{},[15,26554,1200],{"className":26555},[1339,1470,1471],[15,26557,26558,26561],{"style":1474},[15,26559],{"className":26560,"style":1443},[217],[15,26562,26564],{"className":26563},[222,223,224,225],[15,26565,37],{"className":26566},[66,67,225],[15,26568,1114],{"className":26569},[1113],[15,26571,26573],{"className":26572},[206],[15,26574,26576],{"className":26575,"style":1493},[210],[15,26577],{},[15,26579],{"className":26580,"style":697},[71],[15,26582,26584,26587],{"className":26583},[66],[15,26585,8385],{"className":26586,"style":8555},[66,67],[15,26588,26590],{"className":26589},[198],[15,26591,26593,26613],{"className":26592},[202,1090],[15,26594,26596,26610],{"className":26595},[206],[15,26597,26599],{"className":26598,"style":1528},[210],[15,26600,26601,26604],{"style":8570},[15,26602],{"className":26603,"style":218},[217],[15,26605,26607],{"className":26606},[222,223,224,225],[15,26608,1205],{"className":26609},[66,67,225],[15,26611,1114],{"className":26612},[1113],[15,26614,26616],{"className":26615},[206],[15,26617,26619],{"className":26618,"style":1121},[210],[15,26620],{},[15,26622,26624,26627],{"className":26623},[66],[15,26625,1224],{"className":26626,"style":1515},[66,67],[15,26628,26630],{"className":26629},[198],[15,26631,26633,26653],{"className":26632},[202,1090],[15,26634,26636,26650],{"className":26635},[206],[15,26637,26639],{"className":26638,"style":1528},[210],[15,26640,26641,26644],{"style":1531},[15,26642],{"className":26643,"style":218},[217],[15,26645,26647],{"className":26646},[222,223,224,225],[15,26648,1205],{"className":26649},[66,67,225],[15,26651,1114],{"className":26652},[1113],[15,26654,26656],{"className":26655},[206],[15,26657,26659],{"className":26658,"style":1121},[210],[15,26660],{},[15,26662],{"className":26663,"style":697},[71],[15,26665,621],{"className":26666,"style":684},[66,67],[15,26668,120],{"className":26669},[162],[15,26671,123],{"className":26672},[66,67],[15,26674,335],{"className":26675},[422],[15,26677],{"className":26678,"style":697},[71],[15,26680,26682,26685],{"className":26681},[66],[15,26683,123],{"className":26684},[66,67],[15,26686,26688],{"className":26687},[198],[15,26689,26691,26711],{"className":26690},[202,1090],[15,26692,26694,26708],{"className":26693},[206],[15,26695,26697],{"className":26696,"style":1528},[210],[15,26698,26699,26702],{"style":1584},[15,26700],{"className":26701,"style":218},[217],[15,26703,26705],{"className":26704},[222,223,224,225],[15,26706,1205],{"className":26707},[66,67,225],[15,26709,1114],{"className":26710},[1113],[15,26712,26714],{"className":26713},[206],[15,26715,26717],{"className":26716,"style":1121},[210],[15,26718],{},[15,26720,126],{"className":26721},[169],[15,26723],{"className":26724,"style":72},[71],[15,26726,1245],{"className":26727},[76],[15,26729],{"className":26730,"style":72},[71],[15,26732,4585],{"className":26733},[66,67],[15,26735,26737],{"className":26736,"style":1418},[169,1417],[15,26738,126],{"className":26739},[1422,1423],[15,26741],{"className":26742,"style":697},[71],[15,26744,3571],{"className":26745},[66],[11,26747,26748,26749,26752,26753,26811,26812,26840,26841,26910,26911,26980,26981,27009],{},"这里要特别澄清一个常见误解：",[2040,26750,26751],{},"SVM 的稀疏性不是表示定理给的。"," 表示定理只说 ",[15,26754,26756,26773],{"className":26755,"translate":19},[18],[15,26757,26759],{"className":26758},[23],[25,26760,26761],{"xmlns":27},[29,26762,26763,26771],{},[32,26764,26765],{},[131,26766,26767,26769],{},[35,26768,116],{},[39,26770,1162],{},[45,26772,16216],{"encoding":47},[15,26774,26776],{"className":26775,"ariaHidden":53},[52],[15,26777,26779,26782],{"className":26778},[57],[15,26780],{"className":26781,"style":3406},[61],[15,26783,26785,26788],{"className":26784},[66],[15,26786,116],{"className":26787,"style":158},[66,67],[15,26789,26791],{"className":26790},[198],[15,26792,26794],{"className":26793},[202],[15,26795,26797],{"className":26796},[206],[15,26798,26800],{"className":26799,"style":16244},[210],[15,26801,26802,26805],{"style":213},[15,26803],{"className":26804,"style":218},[217],[15,26806,26808],{"className":26807},[222,223,224,225],[15,26809,1162],{"className":26810},[76,225]," 是 ",[15,26813,26815,26828],{"className":26814,"translate":19},[18],[15,26816,26818],{"className":26817},[23],[25,26819,26820],{"xmlns":27},[29,26821,26822,26826],{},[32,26823,26824],{},[35,26825,37],{},[45,26827,37],{"encoding":47},[15,26829,26831],{"className":26830,"ariaHidden":53},[52],[15,26832,26834,26837],{"className":26833},[57],[15,26835],{"className":26836,"style":86},[61],[15,26838,37],{"className":26839},[66,67]," 个核函数的线性组合，",[15,26842,26844,26861],{"className":26843,"translate":19},[18],[15,26845,26847],{"className":26846},[23],[25,26848,26849],{"xmlns":27},[29,26850,26851,26859],{},[32,26852,26853],{},[1057,26854,26855,26857],{},[35,26856,8385],{},[35,26858,1205],{},[45,26860,25211],{"encoding":47},[15,26862,26864],{"className":26863,"ariaHidden":53},[52],[15,26865,26867,26870],{"className":26866},[57],[15,26868],{"className":26869,"style":10192},[61],[15,26871,26873,26876],{"className":26872},[66],[15,26874,8385],{"className":26875,"style":8555},[66,67],[15,26877,26879],{"className":26878},[198],[15,26880,26882,26902],{"className":26881},[202,1090],[15,26883,26885,26899],{"className":26884},[206],[15,26886,26888],{"className":26887,"style":1528},[210],[15,26889,26890,26893],{"style":8570},[15,26891],{"className":26892,"style":218},[217],[15,26894,26896],{"className":26895},[222,223,224,225],[15,26897,1205],{"className":26898},[66,67,225],[15,26900,1114],{"className":26901},[1113],[15,26903,26905],{"className":26904},[206],[15,26906,26908],{"className":26907,"style":1121},[210],[15,26909],{}," 完全可以全部非零。真正让大部分 ",[15,26912,26914,26931],{"className":26913,"translate":19},[18],[15,26915,26917],{"className":26916},[23],[25,26918,26919],{"xmlns":27},[29,26920,26921,26929],{},[32,26922,26923],{},[1057,26924,26925,26927],{},[35,26926,8385],{},[35,26928,1205],{},[45,26930,25211],{"encoding":47},[15,26932,26934],{"className":26933,"ariaHidden":53},[52],[15,26935,26937,26940],{"className":26936},[57],[15,26938],{"className":26939,"style":10192},[61],[15,26941,26943,26946],{"className":26942},[66],[15,26944,8385],{"className":26945,"style":8555},[66,67],[15,26947,26949],{"className":26948},[198],[15,26950,26952,26972],{"className":26951},[202,1090],[15,26953,26955,26969],{"className":26954},[206],[15,26956,26958],{"className":26957,"style":1528},[210],[15,26959,26960,26963],{"style":8570},[15,26961],{"className":26962,"style":218},[217],[15,26964,26966],{"className":26965},[222,223,224,225],[15,26967,1205],{"className":26968},[66,67,225],[15,26970,1114],{"className":26971},[1113],[15,26973,26975],{"className":26974},[206],[15,26976,26978],{"className":26977,"style":1121},[210],[15,26979],{}," 变成零的，是 hinge loss 的 KKT 条件：间隔大于 ",[15,26982,26984,26997],{"className":26983,"translate":19},[18],[15,26985,26987],{"className":26986},[23],[25,26988,26989],{"xmlns":27},[29,26990,26991,26995],{},[32,26992,26993],{},[1209,26994,1211],{},[45,26996,1211],{"encoding":47},[15,26998,27000],{"className":26999,"ariaHidden":53},[52],[15,27001,27003,27006],{"className":27002},[57],[15,27004],{"className":27005,"style":1836},[61],[15,27007,1211],{"className":27008},[66]," 的点对损失没有贡献，对应的乘子为零。",[1904,27011,27013],{"id":27012},"_53-核逻辑回归","5.3 核逻辑回归",[11,27015,27016],{},"取逻辑损失：",[15,27018,27020],{"className":27019,"translate":19},[294],[15,27021,27023,27134],{"className":27022,"translate":19},[18],[15,27024,27026],{"className":27025},[23],[25,27027,27028],{"xmlns":27,"display":303},[29,27029,27030,27131],{},[32,27031,27032,27048,27129],{},[1172,27033,27034,27040],{},[32,27035,27036,27038],{},[35,27037,1178],{},[39,27039,1170],{},[32,27041,27042,27044,27046],{},[35,27043,116],{},[39,27045,1187],{},[35,27047,1061],{"mathvariant":445},[32,27049,27050,27052,27066,27069,27071,27073,27075,27077,27079,27081,27083,27085,27091,27093,27095,27101,27103,27105,27107,27109,27115,27117,27119,27127],{},[39,27051,4305],{"fence":53},[1196,27053,27054,27056,27064],{},[39,27055,1200],{},[32,27057,27058,27060,27062],{},[35,27059,1205],{},[39,27061,129],{},[1209,27063,1211],{},[35,27065,37],{},[35,27067,27068],{},"log",[39,27070,1170],{},[39,27072,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,27074,1211],{},[39,27076,1245],{},[35,27078,6590],{},[39,27080,1170],{},[39,27082,120],{"stretchy":119},[39,27084,6600],{},[1057,27086,27087,27089],{},[35,27088,1224],{},[35,27090,1205],{},[35,27092,116],{},[39,27094,120],{"stretchy":119},[1057,27096,27097,27099],{},[35,27098,123],{},[35,27100,1205],{},[39,27102,126],{"stretchy":119},[39,27104,126],{"stretchy":119},[39,27106,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,27108,1245],{},[6465,27110,27111,27113],{},[35,27112,1248],{},[1209,27114,1263],{},[35,27116,1251],{"mathvariant":138},[35,27118,116],{},[1255,27120,27121,27123,27125],{},[35,27122,1251],{"mathvariant":138},[35,27124,1061],{"mathvariant":445},[1209,27126,1263],{},[39,27128,4328],{"fence":53},[35,27130,3571],{"mathvariant":138},[45,27132,27133],{"encoding":47},"\\min_{f\\in\\mathcal H}\\left\\{\n\\sum_{i=1}^n\\log\\bigl(1+\\exp(-y_i f(x_i))\\bigr)\n+\\frac{\\lambda}{2}\\|f\\|_{\\mathcal H}^2\n\\right\\}.",[15,27135,27137],{"className":27136,"ariaHidden":53},[52],[15,27138,27140,27143,27196,27199,27541,27544],{"className":27139},[57],[15,27141],{"className":27142,"style":1335},[61],[15,27144,27146],{"className":27145},[1339,1351],[15,27147,27149,27188],{"className":27148},[202,1090],[15,27150,27152,27185],{"className":27151},[206],[15,27153,27155,27175],{"className":27154,"style":1361},[210],[15,27156,27157,27160],{"style":1364},[15,27158],{"className":27159,"style":1368},[217],[15,27161,27163],{"className":27162},[222,223,224,225],[15,27164,27166,27169,27172],{"className":27165},[66,225],[15,27167,116],{"className":27168,"style":158},[66,67,225],[15,27170,1187],{"className":27171},[177,225],[15,27173,1061],{"className":27174,"style":1083},[66,494,225],[15,27176,27177,27180],{"style":1386},[15,27178],{"className":27179,"style":1368},[217],[15,27181,27182],{},[15,27183,1178],{"className":27184},[1339],[15,27186,1114],{"className":27187},[1113],[15,27189,27191],{"className":27190},[206],[15,27192,27194],{"className":27193,"style":1404},[210],[15,27195],{},[15,27197],{"className":27198,"style":697},[71],[15,27200,27202,27208,27272,27275,27281,27287,27290,27293,27296,27299,27302,27305,27308,27348,27351,27354,27394,27397,27403,27406,27409,27412,27475,27478,27481,27535],{"className":27201},[1413],[15,27203,27205],{"className":27204,"style":1418},[162,1417],[15,27206,4305],{"className":27207},[1422,1423],[15,27209,27211],{"className":27210},[1339,1351],[15,27212,27214,27264],{"className":27213},[202,1090],[15,27215,27217,27261],{"className":27216},[206],[15,27218,27220,27240,27250],{"className":27219,"style":1436},[210],[15,27221,27222,27225],{"style":1439},[15,27223],{"className":27224,"style":1443},[217],[15,27226,27228],{"className":27227},[222,223,224,225],[15,27229,27231,27234,27237],{"className":27230},[66,225],[15,27232,1205],{"className":27233},[66,67,225],[15,27235,129],{"className":27236},[177,225],[15,27238,1211],{"className":27239},[66,225],[15,27241,27242,27245],{"style":1461},[15,27243],{"className":27244,"style":1443},[217],[15,27246,27247],{},[15,27248,1200],{"className":27249},[1339,1470,1471],[15,27251,27252,27255],{"style":1474},[15,27253],{"className":27254,"style":1443},[217],[15,27256,27258],{"className":27257},[222,223,224,225],[15,27259,37],{"className":27260},[66,67,225],[15,27262,1114],{"className":27263},[1113],[15,27265,27267],{"className":27266},[206],[15,27268,27270],{"className":27269,"style":1493},[210],[15,27271],{},[15,27273],{"className":27274,"style":697},[71],[15,27276,27278,27279],{"className":27277},[1339],"lo",[15,27280,1344],{"style":1343},[15,27282,27284],{"className":27283},[162],[15,27285,120],{"className":27286},[1422,1508],[15,27288,1211],{"className":27289},[66],[15,27291],{"className":27292,"style":72},[71],[15,27294,1245],{"className":27295},[76],[15,27297],{"className":27298,"style":72},[71],[15,27300,6590],{"className":27301},[1339],[15,27303,120],{"className":27304},[162],[15,27306,6600],{"className":27307},[66],[15,27309,27311,27314],{"className":27310},[66],[15,27312,1224],{"className":27313,"style":1515},[66,67],[15,27315,27317],{"className":27316},[198],[15,27318,27320,27340],{"className":27319},[202,1090],[15,27321,27323,27337],{"className":27322},[206],[15,27324,27326],{"className":27325,"style":1528},[210],[15,27327,27328,27331],{"style":1531},[15,27329],{"className":27330,"style":218},[217],[15,27332,27334],{"className":27333},[222,223,224,225],[15,27335,1205],{"className":27336},[66,67,225],[15,27338,1114],{"className":27339},[1113],[15,27341,27343],{"className":27342},[206],[15,27344,27346],{"className":27345,"style":1121},[210],[15,27347],{},[15,27349,116],{"className":27350,"style":158},[66,67],[15,27352,120],{"className":27353},[162],[15,27355,27357,27360],{"className":27356},[66],[15,27358,123],{"className":27359},[66,67],[15,27361,27363],{"className":27362},[198],[15,27364,27366,27386],{"className":27365},[202,1090],[15,27367,27369,27383],{"className":27368},[206],[15,27370,27372],{"className":27371,"style":1528},[210],[15,27373,27374,27377],{"style":1584},[15,27375],{"className":27376,"style":218},[217],[15,27378,27380],{"className":27379},[222,223,224,225],[15,27381,1205],{"className":27382},[66,67,225],[15,27384,1114],{"className":27385},[1113],[15,27387,27389],{"className":27388},[206],[15,27390,27392],{"className":27391,"style":1121},[210],[15,27393],{},[15,27395,18766],{"className":27396},[169],[15,27398,27400],{"className":27399},[169],[15,27401,126],{"className":27402},[1422,1508],[15,27404],{"className":27405,"style":72},[71],[15,27407,1245],{"className":27408},[76],[15,27410],{"className":27411,"style":72},[71],[15,27413,27415,27418,27472],{"className":27414},[66],[15,27416],{"className":27417},[162,23412],[15,27419,27421],{"className":27420},[6465],[15,27422,27424,27464],{"className":27423},[202,1090],[15,27425,27427,27461],{"className":27426},[206],[15,27428,27431,27442,27450],{"className":27429,"style":27430},[210],"height:1.3714em;",[15,27432,27433,27436],{"style":23428},[15,27434],{"className":27435,"style":1368},[217],[15,27437,27439],{"className":27438},[66],[15,27440,1263],{"className":27441},[66],[15,27443,27444,27447],{"style":23440},[15,27445],{"className":27446,"style":1368},[217],[15,27448],{"className":27449,"style":4438},[23447],[15,27451,27452,27455],{"style":23450},[15,27453],{"className":27454,"style":1368},[217],[15,27456,27458],{"className":27457},[66],[15,27459,1248],{"className":27460},[66,67],[15,27462,1114],{"className":27463},[1113],[15,27465,27467],{"className":27466},[206],[15,27468,27470],{"className":27469,"style":23469},[210],[15,27471],{},[15,27473],{"className":27474},[169,23412],[15,27476,1251],{"className":27477},[66],[15,27479,116],{"className":27480,"style":158},[66,67],[15,27482,27484,27487],{"className":27483},[66],[15,27485,1251],{"className":27486},[66],[15,27488,27490],{"className":27489},[198],[15,27491,27493,27527],{"className":27492},[202,1090],[15,27494,27496,27524],{"className":27495},[206],[15,27497,27499,27513],{"className":27498,"style":1650},[210],[15,27500,27501,27504],{"style":1653},[15,27502],{"className":27503,"style":218},[217],[15,27505,27507],{"className":27506},[222,223,224,225],[15,27508,27510],{"className":27509},[66,225],[15,27511,1061],{"className":27512,"style":1083},[66,494,225],[15,27514,27515,27518],{"style":397},[15,27516],{"className":27517,"style":218},[217],[15,27519,27521],{"className":27520},[222,223,224,225],[15,27522,1263],{"className":27523},[66,225],[15,27525,1114],{"className":27526},[1113],[15,27528,27530],{"className":27529},[206],[15,27531,27533],{"className":27532,"style":1686},[210],[15,27534],{},[15,27536,27538],{"className":27537,"style":1418},[169,1417],[15,27539,4328],{"className":27540},[1422,1423],[15,27542],{"className":27543,"style":697},[71],[15,27545,3571],{"className":27546},[66],[11,27548,27549,27550,27794],{},"表示定理同样适用，",[15,27551,27553,27600],{"className":27552,"translate":19},[18],[15,27554,27556],{"className":27555},[23],[25,27557,27558],{"xmlns":27},[29,27559,27560,27598],{},[32,27561,27562,27568,27570,27576,27582,27584,27586,27588,27590,27596],{},[131,27563,27564,27566],{},[35,27565,116],{},[39,27567,1162],{},[39,27569,129],{},[1057,27571,27572,27574],{},[39,27573,1200],{},[35,27575,1205],{},[1057,27577,27578,27580],{},[35,27579,8385],{},[35,27581,1205],{},[35,27583,621],{},[39,27585,120],{"stretchy":119},[39,27587,3266],{},[39,27589,335],{"separator":53},[1057,27591,27592,27594],{},[35,27593,123],{},[35,27595,1205],{},[39,27597,126],{"stretchy":119},[45,27599,16807],{"encoding":47},[15,27601,27603,27647],{"className":27602,"ariaHidden":53},[52],[15,27604,27606,27609,27638,27641,27644],{"className":27605},[57],[15,27607],{"className":27608,"style":3406},[61],[15,27610,27612,27615],{"className":27611},[66],[15,27613,116],{"className":27614,"style":158},[66,67],[15,27616,27618],{"className":27617},[198],[15,27619,27621],{"className":27620},[202],[15,27622,27624],{"className":27623},[206],[15,27625,27627],{"className":27626,"style":16244},[210],[15,27628,27629,27632],{"style":213},[15,27630],{"className":27631,"style":218},[217],[15,27633,27635],{"className":27634},[222,223,224,225],[15,27636,1162],{"className":27637},[76,225],[15,27639],{"className":27640,"style":173},[71],[15,27642,129],{"className":27643},[177],[15,27645],{"className":27646,"style":173},[71],[15,27648,27650,27653,27693,27696,27736,27739,27742,27745,27748,27751,27791],{"className":27649},[57],[15,27651],{"className":27652,"style":16861},[61],[15,27654,27656,27659],{"className":27655},[1339],[15,27657,1200],{"className":27658,"style":13697},[1339,1470,13696],[15,27660,27662],{"className":27661},[198],[15,27663,27665,27685],{"className":27664},[202,1090],[15,27666,27668,27682],{"className":27667},[206],[15,27669,27671],{"className":27670,"style":16880},[210],[15,27672,27673,27676],{"style":13713},[15,27674],{"className":27675,"style":218},[217],[15,27677,27679],{"className":27678},[222,223,224,225],[15,27680,1205],{"className":27681},[66,67,225],[15,27683,1114],{"className":27684},[1113],[15,27686,27688],{"className":27687},[206],[15,27689,27691],{"className":27690,"style":16901},[210],[15,27692],{},[15,27694],{"className":27695,"style":697},[71],[15,27697,27699,27702],{"className":27698},[66],[15,27700,8385],{"className":27701,"style":8555},[66,67],[15,27703,27705],{"className":27704},[198],[15,27706,27708,27728],{"className":27707},[202,1090],[15,27709,27711,27725],{"className":27710},[206],[15,27712,27714],{"className":27713,"style":1528},[210],[15,27715,27716,27719],{"style":8570},[15,27717],{"className":27718,"style":218},[217],[15,27720,27722],{"className":27721},[222,223,224,225],[15,27723,1205],{"className":27724},[66,67,225],[15,27726,1114],{"className":27727},[1113],[15,27729,27731],{"className":27730},[206],[15,27732,27734],{"className":27733,"style":1121},[210],[15,27735],{},[15,27737,621],{"className":27738,"style":684},[66,67],[15,27740,120],{"className":27741},[162],[15,27743,3266],{"className":27744},[66],[15,27746,335],{"className":27747},[422],[15,27749],{"className":27750,"style":697},[71],[15,27752,27754,27757],{"className":27753},[66],[15,27755,123],{"className":27756},[66,67],[15,27758,27760],{"className":27759},[198],[15,27761,27763,27783],{"className":27762},[202,1090],[15,27764,27766,27780],{"className":27765},[206],[15,27767,27769],{"className":27768,"style":1528},[210],[15,27770,27771,27774],{"style":1584},[15,27772],{"className":27773,"style":218},[217],[15,27775,27777],{"className":27776},[222,223,224,225],[15,27778,1205],{"className":27779},[66,67,225],[15,27781,1114],{"className":27782},[1113],[15,27784,27786],{"className":27785},[206],[15,27787,27789],{"className":27788,"style":1121},[210],[15,27790],{},[15,27792,126],{"className":27793},[169],"。有限维问题是",[15,27796,27798],{"className":27797,"translate":19},[294],[15,27799,27801,27913],{"className":27800,"translate":19},[18],[15,27802,27804],{"className":27803},[23],[25,27805,27806],{"xmlns":27,"display":303},[29,27807,27808,27910],{},[32,27809,27810,27830,27908],{},[1172,27811,27812,27818],{},[32,27813,27814,27816],{},[35,27815,1178],{},[39,27817,1170],{},[32,27819,27820,27822,27824],{},[35,27821,8385],{},[39,27823,1187],{},[131,27825,27826,27828],{},[35,27827,455],{"mathvariant":454},[35,27829,37],{},[32,27831,27832,27834,27848,27850,27852,27854,27856,27858,27860,27862,27864,27866,27872,27874,27876,27878,27884,27886,27888,27890,27896,27902,27904,27906],{},[39,27833,4305],{"fence":53},[1196,27835,27836,27838,27846],{},[39,27837,1200],{},[32,27839,27840,27842,27844],{},[35,27841,1205],{},[39,27843,129],{},[1209,27845,1211],{},[35,27847,37],{},[35,27849,27068],{},[39,27851,1170],{},[39,27853,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,27855,1211],{},[39,27857,1245],{},[35,27859,6590],{},[39,27861,1170],{},[39,27863,120],{"stretchy":119},[39,27865,6600],{},[1057,27867,27868,27870],{},[35,27869,1224],{},[35,27871,1205],{},[39,27873,120],{"stretchy":119},[35,27875,2853],{},[35,27877,8385],{},[1057,27879,27880,27882],{},[39,27881,126],{"stretchy":119},[35,27883,1205],{},[39,27885,126],{"stretchy":119},[39,27887,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,27889,1245],{},[6465,27891,27892,27894],{},[35,27893,1248],{},[1209,27895,1263],{},[131,27897,27898,27900],{},[35,27899,8385],{},[35,27901,139],{"mathvariant":138},[35,27903,2853],{},[35,27905,8385],{},[39,27907,4328],{"fence":53},[35,27909,3571],{"mathvariant":138},[45,27911,27912],{"encoding":47},"\\min_{\\alpha\\in\\mathbb R^n}\\left\\{\n\\sum_{i=1}^n\\log\\bigl(1+\\exp(-y_i(K\\alpha)_i)\\bigr)\n+\\frac{\\lambda}{2}\\alpha^\\top K\\alpha\n\\right\\}.",[15,27914,27916],{"className":27915,"ariaHidden":53},[52],[15,27917,27919,27922,28001,28004,28322,28325],{"className":27918},[57],[15,27920],{"className":27921,"style":1335},[61],[15,27923,27925],{"className":27924},[1339,1351],[15,27926,27928,27993],{"className":27927},[202,1090],[15,27929,27931,27990],{"className":27930},[206],[15,27932,27934,27980],{"className":27933,"style":1361},[210],[15,27935,27936,27939],{"style":15659},[15,27937],{"className":27938,"style":1368},[217],[15,27940,27942],{"className":27941},[222,223,224,225],[15,27943,27945,27948,27951],{"className":27944},[66,225],[15,27946,8385],{"className":27947,"style":8555},[66,67,225],[15,27949,1187],{"className":27950},[177,225],[15,27952,27954,27957],{"className":27953},[66,225],[15,27955,455],{"className":27956},[66,517,225],[15,27958,27960],{"className":27959},[198],[15,27961,27963],{"className":27962},[202],[15,27964,27966],{"className":27965},[206],[15,27967,27969],{"className":27968,"style":15693},[210],[15,27970,27971,27974],{"style":15696},[15,27972],{"className":27973,"style":3673},[217],[15,27975,27977],{"className":27976},[222,3677,1508,225],[15,27978,37],{"className":27979},[66,67,225],[15,27981,27982,27985],{"style":1386},[15,27983],{"className":27984,"style":1368},[217],[15,27986,27987],{},[15,27988,1178],{"className":27989},[1339],[15,27991,1114],{"className":27992},[1113],[15,27994,27996],{"className":27995},[206],[15,27997,27999],{"className":27998,"style":15725},[210],[15,28000],{},[15,28002],{"className":28003,"style":697},[71],[15,28005,28007,28013,28077,28080,28085,28091,28094,28097,28100,28103,28106,28109,28112,28152,28155,28158,28161,28201,28204,28210,28213,28216,28219,28281,28310,28313,28316],{"className":28006},[1413],[15,28008,28010],{"className":28009,"style":1418},[162,1417],[15,28011,4305],{"className":28012},[1422,1423],[15,28014,28016],{"className":28015},[1339,1351],[15,28017,28019,28069],{"className":28018},[202,1090],[15,28020,28022,28066],{"className":28021},[206],[15,28023,28025,28045,28055],{"className":28024,"style":1436},[210],[15,28026,28027,28030],{"style":1439},[15,28028],{"className":28029,"style":1443},[217],[15,28031,28033],{"className":28032},[222,223,224,225],[15,28034,28036,28039,28042],{"className":28035},[66,225],[15,28037,1205],{"className":28038},[66,67,225],[15,28040,129],{"className":28041},[177,225],[15,28043,1211],{"className":28044},[66,225],[15,28046,28047,28050],{"style":1461},[15,28048],{"className":28049,"style":1443},[217],[15,28051,28052],{},[15,28053,1200],{"className":28054},[1339,1470,1471],[15,28056,28057,28060],{"style":1474},[15,28058],{"className":28059,"style":1443},[217],[15,28061,28063],{"className":28062},[222,223,224,225],[15,28064,37],{"className":28065},[66,67,225],[15,28067,1114],{"className":28068},[1113],[15,28070,28072],{"className":28071},[206],[15,28073,28075],{"className":28074,"style":1493},[210],[15,28076],{},[15,28078],{"className":28079,"style":697},[71],[15,28081,27278,28083],{"className":28082},[1339],[15,28084,1344],{"style":1343},[15,28086,28088],{"className":28087},[162],[15,28089,120],{"className":28090},[1422,1508],[15,28092,1211],{"className":28093},[66],[15,28095],{"className":28096,"style":72},[71],[15,28098,1245],{"className":28099},[76],[15,28101],{"className":28102,"style":72},[71],[15,28104,6590],{"className":28105},[1339],[15,28107,120],{"className":28108},[162],[15,28110,6600],{"className":28111},[66],[15,28113,28115,28118],{"className":28114},[66],[15,28116,1224],{"className":28117,"style":1515},[66,67],[15,28119,28121],{"className":28120},[198],[15,28122,28124,28144],{"className":28123},[202,1090],[15,28125,28127,28141],{"className":28126},[206],[15,28128,28130],{"className":28129,"style":1528},[210],[15,28131,28132,28135],{"style":1531},[15,28133],{"className":28134,"style":218},[217],[15,28136,28138],{"className":28137},[222,223,224,225],[15,28139,1205],{"className":28140},[66,67,225],[15,28142,1114],{"className":28143},[1113],[15,28145,28147],{"className":28146},[206],[15,28148,28150],{"className":28149,"style":1121},[210],[15,28151],{},[15,28153,120],{"className":28154},[162],[15,28156,2853],{"className":28157,"style":2901},[66,67],[15,28159,8385],{"className":28160,"style":8555},[66,67],[15,28162,28164,28167],{"className":28163},[169],[15,28165,126],{"className":28166},[169],[15,28168,28170],{"className":28169},[198],[15,28171,28173,28193],{"className":28172},[202,1090],[15,28174,28176,28190],{"className":28175},[206],[15,28177,28179],{"className":28178,"style":1528},[210],[15,28180,28181,28184],{"style":1584},[15,28182],{"className":28183,"style":218},[217],[15,28185,28187],{"className":28186},[222,223,224,225],[15,28188,1205],{"className":28189},[66,67,225],[15,28191,1114],{"className":28192},[1113],[15,28194,28196],{"className":28195},[206],[15,28197,28199],{"className":28198,"style":1121},[210],[15,28200],{},[15,28202,126],{"className":28203},[169],[15,28205,28207],{"className":28206},[169],[15,28208,126],{"className":28209},[1422,1508],[15,28211],{"className":28212,"style":72},[71],[15,28214,1245],{"className":28215},[76],[15,28217],{"className":28218,"style":72},[71],[15,28220,28222,28225,28278],{"className":28221},[66],[15,28223],{"className":28224},[162,23412],[15,28226,28228],{"className":28227},[6465],[15,28229,28231,28270],{"className":28230},[202,1090],[15,28232,28234,28267],{"className":28233},[206],[15,28235,28237,28248,28256],{"className":28236,"style":27430},[210],[15,28238,28239,28242],{"style":23428},[15,28240],{"className":28241,"style":1368},[217],[15,28243,28245],{"className":28244},[66],[15,28246,1263],{"className":28247},[66],[15,28249,28250,28253],{"style":23440},[15,28251],{"className":28252,"style":1368},[217],[15,28254],{"className":28255,"style":4438},[23447],[15,28257,28258,28261],{"style":23450},[15,28259],{"className":28260,"style":1368},[217],[15,28262,28264],{"className":28263},[66],[15,28265,1248],{"className":28266},[66,67],[15,28268,1114],{"className":28269},[1113],[15,28271,28273],{"className":28272},[206],[15,28274,28276],{"className":28275,"style":23469},[210],[15,28277],{},[15,28279],{"className":28280},[169,23412],[15,28282,28284,28287],{"className":28283},[66],[15,28285,8385],{"className":28286,"style":8555},[66,67],[15,28288,28290],{"className":28289},[198],[15,28291,28293],{"className":28292},[202],[15,28294,28296],{"className":28295},[206],[15,28297,28299],{"className":28298,"style":394},[210],[15,28300,28301,28304],{"style":397},[15,28302],{"className":28303,"style":218},[217],[15,28305,28307],{"className":28306},[222,223,224,225],[15,28308,139],{"className":28309},[66,225],[15,28311,2853],{"className":28312,"style":2901},[66,67],[15,28314,8385],{"className":28315,"style":8555},[66,67],[15,28317,28319],{"className":28318,"style":1418},[169,1417],[15,28320,4328],{"className":28321},[1422,1423],[15,28323],{"className":28324,"style":697},[71],[15,28326,3571],{"className":28327},[66],[11,28329,28330,28331,28359],{},"这是凸的、可微的，但没有闭式解，通常用 Newton 法、IRLS 或梯度类方法求解。与 SVM 不同，逻辑损失不会产生精确的零系数，",[15,28332,28334,28347],{"className":28333,"translate":19},[18],[15,28335,28337],{"className":28336},[23],[25,28338,28339],{"xmlns":27},[29,28340,28341,28345],{},[32,28342,28343],{},[35,28344,8385],{},[45,28346,16072],{"encoding":47},[15,28348,28350],{"className":28349,"ariaHidden":53},[52],[15,28351,28353,28356],{"className":28352},[57],[15,28354],{"className":28355,"style":86},[61],[15,28357,8385],{"className":28358,"style":8555},[66,67]," 一般是稠密的。",[1904,28361,28363],{"id":28362},"_54-高斯过程","5.4 高斯过程",[11,28365,28366],{},"高斯过程（GP）回归从贝叶斯角度给出了同一个解。设先验",[15,28368,28370],{"className":28369,"translate":19},[294],[15,28371,28373,28410],{"className":28372,"translate":19},[18],[15,28374,28376],{"className":28375},[23],[25,28377,28378],{"xmlns":27,"display":303},[29,28379,28380,28407],{},[32,28381,28382,28384,28387,28395,28397,28399,28401,28403,28405],{},[35,28383,116],{},[39,28385,28386],{},"∼",[32,28388,28389,28392],{},[35,28390,28391],{"mathvariant":445},"G",[35,28393,28394],{"mathvariant":445},"P",[39,28396,120],{"stretchy":119},[1209,28398,1804],{},[39,28400,335],{"separator":53},[35,28402,621],{},[39,28404,126],{"stretchy":119},[39,28406,335],{"separator":53},[45,28408,28409],{"encoding":47},"f\\sim\\mathcal{GP}(0,k),",[15,28411,28413,28431],{"className":28412,"ariaHidden":53},[52],[15,28414,28416,28419,28422,28425,28428],{"className":28415},[57],[15,28417],{"className":28418,"style":3406},[61],[15,28420,116],{"className":28421,"style":158},[66,67],[15,28423],{"className":28424,"style":173},[71],[15,28426,28386],{"className":28427},[177],[15,28429],{"className":28430,"style":173},[71],[15,28432,28434,28437,28448,28451,28454,28457,28460,28463,28466],{"className":28433},[57],[15,28435],{"className":28436,"style":154},[61],[15,28438,28440,28444],{"className":28439},[66],[15,28441,28391],{"className":28442,"style":28443},[66,494],"margin-right:0.0593em;",[15,28445,28394],{"className":28446,"style":28447},[66,494],"margin-right:0.0822em;",[15,28449,120],{"className":28450},[162],[15,28452,1804],{"className":28453},[66],[15,28455,335],{"className":28456},[422],[15,28458],{"className":28459,"style":697},[71],[15,28461,621],{"className":28462,"style":684},[66,67],[15,28464,126],{"className":28465},[169],[15,28467,335],{"className":28468},[422],[11,28470,28471],{},"观测模型",[15,28473,28475],{"className":28474,"translate":19},[294],[15,28476,28478,28550],{"className":28477,"translate":19},[18],[15,28479,28481],{"className":28480},[23],[25,28482,28483],{"xmlns":27,"display":303},[29,28484,28485,28547],{},[32,28486,28487,28493,28495,28497,28499,28505,28507,28509,28516,28518,28520,28526,28528,28531,28533,28535,28537,28543,28545],{},[1057,28488,28489,28491],{},[35,28490,1224],{},[35,28492,1205],{},[39,28494,129],{},[35,28496,116],{},[39,28498,120],{"stretchy":119},[1057,28500,28501,28503],{},[35,28502,123],{},[35,28504,1205],{},[39,28506,126],{"stretchy":119},[39,28508,1245],{},[1057,28510,28511,28514],{},[35,28512,28513],{},"ε",[35,28515,1205],{},[39,28517,335],{"separator":53},[71,28519],{"width":3902},[1057,28521,28522,28524],{},[35,28523,28513],{},[35,28525,1205],{},[39,28527,28386],{},[35,28529,28530],{},"N",[39,28532,120],{"stretchy":119},[1209,28534,1804],{},[39,28536,335],{"separator":53},[131,28538,28539,28541],{},[35,28540,6632],{},[1209,28542,1263],{},[39,28544,126],{"stretchy":119},[39,28546,335],{"separator":53},[45,28548,28549],{"encoding":47},"y_i=f(x_i)+\\varepsilon_i,\\qquad\n\\varepsilon_i\\sim N(0,\\sigma^2),",[15,28551,28553,28608,28672,28776],{"className":28552,"ariaHidden":53},[52],[15,28554,28556,28559,28599,28602,28605],{"className":28555},[57],[15,28557],{"className":28558,"style":471},[61],[15,28560,28562,28565],{"className":28561},[66],[15,28563,1224],{"className":28564,"style":1515},[66,67],[15,28566,28568],{"className":28567},[198],[15,28569,28571,28591],{"className":28570},[202,1090],[15,28572,28574,28588],{"className":28573},[206],[15,28575,28577],{"className":28576,"style":1528},[210],[15,28578,28579,28582],{"style":1531},[15,28580],{"className":28581,"style":218},[217],[15,28583,28585],{"className":28584},[222,223,224,225],[15,28586,1205],{"className":28587},[66,67,225],[15,28589,1114],{"className":28590},[1113],[15,28592,28594],{"className":28593},[206],[15,28595,28597],{"className":28596,"style":1121},[210],[15,28598],{},[15,28600],{"className":28601,"style":173},[71],[15,28603,129],{"className":28604},[177],[15,28606],{"className":28607,"style":173},[71],[15,28609,28611,28614,28617,28620,28660,28663,28666,28669],{"className":28610},[57],[15,28612],{"className":28613,"style":154},[61],[15,28615,116],{"className":28616,"style":158},[66,67],[15,28618,120],{"className":28619},[162],[15,28621,28623,28626],{"className":28622},[66],[15,28624,123],{"className":28625},[66,67],[15,28627,28629],{"className":28628},[198],[15,28630,28632,28652],{"className":28631},[202,1090],[15,28633,28635,28649],{"className":28634},[206],[15,28636,28638],{"className":28637,"style":1528},[210],[15,28639,28640,28643],{"style":1584},[15,28641],{"className":28642,"style":218},[217],[15,28644,28646],{"className":28645},[222,223,224,225],[15,28647,1205],{"className":28648},[66,67,225],[15,28650,1114],{"className":28651},[1113],[15,28653,28655],{"className":28654},[206],[15,28656,28658],{"className":28657,"style":1121},[210],[15,28659],{},[15,28661,126],{"className":28662},[169],[15,28664],{"className":28665,"style":72},[71],[15,28667,1245],{"className":28668},[76],[15,28670],{"className":28671,"style":72},[71],[15,28673,28675,28678,28718,28721,28724,28727,28767,28770,28773],{"className":28674},[57],[15,28676],{"className":28677,"style":471},[61],[15,28679,28681,28684],{"className":28680},[66],[15,28682,28513],{"className":28683},[66,67],[15,28685,28687],{"className":28686},[198],[15,28688,28690,28710],{"className":28689},[202,1090],[15,28691,28693,28707],{"className":28692},[206],[15,28694,28696],{"className":28695,"style":1528},[210],[15,28697,28698,28701],{"style":1584},[15,28699],{"className":28700,"style":218},[217],[15,28702,28704],{"className":28703},[222,223,224,225],[15,28705,1205],{"className":28706},[66,67,225],[15,28708,1114],{"className":28709},[1113],[15,28711,28713],{"className":28712},[206],[15,28714,28716],{"className":28715,"style":1121},[210],[15,28717],{},[15,28719,335],{"className":28720},[422],[15,28722],{"className":28723,"style":4059},[71],[15,28725],{"className":28726,"style":697},[71],[15,28728,28730,28733],{"className":28729},[66],[15,28731,28513],{"className":28732},[66,67],[15,28734,28736],{"className":28735},[198],[15,28737,28739,28759],{"className":28738},[202,1090],[15,28740,28742,28756],{"className":28741},[206],[15,28743,28745],{"className":28744,"style":1528},[210],[15,28746,28747,28750],{"style":1584},[15,28748],{"className":28749,"style":218},[217],[15,28751,28753],{"className":28752},[222,223,224,225],[15,28754,1205],{"className":28755},[66,67,225],[15,28757,1114],{"className":28758},[1113],[15,28760,28762],{"className":28761},[206],[15,28763,28765],{"className":28764,"style":1121},[210],[15,28766],{},[15,28768],{"className":28769,"style":173},[71],[15,28771,28386],{"className":28772},[177],[15,28774],{"className":28775,"style":173},[71],[15,28777,28779,28782,28786,28789,28792,28795,28798,28827,28830],{"className":28778},[57],[15,28780],{"className":28781,"style":11209},[61],[15,28783,28530],{"className":28784,"style":28785},[66,67],"margin-right:0.109em;",[15,28787,120],{"className":28788},[162],[15,28790,1804],{"className":28791},[66],[15,28793,335],{"className":28794},[422],[15,28796],{"className":28797,"style":697},[71],[15,28799,28801,28804],{"className":28800},[66],[15,28802,6632],{"className":28803,"style":1515},[66,67],[15,28805,28807],{"className":28806},[198],[15,28808,28810],{"className":28809},[202],[15,28811,28813],{"className":28812},[206],[15,28814,28816],{"className":28815,"style":1650},[210],[15,28817,28818,28821],{"style":397},[15,28819],{"className":28820,"style":218},[217],[15,28822,28824],{"className":28823},[222,223,224,225],[15,28825,1263],{"className":28826},[66,225],[15,28828,126],{"className":28829},[169],[15,28831,335],{"className":28832},[422],[11,28834,28835],{},"则后验均值是",[15,28837,28839],{"className":28838,"translate":19},[294],[15,28840,28842,28905],{"className":28841,"translate":19},[18],[15,28843,28845],{"className":28844},[23],[25,28846,28847],{"xmlns":27,"display":303},[29,28848,28849,28902],{},[32,28850,28851,28854,28856,28858,28860,28862,288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",[15,29412,29414,29436],{"className":29413,"translate":19},[18],[15,29415,29417],{"className":29416},[23],[25,29418,29419],{"xmlns":27},[29,29420,29421,29433],{},[32,29422,29423,29425,29427],{},[35,29424,1248],{},[39,29426,129],{},[131,29428,29429,29431],{},[35,29430,6632],{},[1209,29432,1263],{},[45,29434,29435],{"encoding":47},"\\lambda=\\sigma^2",[15,29437,29439,29457],{"className":29438,"ariaHidden":53},[52],[15,29440,29442,29445,29448,29451,29454],{"className":29441},[57],[15,29443],{"className":29444,"style":840},[61],[15,29446,1248],{"className":29447},[66,67],[15,29449],{"className":29450,"style":173},[71],[15,29452,129],{"className":29453},[177],[15,29455],{"className":29456,"style":173},[71],[15,29458,29460,29463],{"className":29459},[57],[15,29461],{"className":29462,"style":6738},[61],[15,29464,29466,29469],{"className":29465},[66],[15,29467,6632],{"className":29468,"style":1515},[66,67],[15,29470,29472],{"className":29471},[198],[15,29473,29475],{"className":29474},[202],[15,29476,29478],{"className":29477},[206],[15,29479,29481],{"className":29480,"style":6738},[210],[15,29482,29483,29486],{"style":213},[15,29484],{"className":29485,"style":218},[217],[15,29487,29489],{"className":29488},[222,223,224,225],[15,29490,1263],{"className":29491},[66,225]," 的核岭回归解，也就是表示定理给出的形式。区别在于，GP 还给出了不确定度 ",[15,29494,29496,29520],{"className":29495,"translate":19},[18],[15,29497,29499],{"className":29498},[23],[25,29500,29501],{"xmlns":27},[29,29502,29503,29517],{},[32,29504,29505,29511,29513,29515],{},[131,29506,29507,29509],{},[35,29508,29095],{},[1209,29510,1263],{},[39,29512,120],{"stretchy":119},[35,29514,123],{},[39,29516,126],{"stretchy":119},[45,29518,29519],{"encoding":47},"s^2(x)",[15,29521,29523],{"className":29522,"ariaHidden":53},[52],[15,29524,29526,29529,29558,29561,29564],{"className":29525},[57],[15,29527],{"className":29528,"style":9149},[61],[15,29530,29532,29535],{"className":29531},[66],[15,29533,29095],{"className":29534},[66,67],[15,29536,29538],{"className":29537},[198],[15,29539,29541],{"className":29540},[202],[15,29542,29544],{"className":29543},[206],[15,29545,29547],{"className":29546,"style":6738},[210],[15,29548,29549,29552],{"style":213},[15,29550],{"className":29551,"style":218},[217],[15,29553,29555],{"className":29554},[222,223,224,225],[15,29556,1263],{"className":29557},[66,225],[15,29559,120],{"className":29560},[162],[15,29562,123],{"className":29563},[66,67],[15,29565,126],{"className":29566},[169],"；表示定理只描述均值（或点预测）的结构。",[1904,29569,29571],{"id":29570},"_55-平滑样条","5.5 平滑样条",[11,29573,29574,29575,3247,29649,26811,29677,29705],{},"表示定理最早的重要应用之一是平滑样条。取 ",[15,29576,29578,29604],{"className":29577,"translate":19},[18],[15,29579,29581],{"className":29580},[23],[25,29582,29583],{"xmlns":27},[29,29584,29585,29601],{},[32,29586,29587,29589,29591,29593,29595,29597,29599],{},[35,29588,446],{"mathvariant":445},[39,29590,129],{},[39,29592,1194],{"stretchy":119},[1209,29594,1804],{},[39,29596,335],{"separator":53},[1209,29598,1211],{},[39,29600,1266],{"stretchy":119},[45,29602,29603],{"encoding":47},"\\mathcal X=[0,1]",[15,29605,29607,29625],{"className":29606,"ariaHidden":53},[52],[15,29608,29610,29613,29616,29619,29622],{"className":29609},[57],[15,29611],{"className":29612,"style":490},[61],[15,29614,446],{"className":29615,"style":495},[66,494],[15,29617],{"className":29618,"style":173},[71],[15,29620,129],{"className":29621},[177],[15,29623],{"className":29624,"style":173},[71],[15,29626,29628,29631,29634,29637,29640,29643,29646],{"className":29627},[57],[15,29629],{"className":29630,"style":154},[61],[15,29632,1194],{"className":29633},[162],[15,29635,1804],{"className":29636},[66],[15,29638,335],{"className":29639},[422],[15,29641],{"className":29642,"style":697},[71],[15,29644,1211],{"className":29645},[66],[15,29647,1266],{"className":29648},[169],[15,29650,29652,29665],{"className":29651,"translate":19},[18],[15,29653,29655],{"className":29654},[23],[25,29656,29657],{"xmlns":27},[29,29658,29659,29663],{},[32,29660,29661],{},[35,29662,1061],{"mathvariant":445},[45,29664,1742],{"encoding":47},[15,29666,29668],{"className":29667,"ariaHidden":53},[52],[15,29669,29671,29674],{"className":29670},[57],[15,29672],{"className":29673,"style":490},[61],[15,29675,1061],{"className":29676,"style":1083},[66,494],[15,29678,29680,29693],{"className":29679,"translate":19},[18],[15,29681,29683],{"className":29682},[23],[25,29684,29685],{"xmlns":27},[29,29686,29687,29691],{},[32,29688,29689],{},[35,29690,28853],{},[45,29692,28853],{"encoding":47},[15,29694,29696],{"className":29695,"ariaHidden":53},[52],[15,29697,29699,29702],{"className":29698},[57],[15,29700],{"className":29701,"style":86},[61],[15,29703,28853],{"className":29704},[66,67]," 阶 Sobolev 空间，正则项为",[15,29707,29709],{"className":29708,"translate":19},[294],[15,29710,29712,29769],{"className":29711,"translate":19},[18],[15,29713,29715],{"className":29714},[23],[25,29716,29717],{"xmlns":27,"display":303},[29,29718,29719,29766],{},[32,29720,29721,29723,29732,29734,29746,29748,29750,29752,29758,29760,29762,29764],{},[35,29722,1248],{},[1255,29724,29725,29728,29730],{},[39,29726,29727],{},"∫",[1209,29729,1804],{},[1209,29731,1211],{},[39,29733,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[131,29735,29736,29738],{},[35,29737,116],{},[32,29739,29740,29742,29744],{},[39,29741,120],{"stretchy":119},[35,29743,28853],{},[39,29745,126],{"stretchy":119},[39,29747,120],{"stretchy":119},[35,29749,123],{},[39,29751,126],{"stretchy":119},[131,29753,29754,29756],{},[39,29755,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,29757,1263],{},[1274,29759,2462],{},[35,29761,458],{},[35,29763,123],{},[35,29765,3571],{"mathvariant":138},[45,29767,29768],{"encoding":47},"\\lambda\\int_0^1\\bigl(f^{(m)}(x)\\bigr)^2\\,dx.",[15,29770,29772],{"className":29771,"ariaHidden":53},[52],[15,29773,29775,29779,29782,29785,29841,29847,29886,29889,29892,29895,29927,29930,29933,29936],{"className":29774},[57],[15,29776],{"className":29777,"style":29778},[61],"height:2.476em;vertical-align:-0.9119em;",[15,29780,1248],{"className":29781},[66,67],[15,29783],{"className":29784,"style":697},[71],[15,29786,29788,29792],{"className":29787},[1339],[15,29789,29727],{"className":29790,"style":29791},[1339,1470,1471],"margin-right:0.4445em;position:relative;top:-0.0011em;",[15,29793,29795],{"className":29794},[198],[15,29796,29798,29832],{"className":29797},[202,1090],[15,29799,29801,29829],{"className":29800},[206],[15,29802,29805,29817],{"className":29803,"style":29804},[210],"height:1.564em;",[15,29806,29808,29811],{"style":29807},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[15,29809],{"className":29810,"style":218},[217],[15,29812,29814],{"className":29813},[222,223,224,225],[15,29815,1804],{"className":29816},[66,225],[15,29818,29820,29823],{"style":29819},"top:-3.8129em;margin-right:0.05em;",[15,29821],{"className":29822,"style":218},[217],[15,29824,29826],{"className":29825},[222,223,224,225],[15,29827,1211],{"className":29828},[66,225],[15,29830,1114],{"className":29831},[1113],[15,29833,29835],{"className":29834},[206],[15,29836,29839],{"className":29837,"style":29838},[210],"height:0.9119em;",[15,29840],{},[15,29842,29844],{"className":29843},[162],[15,29845,120],{"className":29846},[1422,1508],[15,29848,29850,29853],{"className":29849},[66],[15,29851,116],{"className":29852,"style":158},[66,67],[15,29854,29856],{"className":29855},[198],[15,29857,29859],{"className":29858},[202],[15,29860,29862],{"className":29861},[206],[15,29863,29866],{"className":29864,"style":29865},[210],"height:0.938em;",[15,29867,29868,29871],{"style":397},[15,29869],{"className":29870,"style":218},[217],[15,29872,29874],{"className":29873},[222,223,224,225],[15,29875,29877,29880,29883],{"className":29876},[66,225],[15,29878,120],{"className":29879},[162,225],[15,29881,28853],{"className":29882},[66,67,225],[15,29884,126],{"className":29885},[169,225],[15,29887,120],{"className":29888},[162],[15,29890,123],{"className":29891},[66,67],[15,29893,126],{"className":29894},[169],[15,29896,29898,29904],{"className":29897},[169],[15,29899,29901],{"className":29900},[169],[15,29902,126],{"className":29903},[1422,1508],[15,29905,29907],{"className":29906},[198],[15,29908,29910],{"className":29909},[202],[15,29911,29913],{"className":29912},[206],[15,29914,29916],{"className":29915,"style":21266},[210],[15,29917,29918,29921],{"style":21269},[15,29919],{"className":29920,"style":218},[217],[15,29922,29924],{"className":29923},[222,223,224,225],[15,29925,1263],{"className":29926},[66,225],[15,29928],{"className":29929,"style":697},[71],[15,29931,458],{"className":29932},[66,67],[15,29934,123],{"className":29935},[66,67],[15,29937,3571],{"className":29938},[66],[11,29940,29941],{},"最小化",[15,29943,29945],{"className":29944,"translate":19},[294],[15,29946,29948,30046],{"className":29947,"translate":19},[18],[15,29949,29951],{"className":29950},[23],[25,29952,29953],{"xmlns":27,"display":303},[29,29954,29955,30043],{},[32,29956,29957,29971,29973,29979,29981,29983,29985,29991,29993,29999,30001,30003,30011,30013,30025,30027,30029,30031,30037,30039,30041],{},[1196,29958,29959,29961,29969],{},[39,29960,1200],{},[32,29962,29963,29965,29967],{},[35,29964,1205],{},[39,29966,129],{},[1209,29968,1211],{},[35,29970,37],{},[39,29972,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,29974,29975,29977],{},[35,29976,1224],{},[35,29978,1205],{},[39,29980,6600],{},[35,29982,116],{},[39,29984,120],{"stretchy":119},[1057,29986,29987,29989],{},[35,29988,123],{},[35,29990,1205],{},[39,29992,126],{"stretchy":119},[131,29994,29995,29997],{},[39,29996,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,29998,1263],{},[39,30000,1245],{},[35,30002,1248],{},[1255,30004,30005,30007,30009],{},[39,30006,29727],{},[1209,30008,1804],{},[1209,30010,1211],{},[39,30012,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[131,30014,30015,30017],{},[35,30016,116],{},[32,30018,30019,30021,30023],{},[39,30020,120],{"stretchy":119},[35,30022,28853],{},[39,30024,126],{"stretchy":119},[39,30026,120],{"stretchy":119},[35,30028,123],{},[39,30030,126],{"stretchy":119},[131,30032,30033,30035],{},[39,30034,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,30036,1263],{},[1274,30038,2462],{},[35,30040,458],{},[35,30042,123],{},[45,30044,30045],{"encoding":47},"\\sum_{i=1}^n\\bigl(y_i-f(x_i)\\bigr)^2\n+\\lambda\\int_0^1\\bigl(f^{(m)}(x)\\bigr)^2\\,dx",[15,30047,30049,30174,30271],{"className":30048,"ariaHidden":53},[52],[15,30050,30052,30055,30119,30125,30165,30168,30171],{"className":30051},[57],[15,30053],{"className":30054,"style":8481},[61],[15,30056,30058],{"className":30057},[1339,1351],[15,30059,30061,30111],{"className":30060},[202,1090],[15,30062,30064,30108],{"className":30063},[206],[15,30065,30067,30087,30097],{"className":30066,"style":1436},[210],[15,30068,30069,30072],{"style":1439},[15,30070],{"className":30071,"style":1443},[217],[15,30073,30075],{"className":30074},[222,223,224,225],[15,30076,30078,30081,30084],{"className":30077},[66,225],[15,30079,1205],{"className":30080},[66,67,225],[15,30082,129],{"className":30083},[177,225],[15,30085,1211],{"className":30086},[66,225],[15,30088,30089,30092],{"style":1461},[15,30090],{"className":30091,"style":1443},[217],[15,30093,30094],{},[15,30095,1200],{"className":30096},[1339,1470,1471],[15,30098,30099,30102],{"style":1474},[15,30100],{"className":30101,"style":1443},[217],[15,30103,30105],{"className":30104},[222,223,224,225],[15,30106,37],{"className":30107},[66,67,225],[15,30109,1114],{"className":30110},[1113],[15,30112,30114],{"className":30113},[206],[15,30115,30117],{"className":30116,"style":1493},[210],[15,30118],{},[15,30120,30122],{"className":30121},[162],[15,30123,120],{"className":30124},[1422,1508],[15,30126,30128,30131],{"className":30127},[66],[15,30129,1224],{"className":30130,"style":1515},[66,67],[15,30132,30134],{"className":30133},[198],[15,30135,30137,30157],{"className":30136},[202,1090],[15,30138,30140,30154],{"className":30139},[206],[15,30141,30143],{"className":30142,"style":1528},[210],[15,30144,30145,30148],{"style":1531},[15,30146],{"className":30147,"style":218},[217],[15,30149,30151],{"className":30150},[222,223,224,225],[15,30152,1205],{"className":30153},[66,67,225],[15,30155,1114],{"className":30156},[1113],[15,30158,30160],{"className":30159},[206],[15,30161,30163],{"className":30162,"style":1121},[210],[15,30164],{},[15,30166],{"className":30167,"style":72},[71],[15,30169,6600],{"className":30170},[76],[15,30172],{"className":30173,"style":72},[71],[15,30175,30177,30181,30184,30187,30227,30230,30262,30265,30268],{"className":30176},[57],[15,30178],{"className":30179,"style":30180},[61],"height:1.404em;vertical-align:-0.35em;",[15,30182,116],{"className":30183,"style":158},[66,67],[15,30185,120],{"className":30186},[162],[15,30188,30190,30193],{"className":30189},[66],[15,30191,123],{"className":30192},[66,67],[15,30194,30196],{"className":30195},[198],[15,30197,30199,30219],{"className":30198},[202,1090],[15,30200,30202,30216],{"className":30201},[206],[15,30203,30205],{"className":30204,"style":1528},[210],[15,30206,30207,30210],{"style":1584},[15,30208],{"className":30209,"style":218},[217],[15,30211,30213],{"className":30212},[222,223,224,225],[15,30214,1205],{"className":30215},[66,67,225],[15,30217,1114],{"className":30218},[1113],[15,30220,30222],{"className":30221},[206],[15,30223,30225],{"className":30224,"style":1121},[210],[15,30226],{},[15,30228,126],{"className":30229},[169],[15,30231,30233,30239],{"className":30232},[169],[15,30234,30236],{"className":30235},[169],[15,30237,126],{"className":30238},[1422,1508],[15,30240,30242],{"className":30241},[198],[15,30243,30245],{"className":30244},[202],[15,30246,30248],{"className":30247},[206],[15,30249,30251],{"className":30250,"style":21266},[210],[15,30252,30253,30256],{"style":21269},[15,30254],{"className":30255,"style":218},[217],[15,30257,30259],{"className":30258},[222,223,224,225],[15,30260,1263],{"className":30261},[66,225],[15,30263],{"className":30264,"style":72},[71],[15,30266,1245],{"className":30267},[76],[15,30269],{"className":30270,"style":72},[71],[15,30272,30274,30277,30280,30283,30334,30340,30378,30381,30384,30387,30419,30422,30425],{"className":30273},[57],[15,30275],{"className":30276,"style":29778},[61],[15,30278,1248],{"className":30279},[66,67],[15,30281],{"className":30282,"style":697},[71],[15,30284,30286,30289],{"className":30285},[1339],[15,30287,29727],{"className":30288,"style":29791},[1339,1470,1471],[15,30290,30292],{"className":30291},[198],[15,30293,30295,30326],{"className":30294},[202,1090],[15,30296,30298,30323],{"className":30297},[206],[15,30299,30301,30312],{"className":30300,"style":29804},[210],[15,30302,30303,30306],{"style":29807},[15,30304],{"className":30305,"style":218},[217],[15,30307,30309],{"className":30308},[222,223,224,225],[15,30310,1804],{"className":30311},[66,225],[15,30313,30314,30317],{"style":29819},[15,30315],{"className":30316,"style":218},[217],[15,30318,30320],{"className":30319},[222,223,224,225],[15,30321,1211],{"className":30322},[66,225],[15,30324,1114],{"className":30325},[1113],[15,30327,30329],{"className":30328},[206],[15,30330,30332],{"className":30331,"style":29838},[210],[15,30333],{},[15,30335,30337],{"className":30336},[162],[15,30338,120],{"className":30339},[1422,1508],[15,30341,30343,30346],{"className":30342},[66],[15,30344,116],{"className":30345,"style":158},[66,67],[15,30347,30349],{"className":30348},[198],[15,30350,30352],{"className":30351},[202],[15,30353,30355],{"className":30354},[206],[15,30356,30358],{"className":30357,"style":29865},[210],[15,30359,30360,30363],{"style":397},[15,30361],{"className":30362,"style":218},[217],[15,30364,30366],{"className":30365},[222,223,224,225],[15,30367,30369,30372,30375],{"className":30368},[66,225],[15,30370,120],{"className":30371},[162,225],[15,30373,28853],{"className":30374},[66,67,225],[15,30376,126],{"className":30377},[169,225],[15,30379,120],{"className":30380},[162],[15,30382,123],{"className":30383},[66,67],[15,30385,126],{"className":30386},[169],[15,30388,30390,30396],{"className":30389},[169],[15,30391,30393],{"className":30392},[169],[15,30394,126],{"className":30395},[1422,1508],[15,30397,30399],{"className":30398},[198],[15,30400,30402],{"className":30401},[202],[15,30403,30405],{"className":30404},[206],[15,30406,30408],{"className":30407,"style":21266},[210],[15,30409,30410,30413],{"style":21269},[15,30411],{"className":30412,"style":218},[217],[15,30414,30416],{"className":30415},[222,223,224,225],[15,30417,1263],{"className":30418},[66,225],[15,30420],{"className":30421,"style":697},[71],[15,30423,458],{"className":30424},[66,67],[15,30426,123],{"className":30427},[66,67],[11,30429,30430,30431,30488,30489,30517,30518,30569],{},"的解是一个以训练点为节点的 ",[15,30432,30434,30454],{"className":30433,"translate":19},[18],[15,30435,30437],{"className":30436},[23],[25,30438,30439],{"xmlns":27},[29,30440,30441,30451],{},[32,30442,30443,30445,30447,30449],{},[1209,30444,1263],{},[35,30446,28853],{},[39,30448,6600],{},[1209,30450,1211],{},[45,30452,30453],{"encoding":47},"2m-1",[15,30455,30457,30479],{"className":30456,"ariaHidden":53},[52],[15,30458,30460,30464,30467,30470,30473,30476],{"className":30459},[57],[15,30461],{"className":30462,"style":30463},[61],"height:0.7278em;vertical-align:-0.0833em;",[15,30465,1263],{"className":30466},[66],[15,30468,28853],{"className":30469},[66,67],[15,30471],{"className":30472,"style":72},[71],[15,30474,6600],{"className":30475},[76],[15,30477],{"className":30478,"style":72},[71],[15,30480,30482,30485],{"className":30481},[57],[15,30483],{"className":30484,"style":1836},[61],[15,30486,1211],{"className":30487},[66]," 次自然多项式样条。表示定理给出了「解在训练点核函数张成空间中」的结构；样条的多项式零空间（",[15,30490,30492,30505],{"className":30491,"translate":19},[18],[15,30493,30495],{"className":30494},[23],[25,30496,30497],{"xmlns":27},[29,30498,30499,30503],{},[32,30500,30501],{},[35,30502,28853],{},[45,30504,28853],{"encoding":47},[15,30506,30508],{"className":30507,"ariaHidden":53},[52],[15,30509,30511,30514],{"className":30510},[57],[15,30512],{"className":30513,"style":86},[61],[15,30515,28853],{"className":30516},[66,67]," 维）则由半参数版本处理。",[15,30519,30521,30539],{"className":30520,"translate":19},[18],[15,30522,30524],{"className":30523},[23],[25,30525,30526],{"xmlns":27},[29,30527,30528,30536],{},[32,30529,30530,30532,30534],{},[35,30531,28853],{},[39,30533,129],{},[1209,30535,1263],{},[45,30537,30538],{"encoding":47},"m=2",[15,30540,30542,30560],{"className":30541,"ariaHidden":53},[52],[15,30543,30545,30548,30551,30554,30557],{"className":30544},[57],[15,30546],{"className":30547,"style":86},[61],[15,30549,28853],{"className":30550},[66,67],[15,30552],{"className":30553,"style":173},[71],[15,30555,129],{"className":30556},[177],[15,30558],{"className":30559,"style":173},[71],[15,30561,30563,30566],{"className":30562},[57],[15,30564],{"className":30565,"style":1836},[61],[15,30567,1263],{"className":30568},[66]," 时就是常见的自然三次样条。",[11,30571,30572],{},"这也是 Kimeldorf–Wahba 定理的内容。它说明表示定理并不是核方法时代才出现的技巧，而是从样条估计中生长出来的结构结论。",[1904,30574,30576],{"id":30575},"_56-一张表","5.6 一张表",[6150,30578,30579,30598],{},[6153,30580,30581],{},[6156,30582,30583,30586,30589,30592,30595],{},[6159,30584,30585],{},"方法",[6159,30587,30588],{},"损失",[6159,30590,30591],{},"正则项",[6159,30593,30594],{},"系数",[6159,30596,30597],{},"稀疏性",[6169,30599,30600,30799,30963,31123,31279],{},[6156,30601,30602,30605,30608,30683,30796],{},[6174,30603,30604],{},"核岭回归",[6174,30606,30607],{},"平方",[6174,30609,30610],{},[15,30611,30613,30636],{"className":30612,"translate":19},[18],[15,30614,30616],{"className":30615},[23],[25,30617,30618],{"xmlns":27},[29,30619,30620,30634],{},[32,30621,30622,30624,30626,30628],{},[35,30623,1248],{},[35,30625,1251],{"mathvariant":138},[35,30627,116],{},[131,30629,30630,30632],{},[35,30631,1251],{"mathvariant":138},[1209,30633,1263],{},[45,30635,9139],{"encoding":47},[15,30637,30639],{"className":30638,"ariaHidden":53},[52],[15,30640,30642,30645,30648,30651,30654],{"className":30641},[57],[15,30643],{"className":30644,"style":9149},[61],[15,30646,1248],{"className":30647},[66,67],[15,30649,1251],{"className":30650},[66],[15,30652,116],{"className":30653,"style":158},[66,67],[15,30655,30657,30660],{"className":30656},[66],[15,30658,1251],{"className":30659},[66],[15,30661,30663],{"className":30662},[198],[15,30664,30666],{"className":30665},[202],[15,30667,30669],{"className":30668},[206],[15,30670,30672],{"className":30671,"style":6738},[210],[15,30673,30674,30677],{"style":213},[15,30675],{"className":30676,"style":218},[217],[15,30678,30680],{"className":30679},[222,223,224,225],[15,30681,1263],{"className":30682},[66,225],[6174,30684,30685],{},[15,30686,30688,30722],{"className":30687,"translate":19},[18],[15,30689,30691],{"className":30690},[23],[25,30692,30693],{"xmlns":27},[29,30694,30695,30719],{},[32,30696,30697,30699,30701,30703,30705,30707,30717],{},[39,30698,120],{"stretchy":119},[35,30700,2853],{},[39,30702,1245],{},[35,30704,1248],{},[35,30706,16422],{},[131,30708,30709,30711],{},[39,30710,126],{"stretchy":119},[32,30712,30713,30715],{},[39,30714,6600],{},[1209,30716,1211],{},[35,30718,1224],{},[45,30720,30721],{"encoding":47},"(K+\\lambda I)^{-1}y",[15,30723,30725,30746],{"className":30724,"ariaHidden":53},[52],[15,30726,30728,30731,30734,30737,30740,30743],{"className":30727},[57],[15,30729],{"className":30730,"style":154},[61],[15,30732,120],{"className":30733},[162],[15,30735,2853],{"className":30736,"style":2901},[66,67],[15,30738],{"className":30739,"style":72},[71],[15,30741,1245],{"className":30742},[76],[15,30744],{"className":30745,"style":72},[71],[15,30747,30749,30752,30755,30758,30793],{"className":30748},[57],[15,30750],{"className":30751,"style":9149},[61],[15,30753,1248],{"className":30754},[66,67],[15,30756,16422],{"className":30757,"style":16492},[66,67],[15,30759,30761,30764],{"className":30760},[169],[15,30762,126],{"className":30763},[169],[15,30765,30767],{"className":30766},[198],[15,30768,30770],{"className":30769},[202],[15,30771,30773],{"className":30772},[206],[15,30774,30776],{"className":30775,"style":6738},[210],[15,30777,30778,30781],{"style":213},[15,30779],{"className":30780,"style":218},[217],[15,30782,30784],{"className":30783},[222,223,224,225],[15,30785,30787,30790],{"className":30786},[66,225],[15,30788,6600],{"className":30789},[66,225],[15,30791,1211],{"className":30792},[66,225],[15,30794,1224],{"className":30795,"style":1515},[66,67],[6174,30797,30798],{},"一般稠密",[6156,30800,30801,30804,30807,30957,30960],{},[6174,30802,30803],{},"SVM",[6174,30805,30806],{},"hinge",[6174,30808,30809],{},[15,30810,30812,30840],{"className":30811,"translate":19},[18],[15,30813,30815],{"className":30814},[23],[25,30816,30817],{"xmlns":27},[29,30818,30819,30837],{},[32,30820,30821,30827,30829,30831],{},[6465,30822,30823,30825],{},[1209,30824,1211],{},[1209,30826,1263],{},[35,30828,1251],{"mathvariant":138},[35,30830,116],{},[131,30832,30833,30835],{},[35,30834,1251],{"mathvariant":138},[1209,30836,1263],{},[45,30838,30839],{"encoding":47},"\\frac12\\|f\\|^2",[15,30841,30843],{"className":30842,"ariaHidden":53},[52],[15,30844,30846,30850,30922,30925,30928],{"className":30845},[57],[15,30847],{"className":30848,"style":30849},[61],"height:1.1901em;vertical-align:-0.345em;",[15,30851,30853,30856,30919],{"className":30852},[66],[15,30854],{"className":30855},[162,23412],[15,30857,30859],{"className":30858},[6465],[15,30860,30862,30910],{"className":30861},[202,1090],[15,30863,30865,30907],{"className":30864},[206],[15,30866,30869,30884,30892],{"className":30867,"style":30868},[210],"height:0.8451em;",[15,30870,30872,30875],{"style":30871},"top:-2.655em;",[15,30873],{"className":30874,"style":1368},[217],[15,30876,30878],{"className":30877},[222,223,224,225],[15,30879,30881],{"className":30880},[66,225],[15,30882,1263],{"className":30883},[66,225],[15,30885,30886,30889],{"style":23440},[15,30887],{"className":30888,"style":1368},[217],[15,30890],{"className":30891,"style":4438},[23447],[15,30893,30895,30898],{"style":30894},"top:-3.394em;",[15,30896],{"className":30897,"style":1368},[217],[15,30899,30901],{"className":30900},[222,223,224,225],[15,30902,30904],{"className":30903},[66,225],[15,30905,1211],{"className":30906},[66,225],[15,30908,1114],{"className":30909},[1113],[15,30911,30913],{"className":30912},[206],[15,30914,30917],{"className":30915,"style":30916},[210],"height:0.345em;",[15,30918],{},[15,30920],{"className":30921},[169,23412],[15,30923,1251],{"className":30924},[66],[15,30926,116],{"className":30927,"style":158},[66,67],[15,30929,30931,30934],{"className":30930},[66],[15,30932,1251],{"className":30933},[66],[15,30935,30937],{"className":30936},[198],[15,30938,30940],{"className":30939},[202],[15,30941,30943],{"className":30942},[206],[15,30944,30946],{"className":30945,"style":6738},[210],[15,30947,30948,30951],{"style":213},[15,30949],{"className":30950,"style":218},[217],[15,30952,30954],{"className":30953},[222,223,224,225],[15,30955,1263],{"className":30956},[66,225],[6174,30958,30959],{},"对偶二次规划",[6174,30961,30962],{},"支持向量稀疏",[6156,30964,30965,30968,30971,31118,31121],{},[6174,30966,30967],{},"核逻辑回归",[6174,30969,30970],{},"logistic",[6174,30972,30973],{},[15,30974,30976,31004],{"className":30975,"translate":19},[18],[15,30977,30979],{"className":30978},[23],[25,30980,30981],{"xmlns":27},[29,30982,30983,31001],{},[32,30984,30985,30991,30993,30995],{},[6465,30986,30987,30989],{},[35,30988,1248],{},[1209,30990,1263],{},[35,30992,1251],{"mathvariant":138},[35,30994,116],{},[131,30996,30997,30999],{},[35,30998,1251],{"mathvariant":138},[1209,31000,1263],{},[45,31002,31003],{"encoding":47},"\\frac{\\lambda}{2}\\|f\\|^2",[15,31005,31007],{"className":31006,"ariaHidden":53},[52],[15,31008,31010,31014,31083,31086,31089],{"className":31009},[57],[15,31011],{"className":31012,"style":31013},[61],"height:1.2251em;vertical-align:-0.345em;",[15,31015,31017,31020,31080],{"className":31016},[66],[15,31018],{"className":31019},[162,23412],[15,31021,31023],{"className":31022},[6465],[15,31024,31026,31072],{"className":31025},[202,1090],[15,31027,31029,31069],{"className":31028},[206],[15,31030,31033,31047,31055],{"className":31031,"style":31032},[210],"height:0.8801em;",[15,31034,31035,31038],{"style":30871},[15,31036],{"className":31037,"style":1368},[217],[15,31039,31041],{"className":31040},[222,223,224,225],[15,31042,31044],{"className":31043},[66,225],[15,31045,1263],{"className":31046},[66,225],[15,31048,31049,31052],{"style":23440},[15,31050],{"className":31051,"style":1368},[217],[15,31053],{"className":31054,"style":4438},[23447],[15,31056,31057,31060],{"style":30894},[15,31058],{"className":31059,"style":1368},[217],[15,31061,31063],{"className":31062},[222,223,224,225],[15,31064,31066],{"className":31065},[66,225],[15,31067,1248],{"className":31068},[66,67,225],[15,31070,1114],{"className":31071},[1113],[15,31073,31075],{"className":31074},[206],[15,31076,31078],{"className":31077,"style":30916},[210],[15,31079],{},[15,31081],{"className":31082},[169,23412],[15,31084,1251],{"className":31085},[66],[15,31087,116],{"className":31088,"style":158},[66,67],[15,31090,31092,31095],{"className":31091},[66],[15,31093,1251],{"className":31094},[66],[15,31096,31098],{"className":31097},[198],[15,31099,31101],{"className":31100},[202],[15,31102,31104],{"className":31103},[206],[15,31105,31107],{"className":31106,"style":6738},[210],[15,31108,31109,31112],{"style":213},[15,31110],{"className":31111,"style":218},[217],[15,31113,31115],{"className":31114},[222,223,224,225],[15,31116,1263],{"className":31117},[66,225],[6174,31119,31120],{},"凸优化",[6174,31122,30798],{},[6156,31124,31125,31128,31131,31134,31277],{},[6174,31126,31127],{},"高斯过程",[6174,31129,31130],{},"平方（高斯似然）",[6174,31132,31133],{},"GP 先验",[6174,31135,31136],{},[15,31137,31139,31177],{"className":31138,"translate":19},[18],[15,31140,31142],{"className":31141},[23],[25,31143,31144],{"xmlns":27},[29,31145,31146,31174],{},[32,31147,31148,31150,31152,31154,31160,31162,31172],{},[39,31149,120],{"stretchy":119},[35,31151,2853],{},[39,31153,1245],{},[131,31155,31156,31158],{},[35,31157,6632],{},[1209,31159,1263],{},[35,31161,16422],{},[131,31163,31164,31166],{},[39,31165,126],{"stretchy":119},[32,31167,31168,31170],{},[39,31169,6600],{},[1209,31171,1211],{},[35,31173,1224],{},[45,31175,31176],{"encoding":47},"(K+\\sigma^2 I)^{-1}y",[15,31178,31180,31201],{"className":31179,"ariaHidden":53},[52],[15,31181,31183,31186,31189,31192,31195,31198],{"className":31182},[57],[15,31184],{"className":31185,"style":154},[61],[15,31187,120],{"className":31188},[162],[15,31190,2853],{"className":31191,"style":2901},[66,67],[15,31193],{"className":31194,"style":72},[71],[15,31196,1245],{"className":31197},[76],[15,31199],{"className":31200,"style":72},[71],[15,31202,31204,31207,31236,31239,31274],{"className":31203},[57],[15,31205],{"className":31206,"style":9149},[61],[15,31208,31210,31213],{"className":31209},[66],[15,31211,6632],{"className":31212,"style":1515},[66,67],[15,31214,31216],{"className":31215},[198],[15,31217,31219],{"className":31218},[202],[15,31220,31222],{"className":31221},[206],[15,31223,31225],{"className":31224,"style":6738},[210],[15,31226,31227,31230],{"style":213},[15,31228],{"className":31229,"style":218},[217],[15,31231,31233],{"className":31232},[222,223,224,225],[15,31234,1263],{"className":31235},[66,225],[15,31237,16422],{"className":31238,"style":16492},[66,67],[15,31240,31242,31245],{"className":31241},[169],[15,31243,126],{"className":31244},[169],[15,31246,31248],{"className":31247},[198],[15,31249,31251],{"className":31250},[202],[15,31252,31254],{"className":31253},[206],[15,31255,31257],{"className":31256,"style":6738},[210],[15,31258,31259,31262],{"style":213},[15,31260],{"className":31261,"style":218},[217],[15,31263,31265],{"className":31264},[222,223,224,225],[15,31266,31268,31271],{"className":31267},[66,225],[15,31269,6600],{"className":31270},[66,225],[15,31272,1211],{"className":31273},[66,225],[15,31275,1224],{"className":31276,"style":1515},[66,67],[6174,31278,30798],{},[6156,31280,31281,31284,31286,31409,31412],{},[6174,31282,31283],{},"平滑样条",[6174,31285,30607],{},[6174,31287,31288],{},[15,31289,31291,31325],{"className":31290,"translate":19},[18],[15,31292,31294],{"className":31293},[23],[25,31295,31296],{"xmlns":27},[29,31297,31298,31322],{},[32,31299,31300,31302,31304,31316],{},[39,31301,29727],{},[39,31303,120],{"stretchy":119},[131,31305,31306,31308],{},[35,31307,116],{},[32,31309,31310,31312,31314],{},[39,31311,120],{"stretchy":119},[35,31313,28853],{},[39,31315,126],{"stretchy":119},[131,31317,31318,31320],{},[39,31319,126],{"stretchy":119},[1209,31321,1263],{},[45,31323,31324],{"encoding":47},"\\int(f^{(m)})^2",[15,31326,31328],{"className":31327,"ariaHidden":53},[52],[15,31329,31331,31335,31339,31342,31380],{"className":31330},[57],[15,31332],{"className":31333,"style":31334},[61],"height:1.1941em;vertical-align:-0.3061em;",[15,31336,29727],{"className":31337,"style":31338},[1339,1470,13696],"margin-right:0.1945em;position:relative;top:-0.0006em;",[15,31340,120],{"className":31341},[162],[15,31343,31345,31348],{"className":31344},[66],[15,31346,116],{"className":31347,"style":158},[66,67],[15,31349,31351],{"className":31350},[198],[15,31352,31354],{"className":31353},[202],[15,31355,31357],{"className":31356},[206],[15,31358,31360],{"className":31359,"style":7064},[210],[15,31361,31362,31365],{"style":213},[15,31363],{"className":31364,"style":218},[217],[15,31366,31368],{"className":31367},[222,223,224,225],[15,31369,31371,31374,31377],{"className":31370},[66,225],[15,31372,120],{"className":31373},[162,225],[15,31375,28853],{"className":31376},[66,67,225],[15,31378,126],{"className":31379},[169,225],[15,31381,31383,31386],{"className":31382},[169],[15,31384,126],{"className":31385},[169],[15,31387,31389],{"className":31388},[198],[15,31390,31392],{"className":31391},[202],[15,31393,31395],{"className":31394},[206],[15,31396,31398],{"className":31397,"style":6738},[210],[15,31399,31400,31403],{"style":213},[15,31401],{"className":31402,"style":218},[217],[15,31404,31406],{"className":31405},[222,223,224,225],[15,31407,1263],{"className":31408},[66,225],[6174,31410,31411],{},"样条系数",[6174,31413,30798],{},[95,31415,31417],{"id":31416},"六计算代价与大规模近似","六、计算代价与大规模近似",[11,31419,31420,31421,31449,31450,31478,31479,31507,31508,31581,31582,31655,31656,31699,31700,3247,31798,31878],{},"表示定理把无穷维问题降到了 ",[15,31422,31424,31437],{"className":31423,"translate":19},[18],[15,31425,31427],{"className":31426},[23],[25,31428,31429],{"xmlns":27},[29,31430,31431,31435],{},[32,31432,31433],{},[35,31434,37],{},[45,31436,37],{"encoding":47},[15,31438,31440],{"className":31439,"ariaHidden":53},[52],[15,31441,31443,31446],{"className":31442},[57],[15,31444],{"className":31445,"style":86},[61],[15,31447,37],{"className":31448},[66,67]," 维，但 ",[15,31451,31453,31466],{"className":31452,"translate":19},[18],[15,31454,31456],{"className":31455},[23],[25,31457,31458],{"xmlns":27},[29,31459,31460,31464],{},[32,31461,31462],{},[35,31463,37],{},[45,31465,37],{"encoding":47},[15,31467,31469],{"className":31468,"ariaHidden":53},[52],[15,31470,31472,31475],{"className":31471},[57],[15,31473],{"className":31474,"style":86},[61],[15,31476,37],{"className":31477},[66,67]," 是训练样本数，可能很大。核矩阵 ",[15,31480,31482,31495],{"className":31481,"translate":19},[18],[15,31483,31485],{"className":31484},[23],[25,31486,31487],{"xmlns":27},[29,31488,31489,31493],{},[32,31490,31491],{},[35,31492,2853],{},[45,31494,2853],{"encoding":47},[15,31496,31498],{"className":31497,"ariaHidden":53},[52],[15,31499,31501,31504],{"className":31500},[57],[15,31502],{"className":31503,"style":490},[61],[15,31505,2853],{"className":31506,"style":2901},[66,67]," 需要 ",[15,31509,31511,31534],{"className":31510,"translate":19},[18],[15,31512,31514],{"className":31513},[23],[25,31515,31516],{"xmlns":27},[29,31517,31518,31532],{},[32,31519,31520,31522,31524,31530],{},[35,31521,22994],{},[39,31523,120],{"stretchy":119},[131,31525,31526,31528],{},[35,31527,37],{},[1209,31529,1263],{},[39,31531,126],{"stretchy":119},[45,31533,23083],{"encoding":47},[15,31535,31537],{"className":31536,"ariaHidden":53},[52],[15,31538,31540,31543,31546,31549,31578],{"className":31539},[57],[15,31541],{"className":31542,"style":9149},[61],[15,31544,22994],{"className":31545,"style":23020},[66,67],[15,31547,120],{"className":31548},[162],[15,31550,31552,31555],{"className":31551},[66],[15,31553,37],{"className":31554},[66,67],[15,31556,31558],{"className":31557},[198],[15,31559,31561],{"className":31560},[202],[15,31562,31564],{"className":31563},[206],[15,31565,31567],{"className":31566,"style":6738},[210],[15,31568,31569,31572],{"style":213},[15,31570],{"className":31571,"style":218},[217],[15,31573,31575],{"className":31574},[222,223,224,225],[15,31576,1263],{"className":31577},[66,225],[15,31579,126],{"className":31580},[169]," 内存；核岭回归的 Cholesky 分解需要 ",[15,31583,31585,31608],{"className":31584,"translate":19},[18],[15,31586,31588],{"className":31587},[23],[25,31589,31590],{"xmlns":27},[29,31591,31592,31606],{},[32,31593,31594,31596,31598,31604],{},[35,31595,22994],{},[39,31597,120],{"stretchy":119},[131,31599,31600,31602],{},[35,31601,37],{},[1209,31603,8673],{},[39,31605,126],{"stretchy":119},[45,31607,23007],{"encoding":47},[15,31609,31611],{"className":31610,"ariaHidden":53},[52],[15,31612,31614,31617,31620,31623,31652],{"className":31613},[57],[15,31615],{"className":31616,"style":9149},[61],[15,31618,22994],{"className":31619,"style":23020},[66,67],[15,31621,120],{"className":31622},[162],[15,31624,31626,31629],{"className":31625},[66],[15,31627,37],{"className":31628},[66,67],[15,31630,31632],{"className":31631},[198],[15,31633,31635],{"className":31634},[202],[15,31636,31638],{"className":31637},[206],[15,31639,31641],{"className":31640,"style":6738},[210],[15,31642,31643,31646],{"style":213},[15,31644],{"className":31645,"style":218},[217],[15,31647,31649],{"className":31648},[222,223,224,225],[15,31650,8673],{"className":31651},[66,225],[15,31653,126],{"className":31654},[169]," 时间；SVM 的通用二次规划更慢。预测一个新点需要计算它与所有训练点的核函数值，代价是 ",[15,31657,31659,31678],{"className":31658,"translate":19},[18],[15,31660,31662],{"className":31661},[23],[25,31663,31664],{"xmlns":27},[29,31665,31666,31676],{},[32,31667,31668,31670,31672,31674],{},[35,31669,22994],{},[39,31671,120],{"stretchy":119},[35,31673,37],{},[39,31675,126],{"stretchy":119},[45,31677,23154],{"encoding":47},[15,31679,31681],{"className":31680,"ariaHidden":53},[52],[15,31682,31684,31687,31690,31693,31696],{"className":31683},[57],[15,31685],{"className":31686,"style":154},[61],[15,31688,22994],{"className":31689,"style":23020},[66,67],[15,31691,120],{"className":31692},[162],[15,31694,37],{"className":31695},[66,67],[15,31697,126],{"className":31698},[169],"（SVM 中是 ",[15,31701,31703,31733],{"className":31702,"translate":19},[18],[15,31704,31706],{"className":31705},[23],[25,31707,31708],{"xmlns":27},[29,31709,31710,31730],{},[32,31711,31712,31714,31716,31728],{},[35,31713,22994],{},[39,31715,120],{"stretchy":119},[1057,31717,31718,31720],{},[35,31719,37],{},[32,31721,31722,31725],{},[35,31723,31724],{"mathvariant":138},"S",[35,31726,31727],{"mathvariant":138},"V",[39,31729,126],{"stretchy":119},[45,31731,31732],{"encoding":47},"O(n_{\\mathrm{SV}})",[15,31734,31736],{"className":31735,"ariaHidden":53},[52],[15,31737,31739,31742,31745,31748,31795],{"className":31738},[57],[15,31740],{"className":31741,"style":154},[61],[15,31743,22994],{"className":31744,"style":23020},[66,67],[15,31746,120],{"className":31747},[162],[15,31749,31751,31754],{"className":31750},[66],[15,31752,37],{"className":31753},[66,67],[15,31755,31757],{"className":31756},[198],[15,31758,31760,31787],{"className":31759},[202,1090],[15,31761,31763,31784],{"className":31762},[206],[15,31764,31766],{"className":31765,"style":3636},[210],[15,31767,31768,31771],{"style":1584},[15,31769],{"className":31770,"style":218},[217],[15,31772,31774],{"className":31773},[222,223,224,225],[15,31775,31777],{"className":31776},[66,225],[15,31778,31780],{"className":31779},[66,225],[15,31781,31783],{"className":31782,"style":1343},[66,4427,225],"SV",[15,31785,1114],{"className":31786},[1113],[15,31788,31790],{"className":31789},[206],[15,31791,31793],{"className":31792,"style":1121},[210],[15,31794],{},[15,31796,126],{"className":31797},[169],[15,31799,31801,31823],{"className":31800,"translate":19},[18],[15,31802,31804],{"className":31803},[23],[25,31805,31806],{"xmlns":27},[29,31807,31808,31820],{},[32,31809,31810],{},[1057,31811,31812,31814],{},[35,31813,37],{},[32,31815,31816,31818],{},[35,31817,31724],{"mathvariant":138},[35,31819,31727],{"mathvariant":138},[45,31821,31822],{"encoding":47},"n_{\\mathrm{SV}}",[15,31824,31826],{"className":31825,"ariaHidden":53},[52],[15,31827,31829,31832],{"className":31828},[57],[15,31830],{"className":31831,"style":10192},[61],[15,31833,31835,31838],{"className":31834},[66],[15,31836,37],{"className":31837},[66,67],[15,31839,31841],{"className":31840},[198],[15,31842,31844,31870],{"className":31843},[202,1090],[15,31845,31847,31867],{"className":31846},[206],[15,31848,31850],{"className":31849,"style":3636},[210],[15,31851,31852,31855],{"style":1584},[15,31853],{"className":31854,"style":218},[217],[15,31856,31858],{"className":31857},[222,223,224,225],[15,31859,31861],{"className":31860},[66,225],[15,31862,31864],{"className":31863},[66,225],[15,31865,31783],{"className":31866,"style":1343},[66,4427,225],[15,31868,1114],{"className":31869},[1113],[15,31871,31873],{"className":31872},[206],[15,31874,31876],{"className":31875,"style":1121},[210],[15,31877],{}," 为支持向量数）。",[11,31880,31881,31882,31910],{},"当 ",[15,31883,31885,31898],{"className":31884,"translate":19},[18],[15,31886,31888],{"className":31887},[23],[25,31889,31890],{"xmlns":27},[29,31891,31892,31896],{},[32,31893,31894],{},[35,31895,37],{},[45,31897,37],{"encoding":47},[15,31899,31901],{"className":31900,"ariaHidden":53},[52],[15,31902,31904,31907],{"className":31903},[57],[15,31905],{"className":31906,"style":86},[61],[15,31908,37],{"className":31909},[66,67]," 很大时，常用的近似方法有：",[16131,31912,31913,32433,32629,32686],{},[3192,31914,31915,31918,31919,31971,31972,32228,32229,7181,32302,3369],{},[2040,31916,31917],{},"Nyström 近似","：选 ",[15,31920,31922,31941],{"className":31921,"translate":19},[18],[15,31923,31925],{"className":31924},[23],[25,31926,31927],{"xmlns":27},[29,31928,31929,31938],{},[32,31930,31931,31933,31936],{},[35,31932,28853],{},[39,31934,31935],{},"≪",[35,31937,37],{},[45,31939,31940],{"encoding":47},"m\\ll n",[15,31942,31944,31962],{"className":31943,"ariaHidden":53},[52],[15,31945,31947,31950,31953,31956,31959],{"className":31946},[57],[15,31948],{"className":31949,"style":3225},[61],[15,31951,28853],{"className":31952},[66,67],[15,31954],{"className":31955,"style":173},[71],[15,31957,31935],{"className":31958},[177],[15,31960],{"className":31961,"style":173},[71],[15,31963,31965,31968],{"className":31964},[57],[15,31966],{"className":31967,"style":86},[61],[15,31969,37],{"className":31970},[66,67]," 个「地标点」，用 ",[15,31973,31975,32034],{"className":31974,"translate":19},[18],[15,31976,31978],{"className":31977},[23],[25,31979,31980],{"xmlns":27},[29,31981,31982,32031],{},[32,31983,31984,31986,31989,32001,32019],{},[35,31985,2853],{},[39,31987,31988],{},"≈",[1057,31990,31991,31993],{},[35,31992,2853],{},[32,31994,31995,31997,31999],{},[35,31996,37],{},[39,31998,335],{"separator":53},[35,32000,28853],{},[1255,32002,32003,32005,32013],{},[35,32004,2853],{},[32,32006,32007,32009,32011],{},[35,32008,28853],{},[39,32010,335],{"separator":53},[35,32012,28853],{},[32,32014,32015,32017],{},[39,32016,6600],{},[1209,32018,1211],{},[1057,32020,32021,32023],{},[35,32022,2853],{},[32,32024,32025,32027,32029],{},[35,32026,28853],{},[39,32028,335],{"separator":53},[35,32030,37],{},[45,32032,32033],{"encoding":47},"K\\approx K_{n,m}K_{m,m}^{-1}K_{m,n}",[15,32035,32037,32055],{"className":32036,"ariaHidden":53},[52],[15,32038,32040,32043,32046,32049,32052],{"className":32039},[57],[15,32041],{"className":32042,"style":490},[61],[15,32044,2853],{"className":32045,"style":2901},[66,67],[15,32047],{"className":32048,"style":173},[71],[15,32050,31988],{"className":32051},[177],[15,32053],{"className":32054,"style":173},[71],[15,32056,32058,32062,32111,32179],{"className":32057},[57],[15,32059],{"className":32060,"style":32061},[61],"height:1.1972em;vertical-align:-0.3831em;",[15,32063,32065,32068],{"className":32064},[66],[15,32066,2853],{"className":32067,"style":2901},[66,67],[15,32069,32071],{"className":32070},[198],[15,32072,32074,32103],{"className":32073},[202,1090],[15,32075,32077,32100],{"className":32076},[206],[15,32078,32080],{"className":32079,"style":2198},[210],[15,32081,32082,32085],{"style":2916},[15,32083],{"className":32084,"style":218},[217],[15,32086,32088],{"className":32087},[222,223,224,225],[15,32089,32091,32094,32097],{"className":32090},[66,225],[15,32092,37],{"className":32093},[66,67,225],[15,32095,335],{"className":32096},[422,225],[15,32098,28853],{"className":32099},[66,67,225],[15,32101,1114],{"className":32102},[1113],[15,32104,32106],{"className":32105},[206],[15,32107,32109],{"className":32108,"style":2715},[210],[15,32110],{},[15,32112,32114,32117],{"className":32113},[66],[15,32115,2853],{"className":32116,"style":2901},[66,67],[15,32118,32120],{"className":32119},[198],[15,32121,32123,32170],{"className":32122},[202,1090],[15,32124,32126,32167],{"className":32125},[206],[15,32127,32129,32150],{"className":32128,"style":6738},[210],[15,32130,32132,32135],{"style":32131},"top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;",[15,32133],{"className":32134,"style":218},[217],[15,32136,32138],{"className":32137},[222,223,224,225],[15,32139,32141,32144,32147],{"className":32140},[66,225],[15,32142,28853],{"className":32143},[66,67,225],[15,32145,335],{"className":32146},[422,225],[15,32148,28853],{"className":32149},[66,67,225],[15,32151,32152,32155],{"style":213},[15,32153],{"className":32154,"style":218},[217],[15,32156,32158],{"className":32157},[222,223,224,225],[15,32159,32161,32164],{"className":32160},[66,225],[15,32162,6600],{"className":32163},[66,225],[15,32165,1211],{"className":32166},[66,225],[15,32168,1114],{"className":32169},[1113],[15,32171,32173],{"className":32172},[206],[15,32174,32177],{"className":32175,"style":32176},[210],"height:0.3831em;",[15,32178],{},[15,32180,32182,32185],{"className":32181},[66],[15,32183,2853],{"className":32184,"style":2901},[66,67],[15,32186,32188],{"className":32187},[198],[15,32189,32191,32220],{"className":32190},[202,1090],[15,32192,32194,32217],{"className":32193},[206],[15,32195,32197],{"className":32196,"style":2198},[210],[15,32198,32199,32202],{"style":2916},[15,32200],{"className":32201,"style":218},[217],[15,32203,32205],{"className":32204},[222,223,224,225],[15,32206,32208,32211,32214],{"className":32207},[66,225],[15,32209,28853],{"className":32210},[66,67,225],[15,32212,335],{"className":32213},[422,225],[15,32215,37],{"className":32216},[66,67,225],[15,32218,1114],{"className":32219},[1113],[15,32221,32223],{"className":32222},[206],[15,32224,32226],{"className":32225,"style":2715},[210],[15,32227],{}," 逼近核矩阵，代价从 ",[15,32230,32232,32255],{"className":32231,"translate":19},[18],[15,32233,32235],{"className":32234},[23],[25,32236,32237],{"xmlns":27},[29,32238,32239,32253],{},[32,32240,32241,32243,32245,32251],{},[35,32242,22994],{},[39,32244,120],{"stretchy":119},[131,32246,32247,32249],{},[35,32248,37],{},[1209,32250,8673],{},[39,32252,126],{"stretchy":119},[45,32254,23007],{"encoding":47},[15,32256,32258],{"className":32257,"ariaHidden":53},[52],[15,32259,32261,32264,32267,32270,32299],{"className":32260},[57],[15,32262],{"className":32263,"style":9149},[61],[15,32265,22994],{"className":32266,"style":23020},[66,67],[15,32268,120],{"className":32269},[162],[15,32271,32273,32276],{"className":32272},[66],[15,32274,37],{"className":32275},[66,67],[15,32277,32279],{"className":32278},[198],[15,32280,32282],{"className":32281},[202],[15,32283,32285],{"className":32284},[206],[15,32286,32288],{"className":32287,"style":6738},[210],[15,32289,32290,32293],{"style":213},[15,32291],{"className":32292,"style":218},[217],[15,32294,32296],{"className":32295},[222,223,224,225],[15,32297,8673],{"className":32298},[66,225],[15,32300,126],{"className":32301},[169],[15,32303,32305,32339],{"className":32304,"translate":19},[18],[15,32306,32308],{"className":32307},[23],[25,32309,32310],{"xmlns":27},[29,32311,32312,32336],{},[32,32313,32314,32316,32318,32320,32326,32328,32334],{},[35,32315,22994],{},[39,32317,120],{"stretchy":119},[35,32319,37],{},[131,32321,32322,32324],{},[35,32323,28853],{},[1209,32325,1263],{},[39,32327,1245],{},[131,32329,32330,32332],{},[35,32331,28853],{},[1209,32333,8673],{},[39,32335,126],{"stretchy":119},[45,32337,32338],{"encoding":47},"O(nm^2+m^3)",[15,32340,32342,32395],{"className":32341,"ariaHidden":53},[52],[15,32343,32345,32348,32351,32354,32357,32386,32389,32392],{"className":32344},[57],[15,32346],{"className":32347,"style":9149},[61],[15,32349,22994],{"className":32350,"style":23020},[66,67],[15,32352,120],{"className":32353},[162],[15,32355,37],{"className":32356},[66,67],[15,32358,32360,32363],{"className":32359},[66],[15,32361,28853],{"className":32362},[66,67],[15,32364,32366],{"className":32365},[198],[15,32367,32369],{"className":32368},[202],[15,32370,32372],{"className":32371},[206],[15,32373,32375],{"className":32374,"style":6738},[210],[15,32376,32377,32380],{"style":213},[15,32378],{"className":32379,"style":218},[217],[15,32381,32383],{"className":32382},[222,223,224,225],[15,32384,1263],{"className":32385},[66,225],[15,32387],{"className":32388,"style":72},[71],[15,32390,1245],{"className":32391},[76],[15,32393],{"className":32394,"style":72},[71],[15,32396,32398,32401,32430],{"className":32397},[57],[15,32399],{"className":32400,"style":9149},[61],[15,32402,32404,32407],{"className":32403},[66],[15,32405,28853],{"className":32406},[66,67],[15,32408,32410],{"className":32409},[198],[15,32411,32413],{"className":32412},[202],[15,32414,32416],{"className":32415},[206],[15,32417,32419],{"className":32418,"style":6738},[210],[15,32420,32421,32424],{"style":213},[15,32422],{"className":32423,"style":218},[217],[15,32425,32427],{"className":32426},[222,223,224,225],[15,32428,8673],{"className":32429},[66,225],[15,32431,126],{"className":32432},[169],[3192,32434,32435,32438,32439,32468,32469,32497,32498,3369],{},[2040,32436,32437],{},"随机傅里叶特征","：用 ",[15,32440,32442,32456],{"className":32441,"translate":19},[18],[15,32443,32445],{"className":32444},[23],[25,32446,32447],{"xmlns":27},[29,32448,32449,32454],{},[32,32450,32451],{},[35,32452,32453],{},"D",[45,32455,32453],{"encoding":47},[15,32457,32459],{"className":32458,"ariaHidden":53},[52],[15,32460,32462,32465],{"className":32461},[57],[15,32463],{"className":32464,"style":490},[61],[15,32466,32453],{"className":32467,"style":23020},[66,67]," 个随机特征近似平移不变核，把核方法转成 ",[15,32470,32472,32485],{"className":32471,"translate":19},[18],[15,32473,32475],{"className":32474},[23],[25,32476,32477],{"xmlns":27},[29,32478,32479,32483],{},[32,32480,32481],{},[35,32482,32453],{},[45,32484,32453],{"encoding":47},[15,32486,32488],{"className":32487,"ariaHidden":53},[52],[15,32489,32491,32494],{"className":32490},[57],[15,32492],{"className":32493,"style":490},[61],[15,32495,32453],{"className":32496,"style":23020},[66,67]," 维的线性方法，代价约为 ",[15,32499,32501,32535],{"className":32500,"translate":19},[18],[15,32502,32504],{"className":32503},[23],[25,32505,32506],{"xmlns":27},[29,32507,32508,32532],{},[32,32509,32510,32512,32514,32516,32522,32524,32530],{},[35,32511,22994],{},[39,32513,120],{"stretchy":119},[35,32515,37],{},[131,32517,32518,32520],{},[35,32519,32453],{},[1209,32521,1263],{},[39,32523,1245],{},[131,32525,32526,32528],{},[35,32527,32453],{},[1209,32529,8673],{},[39,32531,126],{"stretchy":119},[45,32533,32534],{"encoding":47},"O(nD^2+D^3)",[15,32536,32538,32591],{"className":32537,"ariaHidden":53},[52],[15,32539,32541,32544,32547,32550,32553,32582,32585,32588],{"className":32540},[57],[15,32542],{"className":32543,"style":9149},[61],[15,32545,22994],{"className":32546,"style":23020},[66,67],[15,32548,120],{"className":32549},[162],[15,32551,37],{"className":32552},[66,67],[15,32554,32556,32559],{"className":32555},[66],[15,32557,32453],{"className":32558,"style":23020},[66,67],[15,32560,32562],{"className":32561},[198],[15,32563,32565],{"className":32564},[202],[15,32566,32568],{"className":32567},[206],[15,32569,32571],{"className":32570,"style":6738},[210],[15,32572,32573,32576],{"style":213},[15,32574],{"className":32575,"style":218},[217],[15,32577,32579],{"className":32578},[222,223,224,225],[15,32580,1263],{"className":32581},[66,225],[15,32583],{"className":32584,"style":72},[71],[15,32586,1245],{"className":32587},[76],[15,32589],{"className":32590,"style":72},[71],[15,32592,32594,32597,32626],{"className":32593},[57],[15,32595],{"className":32596,"style":9149},[61],[15,32598,32600,32603],{"className":32599},[66],[15,32601,32453],{"className":32602,"style":23020},[66,67],[15,32604,32606],{"className":32605},[198],[15,32607,32609],{"className":32608},[202],[15,32610,32612],{"className":32611},[206],[15,32613,32615],{"className":32614,"style":6738},[210],[15,32616,32617,32620],{"style":213},[15,32618],{"className":32619,"style":218},[217],[15,32621,32623],{"className":32622},[222,223,224,225],[15,32624,8673],{"className":32625},[66,225],[15,32627,126],{"className":32628},[169],[3192,32630,32631,32634,32635,32685],{},[2040,32632,32633],{},"原问题方法","：对线性核或可显式低维特征化的核，直接在原空间优化，避免 ",[15,32636,32638,32655],{"className":32637,"translate":19},[18],[15,32639,32641],{"className":32640},[23],[25,32642,32643],{"xmlns":27},[29,32644,32645,32653],{},[32,32646,32647,32649,32651],{},[35,32648,37],{},[39,32650,41],{},[35,32652,37],{},[45,32654,48],{"encoding":47},[15,32656,32658,32676],{"className":32657,"ariaHidden":53},[52],[15,32659,32661,32664,32667,32670,32673],{"className":32660},[57],[15,32662],{"className":32663,"style":62},[61],[15,32665,37],{"className":32666},[66,67],[15,32668],{"className":32669,"style":72},[71],[15,32671,41],{"className":32672},[76],[15,32674],{"className":32675,"style":72},[71],[15,32677,32679,32682],{"className":32678},[57],[15,32680],{"className":32681,"style":86},[61],[15,32683,37],{"className":32684},[66,67]," 矩阵；",[3192,32687,32688,32691],{},[2040,32689,32690],{},"子集选择 \u002F 预算方法","：只保留一部分训练点作为核中心，控制模型规模。",[11,32693,32694,32695,32723],{},"这些方法改变的是「用多少个核函数」或「如何近似核矩阵」，而不是表示定理本身。表示定理保证精确解在 ",[15,32696,32698,32711],{"className":32697,"translate":19},[18],[15,32699,32701],{"className":32700},[23],[25,32702,32703],{"xmlns":27},[29,32704,32705,32709],{},[32,32706,32707],{},[35,32708,37],{},[45,32710,37],{"encoding":47},[15,32712,32714],{"className":32713,"ariaHidden":53},[52],[15,32715,32717,32720],{"className":32716},[57],[15,32718],{"className":32719,"style":86},[61],[15,32721,37],{"className":32722},[66,67]," 维子空间里；近似方法则用低秩矩阵或低维特征来替代这个子空间的计算。",[95,32725,32727],{"id":32726},"七表示定理与深度学习","七、表示定理与深度学习",[11,32729,32730,32731,32734],{},"表示定理并没有随着深度学习的兴起而过时。相反，它在",[2040,32732,32733],{},"神经正切核","（Neural Tangent Kernel, NTK）的框架里重新出现。",[11,32736,32737],{},"考虑一个无限宽的神经网络，用梯度下降在平方损失上训练。在适当的参数化和训练体制下，网络函数会收敛到核回归的解：",[15,32739,32741],{"className":32740,"translate":19},[294],[15,32742,32744,32823],{"className":32743,"translate":19},[18],[15,32745,32747],{"className":32746},[23],[25,32748,32749],{"xmlns":27,"display":303},[29,32750,32751,32820],{},[32,32752,32753,32755,32757,32759,32761,32763,32776,32778,32780,32782,32784,32786,32788,32800,32802,32804,32806,32816,32818],{},[35,32754,116],{},[39,32756,120],{"stretchy":119},[35,32758,123],{},[39,32760,126],{"stretchy":119},[39,32762,129],{},[1057,32764,32765,32767],{},[35,32766,621],{},[32,32768,32769,32771,32774],{},[35,32770,28530],{"mathvariant":138},[35,32772,32773],{"mathvariant":138},"T",[35,32775,2853],{"mathvariant":138},[39,32777,120],{"stretchy":119},[35,32779,123],{},[39,32781,335],{"separator":53},[35,32783,446],{},[39,32785,126],{"stretchy":119},[39,32787,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,32789,32790,32792],{},[35,32791,2853],{},[32,32793,32794,32796,32798],{},[35,32795,28530],{"mathvariant":138},[35,32797,32773],{"mathvariant":138},[35,32799,2853],{"mathvariant":138},[39,32801,1245],{},[35,32803,1248],{},[35,32805,16422],{},[131,32807,32808,32810],{},[39,32809,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[32,32811,32812,32814],{},[39,32813,6600],{},[1209,32815,1211],{},[35,32817,1224],{},[39,32819,335],{"separator":53},[45,32821,32822],{"encoding":47},"f(x)=k_{\\mathrm{NTK}}(x,X)\\bigl(K_{\\mathrm{NTK}}+\\lambda I\\bigr)^{-1}y,",[15,32824,32826,32853,32986],{"className":32825,"ariaHidden":53},[52],[15,32827,32829,32832,32835,32838,32841,32844,32847,32850],{"className":32828},[57],[15,32830],{"className":32831,"style":154},[61],[15,32833,116],{"className":32834,"style":158},[66,67],[15,32836,120],{"className":32837},[162],[15,32839,123],{"className":32840},[66,67],[15,32842,126],{"className":32843},[169],[15,32845],{"className":32846,"style":173},[71],[15,32848,129],{"className":32849},[177],[15,32851],{"className":32852,"style":173},[71],[15,32854,32856,32859,32907,32910,32913,32916,32919,32922,32925,32931,32977,32980,32983],{"className":32855},[57],[15,32857],{"className":32858,"style":4398},[61],[15,32860,32862,32865],{"className":32861},[66],[15,32863,621],{"className":32864,"style":684},[66,67],[15,32866,32868],{"className":32867},[198],[15,32869,32871,32899],{"className":32870},[202,1090],[15,32872,32874,32896],{"className":32873},[206],[15,32875,32877],{"className":32876,"style":3636},[210],[15,32878,32880,32883],{"style":32879},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[15,32881],{"className":32882,"style":218},[217],[15,32884,32886],{"className":32885},[222,223,224,225],[15,32887,32889],{"className":32888},[66,225],[15,32890,32892],{"className":32891},[66,225],[15,32893,32895],{"className":32894},[66,4427,225],"NTK",[15,32897,1114],{"className":32898},[1113],[15,32900,32902],{"className":32901},[206],[15,32903,32905],{"className":32904,"style":1121},[210],[15,32906],{},[15,32908,120],{"className":32909},[162],[15,32911,123],{"className":32912},[66,67],[15,32914,335],{"className":32915},[422],[15,32917],{"className":32918,"style":697},[71],[15,32920,446],{"className":32921,"style":16492},[66,67],[15,32923,126],{"className":32924},[169],[15,32926,32928],{"className":32927},[162],[15,32929,120],{"className":32930},[1422,1508],[15,32932,32934,32937],{"className":32933},[66],[15,32935,2853],{"className":32936,"style":2901},[66,67],[15,32938,32940],{"className":32939},[198],[15,32941,32943,32969],{"className":32942},[202,1090],[15,32944,32946,32966],{"className":32945},[206],[15,32947,32949],{"className":32948,"style":3636},[210],[15,32950,32951,32954],{"style":2916},[15,32952],{"className":32953,"style":218},[217],[15,32955,32957],{"className":32956},[222,223,224,225],[15,32958,32960],{"className":32959},[66,225],[15,32961,32963],{"className":32962},[66,225],[15,32964,32895],{"className":32965},[66,4427,225],[15,32967,1114],{"className":32968},[1113],[15,32970,32972],{"className":32971},[206],[15,32973,32975],{"className":32974,"style":1121},[210],[15,32976],{},[15,32978],{"className":32979,"style":72},[71],[15,32981,1245],{"className":32982},[76],[15,32984],{"className":32985,"style":72},[71],[15,32987,32989,32992,32995,32998,33036,33039],{"className":32988},[57],[15,32990],{"className":32991,"style":30180},[61],[15,32993,1248],{"className":32994},[66,67],[15,32996,16422],{"className":32997,"style":16492},[66,67],[15,32999,33001,33007],{"className":33000},[169],[15,33002,33004],{"className":33003},[169],[15,33005,126],{"className":33006},[1422,1508],[15,33008,33010],{"className":33009},[198],[15,33011,33013],{"className":33012},[202],[15,33014,33016],{"className":33015},[206],[15,33017,33019],{"className":33018,"style":21266},[210],[15,33020,33021,33024],{"style":21269},[15,33022],{"className":33023,"style":218},[217],[15,33025,33027],{"className":33026},[222,223,224,225],[15,33028,33030,33033],{"className":33029},[66,225],[15,33031,6600],{"className":33032},[66,225],[15,33034,1211],{"className":33035},[66,225],[15,33037,1224],{"className":33038,"style":1515},[66,67],[15,33040,335],{"className":33041},[422],[11,33043,425,33044,33126,33127,33209],{},[15,33045,33047,33071],{"className":33046,"translate":19},[18],[15,33048,33050],{"className":33049},[23],[25,33051,33052],{"xmlns":27},[29,33053,33054,33068],{},[32,33055,33056],{},[1057,33057,33058,33060],{},[35,33059,621],{},[32,33061,33062,33064,33066],{},[35,33063,28530],{"mathvariant":138},[35,33065,32773],{"mathvariant":138},[35,33067,2853],{"mathvariant":138},[45,33069,33070],{"encoding":47},"k_{\\mathrm{NTK}}",[15,33072,33074],{"className":33073,"ariaHidden":53},[52],[15,33075,33077,33080],{"className":33076},[57],[15,33078],{"className":33079,"style":3937},[61],[15,33081,33083,33086],{"className":33082},[66],[15,33084,621],{"className":33085,"style":684},[66,67],[15,33087,33089],{"className":33088},[198],[15,33090,33092,33118],{"className":33091},[202,1090],[15,33093,33095,33115],{"className":33094},[206],[15,33096,33098],{"className":33097,"style":3636},[210],[15,33099,33100,33103],{"style":32879},[15,33101],{"className":33102,"style":218},[217],[15,33104,33106],{"className":33105},[222,223,224,225],[15,33107,33109],{"className":33108},[66,225],[15,33110,33112],{"className":33111},[66,225],[15,33113,32895],{"className":33114},[66,4427,225],[15,33116,1114],{"className":33117},[1113],[15,33119,33121],{"className":33120},[206],[15,33122,33124],{"className":33123,"style":1121},[210],[15,33125],{}," 是网络对应的神经正切核，",[15,33128,33130,33154],{"className":33129,"translate":19},[18],[15,33131,33133],{"className":33132},[23],[25,33134,33135],{"xmlns":27},[29,33136,33137,33151],{},[32,33138,33139],{},[1057,33140,33141,33143],{},[35,33142,2853],{},[32,33144,33145,33147,33149],{},[35,33146,28530],{"mathvariant":138},[35,33148,32773],{"mathvariant":138},[35,33150,2853],{"mathvariant":138},[45,33152,33153],{"encoding":47},"K_{\\mathrm{NTK}}",[15,33155,33157],{"className":33156,"ariaHidden":53},[52],[15,33158,33160,33163],{"className":33159},[57],[15,33161],{"className":33162,"style":1076},[61],[15,33164,33166,33169],{"className":33165},[66],[15,33167,2853],{"className":33168,"style":2901},[66,67],[15,33170,33172],{"className":33171},[198],[15,33173,33175,33201],{"className":33174},[202,1090],[15,33176,33178,33198],{"className":33177},[206],[15,33179,33181],{"className":33180,"style":3636},[210],[15,33182,33183,33186],{"style":2916},[15,33184],{"className":33185,"style":218},[217],[15,33187,33189],{"className":33188},[222,223,224,225],[15,33190,33192],{"className":33191},[66,225],[15,33193,33195],{"className":33194},[66,225],[15,33196,32895],{"className":33197},[66,4427,225],[15,33199,1114],{"className":33200},[1113],[15,33202,33204],{"className":33203},[206],[15,33205,33207],{"className":33206,"style":1121},[210],[15,33208],{}," 是它在训练点上的核矩阵。这个式子与式 (3) 完全同构：预测值是训练点上核函数值的线性组合。",[11,33211,33212],{},"这解释了为什么在 NTK 机制里，深度网络的预测可以写成「训练目标的核加权和」。但也要注意它的边界：NTK 描述的是无限宽、线性化、平方损失、梯度流（或小学习率）的极限；它并不是对一般深度网络特征学习的完整解释。表示定理在这里的作用，是给这个极限提供一个清晰的结构语言。",[95,33214,33216],{"id":33215},"九结语","九、结语",[11,33218,33219,33220,33223],{},"表示定理的核心是一句几何话：",[2040,33221,33222],{},"损失函数看不见训练点之外的正交分量，而正则项会惩罚它。"," 因此，最优解没有理由保留这个分量；在严格递增的正则项下，它必然落在训练点核函数张成的子空间里。",[11,33225,33226,33227,33341],{},"这个论证只用了两件事：再生性 ",[15,33228,33230,33272],{"className":33229,"translate":19},[18],[15,33231,33233],{"className":33232},[23],[25,33234,33235],{"xmlns":27},[29,33236,33237,33269],{},[32,33238,33239,33241,33243,33245,33247,33249,33251,33253,33255,33257,33259,33261,33263,33265,33267],{},[35,33240,116],{},[39,33242,120],{"stretchy":119},[35,33244,123],{},[39,33246,126],{"stretchy":119},[39,33248,129],{},[39,33250,642],{"stretchy":119},[35,33252,116],{},[39,33254,335],{"separator":53},[35,33256,621],{},[39,33258,120],{"stretchy":119},[39,33260,3266],{},[39,33262,335],{"separator":53},[35,33264,123],{},[39,33266,126],{"stretchy":119},[39,33268,667],{"stretchy":119},[45,33270,33271],{"encoding":47},"f(x)=\\langle f,k(\\cdot,x)\\rangle",[15,33273,33275,33302],{"className":33274,"ariaHidden":53},[52],[15,33276,33278,33281,33284,33287,33290,33293,33296,33299],{"className":33277},[57],[15,33279],{"className":33280,"style":154},[61],[15,33282,116],{"className":33283,"style":158},[66,67],[15,33285,120],{"className":33286},[162],[15,33288,123],{"className":33289},[66,67],[15,33291,126],{"className":33292},[169],[15,33294],{"className":33295,"style":173},[71],[15,33297,129],{"className":33298},[177],[15,33300],{"className":33301,"style":173},[71],[15,33303,33305,33308,33311,33314,33317,33320,33323,33326,33329,33332,33335,33338],{"className":33304},[57],[15,33306],{"className":33307,"style":154},[61],[15,33309,642],{"className":33310},[162],[15,33312,116],{"className":33313,"style":158},[66,67],[15,33315,335],{"className":33316},[422],[15,33318],{"className":33319,"style":697},[71],[15,33321,621],{"className":33322,"style":684},[66,67],[15,33324,120],{"className":33325},[162],[15,33327,3266],{"className":33328},[66],[15,33330,335],{"className":33331},[422],[15,33333],{"className":33334,"style":697},[71],[15,33336,123],{"className":33337},[66,67],[15,33339,811],{"className":33340},[169],"，以及范数的勾股分解。它不需要损失凸、不需要可微、不需要核矩阵可逆。也正因如此，表示定理能同时覆盖核岭回归、SVM、核逻辑回归、高斯过程和平滑样条。",[11,33343,33344,33345],{},"但它的边界同样清楚：正则项必须是范数的非减函数；若只是非减而非严格递增，定理给出的是「存在一个」这样的解，而不是「每一个」解都如此。非单调的正则项可以让解离开训练点子空间，4.2 节的反例说明这不是技术细节，而是定理的本质。",[2040,33346,33347,33348,33376,33377,33405],{},"无穷维并没有消失，它只是被 ",[15,33349,33351,33364],{"className":33350,"translate":19},[18],[15,33352,33354],{"className":33353},[23],[25,33355,33356],{"xmlns":27},[29,33357,33358,33362],{},[32,33359,33360],{},[35,33361,37],{},[45,33363,37],{"encoding":47},[15,33365,33367],{"className":33366,"ariaHidden":53},[52],[15,33368,33370,33373],{"className":33369},[57],[15,33371],{"className":33372,"style":86},[61],[15,33374,37],{"className":33375},[66,67]," 个训练点压成了 ",[15,33378,33380,33393],{"className":33379,"translate":19},[18],[15,33381,33383],{"className":33382},[23],[25,33384,33385],{"xmlns":27},[29,33386,33387,33391],{},[32,33388,33389],{},[35,33390,37],{},[45,33392,37],{"encoding":47},[15,33394,33396],{"className":33395,"ariaHidden":53},[52],[15,33397,33399,33402],{"className":33398},[57],[15,33400],{"className":33401,"style":86},[61],[15,33403,37],{"className":33404},[66,67]," 维。",{"title":33407,"searchDepth":33408,"depth":33408,"links":33409},"",2,[33410,33411,33417,33423,33428,33436,33437,33438],{"id":97,"depth":33408,"text":98},{"id":1901,"depth":33408,"text":1902,"children":33412},[33413,33415,33416],{"id":1906,"depth":33414,"text":1907},3,{"id":5307,"depth":33414,"text":5308},{"id":6147,"depth":33414,"text":6148},{"id":7243,"depth":33408,"text":7244,"children":33418},[33419,33420,33421,33422],{"id":7247,"depth":33414,"text":7248},{"id":9354,"depth":33414,"text":9355},{"id":13602,"depth":33414,"text":13603},{"id":16117,"depth":33414,"text":16118},{"id":17686,"depth":33408,"text":17687,"children":33424},[33425,33426,33427],{"id":17690,"depth":33414,"text":17691},{"id":18067,"depth":33414,"text":18068},{"id":20145,"depth":33414,"text":20146},{"id":20773,"depth":33408,"text":20774,"children":33429},[33430,33431,33432,33433,33434,33435],{"id":20777,"depth":33414,"text":20778},{"id":23179,"depth":33414,"text":23180},{"id":27012,"depth":33414,"text":27013},{"id":28362,"depth":33414,"text":28363},{"id":29570,"depth":33414,"text":29571},{"id":30575,"depth":33414,"text":30576},{"id":31416,"depth":33408,"text":31417},{"id":32726,"depth":33408,"text":32727},{"id":33215,"depth":33408,"text":33216},"核方法最反直觉的一点是：你明明在一个可能是无限维的函数空间里做优化，最后却只需要解一个 n×nn\\times nn×n 的线性系统或二次规划。把这件事说清楚的，就是表示定理。",false,true,"md",null,{},"\u002Fblog\u002F2025\u002F2025-09-11-representer-theorem",{"title":6,"description":33439},"blog\u002F2025\u002F2025-09-11-representer-theorem","表示定理是核方法的一个基石性结论。它指出：对于一个在再生核希尔伯特空间（RKHS）中带有范数正则化的经验风险最小化问题，其最优解一定可以写成核函数在训练样本点上的线性组合。",[33450],"machine-learning","2025-09-11T21:00:12+08:00","mTnCagDDEj7M4wOzjAllewJKiowioNxQIPyKgUsW0PI",{"id":33454,"title":33455,"body":33456,"description":33407,"draft":33440,"enableCodeHighlight":33440,"enableCodeHightlight":33440,"enableComment":33441,"enableD2":33440,"enableKatex":33441,"extension":33442,"image":33443,"important":33440,"location":33443,"meta":61569,"navigation":33441,"ogImage":33443,"path":61570,"seo":61571,"stem":61572,"summary":61573,"tags":61574,"time":61576,"__hash__":61577},"blog\u002Fblog\u002F2025\u002F2025-09-18-the-stirling-formula.md","斯特林公式",{"type":8,"value":33457,"toc":61565},[33458,33461,33464,33705,33708,34195,34324,35181,35902,35905,36484,36487,37549,37552,37640,39173,39283,40742,40745,40890,40892,41360,41749,42149,42857,42860,43400,43642,44936,45095,45098,45637,45794,46326,46428,47873,47876,48259,48262,48264,48423,48722,48725,49145,49148,49827,49830,50163,50166,51258,51261,51263,51510,51513,51776,51779,51782,52325,52837,52969,52972,53291,53364,54061,54637,54640,55153,55184,55187,55432,55435,55740,55743,56073,56252,56634,56735,57213,57215,57806,57972,57975,58320,58323,58326,58708,58906,59251,59681,59684,60116,60119,60344,60347,60898,60901,61012,61044,61321,61324],[95,33459,33460],{"id":33460},"沃利斯公式",[11,33462,33463],{},"我们先从一个定积分开始：",[15,33465,33467],{"className":33466,"translate":19},[294],[15,33468,33470,33514],{"className":33469,"translate":19},[18],[15,33471,33473],{"className":33472},[23],[25,33474,33475],{"xmlns":27,"display":303},[29,33476,33477,33511],{},[32,33478,33479,33492,33503,33505,33507,33509],{},[1255,33480,33481,33483,33485],{},[39,33482,29727],{},[1209,33484,1804],{},[6465,33486,33487,33490],{},[35,33488,33489],{},"π",[1209,33491,1263],{},[131,33493,33494,33501],{},[32,33495,33496,33499],{},[35,33497,33498],{},"sin",[39,33500,1170],{},[35,33502,37],{},[35,33504,123],{},[1274,33506,2462],{},[35,33508,458],{},[35,33510,123],{},[45,33512,33513],{"encoding":47},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx",[15,33515,33517],{"className":33516,"ariaHidden":53},[52],[15,33518,33520,33524,33656,33659,33690,33693,33696,33699,33702],{"className":33519},[57],[15,33521],{"className":33522,"style":33523},[61],"height:2.5089em;vertical-align:-0.9119em;",[15,33525,33527,33530],{"className":33526},[1339],[15,33528,29727],{"className":33529,"style":29791},[1339,1470,1471],[15,33531,33533],{"className":33532},[198],[15,33534,33536,33648],{"className":33535},[202,1090],[15,33537,33539,33645],{"className":33538},[206],[15,33540,33543,33558],{"className":33541,"style":33542},[210],"height:1.597em;",[15,33544,33546,33549],{"style":33545},"top:-2.088em;margin-left:-0.4445em;margin-right:0.05em;",[15,33547],{"className":33548,"style":1368},[217],[15,33550,33552],{"className":33551},[222,223,224,225],[15,33553,33555],{"className":33554},[66,225],[15,33556,1804],{"className":33557},[66,225],[15,33559,33561,33564],{"style":33560},"top:-4.1129em;margin-right:0.05em;",[15,33562],{"className":33563,"style":1368},[217],[15,33565,33567],{"className":33566},[222,223,224,225],[15,33568,33570],{"className":33569},[66,225],[15,33571,33573,33577,33642],{"className":33572},[66,225],[15,33574],{"className":33575},[162,23412,222,3677,33576],"size6",[15,33578,33580],{"className":33579},[6465],[15,33581,33583,33633],{"className":33582},[202,1090],[15,33584,33586,33630],{"className":33585},[206],[15,33587,33590,33605,33615],{"className":33588,"style":33589},[210],"height:0.6915em;",[15,33591,33593,33596],{"style":33592},"top:-2.656em;",[15,33594],{"className":33595,"style":1368},[217],[15,33597,33599],{"className":33598},[222,3677,1508,225],[15,33600,33602],{"className":33601},[66,225],[15,33603,1263],{"className":33604},[66,225],[15,33606,33608,33611],{"style":33607},"top:-3.2255em;",[15,33609],{"className":33610,"style":1368},[217],[15,33612],{"className":33613,"style":33614},[23447,225],"border-bottom-width:0.049em;",[15,33616,33618,33621],{"style":33617},"top:-3.384em;",[15,33619],{"className":33620,"style":1368},[217],[15,33622,33624],{"className":33623},[222,3677,1508,225],[15,33625,33627],{"className":33626},[66,225],[15,33628,33489],{"className":33629,"style":1515},[66,67,225],[15,33631,1114],{"className":33632},[1113],[15,33634,33636],{"className":33635},[206],[15,33637,33640],{"className":33638,"style":33639},[210],"height:0.344em;",[15,33641],{},[15,33643],{"className":33644},[169,23412,222,3677,33576],[15,33646,1114],{"className":33647},[1113],[15,33649,33651],{"className":33650},[206],[15,33652,33654],{"className":33653,"style":29838},[210],[15,33655],{},[15,33657],{"className":33658,"style":697},[71],[15,33660,33662,33665],{"className":33661},[1339],[15,33663,33498],{"className":33664},[1339],[15,33666,33668],{"className":33667},[198],[15,33669,33671],{"className":33670},[202],[15,33672,33674],{"className":33673},[206],[15,33675,33678],{"className":33676,"style":33677},[210],"height:0.7222em;",[15,33679,33681,33684],{"style":33680},"top:-3.1208em;margin-right:0.05em;",[15,33682],{"className":33683,"style":218},[217],[15,33685,33687],{"className":33686},[222,223,224,225],[15,33688,37],{"className":33689},[66,67,225],[15,33691],{"className":33692,"style":697},[71],[15,33694,123],{"className":33695},[66,67],[15,33697],{"className":33698,"style":697},[71],[15,33700,458],{"className":33701},[66,67],[15,33703,123],{"className":33704},[66,67],[11,33706,33707],{},"我们可以推导出如下结果：",[15,33709,33711],{"className":33710,"translate":19},[294],[15,33712,33714,33805],{"className":33713,"translate":19},[18],[15,33715,33717],{"className":33716},[23],[25,33718,33719],{"xmlns":27,"display":303},[29,33720,33721,33802],{},[32,33722,33723,33735,33745,33747,33749,33751,33753,33755,33757,33769,33785,33787,33789,33791,33793,33796,33798,33800],{},[1255,33724,33725,33727,33729],{},[39,33726,29727],{},[1209,33728,1804],{},[6465,33730,33731,33733],{},[35,33732,33489],{},[1209,33734,1263],{},[131,33736,33737,33743],{},[32,33738,33739,33741],{},[35,33740,33498],{},[39,33742,1170],{},[35,33744,37],{},[35,33746,123],{},[1274,33748,2462],{},[35,33750,458],{},[35,33752,123],{},[39,33754,129],{},[39,33756,6600],{},[1255,33758,33759,33761,33763],{},[39,33760,29727],{},[1209,33762,1804],{},[6465,33764,33765,33767],{},[35,33766,33489],{},[1209,33768,1263],{},[131,33770,33771,33777],{},[32,33772,33773,33775],{},[35,33774,33498],{},[39,33776,1170],{},[32,33778,33779,33781,33783],{},[35,33780,37],{},[39,33782,6600],{},[1209,33784,1211],{},[35,33786,123],{},[1274,33788,2462],{},[35,33790,458],{},[39,33792,120],{"stretchy":119},[35,33794,33795],{},"cos",[39,33797,1170],{},[35,33799,123],{},[39,33801,126],{"stretchy":119},[45,33803,33804],{"encoding":47},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx = -\\int_{0}^{\\frac{\\pi}{2}} \\sin^{n-1} x \\, d(\\cos x)",[15,33806,33808,33992],{"className":33807,"ariaHidden":53},[52],[15,33809,33811,33814,33936,33939,33968,33971,33974,33977,33980,33983,33986,33989],{"className":33810},[57],[15,33812],{"className":33813,"style":33523},[61],[15,33815,33817,33820],{"className":33816},[1339],[15,33818,29727],{"className":33819,"style":29791},[1339,1470,1471],[15,33821,33823],{"className":33822},[198],[15,33824,33826,33928],{"className":33825},[202,1090],[15,33827,33829,33925],{"className":33828},[206],[15,33830,33832,33846],{"className":33831,"style":33542},[210],[15,33833,33834,33837],{"style":33545},[15,33835],{"className":33836,"style":1368},[217],[15,33838,33840],{"className":33839},[222,223,224,225],[15,33841,33843],{"className":33842},[66,225],[15,33844,1804],{"className":33845},[66,225],[15,33847,33848,33851],{"style":33560},[15,33849],{"className":33850,"style":1368},[217],[15,33852,33854],{"className":33853},[222,223,224,225],[15,33855,33857],{"className":33856},[66,225],[15,33858,33860,33863,33922],{"className":33859},[66,225],[15,33861],{"className":33862},[162,23412,222,3677,33576],[15,33864,33866],{"className":33865},[6465],[15,33867,33869,33914],{"className":33868},[202,1090],[15,33870,33872,33911],{"className":33871},[206],[15,33873,33875,33889,33897],{"className":33874,"style":33589},[210],[15,33876,33877,33880],{"style":33592},[15,33878],{"className":33879,"style":1368},[217],[15,33881,33883],{"className":33882},[222,3677,1508,225],[15,33884,33886],{"className":33885},[66,225],[15,33887,1263],{"className":33888},[66,225],[15,33890,33891,33894],{"style":33607},[15,33892],{"className":33893,"style":1368},[217],[15,33895],{"className":33896,"style":33614},[23447,225],[15,33898,33899,33902],{"style":33617},[15,33900],{"className":33901,"style":1368},[217],[15,33903,33905],{"className":33904},[222,3677,1508,225],[15,33906,33908],{"className":33907},[66,225],[15,33909,33489],{"className":33910,"style":1515},[66,67,225],[15,33912,1114],{"className":33913},[1113],[15,33915,33917],{"className":33916},[206],[15,33918,33920],{"className":33919,"style":33639},[210],[15,33921],{},[15,33923],{"className":33924},[169,23412,222,3677,33576],[15,33926,1114],{"className":33927},[1113],[15,33929,33931],{"className":33930},[206],[15,33932,33934],{"className":33933,"style":29838},[210],[15,33935],{},[15,33937],{"className":33938,"style":697},[71],[15,33940,33942,33945],{"className":33941},[1339],[15,33943,33498],{"className":33944},[1339],[15,33946,33948],{"className":33947},[198],[15,33949,33951],{"className":33950},[202],[15,33952,33954],{"className":33953},[206],[15,33955,33957],{"className":33956,"style":33677},[210],[15,33958,33959,33962],{"style":33680},[15,33960],{"className":33961,"style":218},[217],[15,33963,33965],{"className":33964},[222,223,224,225],[15,33966,37],{"className":33967},[66,67,225],[15,33969],{"className":33970,"style":697},[71],[15,33972,123],{"className":33973},[66,67],[15,33975],{"className":33976,"style":697},[71],[15,33978,458],{"className":33979},[66,67],[15,33981,123],{"className":33982},[66,67],[15,33984],{"className":33985,"style":173},[71],[15,33987,129],{"className":33988},[177],[15,33990],{"className":33991,"style":173},[71],[15,33993,33995,33998,34001,34004,34126,34129,34168,34171,34174,34177,34180,34183,34186,34189,34192],{"className":33994},[57],[15,33996],{"className":33997,"style":33523},[61],[15,33999,6600],{"className":34000},[66],[15,34002],{"className":34003,"style":697},[71],[15,34005,34007,34010],{"className":34006},[1339],[15,34008,29727],{"className":34009,"style":29791},[1339,1470,1471],[15,34011,34013],{"className":34012},[198],[15,34014,34016,34118],{"className":34015},[202,1090],[15,34017,34019,34115],{"className":34018},[206],[15,34020,34022,34036],{"className":34021,"style":33542},[210],[15,34023,34024,34027],{"style":33545},[15,34025],{"className":34026,"style":1368},[217],[15,34028,34030],{"className":34029},[222,223,224,225],[15,34031,34033],{"className":34032},[66,225],[15,34034,1804],{"className":34035},[66,225],[15,34037,34038,34041],{"style":33560},[15,34039],{"className":34040,"style":1368},[217],[15,34042,34044],{"className":34043},[222,223,224,225],[15,34045,34047],{"className":34046},[66,225],[15,34048,34050,34053,34112],{"className":34049},[66,225],[15,34051],{"className":34052},[162,23412,222,3677,33576],[15,34054,34056],{"className":34055},[6465],[15,34057,34059,34104],{"className":34058},[202,1090],[15,34060,34062,34101],{"className":34061},[206],[15,34063,34065,34079,34087],{"className":34064,"style":33589},[210],[15,34066,34067,34070],{"style":33592},[15,34068],{"className":34069,"style":1368},[217],[15,34071,34073],{"className":34072},[222,3677,1508,225],[15,34074,34076],{"className":34075},[66,225],[15,34077,1263],{"className":34078},[66,225],[15,34080,34081,34084],{"style":33607},[15,34082],{"className":34083,"style":1368},[217],[15,34085],{"className":34086,"style":33614},[23447,225],[15,34088,34089,34092],{"style":33617},[15,34090],{"className":34091,"style":1368},[217],[15,34093,34095],{"className":34094},[222,3677,1508,225],[15,34096,34098],{"className":34097},[66,225],[15,34099,33489],{"className":34100,"style":1515},[66,67,225],[15,34102,1114],{"className":34103},[1113],[15,34105,34107],{"className":34106},[206],[15,34108,34110],{"className":34109,"style":33639},[210],[15,34111],{},[15,34113],{"className":34114},[169,23412,222,3677,33576],[15,34116,1114],{"className":34117},[1113],[15,34119,34121],{"className":34120},[206],[15,34122,34124],{"className":34123,"style":29838},[210],[15,34125],{},[15,34127],{"className":34128,"style":697},[71],[15,34130,34132,34135],{"className":34131},[1339],[15,34133,33498],{"className":34134},[1339],[15,34136,34138],{"className":34137},[198],[15,34139,34141],{"className":34140},[202],[15,34142,34144],{"className":34143},[206],[15,34145,34148],{"className":34146,"style":34147},[210],"height:0.8719em;",[15,34149,34150,34153],{"style":33680},[15,34151],{"className":34152,"style":218},[217],[15,34154,34156],{"className":34155},[222,223,224,225],[15,34157,34159,34162,34165],{"className":34158},[66,225],[15,34160,37],{"className":34161},[66,67,225],[15,34163,6600],{"className":34164},[76,225],[15,34166,1211],{"className":34167},[66,225],[15,34169],{"className":34170,"style":697},[71],[15,34172,123],{"className":34173},[66,67],[15,34175],{"className":34176,"style":697},[71],[15,34178,458],{"className":34179},[66,67],[15,34181,120],{"className":34182},[162],[15,34184,33795],{"className":34185},[1339],[15,34187],{"className":34188,"style":697},[71],[15,34190,123],{"className":34191},[66,67],[15,34193,126],{"className":34194},[169],[11,34196,34197,34198,34323],{},"利用分部积分法，其中 ",[15,34199,34201,34243],{"className":34200,"translate":19},[18],[15,34202,34204],{"className":34203},[23],[25,34205,34206],{"xmlns":27},[29,34207,34208,34240],{},[32,34209,34210,34212,34215,34217,34219,34222,34224,34226,34228,34230,34232,34234,34236,34238],{},[39,34211,29727],{},[35,34213,34214],{},"u",[1274,34216,2462],{},[35,34218,458],{},[35,34220,34221],{},"v",[39,34223,129],{},[35,34225,34214],{},[35,34227,34221],{},[39,34229,6600],{},[39,34231,29727],{},[35,34233,34221],{},[1274,34235,2462],{},[35,34237,458],{},[35,34239,34214],{},[45,34241,34242],{"encoding":47},"\\int u \\, dv = uv - \\int v \\, du",[15,34244,34246,34280,34299],{"className":34245,"ariaHidden":53},[52],[15,34247,34249,34253,34256,34259,34262,34265,34268,34271,34274,34277],{"className":34248},[57],[15,34250],{"className":34251,"style":34252},[61],"height:1.1111em;vertical-align:-0.3061em;",[15,34254,29727],{"className":34255,"style":31338},[1339,1470,13696],[15,34257],{"className":34258,"style":697},[71],[15,34260,34214],{"className":34261},[66,67],[15,34263],{"className":34264,"style":697},[71],[15,34266,458],{"className":34267},[66,67],[15,34269,34221],{"className":34270,"style":1515},[66,67],[15,34272],{"className":34273,"style":173},[71],[15,34275,129],{"className":34276},[177],[15,34278],{"className":34279,"style":173},[71],[15,34281,34283,34286,34290,34293,34296],{"className":34282},[57],[15,34284],{"className":34285,"style":62},[61],[15,34287,34289],{"className":34288,"style":1515},[66,67],"uv",[15,34291],{"className":34292,"style":72},[71],[15,34294,6600],{"className":34295},[76],[15,34297],{"className":34298,"style":72},[71],[15,34300,34302,34305,34308,34311,34314,34317,34320],{"className":34301},[57],[15,34303],{"className":34304,"style":34252},[61],[15,34306,29727],{"className":34307,"style":31338},[1339,1470,13696],[15,34309],{"className":34310,"style":697},[71],[15,34312,34221],{"className":34313,"style":1515},[66,67],[15,34315],{"className":34316,"style":697},[71],[15,34318,458],{"className":34319},[66,67],[15,34321,34214],{"className":34322},[66,67],"，可得：",[15,34325,34327],{"className":34326,"translate":19},[294],[15,34328,34330,34485],{"className":34329,"translate":19},[18],[15,34331,34333],{"className":34332},[23],[25,34334,34335],{"xmlns":27,"display":303},[29,34336,34337,34482],{},[32,34338,34339,34341,34353,34369,34371,34373,34375,34377,34379,34381,34383,34385,34387,34389,34391,34421,34434,34436,34448,34450,34452,34454,34456,34458],{},[39,34340,6600],{},[1255,34342,34343,34345,34347],{},[39,34344,29727],{},[1209,34346,1804],{},[6465,34348,34349,34351],{},[35,34350,33489],{},[1209,34352,1263],{},[131,34354,34355,34361],{},[32,34356,34357,34359],{},[35,34358,33498],{},[39,34360,1170],{},[32,34362,34363,34365,34367],{},[35,34364,37],{},[39,34366,6600],{},[1209,34368,1211],{},[35,34370,123],{},[1274,34372,2462],{},[35,34374,458],{},[39,34376,120],{"stretchy":119},[35,34378,33795],{},[39,34380,1170],{},[35,34382,123],{},[39,34384,126],{"stretchy":119},[71,34386],{"linebreak":10451},[39,34388,129],{},[39,34390,6600],{},[32,34392,34393,34395,34411,34413,34415,34417,34419],{},[39,34394,120],{"fence":53},[131,34396,34397,34403],{},[32,34398,34399,34401],{},[35,34400,33498],{},[39,34402,1170],{},[32,34404,34405,34407,34409],{},[35,34406,37],{},[39,34408,6600],{},[1209,34410,1211],{},[35,34412,123],{},[35,34414,33795],{},[39,34416,1170],{},[35,34418,123],{},[39,34420,126],{"fence":53},[1255,34422,34423,34426,34428],{},[39,34424,34425],{"fence":119,"stretchy":53,"minsize":4543,"maxsize":4543},"∣",[1209,34427,1804],{},[6465,34429,34430,34432],{},[35,34431,33489],{},[1209,34433,1263],{},[39,34435,1245],{},[1255,34437,34438,34440,34442],{},[39,34439,29727],{},[1209,34441,1804],{},[6465,34443,34444,34446],{},[35,34445,33489],{},[1209,34447,1263],{},[35,34449,33795],{},[39,34451,1170],{},[35,34453,123],{},[1274,34455,2462],{},[35,34457,458],{},[32,34459,34460,34462,34478,34480],{},[39,34461,120],{"fence":53},[131,34463,34464,34470],{},[32,34465,34466,34468],{},[35,34467,33498],{},[39,34469,1170],{},[32,34471,34472,34474,34476],{},[35,34473,37],{},[39,34475,6600],{},[1209,34477,1211],{},[35,34479,123],{},[39,34481,126],{"fence":53},[45,34483,34484],{"encoding":47},"-\\int_{0}^{\\frac{\\pi}{2}} \\sin^{n-1} x \\, d(\\cos x) \\\\ = -\\left( \\sin^{n-1} x \\cos x \\right) \\Big|_{0}^{\\frac{\\pi}{2}} + \\int_{0}^{\\frac{\\pi}{2}} \\cos x \\, d\\left(\\sin^{n-1} x 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\\sin^n x \\, dx = (n-1) \\int_{0}^{\\frac{\\pi}{2}} \\left( \\sin^{n-2} x - \\sin^n x \\right) \\, 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\\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx = (n-1) \\int_{0}^{\\frac{\\pi}{2}} \\sin^{n-2} x \\, dx \\\\ \\implies \\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx = \\frac{n-1}{n} \\int_{0}^{\\frac{\\pi}{2}} \\sin^{n-2} x \\, 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为偶数时，即 ",[15,37584,37586,37606],{"className":37585,"translate":19},[18],[15,37587,37589],{"className":37588},[23],[25,37590,37591],{"xmlns":27},[29,37592,37593,37603],{},[32,37594,37595,37597,37599,37601],{},[35,37596,37],{},[39,37598,129],{},[1209,37600,1263],{},[35,37602,621],{},[45,37604,37605],{"encoding":47},"n = 2k",[15,37607,37609,37627],{"className":37608,"ariaHidden":53},[52],[15,37610,37612,37615,37618,37621,37624],{"className":37611},[57],[15,37613],{"className":37614,"style":86},[61],[15,37616,37],{"className":37617},[66,67],[15,37619],{"className":37620,"style":173},[71],[15,37622,129],{"className":37623},[177],[15,37625],{"className":37626,"style":173},[71],[15,37628,37630,37633,37636],{"className":37629},[57],[15,37631],{"className":37632,"style":840},[61],[15,37634,1263],{"className":37635},[66],[15,37637,621],{"className":37638,"style":684},[66,67],"：",[15,37641,37643],{"className":37642,"translate":19},[294],[15,37644,37646,37897],{"className":37645,"translate":19},[18],[15,37647,37649],{"className":37648},[23],[25,37650,37651],{"xmlns":27,"display":303},[29,37652,37653,37894],{},[32,37654,37655,37667,37681,37683,37685,37687,37689,37691,37709,37721,37739,37741,37743,37745,37747,37749,37751,37769,37771,37793,37796,37802,37814,37824,37826,37828,37830,37832,37834,37836,37854,37856,37878,37880,37886,37888],{},[1255,37656,37657,37659,37661],{},[39,37658,29727],{},[1209,37660,1804],{},[6465,37662,37663,37665],{},[35,37664,33489],{},[1209,37666,1263],{},[131,37668,37669,37675],{},[32,37670,37671,37673],{},[35,37672,33498],{},[39,37674,1170],{},[32,37676,37677,37679],{},[1209,37678,1263],{},[35,37680,621],{},[35,37682,123],{},[1274,37684,2462],{},[35,37686,458],{},[35,37688,123],{},[39,37690,129],{},[6465,37692,37693,37703],{},[32,37694,37695,37697,37699,37701],{},[1209,37696,1263],{},[35,37698,621],{},[39,37700,6600],{},[1209,37702,1211],{},[32,37704,37705,37707],{},[1209,37706,1263],{},[35,37708,621],{},[1255,37710,37711,37713,37715],{},[39,37712,29727],{},[1209,37714,1804],{},[6465,37716,37717,37719],{},[35,37718,33489],{},[1209,37720,1263],{},[131,37722,37723,37729],{},[32,37724,37725,37727],{},[35,37726,33498],{},[39,37728,1170],{},[32,37730,37731,37733,37735,37737],{},[1209,37732,1263],{},[35,37734,621],{},[39,37736,6600],{},[1209,37738,1263],{},[35,37740,123],{},[1274,37742,2462],{},[35,37744,458],{},[35,37746,123],{},[71,37748],{"linebreak":10451},[39,37750,129],{},[6465,37752,37753,37763],{},[32,37754,37755,37757,37759,37761],{},[1209,37756,1263],{},[35,37758,621],{},[39,37760,6600],{},[1209,37762,1211],{},[32,37764,37765,37767],{},[1209,37766,1263],{},[35,37768,621],{},[39,37770,3266],{},[6465,37772,37773,37783],{},[32,37774,37775,37777,37779,37781],{},[1209,37776,1263],{},[35,37778,621],{},[39,37780,6600],{},[1209,37782,8673],{},[32,37784,37785,37787,37789,37791],{},[1209,37786,1263],{},[35,37788,621],{},[39,37790,6600],{},[1209,37792,1263],{},[39,37794,37795],{},"⋯",[6465,37797,37798,37800],{},[1209,37799,1211],{},[1209,37801,1263],{},[1255,37803,37804,37806,37808],{},[39,37805,29727],{},[1209,37807,1804],{},[6465,37809,37810,37812],{},[35,37811,33489],{},[1209,37813,1263],{},[131,37815,37816,37822],{},[32,37817,37818,37820],{},[35,37819,33498],{},[39,37821,1170],{},[1209,37823,1804],{},[35,37825,123],{},[1274,37827,2462],{},[35,37829,458],{},[35,37831,123],{},[71,37833],{"linebreak":10451},[39,37835,129],{},[6465,37837,37838,37848],{},[32,37839,37840,37842,37844,37846],{},[1209,37841,1263],{},[35,37843,621],{},[39,37845,6600],{},[1209,37847,1211],{},[32,37849,37850,37852],{},[1209,37851,1263],{},[35,37853,621],{},[39,37855,3266],{},[6465,37857,37858,37868],{},[32,37859,37860,37862,37864,37866],{},[1209,37861,1263],{},[35,37863,621],{},[39,37865,6600],{},[1209,37867,8673],{},[32,37869,37870,37872,37874,37876],{},[1209,37871,1263],{},[35,37873,621],{},[39,37875,6600],{},[1209,37877,1263],{},[39,37879,37795],{},[6465,37881,37882,37884],{},[1209,37883,1211],{},[1209,37885,1263],{},[39,37887,3266],{},[6465,37889,37890,37892],{},[35,37891,33489],{},[1209,37893,1263],{},[45,37895,37896],{"encoding":47},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^{2k} x \\, dx = \\frac{2k-1}{2k} \\int_{0}^{\\frac{\\pi}{2}} \\sin^{2k-2} x \\, dx \\\\ = \\frac{2k-1}{2k} \\cdot \\frac{2k-3}{2k-2} \\cdots \\frac{1}{2} \\int_{0}^{\\frac{\\pi}{2}} \\sin^0 x \\, dx \\\\= \\frac{2k-1}{2k} \\cdot \\frac{2k-3}{2k-2} \\cdots \\frac{1}{2} \\cdot 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为奇数时，即 ",[15,39205,39207,39231],{"className":39206,"translate":19},[18],[15,39208,39210],{"className":39209},[23],[25,39211,39212],{"xmlns":27},[29,39213,39214,39228],{},[32,39215,39216,39218,39220,39222,39224,39226],{},[35,39217,37],{},[39,39219,129],{},[1209,39221,1263],{},[35,39223,621],{},[39,39225,1245],{},[1209,39227,1211],{},[45,39229,39230],{"encoding":47},"n = 2k + 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= m (m-2) (m-4) 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上有 ",[15,41899,41901,41973],{"className":41900,"translate":19},[18],[15,41902,41904],{"className":41903},[23],[25,41905,41906],{"xmlns":27},[29,41907,41908,41970],{},[32,41909,41910,41928,41930,41932,41946,41948,41950,41968],{},[131,41911,41912,41918],{},[32,41913,41914,41916],{},[35,41915,33498],{},[39,41917,1170],{},[32,41919,41920,41922,41924,41926],{},[1209,41921,1263],{},[35,41923,621],{},[39,41925,1245],{},[1209,41927,1211],{},[35,41929,123],{},[39,41931,25767],{},[131,41933,41934,41940],{},[32,41935,41936,41938],{},[35,41937,33498],{},[39,41939,1170],{},[32,41941,41942,41944],{},[1209,41943,1263],{},[35,41945,621],{},[35,41947,123],{},[39,41949,25767],{},[131,41951,41952,41958],{},[32,41953,41954,41956],{},[35,41955,33498],{},[39,41957,1170],{},[32,41959,41960,41962,41964,41966],{},[1209,41961,1263],{},[35,41963,621],{},[39,41965,6600],{},[1209,41967,1211],{},[35,41969,123],{},[45,41971,41972],{"encoding":47},"\\sin^{2k+1} x \u003C \\sin^{2k} x \u003C \\sin^{2k-1} x",[15,41974,41976,42039,42095],{"className":41975,"ariaHidden":53},[52],[15,41977,41979,41983,42024,42027,42030,42033,42036],{"className":41978},[57],[15,41980],{"className":41981,"style":41982},[61],"height:0.946em;vertical-align:-0.0391em;",[15,41984,41986,41989],{"className":41985},[1339],[15,41987,33498],{"className":41988},[1339],[15,41990,41992],{"className":41991},[198],[15,41993,41995],{"className":41994},[202],[15,41996,41998],{"className":41997},[206],[15,41999,42001],{"className":42000,"style":38049},[210],[15,42002,42003,42006],{"style":33680},[15,42004],{"className":42005,"style":218},[217],[15,42007,42009],{"className":42008},[222,223,224,225],[15,42010,42012,42015,42018,42021],{"className":42011},[66,225],[15,42013,1263],{"className":42014},[66,225],[15,42016,621],{"className":42017,"style":684},[66,67,225],[15,42019,1245],{"className":42020},[76,225],[15,42022,1211],{"className":42023},[66,225],[15,42025],{"className":42026,"style":697},[71],[15,42028,123],{"className":42029},[66,67],[15,42031],{"className":42032,"style":173},[71],[15,42034,25767],{"className":42035},[177],[15,42037],{"className":42038,"style":173},[71],[15,42040,42042,42045,42080,42083,42086,42089,42092],{"className":42041},[57],[15,42043],{"className":42044,"style":41982},[61],[15,42046,42048,42051],{"className":42047},[1339],[15,42049,33498],{"className":42050},[1339],[15,42052,42054],{"className":42053},[198],[15,42055,42057],{"className":42056},[202],[15,42058,42060],{"className":42059},[206],[15,42061,42063],{"className":42062,"style":38049},[210],[15,42064,42065,42068],{"style":33680},[15,42066],{"className":42067,"style":218},[217],[15,42069,42071],{"className":42070},[222,223,224,225],[15,42072,42074,42077],{"className":42073},[66,225],[15,42075,1263],{"className":42076},[66,225],[15,42078,621],{"className":42079,"style":684},[66,67,225],[15,42081],{"className":42082,"style":697},[71],[15,42084,123],{"className":42085},[66,67],[15,42087],{"className":42088,"style":173},[71],[15,42090,25767],{"className":42091},[177],[15,42093],{"className":42094,"style":173},[71],[15,42096,42098,42101,42142,42145],{"className":42097},[57],[15,42099],{"className":42100,"style":38049},[61],[15,42102,42104,42107],{"className":42103},[1339],[15,42105,33498],{"className":42106},[1339],[15,42108,42110],{"className":42109},[198],[15,42111,42113],{"className":42112},[202],[15,42114,42116],{"className":42115},[206],[15,42117,42119],{"className":42118,"style":38049},[210],[15,42120,42121,42124],{"style":33680},[15,42122],{"className":42123,"style":218},[217],[15,42125,42127],{"className":42126},[222,223,224,225],[15,42128,42130,42133,42136,42139],{"className":42129},[66,225],[15,42131,1263],{"className":42132},[66,225],[15,42134,621],{"className":42135,"style":684},[66,67,225],[15,42137,6600],{"className":42138},[76,225],[15,42140,1211],{"className":42141},[66,225],[15,42143],{"className":42144,"style":697},[71],[15,42146,123],{"className":42147},[66,67],"，因此有：",[15,42150,42152],{"className":42151,"translate":19},[294],[15,42153,42155,42281],{"className":42154,"translate":19},[18],[15,42156,42158],{"className":42157},[23],[25,42159,42160],{"xmlns":27,"display":303},[29,42161,42162,42278],{},[32,42163,42164,42176,42194,42196,42198,42200,42202,42204,42216,42230,42232,42234,42236,42238,42240,42252,42270,42272,42274,42276],{},[1255,42165,42166,42168,42170],{},[39,42167,29727],{},[1209,42169,1804],{},[6465,42171,42172,42174],{},[35,42173,33489],{},[1209,42175,1263],{},[131,42177,42178,42184],{},[32,42179,42180,42182],{},[35,42181,33498],{},[39,42183,1170],{},[32,42185,42186,42188,42190,42192],{},[1209,42187,1263],{},[35,42189,621],{},[39,42191,1245],{},[1209,42193,1211],{},[35,42195,123],{},[1274,42197,2462],{},[35,42199,458],{},[35,42201,123],{},[39,42203,25767],{},[1255,42205,42206,42208,42210],{},[39,42207,29727],{},[1209,42209,1804],{},[6465,42211,42212,42214],{},[35,42213,33489],{},[1209,42215,1263],{},[131,42217,42218,42224],{},[32,42219,42220,42222],{},[35,42221,33498],{},[39,42223,1170],{},[32,42225,42226,42228],{},[1209,42227,1263],{},[35,42229,621],{},[35,42231,123],{},[1274,42233,2462],{},[35,42235,458],{},[35,42237,123],{},[39,42239,25767],{},[1255,42241,42242,42244,42246],{},[39,42243,29727],{},[1209,42245,1804],{},[6465,42247,42248,42250],{},[35,42249,33489],{},[1209,42251,1263],{},[131,42253,42254,42260],{},[32,42255,42256,42258],{},[35,42257,33498],{},[39,42259,1170],{},[32,42261,42262,42264,42266,42268],{},[1209,42263,1263],{},[35,42265,621],{},[39,42267,6600],{},[1209,42269,1211],{},[35,42271,123],{},[1274,42273,2462],{},[35,42275,458],{},[35,42277,123],{},[45,42279,42280],{"encoding":47},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^{2k+1} x \\, dx \u003C \\int_{0}^{\\frac{\\pi}{2}} \\sin^{2k} x \\, dx \u003C \\int_{0}^{\\frac{\\pi}{2}} \\sin^{2k-1} x \\, dx",[15,42282,42284,42480,42670],{"className":42283,"ariaHidden":53},[52],[15,42285,42287,42290,42412,42415,42456,42459,42462,42465,42468,42471,42474,42477],{"className":42286},[57],[15,42288],{"className":42289,"style":33523},[61],[15,42291,42293,42296],{"className":42292},[1339],[15,42294,29727],{"className":42295,"style":29791},[1339,1470,1471],[15,42297,42299],{"className":42298},[198],[15,42300,42302,42404],{"className":42301},[202,1090],[15,42303,42305,42401],{"className":42304},[206],[15,42306,42308,42322],{"className":42307,"style":33542},[210],[15,42309,42310,42313],{"style":33545},[15,42311],{"className":42312,"style":1368},[217],[15,42314,42316],{"className":42315},[222,223,224,225],[15,42317,42319],{"className":42318},[66,225],[15,42320,1804],{"className":42321},[66,225],[15,42323,42324,42327],{"style":33560},[15,42325],{"className":42326,"style":1368},[217],[15,42328,42330],{"className":42329},[222,223,224,225],[15,42331,42333],{"className":42332},[66,225],[15,42334,42336,42339,42398],{"className":42335},[66,225],[15,42337],{"className":42338},[162,23412,222,3677,33576],[15,42340,42342],{"className":42341},[6465],[15,42343,42345,42390],{"className":42344},[202,1090],[15,42346,42348,42387],{"className":42347},[206],[15,42349,42351,42365,42373],{"className":42350,"style":33589},[210],[15,42352,42353,42356],{"style":33592},[15,42354],{"className":42355,"style":1368},[217],[15,42357,42359],{"className":42358},[222,3677,1508,225],[15,42360,42362],{"className":42361},[66,225],[15,42363,1263],{"className":42364},[66,225],[15,42366,42367,42370],{"style":33607},[15,42368],{"className":42369,"style":1368},[217],[15,42371],{"className":42372,"style":33614},[23447,225],[15,42374,42375,42378],{"style":33617},[15,42376],{"className":42377,"style":1368},[217],[15,42379,42381],{"className":42380},[222,3677,1508,225],[15,42382,42384],{"className":42383},[66,225],[15,42385,33489],{"className":42386,"style":1515},[66,67,225],[15,42388,1114],{"className":42389},[1113],[15,42391,42393],{"className":42392},[206],[15,42394,42396],{"className":42395,"style":33639},[210],[15,42397],{},[15,42399],{"className":42400},[169,23412,222,3677,33576],[15,42402,1114],{"className":42403},[1113],[15,42405,42407],{"className":42406},[206],[15,42408,42410],{"className":42409,"style":29838},[210],[15,42411],{},[15,42413],{"className":42414,"style":697},[71],[15,42416,42418,42421],{"className":42417},[1339],[15,42419,33498],{"className":42420},[1339],[15,42422,42424],{"className":42423},[198],[15,42425,42427],{"className":42426},[202],[15,42428,42430],{"className":42429},[206],[15,42431,42433],{"className":42432,"style":38049},[210],[15,42434,42435,42438],{"style":33680},[15,42436],{"className":42437,"style":218},[217],[15,42439,42441],{"className":42440},[222,223,224,225],[15,42442,42444,42447,42450,42453],{"className":42443},[66,225],[15,42445,1263],{"className":42446},[66,225],[15,42448,621],{"className":42449,"style":684},[66,67,225],[15,42451,1245],{"className":42452},[76,225],[15,42454,1211],{"className":42455},[66,225],[15,42457],{"className":42458,"style":697},[71],[15,42460,123],{"className":42461},[66,67],[15,42463],{"className":42464,"style":697},[71],[15,42466,458],{"className":42467},[66,67],[15,42469,123],{"className":42470},[66,67],[15,42472],{"className":42473,"style":173},[71],[15,42475,25767],{"className":42476},[177],[15,42478],{"className":42479,"style":173},[71],[15,42481,42483,42486,42608,42611,42646,42649,42652,42655,42658,42661,42664,42667],{"className":42482},[57],[15,42484],{"className":42485,"style":33523},[61],[15,42487,42489,42492],{"className":42488},[1339],[15,42490,29727],{"className":42491,"style":29791},[1339,1470,1471],[15,42493,42495],{"className":42494},[198],[15,42496,42498,42600],{"className":42497},[202,1090],[15,42499,42501,42597],{"className":42500},[206],[15,42502,42504,42518],{"className":42503,"style":33542},[210],[15,42505,42506,42509],{"style":33545},[15,42507],{"className":42508,"style":1368},[217],[15,42510,42512],{"className":42511},[222,223,224,225],[15,42513,42515],{"className":42514},[66,225],[15,42516,1804],{"className":42517},[66,225],[15,42519,42520,42523],{"style":33560},[15,42521],{"className":42522,"style":1368},[217],[15,42524,42526],{"className":42525},[222,223,224,225],[15,42527,42529],{"className":42528},[66,225],[15,42530,42532,42535,42594],{"className":42531},[66,225],[15,42533],{"className":42534},[162,23412,222,3677,33576],[15,42536,42538],{"className":42537},[6465],[15,42539,42541,42586],{"className":42540},[202,1090],[15,42542,42544,42583],{"className":42543},[206],[15,42545,42547,42561,42569],{"className":42546,"style":33589},[210],[15,42548,42549,42552],{"style":33592},[15,42550],{"className":42551,"style":1368},[217],[15,42553,42555],{"className":42554},[222,3677,1508,225],[15,42556,42558],{"className":42557},[66,225],[15,42559,1263],{"className":42560},[66,225],[15,42562,42563,42566],{"style":33607},[15,42564],{"className":42565,"style":1368},[217],[15,42567],{"className":42568,"style":33614},[23447,225],[15,42570,42571,42574],{"style":33617},[15,42572],{"className":42573,"style":1368},[217],[15,42575,42577],{"className":42576},[222,3677,1508,225],[15,42578,42580],{"className":42579},[66,225],[15,42581,33489],{"className":42582,"style":1515},[66,67,225],[15,42584,1114],{"className":42585},[1113],[15,42587,42589],{"className":42588},[206],[15,42590,42592],{"className":42591,"style":33639},[210],[15,42593],{},[15,42595],{"className":42596},[169,23412,222,3677,33576],[15,42598,1114],{"className":42599},[1113],[15,42601,42603],{"className":42602},[206],[15,42604,42606],{"className":42605,"style":29838},[210],[15,42607],{},[15,42609],{"className":42610,"style":697},[71],[15,42612,42614,42617],{"className":42613},[1339],[15,42615,33498],{"className":42616},[1339],[15,42618,42620],{"className":42619},[198],[15,42621,42623],{"className":42622},[202],[15,42624,42626],{"className":42625},[206],[15,42627,42629],{"className":42628,"style":38049},[210],[15,42630,42631,42634],{"style":33680},[15,42632],{"className":42633,"style":218},[217],[15,42635,42637],{"className":42636},[222,223,224,225],[15,42638,42640,42643],{"className":42639},[66,225],[15,42641,1263],{"className":42642},[66,225],[15,42644,621],{"className":42645,"style":684},[66,67,225],[15,42647],{"className":42648,"style":697},[71],[15,42650,123],{"className":42651},[66,67],[15,42653],{"className":42654,"style":697},[71],[15,42656,458],{"className":42657},[66,67],[15,42659,123],{"className":42660},[66,67],[15,42662],{"className":42663,"style":173},[71],[15,42665,25767],{"className":42666},[177],[15,42668],{"className":42669,"style":173},[71],[15,42671,42673,42676,42798,42801,42842,42845,42848,42851,42854],{"className":42672},[57],[15,42674],{"className":42675,"style":33523},[61],[15,42677,42679,42682],{"className":42678},[1339],[15,42680,29727],{"className":42681,"style":29791},[1339,1470,1471],[15,42683,42685],{"className":42684},[198],[15,42686,42688,42790],{"className":42687},[202,1090],[15,42689,42691,42787],{"className":42690},[206],[15,42692,42694,42708],{"className":42693,"style":33542},[210],[15,42695,42696,42699],{"style":33545},[15,42697],{"className":42698,"style":1368},[217],[15,42700,42702],{"className":42701},[222,223,224,225],[15,42703,42705],{"className":42704},[66,225],[15,42706,1804],{"className":42707},[66,225],[15,42709,42710,42713],{"style":33560},[15,42711],{"className":42712,"style":1368},[217],[15,42714,42716],{"className":42715},[222,223,224,225],[15,42717,42719],{"className":42718},[66,225],[15,42720,42722,42725,42784],{"className":42721},[66,225],[15,42723],{"className":42724},[162,23412,222,3677,33576],[15,42726,42728],{"className":42727},[6465],[15,42729,42731,42776],{"className":42730},[202,1090],[15,42732,42734,42773],{"className":42733},[206],[15,42735,42737,42751,42759],{"className":42736,"style":33589},[210],[15,42738,42739,42742],{"style":33592},[15,42740],{"className":42741,"style":1368},[217],[15,42743,42745],{"className":42744},[222,3677,1508,225],[15,42746,42748],{"className":42747},[66,225],[15,42749,1263],{"className":42750},[66,225],[15,42752,42753,42756],{"style":33607},[15,42754],{"className":42755,"style":1368},[217],[15,42757],{"className":42758,"style":33614},[23447,225],[15,42760,42761,42764],{"style":33617},[15,42762],{"className":42763,"style":1368},[217],[15,42765,42767],{"className":42766},[222,3677,1508,225],[15,42768,42770],{"className":42769},[66,225],[15,42771,33489],{"className":42772,"style":1515},[66,67,225],[15,42774,1114],{"className":42775},[1113],[15,42777,42779],{"className":42778},[206],[15,42780,42782],{"className":42781,"style":33639},[210],[15,42783],{},[15,42785],{"className":42786},[169,23412,222,3677,33576],[15,42788,1114],{"className":42789},[1113],[15,42791,42793],{"className":42792},[206],[15,42794,42796],{"className":42795,"style":29838},[210],[15,42797],{},[15,42799],{"className":42800,"style":697},[71],[15,42802,42804,42807],{"className":42803},[1339],[15,42805,33498],{"className":42806},[1339],[15,42808,42810],{"className":42809},[198],[15,42811,42813],{"className":42812},[202],[15,42814,42816],{"className":42815},[206],[15,42817,42819],{"className":42818,"style":38049},[210],[15,42820,42821,42824],{"style":33680},[15,42822],{"className":42823,"style":218},[217],[15,42825,42827],{"className":42826},[222,223,224,225],[15,42828,42830,42833,42836,42839],{"className":42829},[66,225],[15,42831,1263],{"className":42832},[66,225],[15,42834,621],{"className":42835,"style":684},[66,67,225],[15,42837,6600],{"className":42838},[76,225],[15,42840,1211],{"className":42841},[66,225],[15,42843],{"className":42844,"style":697},[71],[15,42846,123],{"className":42847},[66,67],[15,42849],{"className":42850,"style":697},[71],[15,42852,458],{"className":42853},[66,67],[15,42855,123],{"className":42856},[66,67],[11,42858,42859],{},"代入上面的公式：",[15,42861,42863],{"className":42862,"translate":19},[294],[15,42864,42866,42996],{"className":42865,"translate":19},[18],[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\u003C \\frac{(2k-1)!!}{(2k)!!} \\cdot \\frac{\\pi}{2} \u003C 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上下界之间的差：",[15,43643,43645],{"className":43644,"translate":19},[294],[15,43646,43648,43959],{"className":43647,"translate":19},[18],[15,43649,43651],{"className":43650},[23],[25,43652,43653],{"xmlns":27,"display":303},[29,43654,43655,43956],{},[32,43656,43657,43674,43754,43756,43758,43774,43878,43880,43882,43898,43912,43914,43952,43954],{},[1172,43658,43659,43666],{},[32,43660,43661,43664],{},[35,43662,43663],{},"lim",[39,43665,1170],{},[32,43667,43668,43670,43672],{},[35,43669,621],{},[39,43671,449],{},[35,43673,7615],{"mathvariant":138},[32,43675,43676,43678,43716,43718,43752],{},[39,43677,120],{"fence":53},[6465,43679,43680,43698],{},[32,43681,43682,43684,43686,43688,43690,43692,43694,43696],{},[39,43683,120],{"stretchy":119},[1209,43685,1263],{},[35,43687,621],{},[39,43689,6600],{},[1209,43691,1263],{},[39,43693,126],{"stretchy":119},[39,43695,40764],{"stretchy":119},[39,43697,40764],{"stretchy":119},[32,43699,43700,43702,43704,43706,43708,43710,43712,43714],{},[39,43701,120],{"stretchy":119},[1209,43703,1263],{},[35,43705,621],{},[39,43707,6600],{},[1209,43709,1211],{},[39,43711,126],{"stretchy":119},[39,43713,40764],{"stretchy":119},[39,43715,40764],{"stretchy":119},[39,43717,6600],{},[6465,43719,43720,43734],{},[32,43721,43722,43724,43726,43728,43730,43732],{},[39,43723,120],{"stretchy":119},[1209,43725,1263],{},[35,43727,621],{},[39,43729,126],{"stretchy":119},[39,43731,40764],{"stretchy":119},[39,43733,40764],{"stretchy":119},[32,43735,43736,43738,43740,43742,43744,43746,43748,43750],{},[39,43737,120],{"stretchy":119},[1209,43739,1263],{},[35,43741,621],{},[39,43743,1245],{},[1209,43745,1211],{},[39,43747,126],{"stretchy":119},[39,43749,40764],{"stretchy":119},[39,43751,40764],{"stretchy":119},[39,43753,126],{"fence":53},[71,43755],{"linebreak":10451},[39,43757,129],{},[1172,43759,43760,43766],{},[32,43761,43762,43764],{},[35,43763,43663],{},[39,43765,1170],{},[32,43767,43768,43770,43772],{},[35,43769,621],{},[39,43771,449],{},[35,43773,7615],{"mathvariant":138},[32,43775,43776,43778,43816,43818,43836,43838,43876],{},[39,43777,120],{"fence":53},[6465,43779,43780,43798],{},[32,43781,43782,43784,43786,43788,43790,43792,43794,43796],{},[39,43783,120],{"stretchy":119},[1209,43785,1263],{},[35,43787,621],{},[39,43789,6600],{},[1209,43791,1263],{},[39,43793,126],{"stretchy":119},[39,43795,40764],{"stretchy":119},[39,43797,40764],{"stretchy":119},[32,43799,43800,43802,43804,43806,43808,43810,43812,43814],{},[39,43801,120],{"stretchy":119},[1209,43803,1263],{},[35,43805,621],{},[39,43807,6600],{},[1209,43809,1211],{},[39,43811,126],{"stretchy":119},[39,43813,40764],{"stretchy":119},[39,43815,40764],{"stretchy":119},[39,43817,6600],{},[6465,43819,43820,43826],{},[32,43821,43822,43824],{},[1209,43823,1263],{},[35,43825,621],{},[32,43827,43828,43830,43832,43834],{},[1209,43829,1263],{},[35,43831,621],{},[39,43833,1245],{},[1209,43835,1211],{},[39,43837,3266],{},[6465,43839,43840,43858],{},[32,43841,43842,43844,43846,43848,43850,43852,43854,43856],{},[39,43843,120],{"stretchy":119},[1209,43845,1263],{},[35,43847,621],{},[39,43849,6600],{},[1209,43851,1263],{},[39,43853,126],{"stretchy":119},[39,43855,40764],{"stretchy":119},[39,43857,40764],{"stretchy":119},[32,43859,43860,43862,43864,43866,43868,43870,43872,43874],{},[39,43861,120],{"stretchy":119},[1209,43863,1263],{},[35,43865,621],{},[39,43867,6600],{},[1209,43869,1211],{},[39,43871,126],{"stretchy":119},[39,43873,40764],{"stretchy":119},[39,43875,40764],{"stretchy":119},[39,43877,126],{"fence":53},[71,43879],{"linebreak":10451},[39,43881,129],{},[1172,43883,43884,43890],{},[32,43885,43886,43888],{},[35,43887,43663],{},[39,43889,1170],{},[32,43891,43892,43894,43896],{},[35,43893,621],{},[39,43895,449],{},[35,43897,7615],{"mathvariant":138},[6465,43899,43900,43902],{},[1209,43901,1211],{},[32,43903,43904,43906,43908,43910],{},[1209,43905,1263],{},[35,43907,621],{},[39,43909,1245],{},[1209,43911,1211],{},[39,43913,3266],{},[6465,43915,43916,43934],{},[32,43917,43918,43920,43922,43924,43926,43928,43930,43932],{},[39,43919,120],{"stretchy":119},[1209,43921,1263],{},[35,43923,621],{},[39,43925,6600],{},[1209,43927,1263],{},[39,43929,126],{"stretchy":119},[39,43931,40764],{"stretchy":119},[39,43933,40764],{"stretchy":119},[32,43935,43936,43938,43940,43942,43944,43946,43948,43950],{},[39,43937,120],{"stretchy":119},[1209,43939,1263],{},[35,43941,621],{},[39,43943,6600],{},[1209,43945,1211],{},[39,43947,126],{"stretchy":119},[39,43949,40764],{"stretchy":119},[39,43951,40764],{"stretchy":119},[39,43953,129],{},[1209,43955,1804],{},[45,43957,43958],{"encoding":47},"\\lim_{k \\rightarrow \\infty} \\left( \\frac{(2k-2)!!}{(2k-1)!!} - \\frac{(2k)!!}{(2k+1)!!} \\right) \\\\ = \\lim_{k \\rightarrow \\infty} \\left( \\frac{(2k-2)!!}{(2k-1)!!} - \\frac{2k}{2k+1} \\cdot \\frac{(2k-2)!!}{(2k-1)!!} \\right) \\\\ = \\lim_{k \\rightarrow \\infty} \\frac{1}{2k+1} \\cdot \\frac{(2k-2)!!}{(2k-1)!!} = 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1",[15,44981,44983,45086],{"className":44982,"ariaHidden":53},[52],[15,44984,44986,44990,45077,45080,45083],{"className":44985},[57],[15,44987],{"className":44988,"style":44989},[61],"height:1.2834em;vertical-align:-0.4033em;",[15,44991,44993,44996,45074],{"className":44992},[66],[15,44994],{"className":44995},[162,23412],[15,44997,44999],{"className":44998},[6465],[15,45000,45002,45065],{"className":45001},[202,1090],[15,45003,45005,45062],{"className":45004},[206],[15,45006,45008,45031,45039],{"className":45007,"style":31032},[210],[15,45009,45010,45013],{"style":30871},[15,45011],{"className":45012,"style":1368},[217],[15,45014,45016],{"className":45015},[222,223,224,225],[15,45017,45019,45022,45025,45028],{"className":45018},[66,225],[15,45020,1263],{"className":45021},[66,225],[15,45023,621],{"className":45024,"style":684},[66,67,225],[15,45026,6600],{"className":45027},[76,225],[15,45029,1211],{"className":45030},[66,225],[15,45032,45033,45036],{"style":23440},[15,45034],{"className":45035,"style":1368},[217],[15,45037],{"className":45038,"style":4438},[23447],[15,45040,45041,45044],{"style":30894},[15,45042],{"className":45043,"style":1368},[217],[15,45045,45047],{"className":45046},[222,223,224,225],[15,45048,45050,45053,45056,45059],{"className":45049},[66,225],[15,45051,1263],{"className":45052},[66,225],[15,45054,621],{"className":45055,"style":684},[66,67,225],[15,45057,6600],{"className":45058},[76,225],[15,45060,1263],{"className":45061},[66,225],[15,45063,1114],{"className":45064},[1113],[15,45066,45068],{"className":45067},[206],[15,45069,45072],{"className":45070,"style":45071},[210],"height:0.4033em;",[15,45073],{},[15,45075],{"className":45076},[169,23412],[15,45078],{"className":45079,"style":173},[71],[15,45081,25767],{"className":45082},[177],[15,45084],{"className":45085,"style":173},[71],[15,45087,45089,45092],{"className":45088},[57],[15,45090],{"className":45091,"style":1836},[61],[15,45093,1211],{"className":45094},[66],[11,45096,45097],{},"对于不等式：",[15,45099,45101],{"className":45100,"translate":19},[294],[15,45102,45104,45233],{"className":45103,"translate":19},[18],[15,45105,45107],{"className":45106},[23],[25,45108,45109],{"xmlns":27,"display":303},[29,45110,45111,45231],{},[32,45112,45113,45147,45149,45183,45185,45191,45193],{},[6465,45114,45115,45129],{},[32,45116,45117,45119,45121,45123,45125,45127],{},[39,45118,120],{"stretchy":119},[1209,45120,1263],{},[35,45122,621],{},[39,45124,126],{"stretchy":119},[39,45126,40764],{"stretchy":119},[39,45128,40764],{"stretchy":119},[32,45130,45131,45133,45135,45137,45139,45141,45143,45145],{},[39,45132,120],{"stretchy":119},[1209,45134,1263],{},[35,45136,621],{},[39,45138,1245],{},[1209,45140,1211],{},[39,45142,126],{"stretchy":119},[39,45144,40764],{"stretchy":119},[39,45146,40764],{"stretchy":119},[39,45148,25767],{},[6465,45150,45151,45169],{},[32,45152,45153,45155,45157,45159,45161,45163,45165,45167],{},[39,45154,120],{"stretchy":119},[1209,45156,1263],{},[35,45158,621],{},[39,45160,6600],{},[1209,45162,1211],{},[39,45164,126],{"stretchy":119},[39,45166,40764],{"stretchy":119},[39,45168,40764],{"stretchy":119},[32,45170,45171,45173,45175,45177,45179,45181],{},[39,45172,120],{"stretchy":119},[1209,45174,1263],{},[35,45176,621],{},[39,45178,126],{"stretchy":119},[39,45180,40764],{"stretchy":119},[39,45182,40764],{"stretchy":119},[39,45184,3266],{},[6465,45186,45187,45189],{},[35,45188,33489],{},[1209,45190,1263],{},[39,45192,25767],{},[6465,45194,45195,45213],{},[32,45196,45197,45199,45201,45203,45205,45207,45209,45211],{},[39,45198,120],{"stretchy":119},[1209,45200,1263],{},[35,45202,621],{},[39,45204,6600],{},[1209,45206,1263],{},[39,45208,126],{"stretchy":119},[39,45210,40764],{"stretchy":119},[39,45212,40764],{"stretchy":119},[32,45214,45215,45217,45219,45221,45223,45225,45227,45229],{},[39,45216,120],{"stretchy":119},[1209,45218,1263],{},[35,45220,621],{},[39,45222,6600],{},[1209,45224,1211],{},[39,45226,126],{"stretchy":119},[39,45228,40764],{"stretchy":119},[39,45230,40764],{"stretchy":119},[45,45232,42995],{"encoding":47},[15,45234,45236,45343,45450,45527],{"className":45235,"ariaHidden":53},[52],[15,45237,45239,45242,45334,45337,45340],{"className":45238},[57],[15,45240],{"className":45241,"style":41187},[61],[15,45243,45245,45248,45331],{"className":45244},[66],[15,45246],{"className":45247},[162,23412],[15,45249,45251],{"className":45250},[6465],[15,45252,45254,45323],{"className":45253},[202,1090],[15,45255,45257,45320],{"className":45256},[206],[15,45258,45260,45292,45300],{"className":45259,"style":41206},[210],[15,45261,45262,45265],{"style":23428},[15,45263],{"className":45264,"style":1368},[217],[15,45266,45268,45271,45274,45277,45280,45283,45286,45289],{"className":45267},[66],[15,45269,120],{"className":45270},[162],[15,45272,1263],{"className":45273},[66],[15,45275,621],{"className":45276,"style":684},[66,67],[15,45278],{"className":45279,"style":72},[71],[15,45281,1245],{"className":45282},[76],[15,45284],{"className":45285,"style":72},[71],[15,45287,1211],{"className":45288},[66],[15,45290,41227],{"className":45291},[169],[15,45293,45294,45297],{"style":23440},[15,45295],{"className":45296,"style":1368},[217],[15,45298],{"className":45299,"style":4438},[23447],[15,45301,45302,45305],{"style":23450},[15,45303],{"className":45304,"style":1368},[217],[15,45306,45308,45311,45314,45317],{"className":45307},[66],[15,45309,120],{"className":45310},[162],[15,45312,1263],{"className":45313},[66],[15,45315,621],{"className":45316,"style":684},[66,67],[15,45318,41227],{"className":45319},[169],[15,45321,1114],{"className":45322},[1113],[15,45324,45326],{"className":45325},[206],[15,45327,45329],{"className":45328,"style":41277},[210],[15,45330],{},[15,45332],{"className":45333},[169,23412],[15,45335],{"className":45336,"style":173},[71],[15,45338,25767],{"className":45339},[177],[15,45341],{"className":45342,"style":173},[71],[15,45344,45346,45349,45441,45444,45447],{"className":45345},[57],[15,45347],{"className":45348,"style":41187},[61],[15,45350,45352,45355,45438],{"className":45351},[66],[15,45353],{"className":45354},[162,23412],[15,45356,45358],{"className":45357},[6465],[15,45359,45361,45430],{"className":45360},[202,1090],[15,45362,45364,45427],{"className":45363},[206],[15,45365,45367,45387,45395],{"className":45366,"style":41206},[210],[15,45368,45369,45372],{"style":23428},[15,45370],{"className":45371,"style":1368},[217],[15,45373,45375,45378,45381,45384],{"className":45374},[66],[15,45376,120],{"className":45377},[162],[15,45379,1263],{"className":45380},[66],[15,45382,621],{"className":45383,"style":684},[66,67],[15,45385,41227],{"className":45386},[169],[15,45388,45389,45392],{"style":23440},[15,45390],{"className":45391,"style":1368},[217],[15,45393],{"className":45394,"style":4438},[23447],[15,45396,45397,45400],{"style":23450},[15,45398],{"className":45399,"style":1368},[217],[15,45401,45403,45406,45409,45412,45415,45418,45421,45424],{"className":45402},[66],[15,45404,120],{"className":45405},[162],[15,45407,1263],{"className":45408},[66],[15,45410,621],{"className":45411,"style":684},[66,67],[15,45413],{"className":45414,"style":72},[71],[15,45416,6600],{"className":45417},[76],[15,45419],{"className":45420,"style":72},[71],[15,45422,1211],{"className":45423},[66],[15,45425,41227],{"className":45426},[169],[15,45428,1114],{"className":45429},[1113],[15,45431,45433],{"className":45432},[206],[15,45434,45436],{"className":45435,"style":41277},[210],[15,45437],{},[15,45439],{"className":45440},[169,23412],[15,45442],{"className":45443,"style":72},[71],[15,45445,3266],{"className":45446},[76],[15,45448],{"className":45449,"style":72},[71],[15,45451,45453,45456,45518,45521,45524],{"className":45452},[57],[15,45454],{"className":45455,"style":39109},[61],[15,45457,45459,45462,45515],{"className":45458},[66],[15,45460],{"className":45461},[162,23412],[15,45463,45465],{"className":45464},[6465],[15,45466,45468,45507],{"className":45467},[202,1090],[15,45469,45471,45504],{"className":45470},[206],[15,45472,45474,45485,45493],{"className":45473,"style":39128},[210],[15,45475,45476,45479],{"style":23428},[15,45477],{"className":45478,"style":1368},[217],[15,45480,45482],{"className":45481},[66],[15,45483,1263],{"className":45484},[66],[15,45486,45487,45490],{"style":23440},[15,45488],{"className":45489,"style":1368},[217],[15,45491],{"className":45492,"style":4438},[23447],[15,45494,45495,45498],{"style":23450},[15,45496],{"className":45497,"style":1368},[217],[15,45499,45501],{"className":45500},[66],[15,45502,33489],{"className":45503,"style":1515},[66,67],[15,45505,1114],{"className":45506},[1113],[15,45508,45510],{"className":45509},[206],[15,45511,45513],{"className":45512,"style":23469},[210],[15,45514],{},[15,45516],{"className":45517},[169,23412],[15,45519],{"className":45520,"style":173},[71],[15,45522,25767],{"className":45523},[177],[15,45525],{"className":45526,"style":173},[71],[15,45528,45530,45533],{"className":45529},[57],[15,45531],{"className":45532,"style":41187},[61],[15,45534,45536,45539,45634],{"className":45535},[66],[15,45537],{"className":45538},[162,23412],[15,45540,45542],{"className":45541},[6465],[15,45543,45545,45626],{"className":45544},[202,1090],[15,45546,45548,45623],{"className":45547},[206],[15,45549,45551,45583,45591],{"className":45550,"style":41206},[210],[15,45552,45553,45556],{"style":23428},[15,45554],{"className":45555,"style":1368},[217],[15,45557,45559,45562,45565,45568,45571,45574,45577,45580],{"className":45558},[66],[15,45560,120],{"className":45561},[162],[15,45563,1263],{"className":45564},[66],[15,45566,621],{"className":45567,"style":684},[66,67],[15,45569],{"className":45570,"style":72},[71],[15,45572,6600],{"className":45573},[76],[15,45575],{"className":45576,"style":72},[71],[15,45578,1211],{"className":45579},[66],[15,45581,41227],{"className":45582},[169],[15,45584,45585,45588],{"style":23440},[15,45586],{"className":45587,"style":1368},[217],[15,45589],{"className":45590,"style":4438},[23447],[15,45592,45593,45596],{"style":23450},[15,45594],{"className":45595,"style":1368},[217],[15,45597,45599,45602,45605,45608,45611,45614,45617,45620],{"className":45598},[66],[15,45600,120],{"className":45601},[162],[15,45603,1263],{"className":45604},[66],[15,45606,621],{"className":45607,"style":684},[66,67],[15,45609],{"className":45610,"style":72},[71],[15,45612,6600],{"className":45613},[76],[15,45615],{"className":45616,"style":72},[71],[15,45618,1263],{"className":45619},[66],[15,45621,41227],{"className":45622},[169],[15,45624,1114],{"className":45625},[1113],[15,45627,45629],{"className":45628},[206],[15,45630,45632],{"className":45631,"style":41277},[210],[15,45633],{},[15,45635],{"className":45636},[169,23412],[11,45638,45639,45640,37639],{},"将所有项乘以 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(2k-1)!!} \u003C \\frac{\\pi}{2} \u003C \\frac{(2k-2)!! 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\\rightarrow \\infty} \\left( \\frac{(2k-2)!! 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(2k-1)!!} \\\\= \\lim_{k \\rightarrow \\infty} \\frac{1}{2k} \\cdot \\frac{(2k)!!^2}{(2k+1)!!^2} = 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\\rightarrow \\infty} \\frac{(2k)!!^2}{(2k+1)!! (2k-1)!!} = \\frac{\\pi}{2}",[15,47973,47975,48191],{"className":47974,"ariaHidden":53},[52],[15,47976,47978,47981,48034,48037,48182,48185,48188],{"className":47977},[57],[15,47979],{"className":47980,"style":45939},[61],[15,47982,47984],{"className":47983},[1339,1351],[15,47985,47987,48026],{"className":47986},[202,1090],[15,47988,47990,48023],{"className":47989},[206],[15,47991,47993,48013],{"className":47992,"style":840},[210],[15,47994,47995,47998],{"style":1364},[15,47996],{"className":47997,"style":1368},[217],[15,47999,48001],{"className":48000},[222,223,224,225],[15,48002,48004,48007,48010],{"className":48003},[66,225],[15,48005,621],{"className":48006,"style":684},[66,67,225],[15,48008,449],{"className":48009},[177,225],[15,48011,7615],{"className":48012},[66,225],[15,48014,48015,48018],{"style":1386},[15,48016],{"className":48017,"style":1368},[217],[15,48019,48020],{},[15,48021,43663],{"className":48022},[1339],[15,48024,1114],{"className":48025},[1113],[15,48027,48029],{"className":48028},[206],[15,48030,48032],{"className":48031,"style":44020},[210],[15,48033],{},[15,48035],{"className":48036,"style":697},[71],[15,48038,48040,48043,48179],{"className":48039},[66],[15,48041],{"className":48042},[162,23412],[15,48044,48046],{"className":48045},[6465],[15,48047,48049,48171],{"className":48048},[202,1090],[15,48050,48052,48168],{"className":48051},[206],[15,48053,48055,48111,48119],{"className":48054,"style":45958},[210],[15,48056,48057,48060],{"style":23428},[15,48058],{"className":48059,"style":1368},[217],[15,48061,48063,48066,48069,48072,48075,48078,48081,48084,48087,48090,48093,48096,48099,48102,48105,48108],{"className":48062},[66],[15,48064,120],{"className":48065},[162],[15,48067,1263],{"className":48068},[66],[15,48070,621],{"className":48071,"style":684},[66,67],[15,48073],{"className":48074,"style":72},[71],[15,48076,1245],{"className":48077},[76],[15,48079],{"className":48080,"style":72},[71],[15,48082,1211],{"className":48083},[66],[15,48085,41227],{"className":48086},[169],[15,48088,120],{"className":48089},[162],[15,48091,1263],{"className":48092},[66],[15,48094,621],{"className":48095,"style":684},[66,67],[15,48097],{"className":48098,"style":72},[71],[15,48100,6600],{"className":48101},[76],[15,48103],{"className":48104,"style":72},[71],[15,48106,1211],{"className":48107},[66],[15,48109,41227],{"className":48110},[169],[15,48112,48113,48116],{"style":23440},[15,48114],{"className":48115,"style":1368},[217],[15,48117],{"className":48118,"style":4438},[23447],[15,48120,48121,48124],{"style":23450},[15,48122],{"className":48123,"style":1368},[217],[15,48125,48127,48130,48133,48136,48139],{"className":48126},[66],[15,48128,120],{"className":48129},[162],[15,48131,1263],{"className":48132},[66],[15,48134,621],{"className":48135,"style":684},[66,67],[15,48137,46043],{"className":48138},[169],[15,48140,48142,48145],{"className":48141},[169],[15,48143,40764],{"className":48144},[169],[15,48146,48148],{"className":48147},[198],[15,48149,48151],{"className":48150},[202],[15,48152,48154],{"className":48153},[206],[15,48155,48157],{"className":48156,"style":6738},[210],[15,48158,48159,48162],{"style":213},[15,48160],{"className":48161,"style":218},[217],[15,48163,48165],{"className":48164},[222,223,224,225],[15,48166,1263],{"className":48167},[66,225],[15,48169,1114],{"className":48170},[1113],[15,48172,48174],{"className":48173},[206],[15,48175,48177],{"className":48176,"style":41277},[210],[15,48178],{},[15,48180],{"className":48181},[169,23412],[15,48183],{"className":48184,"style":173},[71],[15,48186,129],{"className":48187},[177],[15,48189],{"className":48190,"style":173},[71],[15,48192,48194,48197],{"className":48193},[57],[15,48195],{"className":48196,"style":39109},[61],[15,48198,48200,48203,48256],{"className":48199},[66],[15,48201],{"className":48202},[162,23412],[15,48204,48206],{"className":48205},[6465],[15,48207,48209,48248],{"className":48208},[202,1090],[15,48210,48212,48245],{"className":48211},[206],[15,48213,48215,48226,48234],{"className":48214,"style":39128},[210],[15,48216,48217,48220],{"style":23428},[15,48218],{"className":48219,"style":1368},[217],[15,48221,48223],{"className":48222},[66],[15,48224,1263],{"className":48225},[66],[15,48227,48228,48231],{"style":23440},[15,48229],{"className":48230,"style":1368},[217],[15,48232],{"className":48233,"style":4438},[23447],[15,48235,48236,48239],{"style":23450},[15,48237],{"className":48238,"style":1368},[217],[15,48240,48242],{"className":48241},[66],[15,48243,33489],{"className":48244,"style":1515},[66,67],[15,48246,1114],{"className":48247},[1113],[15,48249,48251],{"className":48250},[206],[15,48252,48254],{"className":48253,"style":23469},[210],[15,48255],{},[15,48257],{"className":48258},[169,23412],[11,48260,48261],{},"该公式引入了沃利斯的结果，为推导斯特林公式做准备。",[95,48263,33455],{"id":33455},[11,48265,48266,48267,48301,48302,48361,48362,48422],{},"斯特林公式揭示了 ",[15,48268,48270,48286],{"className":48269,"translate":19},[18],[15,48271,48273],{"className":48272},[23],[25,48274,48275],{"xmlns":27},[29,48276,48277,48283],{},[32,48278,48279,48281],{},[35,48280,37],{},[39,48282,40764],{"stretchy":119},[45,48284,48285],{"encoding":47},"n!",[15,48287,48289],{"className":48288,"ariaHidden":53},[52],[15,48290,48292,48295,48298],{"className":48291},[57],[15,48293],{"className":48294,"style":840},[61],[15,48296,37],{"className":48297},[66,67],[15,48299,40764],{"className":48300},[169],"、",[15,48303,48305,48323],{"className":48304,"translate":19},[18],[15,48306,48308],{"className":48307},[23],[25,48309,48310],{"xmlns":27},[29,48311,48312,48320],{},[32,48313,48314],{},[131,48315,48316,48318],{},[35,48317,37],{},[35,48319,37],{},[45,48321,48322],{"encoding":47},"n^n",[15,48324,48326],{"className":48325,"ariaHidden":53},[52],[15,48327,48329,48332],{"className":48328},[57],[15,48330],{"className":48331,"style":6436},[61],[15,48333,48335,48338],{"className":48334},[66],[15,48336,37],{"className":48337},[66,67],[15,48339,48341],{"className":48340},[198],[15,48342,48344],{"className":48343},[202],[15,48345,48347],{"className":48346},[206],[15,48348,48350],{"className":48349,"style":6436},[210],[15,48351,48352,48355],{"style":213},[15,48353],{"className":48354,"style":218},[217],[15,48356,48358],{"className":48357},[222,223,224,225],[15,48359,37],{"className":48360},[66,67,225]," 与 ",[15,48363,48365,48384],{"className":48364,"translate":19},[18],[15,48366,48368],{"className":48367},[23],[25,48369,48370],{"xmlns":27},[29,48371,48372,48381],{},[32,48373,48374],{},[131,48375,48376,48379],{},[35,48377,48378],{},"e",[35,48380,37],{},[45,48382,48383],{"encoding":47},"e^n",[15,48385,48387],{"className":48386,"ariaHidden":53},[52],[15,48388,48390,48393],{"className":48389},[57],[15,48391],{"className":48392,"style":6436},[61],[15,48394,48396,48399],{"className":48395},[66],[15,48397,48378],{"className":48398},[66,67],[15,48400,48402],{"className":48401},[198],[15,48403,48405],{"className":48404},[202],[15,48406,48408],{"className":48407},[206],[15,48409,48411],{"className":48410,"style":6436},[210],[15,48412,48413,48416],{"style":213},[15,48414],{"className":48415,"style":218},[217],[15,48417,48419],{"className":48418},[222,223,224,225],[15,48420,37],{"className":48421},[66,67,225]," 之间的关系。",[48424,48425,48426],"blockquote",{},[11,48427,31881,48428,48456,48457,48490,48491,1124],{},[15,48429,48431,48444],{"className":48430,"translate":19},[18],[15,48432,48434],{"className":48433},[23],[25,48435,48436],{"xmlns":27},[29,48437,48438,48442],{},[32,48439,48440],{},[35,48441,37],{},[45,48443,37],{"encoding":47},[15,48445,48447],{"className":48446,"ariaHidden":53},[52],[15,48448,48450,48453],{"className":48449},[57],[15,48451],{"className":48452,"style":86},[61],[15,48454,37],{"className":48455},[66,67]," 趋近无穷大时，",[15,48458,48460,48475],{"className":48459,"translate":19},[18],[15,48461,48463],{"className":48462},[23],[25,48464,48465],{"xmlns":27},[29,48466,48467,48473],{},[32,48468,48469,48471],{},[35,48470,37],{},[39,48472,40764],{"stretchy":119},[45,48474,48285],{"encoding":47},[15,48476,48478],{"className":48477,"ariaHidden":53},[52],[15,48479,48481,48484,48487],{"className":48480},[57],[15,48482],{"className":48483,"style":840},[61],[15,48485,37],{"className":48486},[66,67],[15,48488,40764],{"className":48489},[169]," 约等于 ",[15,48492,48494,48533],{"className":48493,"translate":19},[18],[15,48495,48497],{"className":48496},[23],[25,48498,48499],{"xmlns":27},[29,48500,48501,48530],{},[32,48502,48503,48514],{},[48504,48505,48506],"msqrt",{},[32,48507,48508,48510,48512],{},[1209,48509,1263],{},[35,48511,33489],{},[35,48513,37],{},[131,48515,48516,48528],{},[32,48517,48518,48520,48526],{},[39,48519,120],{"fence":53},[6465,48521,48522,48524],{},[35,48523,37],{},[35,48525,48378],{},[39,48527,126],{"fence":53},[35,48529,37],{},[45,48531,48532],{"encoding":47},"\\sqrt{2 \\pi n} \\left(\\frac{n}{e}\\right)^n",[15,48534,48536],{"className":48535,"ariaHidden":53},[52],[15,48537,48539,48543,48609,48612],{"className":48538},[57],[15,48540],{"className":48541,"style":48542},[61],"height:1.2572em;vertical-align:-0.35em;",[15,48544,48547],{"className":48545},[66,48546],"sqrt",[15,48548,48550,48600],{"className":48549},[202,1090],[15,48551,48553,48597],{"className":48552},[206],[15,48554,48557,48577],{"className":48555,"style":48556},[210],"height:0.9072em;",[15,48558,48561,48564],{"className":48559,"style":1386},[48560],"svg-align",[15,48562],{"className":48563,"style":1368},[217],[15,48565,48568,48571,48574],{"className":48566,"style":48567},[66],"padding-left:0.833em;",[15,48569,1263],{"className":48570},[66],[15,48572,33489],{"className":48573,"style":1515},[66,67],[15,48575,37],{"className":48576},[66,67],[15,48578,48580,48583],{"style":48579},"top:-2.8672em;",[15,48581],{"className":48582,"style":1368},[217],[15,48584,48588],{"className":48585,"style":48587},[48586],"hide-tail","min-width:0.853em;height:1.08em;",[34823,48589,48594],{"xmlns":34825,"width":48590,"height":48591,"viewBox":48592,"preserveAspectRatio":48593},"400em","1.08em","0 0 400000 1080","xMinYMin slice",[34829,48595],{"d":48596},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[15,48598,1114],{"className":48599},[1113],[15,48601,48603],{"className":48602},[206],[15,48604,48607],{"className":48605,"style":48606},[210],"height:0.1328em;",[15,48608],{},[15,48610],{"className":48611,"style":697},[71],[15,48613,48615,48698],{"className":48614},[1413],[15,48616,48618,48624,48692],{"className":48617},[1413],[15,48619,48621],{"className":48620,"style":1418},[162,1417],[15,48622,120],{"className":48623},[1422,1508],[15,48625,48627,48630,48689],{"className":48626},[66],[15,48628],{"className":48629},[162,23412],[15,48631,48633],{"className":48632},[6465],[15,48634,48636,48681],{"className":48635},[202,1090],[15,48637,48639,48678],{"className":48638},[206],[15,48640,48642,48656,48664],{"className":48641,"style":41843},[210],[15,48643,48644,48647],{"style":30871},[15,48645],{"className":48646,"style":1368},[217],[15,48648,48650],{"className":48649},[222,223,224,225],[15,48651,48653],{"className":48652},[66,225],[15,48654,48378],{"className":48655},[66,67,225],[15,48657,48658,48661],{"style":23440},[15,48659],{"className":48660,"style":1368},[217],[15,48662],{"className":48663,"style":4438},[23447],[15,48665,48666,48669],{"style":30894},[15,48667],{"className":48668,"style":1368},[217],[15,48670,48672],{"className":48671},[222,223,224,225],[15,48673,48675],{"className":48674},[66,225],[15,48676,37],{"className":48677},[66,67,225],[15,48679,1114],{"className":48680},[1113],[15,48682,48684],{"className":48683},[206],[15,48685,48687],{"className":48686,"style":30916},[210],[15,48688],{},[15,48690],{"className":48691},[169,23412],[15,48693,48695],{"className":48694,"style":1418},[169,1417],[15,48696,126],{"className":48697},[1422,1508],[15,48699,48701],{"className":48700},[198],[15,48702,48704],{"className":48703},[202],[15,48705,48707],{"className":48706},[206],[15,48708,48711],{"className":48709,"style":48710},[210],"height:0.9043em;",[15,48712,48713,48716],{"style":21269},[15,48714],{"className":48715,"style":218},[217],[15,48717,48719],{"className":48718},[222,223,224,225],[15,48720,37],{"className":48721},[66,67,225],[11,48723,48724],{},"首先引入一个极限：",[15,48726,48728],{"className":48727,"translate":19},[294],[15,48729,48731,48795],{"className":48730,"translate":19},[18],[15,48732,48734],{"className":48733},[23],[25,48735,48736],{"xmlns":27,"display":303},[29,48737,48738,48792],{},[32,48739,48740,48756,48784,48786],{},[1172,48741,48742,48748],{},[32,48743,48744,48746],{},[35,48745,43663],{},[39,48747,1170],{},[32,48749,48750,48752,48754],{},[35,48751,37],{},[39,48753,449],{},[35,48755,7615],{"mathvariant":138},[131,48757,48758,48778],{},[32,48759,48760,48762,48776],{},[39,48761,120],{"fence":53},[6465,48763,48764,48770],{},[32,48765,48766,48768],{},[35,48767,37],{},[39,48769,40764],{"stretchy":119},[131,48771,48772,48774],{},[35,48773,37],{},[35,48775,37],{},[39,48777,126],{"fence":53},[6465,48779,48780,48782],{},[1209,48781,1211],{},[35,48783,37],{},[39,48785,129],{},[6465,48787,48788,48790],{},[1209,48789,1211],{},[35,48791,48378],{},[45,48793,48794],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\left( \\frac{n!}{n^n} \\right)^{\\frac{1}{n}} = \\frac{1}{e}",[15,48796,48798,49076],{"className":48797,"ariaHidden":53},[52],[15,48799,48801,48805,48860,48863,49067,49070,49073],{"className":48800},[57],[15,48802],{"className":48803,"style":48804},[61],"height:2.744em;vertical-align:-0.95em;",[15,48806,48808],{"className":48807},[1339,1351],[15,48809,48811,48851],{"className":48810},[202,1090],[15,48812,48814,48848],{"className":48813},[206],[15,48815,48817,48838],{"className":48816,"style":840},[210],[15,48818,48820,48823],{"style":48819},"top:-2.4em;margin-left:0em;",[15,48821],{"className":48822,"style":1368},[217],[15,48824,48826],{"className":48825},[222,223,224,225],[15,48827,48829,48832,48835],{"className":48828},[66,225],[15,48830,37],{"className":48831},[66,67,225],[15,48833,449],{"className":48834},[177,225],[15,48836,7615],{"className":48837},[66,225],[15,48839,48840,48843],{"style":1386},[15,48841],{"className":48842,"style":1368},[217],[15,48844,48845],{},[15,48846,43663],{"className":48847},[1339],[15,48849,1114],{"className":48850},[1113],[15,48852,48854],{"className":48853},[206],[15,48855,48858],{"className":48856,"style":48857},[210],"height:0.7em;",[15,48859],{},[15,48861],{"className":48862,"style":697},[71],[15,48864,48866,48973],{"className":48865},[1413],[15,48867,48869,48875,48967],{"className":48868},[1413],[15,48870,48872],{"className":48871,"style":1418},[162,1417],[15,48873,120],{"className":48874},[1422,224],[15,48876,48878,48881,48964],{"className":48877},[66],[15,48879],{"className":48880},[162,23412],[15,48882,48884],{"className":48883},[6465],[15,48885,48887,48956],{"className":48886},[202,1090],[15,48888,48890,48953],{"className":48889},[206],[15,48891,48893,48931,48939],{"className":48892,"style":27430},[210],[15,48894,48895,48898],{"style":23428},[15,48896],{"className":48897,"style":1368},[217],[15,48899,48901],{"className":48900},[66],[15,48902,48904,48907],{"className":48903},[66],[15,48905,37],{"className":48906},[66,67],[15,48908,48910],{"className":48909},[198],[15,48911,48913],{"className":48912},[202],[15,48914,48916],{"className":48915},[206],[15,48917,48920],{"className":48918,"style":48919},[210],"height:0.5904em;",[15,48921,48922,48925],{"style":46250},[15,48923],{"className":48924,"style":218},[217],[15,48926,48928],{"className":48927},[222,223,224,225],[15,48929,37],{"className":48930},[66,67,225],[15,48932,48933,48936],{"style":23440},[15,48934],{"className":48935,"style":1368},[217],[15,48937],{"className":48938,"style":4438},[23447],[15,48940,48941,48944],{"style":23450},[15,48942],{"className":48943,"style":1368},[217],[15,48945,48947,48950],{"className":48946},[66],[15,48948,37],{"className":48949},[66,67],[15,48951,40764],{"className":48952},[169],[15,48954,1114],{"className":48955},[1113],[15,48957,48959],{"className":48958},[206],[15,48960,48962],{"className":48961,"style":23469},[210],[15,48963],{},[15,48965],{"className":48966},[169,23412],[15,48968,48970],{"className":48969,"style":1418},[169,1417],[15,48971,126],{"className":48972},[1422,224],[15,48974,48976],{"className":48975},[198],[15,48977,48979],{"className":48978},[202],[15,48980,48982],{"className":48981},[206],[15,48983,48986],{"className":48984,"style":48985},[210],"height:1.7939em;",[15,48987,48989,48992],{"style":48988},"top:-4.2029em;margin-right:0.05em;",[15,48990],{"className":48991,"style":1368},[217],[15,48993,48995],{"className":48994},[222,223,224,225],[15,48996,48998],{"className":48997},[66,225],[15,48999,49001,49004,49064],{"className":49000},[66,225],[15,49002],{"className":49003},[162,23412,222,3677,33576],[15,49005,49007],{"className":49006},[6465],[15,49008,49010,49056],{"className":49009},[202,1090],[15,49011,49013,49053],{"className":49012},[206],[15,49014,49017,49031,49039],{"className":49015,"style":49016},[210],"height:0.8443em;",[15,49018,49019,49022],{"style":33592},[15,49020],{"className":49021,"style":1368},[217],[15,49023,49025],{"className":49024},[222,3677,1508,225],[15,49026,49028],{"className":49027},[66,225],[15,49029,37],{"className":49030},[66,67,225],[15,49032,49033,49036],{"style":33607},[15,49034],{"className":49035,"style":1368},[217],[15,49037],{"className":49038,"style":33614},[23447,225],[15,49040,49041,49044],{"style":33617},[15,49042],{"className":49043,"style":1368},[217],[15,49045,49047],{"className":49046},[222,3677,1508,225],[15,49048,49050],{"className":49049},[66,225],[15,49051,1211],{"className":49052},[66,225],[15,49054,1114],{"className":49055},[1113],[15,49057,49059],{"className":49058},[206],[15,49060,49062],{"className":49061,"style":33639},[210],[15,49063],{},[15,49065],{"className":49066},[169,23412,222,3677,33576],[15,49068],{"className":49069,"style":173},[71],[15,49071,129],{"className":49072},[177],[15,49074],{"className":49075,"style":173},[71],[15,49077,49079,49083],{"className":49078},[57],[15,49080],{"className":49081,"style":49082},[61],"height:2.0074em;vertical-align:-0.686em;",[15,49084,49086,49089,49142],{"className":49085},[66],[15,49087],{"className":49088},[162,23412],[15,49090,49092],{"className":49091},[6465],[15,49093,49095,49134],{"className":49094},[202,1090],[15,49096,49098,49131],{"className":49097},[206],[15,49099,49101,49112,49120],{"className":49100,"style":23425},[210],[15,49102,49103,49106],{"style":23428},[15,49104],{"className":49105,"style":1368},[217],[15,49107,49109],{"className":49108},[66],[15,49110,48378],{"className":49111},[66,67],[15,49113,49114,49117],{"style":23440},[15,49115],{"className":49116,"style":1368},[217],[15,49118],{"className":49119,"style":4438},[23447],[15,49121,49122,49125],{"style":23450},[15,49123],{"className":49124,"style":1368},[217],[15,49126,49128],{"className":49127},[66],[15,49129,1211],{"className":49130},[66],[15,49132,1114],{"className":49133},[1113],[15,49135,49137],{"className":49136},[206],[15,49138,49140],{"className":49139,"style":23469},[210],[15,49141],{},[15,49143],{"className":49144},[169,23412],[11,49146,49147],{},"证明：",[15,49149,49151],{"className":49150,"translate":19},[294],[15,49152,49154,49265],{"className":49153,"translate":19},[18],[15,49155,49157],{"className":49156},[23],[25,49158,49159],{"xmlns":27,"display":303},[29,49160,49161,49262],{},[32,49162,49163,49179,49207,49209,49225],{},[1172,49164,49165,49171],{},[32,49166,49167,49169],{},[35,49168,43663],{},[39,49170,1170],{},[32,49172,49173,49175,49177],{},[35,49174,37],{},[39,49176,449],{},[35,49178,7615],{"mathvariant":138},[131,49180,49181,49201],{},[32,49182,49183,49185,49199],{},[39,49184,120],{"fence":53},[6465,49186,49187,49193],{},[32,49188,49189,49191],{},[35,49190,37],{},[39,49192,40764],{"stretchy":119},[131,49194,49195,49197],{},[35,49196,37],{},[35,49198,37],{},[39,49200,126],{"fence":53},[6465,49202,49203,49205],{},[1209,49204,1211],{},[35,49206,37],{},[39,49208,129],{},[1172,49210,49211,49217],{},[32,49212,49213,49215],{},[35,49214,43663],{},[39,49216,1170],{},[32,49218,49219,49221,49223],{},[35,49220,37],{},[39,49222,449],{},[35,49224,7615],{"mathvariant":138},[131,49226,49227,49229],{},[35,49228,48378],{},[32,49230,49231,49237,49240,49242],{},[6465,49232,49233,49235],{},[1209,49234,1211],{},[35,49236,37],{},[35,49238,49239],{},"ln",[39,49241,1170],{},[32,49243,49244,49246,49260],{},[39,49245,120],{"fence":53},[6465,49247,49248,49254],{},[32,49249,49250,49252],{},[35,49251,37],{},[39,49253,40764],{"stretchy":119},[131,49255,49256,49258],{},[35,49257,37],{},[35,49259,37],{},[39,49261,126],{"fence":53},[45,49263,49264],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\left( \\frac{n!}{n^n} \\right)^{\\frac{1}{n}} = \\lim_{n \\rightarrow \\infty} e^{\\frac{1}{n} \\ln \\left( \\frac{n!}{n^n} \\right)}",[15,49266,49268,49539],{"className":49267,"ariaHidden":53},[52],[15,49269,49271,49274,49327,49330,49530,49533,49536],{"className":49270},[57],[15,49272],{"className":49273,"style":48804},[61],[15,49275,49277],{"className":49276},[1339,1351],[15,49278,49280,49319],{"className":49279},[202,1090],[15,49281,49283,49316],{"className":49282},[206],[15,49284,49286,49306],{"className":49285,"style":840},[210],[15,49287,49288,49291],{"style":48819},[15,49289],{"className":49290,"style":1368},[217],[15,49292,49294],{"className":49293},[222,223,224,225],[15,49295,49297,49300,49303],{"className":49296},[66,225],[15,49298,37],{"className":49299},[66,67,225],[15,49301,449],{"className":49302},[177,225],[15,49304,7615],{"className":49305},[66,225],[15,49307,49308,49311],{"style":1386},[15,49309],{"className":49310,"style":1368},[217],[15,49312,49313],{},[15,49314,43663],{"className":49315},[1339],[15,49317,1114],{"className":49318},[1113],[15,49320,49322],{"className":49321},[206],[15,49323,49325],{"className":49324,"style":48857},[210],[15,49326],{},[15,49328],{"className":49329,"style":697},[71],[15,49331,49333,49439],{"className":49332},[1413],[15,49334,49336,49342,49433],{"className":49335},[1413],[15,49337,49339],{"className":49338,"style":1418},[162,1417],[15,49340,120],{"className":49341},[1422,224],[15,49343,49345,49348,49430],{"className":49344},[66],[15,49346],{"className":49347},[162,23412],[15,49349,49351],{"className":49350},[6465],[15,49352,49354,49422],{"className":49353},[202,1090],[15,49355,49357,49419],{"className":49356},[206],[15,49358,49360,49397,49405],{"className":49359,"style":27430},[210],[15,49361,49362,49365],{"style":23428},[15,49363],{"className":49364,"style":1368},[217],[15,49366,49368],{"className":49367},[66],[15,49369,49371,49374],{"className":49370},[66],[15,49372,37],{"className":49373},[66,67],[15,49375,49377],{"className":49376},[198],[15,49378,49380],{"className":49379},[202],[15,49381,49383],{"className":49382},[206],[15,49384,49386],{"className":49385,"style":48919},[210],[15,49387,49388,49391],{"style":46250},[15,49389],{"className":49390,"style":218},[217],[15,49392,49394],{"className":49393},[222,223,224,225],[15,49395,37],{"className":49396},[66,67,225],[15,49398,49399,49402],{"style":23440},[15,49400],{"className":49401,"style":1368},[217],[15,49403],{"className":49404,"style":4438},[23447],[15,49406,49407,49410],{"style":23450},[15,49408],{"className":49409,"style":1368},[217],[15,49411,49413,49416],{"className":49412},[66],[15,49414,37],{"className":49415},[66,67],[15,49417,40764],{"className":49418},[169],[15,49420,1114],{"className":49421},[1113],[15,49423,49425],{"className":49424},[206],[15,49426,49428],{"className":49427,"style":23469},[210],[15,49429],{},[15,49431],{"className":49432},[169,23412],[15,49434,49436],{"className":49435,"style":1418},[169,1417],[15,49437,126],{"className":49438},[1422,224],[15,49440,49442],{"className":49441},[198],[15,49443,49445],{"className":49444},[202],[15,49446,49448],{"className":49447},[206],[15,49449,49451],{"className":49450,"style":48985},[210],[15,49452,49453,49456],{"style":48988},[15,49454],{"className":49455,"style":1368},[217],[15,49457,49459],{"className":49458},[222,223,224,225],[15,49460,49462],{"className":49461},[66,225],[15,49463,49465,49468,49527],{"className":49464},[66,225],[15,49466],{"className":49467},[162,23412,222,3677,33576],[15,49469,49471],{"className":49470},[6465],[15,49472,49474,49519],{"className":49473},[202,1090],[15,49475,49477,49516],{"className":49476},[206],[15,49478,49480,49494,49502],{"className":49479,"style":49016},[210],[15,49481,49482,49485],{"style":33592},[15,49483],{"className":49484,"style":1368},[217],[15,49486,49488],{"className":49487},[222,3677,1508,225],[15,49489,49491],{"className":49490},[66,225],[15,49492,37],{"className":49493},[66,67,225],[15,49495,49496,49499],{"style":33607},[15,49497],{"className":49498,"style":1368},[217],[15,49500],{"className":49501,"style":33614},[23447,225],[15,49503,49504,49507],{"style":33617},[15,49505],{"className":49506,"style":1368},[217],[15,49508,49510],{"className":49509},[222,3677,1508,225],[15,49511,49513],{"className":49512},[66,225],[15,49514,1211],{"className":49515},[66,225],[15,49517,1114],{"className":49518},[1113],[15,49520,49522],{"className":49521},[206],[15,49523,49525],{"className":49524,"style":33639},[210],[15,49526],{},[15,49528],{"className":49529},[169,23412,222,3677,33576],[15,49531],{"className":49532,"style":173},[71],[15,49534,129],{"className":49535},[177],[15,49537],{"className":49538,"style":173},[71],[15,49540,49542,49546,49599,49602],{"className":49541},[57],[15,49543],{"className":49544,"style":49545},[61],"height:1.8105em;vertical-align:-0.7em;",[15,49547,49549],{"className":49548},[1339,1351],[15,49550,49552,49591],{"className":49551},[202,1090],[15,49553,49555,49588],{"className":49554},[206],[15,49556,49558,49578],{"className":49557,"style":840},[210],[15,49559,49560,49563],{"style":48819},[15,49561],{"className":49562,"style":1368},[217],[15,49564,49566],{"className":49565},[222,223,224,225],[15,49567,49569,49572,49575],{"className":49568},[66,225],[15,49570,37],{"className":49571},[66,67,225],[15,49573,449],{"className":49574},[177,225],[15,49576,7615],{"className":49577},[66,225],[15,49579,49580,49583],{"style":1386},[15,49581],{"className":49582,"style":1368},[217],[15,49584,49585],{},[15,49586,43663],{"className":49587},[1339],[15,49589,1114],{"className":49590},[1113],[15,49592,49594],{"className":49593},[206],[15,49595,49597],{"className":49596,"style":48857},[210],[15,49598],{},[15,49600],{"className":49601,"style":697},[71],[15,49603,49605,49608],{"className":49604},[66],[15,49606,48378],{"className":49607},[66,67],[15,49609,49611],{"className":49610},[198],[15,49612,49614],{"className":49613},[202],[15,49615,49617],{"className":49616},[206],[15,49618,49621],{"className":49619,"style":49620},[210],"height:1.1105em;",[15,49622,49624,49627],{"style":49623},"top:-3.413em;margin-right:0.05em;",[15,49625],{"className":49626,"style":1368},[217],[15,49628,49630],{"className":49629},[222,223,224,225],[15,49631,49633,49701,49704,49714],{"className":49632},[66,225],[15,49634,49636,49639,49698],{"className":49635},[66,225],[15,49637],{"className":49638},[162,23412,222,3677,33576],[15,49640,49642],{"className":49641},[6465],[15,49643,49645,49690],{"className":49644},[202,1090],[15,49646,49648,49687],{"className":49647},[206],[15,49649,49651,49665,49673],{"className":49650,"style":49016},[210],[15,49652,49653,49656],{"style":33592},[15,49654],{"className":49655,"style":1368},[217],[15,49657,49659],{"className":49658},[222,3677,1508,225],[15,49660,49662],{"className":49661},[66,225],[15,49663,37],{"className":49664},[66,67,225],[15,49666,49667,49670],{"style":33607},[15,49668],{"className":49669,"style":1368},[217],[15,49671],{"className":49672,"style":33614},[23447,225],[15,49674,49675,49678],{"style":33617},[15,49676],{"className":49677,"style":1368},[217],[15,49679,49681],{"className":49680},[222,3677,1508,225],[15,49682,49684],{"className":49683},[66,225],[15,49685,1211],{"className":49686},[66,225],[15,49688,1114],{"className":49689},[1113],[15,49691,49693],{"className":49692},[206],[15,49694,49696],{"className":49695,"style":33639},[210],[15,49697],{},[15,49699],{"className":49700},[169,23412,222,3677,33576],[15,49702],{"className":49703,"style":23363},[71,225],[15,49705,49707,49711],{"className":49706},[1339,225],[15,49708,49710],{"className":49709},[225],"l",[15,49712,37],{"className":49713},[225],[15,49715,49717,49724,49821],{"className":49716},[1413,225],[15,49718,49721],{"className":49719,"style":49720},[162,222,3677,33576,225,1417],"top:0.075em;",[15,49722,120],{"className":49723},[225],[15,49725,49727,49730,49818],{"className":49726},[66,225],[15,49728],{"className":49729},[162,23412,222,3677,33576],[15,49731,49733],{"className":49732},[6465],[15,49734,49736,49810],{"className":49735},[202,1090],[15,49737,49739,49807],{"className":49738},[206],[15,49740,49743,49782,49790],{"className":49741,"style":49742},[210],"height:0.88em;",[15,49744,49745,49748],{"style":33592},[15,49746],{"className":49747,"style":1368},[217],[15,49749,49751],{"className":49750},[222,3677,1508,225],[15,49752,49754],{"className":49753},[66,225],[15,49755,49757,49760],{"className":49756},[66,225],[15,49758,37],{"className":49759},[66,67,225],[15,49761,49763],{"className":49762},[198],[15,49764,49766],{"className":49765},[202],[15,49767,49769],{"className":49768},[206],[15,49770,49773],{"className":49771,"style":49772},[210],"height:0.7246em;",[15,49774,49776,49779],{"style":49775},"top:-2.794em;margin-right:0.1em;",[15,49777],{"className":49778,"style":3673},[217],[15,49780,37],{"className":49781},[66,67,225],[15,49783,49784,49787],{"style":33607},[15,49785],{"className":49786,"style":1368},[217],[15,49788],{"className":49789,"style":33614},[23447,225],[15,49791,49792,49795],{"style":33617},[15,49793],{"className":49794,"style":1368},[217],[15,49796,49798],{"className":49797},[222,3677,1508,225],[15,49799,49801,49804],{"className":49800},[66,225],[15,49802,37],{"className":49803},[66,67,225],[15,49805,40764],{"className":49806},[169,225],[15,49808,1114],{"className":49809},[1113],[15,49811,49813],{"className":49812},[206],[15,49814,49816],{"className":49815,"style":33639},[210],[15,49817],{},[15,49819],{"className":49820},[169,23412,222,3677,33576],[15,49822,49824],{"className":49823,"style":49720},[169,222,3677,33576,225,1417],[15,49825,126],{"className":49826},[225],[11,49828,49829],{},"等价于证明：",[15,49831,49833],{"className":49832,"translate":19},[294],[15,49834,49836,49900],{"className":49835,"translate":19},[18],[15,49837,49839],{"className":49838},[23],[25,49840,49841],{"xmlns":27,"display":303},[29,49842,49843,49897],{},[32,49844,49845,49861,49867,49869,49871,49891,49893,49895],{},[1172,49846,49847,49853],{},[32,49848,49849,49851],{},[35,49850,43663],{},[39,49852,1170],{},[32,49854,49855,49857,49859],{},[35,49856,37],{},[39,49858,449],{},[35,49860,7615],{"mathvariant":138},[6465,49862,49863,49865],{},[1209,49864,1211],{},[35,49866,37],{},[35,49868,49239],{},[39,49870,1170],{},[32,49872,49873,49875,49889],{},[39,49874,120],{"fence":53},[6465,49876,49877,49883],{},[32,49878,49879,49881],{},[35,49880,37],{},[39,49882,40764],{"stretchy":119},[131,49884,49885,49887],{},[35,49886,37],{},[35,49888,37],{},[39,49890,126],{"fence":53},[39,49892,129],{},[39,49894,6600],{},[1209,49896,1211],{},[45,49898,49899],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{1}{n} \\ln \\left( \\frac{n!}{n^n} \\right) = 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\\rightarrow \\infty} \\frac{1}{n} \\ln \\left( \\frac{n!}{n^n} \\right) \\\\ = \\lim_{n \\rightarrow \\infty} \\frac{1}{n} \\left( \\ln \\frac{1}{n} + \\ln \\frac{2}{n} + \\cdots + \\ln \\frac{n}{n} \\right) \\\\ = \\int_{0}^{1} \\ln x \\, dx = (x \\ln x - x) \\Big|_{0}^{1} = 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260],{},"证毕。",[11,51262,35904],{},[15,51264,51266],{"className":51265,"translate":19},[294],[15,51267,51269,51307],{"className":51268,"translate":19},[18],[15,51270,51272],{"className":51271},[23],[25,51273,51274],{"xmlns":27,"display":303},[29,51275,51276,51304],{},[32,51277,51278,51292,51294],{},[6465,51279,51280,51286],{},[32,51281,51282,51284],{},[35,51283,37],{},[39,51285,40764],{"stretchy":119},[131,51287,51288,51290],{},[35,51289,37],{},[35,51291,37],{},[39,51293,28386],{},[6465,51295,51296,51298],{},[1209,51297,1211],{},[131,51299,51300,51302],{},[35,51301,48378],{},[35,51303,37],{},[45,51305,51306],{"encoding":47},"\\frac{n!}{n^n} \\sim \\frac{1}{e^n}",[15,51308,51310,51416],{"className":51309,"ariaHidden":53},[52],[15,51311,51313,51316,51407,51410,51413],{"className":51312},[57],[15,51314],{"className":51315,"style":38382},[61],[15,51317,51319,51322,51404],{"className":51318},[66],[15,51320],{"className":51321},[162,23412],[15,51323,51325],{"className":51324},[6465],[15,51326,51328,51396],{"className":51327},[202,1090],[15,51329,51331,51393],{"className":51330},[206],[15,51332,51334,51371,51379],{"className":51333,"style":27430},[210],[15,51335,51336,51339],{"style":23428},[15,51337],{"className":51338,"style":1368},[217],[15,51340,51342],{"className":51341},[66],[15,51343,51345,51348],{"className":51344},[66],[15,51346,37],{"className":51347},[66,67],[15,51349,51351],{"className":51350},[198],[15,51352,51354],{"className":51353},[202],[15,51355,51357],{"className":51356},[206],[15,51358,51360],{"className":51359,"style":48919},[210],[15,51361,51362,51365],{"style":46250},[15,51363],{"className":51364,"style":218},[217],[15,51366,51368],{"className":51367},[222,223,224,225],[15,51369,37],{"className":51370},[66,67,225],[15,51372,51373,51376],{"style":23440},[15,51374],{"className":51375,"style":1368},[217],[15,51377],{"className":51378,"style":4438},[23447],[15,51380,51381,51384],{"style":23450},[15,51382],{"className":51383,"style":1368},[217],[15,51385,51387,51390],{"className":51386},[66],[15,51388,37],{"className":51389},[66,67],[15,51391,40764],{"className":51392},[169],[15,51394,1114],{"className":51395},[1113],[15,51397,51399],{"className":51398},[206],[15,51400,51402],{"className":51401,"style":23469},[210],[15,51403],{},[15,51405],{"className":51406},[169,23412],[15,51408],{"className":51409,"style":173},[71],[15,51411,28386],{"className":51412},[177],[15,51414],{"className":51415,"style":173},[71],[15,51417,51419,51422],{"className":51418},[57],[15,51420],{"className":51421,"style":49082},[61],[15,51423,51425,51428,51507],{"className":51424},[66],[15,51426],{"className":51427},[162,23412],[15,51429,51431],{"className":51430},[6465],[15,51432,51434,51499],{"className":51433},[202,1090],[15,51435,51437,51496],{"className":51436},[206],[15,51438,51440,51477,51485],{"className":51439,"style":23425},[210],[15,51441,51442,51445],{"style":23428},[15,51443],{"className":51444,"style":1368},[217],[15,51446,51448],{"className":51447},[66],[15,51449,51451,51454],{"className":51450},[66],[15,51452,48378],{"className":51453},[66,67],[15,51455,51457],{"className":51456},[198],[15,51458,51460],{"className":51459},[202],[15,51461,51463],{"className":51462},[206],[15,51464,51466],{"className":51465,"style":48919},[210],[15,51467,51468,51471],{"style":46250},[15,51469],{"className":51470,"style":218},[217],[15,51472,51474],{"className":51473},[222,223,224,225],[15,51475,37],{"className":51476},[66,67,225],[15,51478,51479,51482],{"style":23440},[15,51480],{"className":51481,"style":1368},[217],[15,51483],{"className":51484,"style":4438},[23447],[15,51486,51487,51490],{"style":23450},[15,51488],{"className":51489,"style":1368},[217],[15,51491,51493],{"className":51492},[66],[15,51494,1211],{"className":51495},[66],[15,51497,1114],{"className":51498},[1113],[15,51500,51502],{"className":51501},[206],[15,51503,51505],{"className":51504,"style":23469},[210],[15,51506],{},[15,51508],{"className":51509},[169,23412],[11,51511,51512],{},"我们希望验证：",[15,51514,51516],{"className":51515,"translate":19},[294],[15,51517,51519,51571],{"className":51518,"translate":19},[18],[15,51520,51522],{"className":51521},[23],[25,51523,51524],{"xmlns":27,"display":303},[29,51525,51526,51568],{},[32,51527,51528,51544,51558,51564,51566],{},[1172,51529,51530,51536],{},[32,51531,51532,51534],{},[35,51533,43663],{},[39,51535,1170],{},[32,51537,51538,51540,51542],{},[35,51539,37],{},[39,51541,449],{},[35,51543,7615],{"mathvariant":138},[6465,51545,51546,51552],{},[32,51547,51548,51550],{},[35,51549,37],{},[39,51551,40764],{"stretchy":119},[131,51553,51554,51556],{},[35,51555,37],{},[35,51557,37],{},[131,51559,51560,51562],{},[35,51561,48378],{},[35,51563,37],{},[39,51565,129],{},[1209,51567,1211],{},[45,51569,51570],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{n!}{n^n} e^n = 1",[15,51572,51574,51767],{"className":51573,"ariaHidden":53},[52],[15,51575,51577,51581,51634,51637,51728,51758,51761,51764],{"className":51576},[57],[15,51578],{"className":51579,"style":51580},[61],"height:2.0714em;vertical-align:-0.7em;",[15,51582,51584],{"className":51583},[1339,1351],[15,51585,51587,51626],{"className":51586},[202,1090],[15,51588,51590,51623],{"className":51589},[206],[15,51591,51593,51613],{"className":51592,"style":840},[210],[15,51594,51595,51598],{"style":48819},[15,51596],{"className":51597,"style":1368},[217],[15,51599,51601],{"className":51600},[222,223,224,225],[15,51602,51604,51607,51610],{"className":51603},[66,225],[15,51605,37],{"className":51606},[66,67,225],[15,51608,449],{"className":51609},[177,225],[15,51611,7615],{"className":51612},[66,225],[15,51614,51615,51618],{"style":1386},[15,51616],{"className":51617,"style":1368},[217],[15,51619,51620],{},[15,51621,43663],{"className":51622},[1339],[15,51624,1114],{"className":51625},[1113],[15,51627,51629],{"className":51628},[206],[15,51630,51632],{"className":51631,"style":48857},[210],[15,51633],{},[15,51635],{"className":51636,"style":697},[71],[15,51638,51640,51643,51725],{"className":51639},[66],[15,51641],{"className":51642},[162,23412],[15,51644,51646],{"className":51645},[6465],[15,51647,51649,51717],{"className":51648},[202,1090],[15,51650,51652,51714],{"className":51651},[206],[15,51653,51655,51692,51700],{"className":51654,"style":27430},[210],[15,51656,51657,51660],{"style":23428},[15,51658],{"className":51659,"style":1368},[217],[15,51661,51663],{"className":51662},[66],[15,51664,51666,51669],{"className":51665},[66],[15,51667,37],{"className":51668},[66,67],[15,51670,51672],{"className":51671},[198],[15,51673,51675],{"className":51674},[202],[15,51676,51678],{"className":51677},[206],[15,51679,51681],{"className":51680,"style":48919},[210],[15,51682,51683,51686],{"style":46250},[15,51684],{"className":51685,"style":218},[217],[15,51687,51689],{"className":51688},[222,223,224,225],[15,51690,37],{"className":51691},[66,67,225],[15,51693,51694,51697],{"style":23440},[15,51695],{"className":51696,"style":1368},[217],[15,51698],{"className":51699,"style":4438},[23447],[15,51701,51702,51705],{"style":23450},[15,51703],{"className":51704,"style":1368},[217],[15,51706,51708,51711],{"className":51707},[66],[15,51709,37],{"className":51710},[66,67],[15,51712,40764],{"className":51713},[169],[15,51715,1114],{"className":51716},[1113],[15,51718,51720],{"className":51719},[206],[15,51721,51723],{"className":51722,"style":23469},[210],[15,51724],{},[15,51726],{"className":51727},[169,23412],[15,51729,51731,51734],{"className":51730},[66],[15,51732,48378],{"className":51733},[66,67],[15,51735,51737],{"className":51736},[198],[15,51738,51740],{"className":51739},[202],[15,51741,51743],{"className":51742},[206],[15,51744,51747],{"className":51745,"style":51746},[210],"height:0.7144em;",[15,51748,51749,51752],{"style":397},[15,51750],{"className":51751,"style":218},[217],[15,51753,51755],{"className":51754},[222,223,224,225],[15,51756,37],{"className":51757},[66,67,225],[15,51759],{"className":51760,"style":173},[71],[15,51762,129],{"className":51763},[177],[15,51765],{"className":51766,"style":173},[71],[15,51768,51770,51773],{"className":51769},[57],[15,51771],{"className":51772,"style":1836},[61],[15,51774,1211],{"className":51775},[66],[11,51777,51778],{},"我们试一下。",[11,51780,51781],{},"使用比值判别法：",[15,51783,51785],{"className":51784,"translate":19},[294],[15,51786,51788,51896],{"className":51787,"translate":19},[18],[15,51789,51791],{"className":51790},[23],[25,51792,51793],{"xmlns":27,"display":303},[29,51794,51795,51893],{},[32,51796,51797,51803,51805,51819,51825,51827,51829,51841,51843,51881],{},[1057,51798,51799,51801],{},[35,51800,4554],{},[35,51802,37],{},[39,51804,129],{},[6465,51806,51807,51813],{},[32,51808,51809,51811],{},[35,51810,37],{},[39,51812,40764],{"stretchy":119},[131,51814,51815,51817],{},[35,51816,37],{},[35,51818,37],{},[131,51820,51821,51823],{},[35,51822,48378],{},[35,51824,37],{},[39,51826,335],{"separator":53},[71,51828],{"width":9786},[1057,51830,51831,51833],{},[35,51832,4554],{},[32,51834,51835,51837,51839],{},[35,51836,37],{},[39,51838,1245],{},[1209,51840,1211],{},[39,51842,129],{},[6465,51844,51845,51859],{},[32,51846,51847,51849,51851,51853,51855,51857],{},[39,51848,120],{"stretchy":119},[35,51850,37],{},[39,51852,1245],{},[1209,51854,1211],{},[39,51856,126],{"stretchy":119},[39,51858,40764],{"stretchy":119},[32,51860,51861,51863,51865,51867,51869],{},[39,51862,120],{"stretchy":119},[35,51864,37],{},[39,51866,1245],{},[1209,51868,1211],{},[131,51870,51871,51873],{},[39,51872,126],{"stretchy":119},[32,51874,51875,51877,51879],{},[35,51876,37],{},[39,51878,1245],{},[1209,51880,1211],{},[131,51882,51883,51885],{},[35,51884,48378],{},[32,51886,51887,51889,51891],{},[35,51888,37],{},[39,51890,1245],{},[1209,51892,1211],{},[45,51894,51895],{"encoding":47},"a_n = \\frac{n!}{n^n} e^n, \\quad a_{n+1} = \\frac{(n+1)!}{(n+1)^{n+1}} e^{n+1}",[15,51897,51899,51954,52148],{"className":51898,"ariaHidden":53},[52],[15,51900,51902,51905,51945,51948,51951],{"className":51901},[57],[15,51903],{"className":51904,"style":10192},[61],[15,51906,51908,51911],{"className":51907},[66],[15,51909,4554],{"className":51910},[66,67],[15,51912,51914],{"className":51913},[198],[15,51915,51917,51937],{"className":51916},[202,1090],[15,51918,51920,51934],{"className":51919},[206],[15,51921,51923],{"className":51922,"style":2198},[210],[15,51924,51925,51928],{"style":1584},[15,51926],{"className":51927,"style":218},[217],[15,51929,51931],{"className":51930},[222,223,224,225],[15,51932,37],{"className":51933},[66,67,225],[15,51935,1114],{"className":51936},[1113],[15,51938,51940],{"className":51939},[206],[15,51941,51943],{"className":51942,"style":1121},[210],[15,51944],{},[15,51946],{"className":51947,"style":173},[71],[15,51949,129],{"className":51950},[177],[15,51952],{"className":51953,"style":173},[71],[15,51955,51957,51960,52051,52080,52083,52086,52089,52139,52142,52145],{"className":51956},[57],[15,51958],{"className":51959,"style":38382},[61],[15,51961,51963,51966,52048],{"className":51962},[66],[15,51964],{"className":51965},[162,23412],[15,51967,51969],{"className":51968},[6465],[15,51970,51972,52040],{"className":51971},[202,1090],[15,51973,51975,52037],{"className":51974},[206],[15,51976,51978,52015,52023],{"className":51977,"style":27430},[210],[15,51979,51980,51983],{"style":23428},[15,51981],{"className":51982,"style":1368},[217],[15,51984,51986],{"className":51985},[66],[15,51987,51989,51992],{"className":51988},[66],[15,51990,37],{"className":51991},[66,67],[15,51993,51995],{"className":51994},[198],[15,51996,51998],{"className":51997},[202],[15,51999,52001],{"className":52000},[206],[15,52002,52004],{"className":52003,"style":48919},[210],[15,52005,52006,52009],{"style":46250},[15,52007],{"className":52008,"style":218},[217],[15,52010,52012],{"className":52011},[222,223,224,225],[15,52013,37],{"className":52014},[66,67,225],[15,52016,52017,52020],{"style":23440},[15,52018],{"className":52019,"style":1368},[217],[15,52021],{"className":52022,"style":4438},[23447],[15,52024,52025,52028],{"style":23450},[15,52026],{"className":52027,"style":1368},[217],[15,52029,52031,52034],{"className":52030},[66],[15,52032,37],{"className":52033},[66,67],[15,52035,40764],{"className":52036},[169],[15,52038,1114],{"className":52039},[1113],[15,52041,52043],{"className":52042},[206],[15,52044,52046],{"className":52045,"style":23469},[210],[15,52047],{},[15,52049],{"className":52050},[169,23412],[15,52052,52054,52057],{"className":52053},[66],[15,52055,48378],{"className":52056},[66,67],[15,52058,52060],{"className":52059},[198],[15,52061,52063],{"className":52062},[202],[15,52064,52066],{"className":52065},[206],[15,52067,52069],{"className":52068,"style":51746},[210],[15,52070,52071,52074],{"style":397},[15,52072],{"className":52073,"style":218},[217],[15,52075,52077],{"className":52076},[222,223,224,225],[15,52078,37],{"className":52079},[66,67,225],[15,52081,335],{"className":52082},[422],[15,52084],{"className":52085,"style":10044},[71],[15,52087],{"className":52088,"style":697},[71],[15,52090,52092,52095],{"className":52091},[66],[15,52093,4554],{"className":52094},[66,67],[15,52096,52098],{"className":52097},[198],[15,52099,52101,52130],{"className":52100},[202,1090],[15,52102,52104,52127],{"className":52103},[206],[15,52105,52107],{"className":52106,"style":2139},[210],[15,52108,52109,52112],{"style":1584},[15,52110],{"className":52111,"style":218},[217],[15,52113,52115],{"className":52114},[222,223,224,225],[15,52116,52118,52121,52124],{"className":52117},[66,225],[15,52119,37],{"className":52120},[66,67,225],[15,52122,1245],{"className":52123},[76,225],[15,52125,1211],{"className":52126},[66,225],[15,52128,1114],{"className":52129},[1113],[15,52131,52133],{"className":52132},[206],[15,52134,52137],{"className":52135,"style":52136},[210],"height:0.2083em;",[15,52138],{},[15,52140],{"className":52141,"style":173},[71],[15,52143,129],{"className":52144},[177],[15,52146],{"className":52147,"style":173},[71],[15,52149,52151,52154,52287],{"className":52150},[57],[15,52152],{"className":52153,"style":41187},[61],[15,52155,52157,52160,52284],{"className":52156},[66],[15,52158],{"className":52159},[162,23412],[15,52161,52163],{"className":52162},[6465],[15,52164,52166,52276],{"className":52165},[202,1090],[15,52167,52169,52273],{"className":52168},[206],[15,52170,52172,52236,52244],{"className":52171,"style":41206},[210],[15,52173,52174,52177],{"style":23428},[15,52175],{"className":52176,"style":1368},[217],[15,52178,52180,52183,52186,52189,52192,52195,52198],{"className":52179},[66],[15,52181,120],{"className":52182},[162],[15,52184,37],{"className":52185},[66,67],[15,52187],{"className":52188,"style":72},[71],[15,52190,1245],{"className":52191},[76],[15,52193],{"className":52194,"style":72},[71],[15,52196,1211],{"className":52197},[66],[15,52199,52201,52204],{"className":52200},[169],[15,52202,126],{"className":52203},[169],[15,52205,52207],{"className":52206},[198],[15,52208,52210],{"className":52209},[202],[15,52211,52213],{"className":52212},[206],[15,52214,52216],{"className":52215,"style":46247},[210],[15,52217,52218,52221],{"style":46250},[15,52219],{"className":52220,"style":218},[217],[15,52222,52224],{"className":52223},[222,223,224,225],[15,52225,52227,52230,52233],{"className":52226},[66,225],[15,52228,37],{"className":52229},[66,67,225],[15,52231,1245],{"className":52232},[76,225],[15,52234,1211],{"className":52235},[66,225],[15,52237,52238,52241],{"style":23440},[15,52239],{"className":52240,"style":1368},[217],[15,52242],{"className":52243,"style":4438},[23447],[15,52245,52246,52249],{"style":23450},[15,52247],{"className":52248,"style":1368},[217],[15,52250,52252,52255,52258,52261,52264,52267,52270],{"className":52251},[66],[15,52253,120],{"className":52254},[162],[15,52256,37],{"className":52257},[66,67],[15,52259],{"className":52260,"style":72},[71],[15,52262,1245],{"className":52263},[76],[15,52265],{"className":52266,"style":72},[71],[15,52268,1211],{"className":52269},[66],[15,52271,46043],{"className":52272},[169],[15,52274,1114],{"className":52275},[1113],[15,52277,52279],{"className":52278},[206],[15,52280,52282],{"className":52281,"style":41277},[210],[15,52283],{},[15,52285],{"className":52286},[169,23412],[15,52288,52290,52293],{"className":52289},[66],[15,52291,48378],{"className":52292},[66,67],[15,52294,52296],{"className":52295},[198],[15,52297,52299],{"className":52298},[202],[15,52300,52302],{"className":52301},[206],[15,52303,52305],{"className":52304,"style":1650},[210],[15,52306,52307,52310],{"style":397},[15,52308],{"className":52309,"style":218},[217],[15,52311,52313],{"className":52312},[222,223,224,225],[15,52314,52316,52319,52322],{"className":52315},[66,225],[15,52317,37],{"className":52318},[66,67,225],[15,52320,1245],{"className":52321},[76,225],[15,52323,1211],{"className":52324},[66,225],[15,52326,52328],{"className":52327,"translate":19},[294],[15,52329,52331,52409],{"className":52330,"translate":19},[18],[15,52332,52334],{"className":52333},[23],[25,52335,52336],{"xmlns":27,"display":303},[29,52337,52338,52406],{},[32,52339,52340,52356,52376,52378,52402,52404],{},[1172,52341,52342,52348],{},[32,52343,52344,52346],{},[35,52345,43663],{},[39,52347,1170],{},[32,52349,52350,52352,52354],{},[35,52351,37],{},[39,52353,449],{},[35,52355,7615],{"mathvariant":138},[6465,52357,52358,52370],{},[1057,52359,52360,52362],{},[35,52361,4554],{},[32,52363,52364,52366,52368],{},[35,52365,37],{},[39,52367,1245],{},[1209,52369,1211],{},[1057,52371,52372,52374],{},[35,52373,4554],{},[35,52375,37],{},[39,52377,129],{},[6465,52379,52380,52382],{},[35,52381,48378],{},[131,52383,52384,52400],{},[32,52385,52386,52388,52390,52392,52398],{},[39,52387,120],{"fence":53},[1209,52389,1211],{},[39,52391,1245],{},[6465,52393,52394,52396],{},[1209,52395,1211],{},[35,52397,37],{},[39,52399,126],{"fence":53},[35,52401,37],{},[39,52403,9278],{},[1209,52405,1211],{},[45,52407,52408],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^n} > 1",[15,52410,52412,52630,52828],{"className":52411,"ariaHidden":53},[52],[15,52413,52415,52419,52472,52475,52621,52624,52627],{"className":52414},[57],[15,52416],{"className":52417,"style":52418},[61],"height:1.9436em;vertical-align:-0.836em;",[15,52420,52422],{"className":52421},[1339,1351],[15,52423,52425,52464],{"className":52424},[202,1090],[15,52426,52428,52461],{"className":52427},[206],[15,52429,52431,52451],{"className":52430,"style":840},[210],[15,52432,52433,52436],{"style":48819},[15,52434],{"className":52435,"style":1368},[217],[15,52437,52439],{"className":52438},[222,223,224,225],[15,52440,52442,52445,52448],{"className":52441},[66,225],[15,52443,37],{"className":52444},[66,67,225],[15,52446,449],{"className":52447},[177,225],[15,52449,7615],{"className":52450},[66,225],[15,52452,52453,52456],{"style":1386},[15,52454],{"className":52455,"style":1368},[217],[15,52457,52458],{},[15,52459,43663],{"className":52460},[1339],[15,52462,1114],{"className":52463},[1113],[15,52465,52467],{"className":52466},[206],[15,52468,52470],{"className":52469,"style":48857},[210],[15,52471],{},[15,52473],{"className":52474,"style":697},[71],[15,52476,52478,52481,52618],{"className":52477},[66],[15,52479],{"className":52480},[162,23412],[15,52482,52484],{"className":52483},[6465],[15,52485,52487,52609],{"className":52486},[202,1090],[15,52488,52490,52606],{"className":52489},[206],[15,52491,52493,52541,52549],{"className":52492,"style":39128},[210],[15,52494,52495,52498],{"style":23428},[15,52496],{"className":52497,"style":1368},[217],[15,52499,52501],{"className":52500},[66],[15,52502,52504,52507],{"className":52503},[66],[15,52505,4554],{"className":52506},[66,67],[15,52508,52510],{"className":52509},[198],[15,52511,52513,52533],{"className":52512},[202,1090],[15,52514,52516,52530],{"className":52515},[206],[15,52517,52519],{"className":52518,"style":2198},[210],[15,52520,52521,52524],{"style":1584},[15,52522],{"className":52523,"style":218},[217],[15,52525,52527],{"className":52526},[222,223,224,225],[15,52528,37],{"className":52529},[66,67,225],[15,52531,1114],{"className":52532},[1113],[15,52534,52536],{"className":52535},[206],[15,52537,52539],{"className":52538,"style":1121},[210],[15,52540],{},[15,52542,52543,52546],{"style":23440},[15,52544],{"className":52545,"style":1368},[217],[15,52547],{"className":52548,"style":4438},[23447],[15,52550,52551,52554],{"style":23450},[15,52552],{"className":52553,"style":1368},[217],[15,52555,52557],{"className":52556},[66],[15,52558,52560,52563],{"className":52559},[66],[15,52561,4554],{"className":52562},[66,67],[15,52564,52566],{"className":52565},[198],[15,52567,52569,52598],{"className":52568},[202,1090],[15,52570,52572,52595],{"className":52571},[206],[15,52573,52575],{"className":52574,"style":2139},[210],[15,52576,52577,52580],{"style":1584},[15,52578],{"className":52579,"style":218},[217],[15,52581,52583],{"className":52582},[222,223,224,225],[15,52584,52586,52589,52592],{"className":52585},[66,225],[15,52587,37],{"className":52588},[66,67,225],[15,52590,1245],{"className":52591},[76,225],[15,52593,1211],{"className":52594},[66,225],[15,52596,1114],{"className":52597},[1113],[15,52599,52601],{"className":52600},[206],[15,52602,52604],{"className":52603,"style":52136},[210],[15,52605],{},[15,52607,1114],{"className":52608},[1113],[15,52610,52612],{"className":52611},[206],[15,52613,52616],{"className":52614,"style":52615},[210],"height:0.836em;",[15,52617],{},[15,52619],{"className":52620},[169,23412],[15,52622],{"className":52623,"style":173},[71],[15,52625,129],{"className":52626},[177],[15,52628],{"className":52629,"style":173},[71],[15,52631,52633,52637,52819,52822,52825],{"className":52632},[57],[15,52634],{"className":52635,"style":52636},[61],"height:2.2519em;vertical-align:-1.1443em;",[15,52638,52640,52643,52816],{"className":52639},[66],[15,52641],{"className":52642},[162,23412],[15,52644,52646],{"className":52645},[6465],[15,52647,52649,52807],{"className":52648},[202,1090],[15,52650,52652,52804],{"className":52651},[206],[15,52653,52655,52785,52793],{"className":52654,"style":39128},[210],[15,52656,52658,52661],{"style":52657},"top:-2.2057em;",[15,52659],{"className":52660,"style":1368},[217],[15,52662,52664],{"className":52663},[66],[15,52665,52667,52762],{"className":52666},[1413],[15,52668,52670,52676,52679,52682,52685,52688,52756],{"className":52669},[1413],[15,52671,52673],{"className":52672,"style":1418},[162,1417],[15,52674,120],{"className":52675},[1422,1508],[15,52677,1211],{"className":52678},[66],[15,52680],{"className":52681,"style":72},[71],[15,52683,1245],{"className":52684},[76],[15,52686],{"className":52687,"style":72},[71],[15,52689,52691,52694,52753],{"className":52690},[66],[15,52692],{"className":52693},[162,23412],[15,52695,52697],{"className":52696},[6465],[15,52698,52700,52745],{"className":52699},[202,1090],[15,52701,52703,52742],{"className":52702},[206],[15,52704,52706,52720,52728],{"className":52705,"style":30868},[210],[15,52707,52708,52711],{"style":30871},[15,52709],{"className":52710,"style":1368},[217],[15,52712,52714],{"className":52713},[222,223,224,225],[15,52715,52717],{"className":52716},[66,225],[15,52718,37],{"className":52719},[66,67,225],[15,52721,52722,52725],{"style":23440},[15,52723],{"className":52724,"style":1368},[217],[15,52726],{"className":52727,"style":4438},[23447],[15,52729,52730,52733],{"style":30894},[15,52731],{"className":52732,"style":1368},[217],[15,52734,52736],{"className":52735},[222,223,224,225],[15,52737,52739],{"className":52738},[66,225],[15,52740,1211],{"className":52741},[66,225],[15,52743,1114],{"className":52744},[1113],[15,52746,52748],{"className":52747},[206],[15,52749,52751],{"className":52750,"style":30916},[210],[15,52752],{},[15,52754],{"className":52755},[169,23412],[15,52757,52759],{"className":52758,"style":1418},[169,1417],[15,52760,126],{"className":52761},[1422,1508],[15,52763,52765],{"className":52764},[198],[15,52766,52768],{"className":52767},[202],[15,52769,52771],{"className":52770},[206],[15,52772,52774],{"className":52773,"style":48710},[210],[15,52775,52776,52779],{"style":21269},[15,52777],{"className":52778,"style":218},[217],[15,52780,52782],{"className":52781},[222,223,224,225],[15,52783,37],{"className":52784},[66,67,225],[15,52786,52787,52790],{"style":23440},[15,52788],{"className":52789,"style":1368},[217],[15,52791],{"className":52792,"style":4438},[23447],[15,52794,52795,52798],{"style":23450},[15,52796],{"className":52797,"style":1368},[217],[15,52799,52801],{"className":52800},[66],[15,52802,48378],{"className":52803},[66,67],[15,52805,1114],{"className":52806},[1113],[15,52808,52810],{"className":52809},[206],[15,52811,52814],{"className":52812,"style":52813},[210],"height:1.1443em;",[15,52815],{},[15,52817],{"className":52818},[169,23412],[15,52820],{"className":52821,"style":173},[71],[15,52823,9278],{"className":52824},[177],[15,52826],{"className":52827,"style":173},[71],[15,52829,52831,52834],{"className":52830},[57],[15,52832],{"className":52833,"style":1836},[61],[15,52835,1211],{"className":52836},[66],[11,52838,52839,52840,52868,52869,52897,52898,52968],{},"虽然该表达式的极限是 ",[15,52841,52843,52856],{"className":52842,"translate":19},[18],[15,52844,52846],{"className":52845},[23],[25,52847,52848],{"xmlns":27},[29,52849,52850,52854],{},[32,52851,52852],{},[35,52853,48378],{},[45,52855,48378],{"encoding":47},[15,52857,52859],{"className":52858,"ariaHidden":53},[52],[15,52860,52862,52865],{"className":52861},[57],[15,52863],{"className":52864,"style":86},[61],[15,52866,48378],{"className":52867},[66,67],"，但它总是从下方趋近 ",[15,52870,52872,52885],{"className":52871,"translate":19},[18],[15,52873,52875],{"className":52874},[23],[25,52876,52877],{"xmlns":27},[29,52878,52879,52883],{},[32,52880,52881],{},[35,52882,48378],{},[45,52884,48378],{"encoding":47},[15,52886,52888],{"className":52887,"ariaHidden":53},[52],[15,52889,52891,52894],{"className":52890},[57],[15,52892],{"className":52893,"style":86},[61],[15,52895,48378],{"className":52896},[66,67],"，因此比值仍大于 1，说明 ",[15,52899,52901,52919],{"className":52900,"translate":19},[18],[15,52902,52904],{"className":52903},[23],[25,52905,52906],{"xmlns":27},[29,52907,52908,52916],{},[32,52909,52910],{},[1057,52911,52912,52914],{},[35,52913,4554],{},[35,52915,37],{},[45,52917,52918],{"encoding":47},"a_n",[15,52920,52922],{"className":52921,"ariaHidden":53},[52],[15,52923,52925,52928],{"className":52924},[57],[15,52926],{"className":52927,"style":10192},[61],[15,52929,52931,52934],{"className":52930},[66],[15,52932,4554],{"className":52933},[66,67],[15,52935,52937],{"className":52936},[198],[15,52938,52940,52960],{"className":52939},[202,1090],[15,52941,52943,52957],{"className":52942},[206],[15,52944,52946],{"className":52945,"style":2198},[210],[15,52947,52948,52951],{"style":1584},[15,52949],{"className":52950,"style":218},[217],[15,52952,52954],{"className":52953},[222,223,224,225],[15,52955,37],{"className":52956},[66,67,225],[15,52958,1114],{"className":52959},[1113],[15,52961,52963],{"className":52962},[206],[15,52964,52966],{"className":52965,"style":1121},[210],[15,52967],{}," 是发散的。",[11,52970,52971],{},"数学家们尝试了各种方法，发现：",[15,52973,52975],{"className":52974,"translate":19},[294],[15,52976,52978,53028],{"className":52977,"translate":19},[18],[15,52979,52981],{"className":52980},[23],[25,52982,52983],{"xmlns":27,"display":303},[29,52984,52985,53025],{},[32,52986,52987,52993,52995,53019],{},[1057,52988,52989,52991],{},[35,52990,4554],{},[35,52992,37],{},[39,52994,129],{},[6465,52996,52997,53003],{},[32,52998,52999,53001],{},[35,53000,37],{},[39,53002,40764],{"stretchy":119},[131,53004,53005,53007],{},[35,53006,37],{},[32,53008,53009,53011,53013],{},[35,53010,37],{},[39,53012,1245],{},[6465,53014,53015,53017],{},[1209,53016,1211],{},[1209,53018,1263],{},[131,53020,53021,53023],{},[35,53022,48378],{},[35,53024,37],{},[45,53026,53027],{"encoding":47},"a_n = \\frac{n!}{n^{n + \\frac{1}{2}}} e^n",[15,53029,53031,53086],{"className":53030,"ariaHidden":53},[52],[15,53032,53034,53037,53077,53080,53083],{"className":53033},[57],[15,53035],{"className":53036,"style":10192},[61],[15,53038,53040,53043],{"className":53039},[66],[15,53041,4554],{"className":53042},[66,67],[15,53044,53046],{"className":53045},[198],[15,53047,53049,53069],{"className":53048},[202,1090],[15,53050,53052,53066],{"className":53051},[206],[15,53053,53055],{"className":53054,"style":2198},[210],[15,53056,53057,53060],{"style":1584},[15,53058],{"className":53059,"style":218},[217],[15,53061,53063],{"className":53062},[222,223,224,225],[15,53064,37],{"className":53065},[66,67,225],[15,53067,1114],{"className":53068},[1113],[15,53070,53072],{"className":53071},[206],[15,53073,53075],{"className":53074,"style":1121},[210],[15,53076],{},[15,53078],{"className":53079,"style":173},[71],[15,53081,129],{"className":53082},[177],[15,53084],{"className":53085,"style":173},[71],[15,53087,53089,53093,53262],{"className":53088},[57],[15,53090],{"className":53091,"style":53092},[61],"height:2.201em;vertical-align:-0.8296em;",[15,53094,53096,53099,53259],{"className":53095},[66],[15,53097],{"className":53098},[162,23412],[15,53100,53102],{"className":53101},[6465],[15,53103,53105,53250],{"className":53104},[202,1090],[15,53106,53108,53247],{"className":53107},[206],[15,53109,53111,53225,53233],{"className":53110,"style":27430},[210],[15,53112,53114,53117],{"style":53113},"top:-2.1704em;",[15,53115],{"className":53116,"style":1368},[217],[15,53118,53120],{"className":53119},[66],[15,53121,53123,53126],{"className":53122},[66],[15,53124,37],{"className":53125},[66,67],[15,53127,53129],{"className":53128},[198],[15,53130,53132],{"className":53131},[202],[15,53133,53135],{"className":53134},[206],[15,53136,53139],{"className":53137,"style":53138},[210],"height:0.9396em;",[15,53140,53142,53145],{"style":53141},"top:-3.3486em;margin-right:0.05em;",[15,53143],{"className":53144,"style":1368},[217],[15,53146,53148],{"className":53147},[222,223,224,225],[15,53149,53151,53154,53157],{"className":53150},[66,225],[15,53152,37],{"className":53153},[66,67,225],[15,53155,1245],{"className":53156},[76,225],[15,53158,53160,53163,53222],{"className":53159},[66,225],[15,53161],{"className":53162},[162,23412,222,3677,33576],[15,53164,53166],{"className":53165},[6465],[15,53167,53169,53214],{"className":53168},[202,1090],[15,53170,53172,53211],{"className":53171},[206],[15,53173,53175,53189,53197],{"className":53174,"style":49016},[210],[15,53176,53177,53180],{"style":33592},[15,53178],{"className":53179,"style":1368},[217],[15,53181,53183],{"className":53182},[222,3677,1508,225],[15,53184,53186],{"className":53185},[66,225],[15,53187,1263],{"className":53188},[66,225],[15,53190,53191,53194],{"style":33607},[15,53192],{"className":53193,"style":1368},[217],[15,53195],{"className":53196,"style":33614},[23447,225],[15,53198,53199,53202],{"style":33617},[15,53200],{"className":53201,"style":1368},[217],[15,53203,53205],{"className":53204},[222,3677,1508,225],[15,53206,53208],{"className":53207},[66,225],[15,53209,1211],{"className":53210},[66,225],[15,53212,1114],{"className":53213},[1113],[15,53215,53217],{"className":53216},[206],[15,53218,53220],{"className":53219,"style":33639},[210],[15,53221],{},[15,53223],{"className":53224},[169,23412,222,3677,33576],[15,53226,53227,53230],{"style":23440},[15,53228],{"className":53229,"style":1368},[217],[15,53231],{"className":53232,"style":4438},[23447],[15,53234,53235,53238],{"style":23450},[15,53236],{"className":53237,"style":1368},[217],[15,53239,53241,53244],{"className":53240},[66],[15,53242,37],{"className":53243},[66,67],[15,53245,40764],{"className":53246},[169],[15,53248,1114],{"className":53249},[1113],[15,53251,53253],{"className":53252},[206],[15,53254,53257],{"className":53255,"style":53256},[210],"height:0.8296em;",[15,53258],{},[15,53260],{"className":53261},[169,23412],[15,53263,53265,53268],{"className":53264},[66],[15,53266,48378],{"className":53267},[66,67],[15,53269,53271],{"className":53270},[198],[15,53272,53274],{"className":53273},[202],[15,53275,53277],{"className":53276},[206],[15,53278,53280],{"className":53279,"style":51746},[210],[15,53281,53282,53285],{"style":397},[15,53283],{"className":53284,"style":218},[217],[15,53286,53288],{"className":53287},[222,223,224,225],[15,53289,37],{"className":53290},[66,67,225],[11,53292,53293,53294,53363],{},"为了证明 ",[15,53295,53297,53314],{"className":53296,"translate":19},[18],[15,53298,53300],{"className":53299},[23],[25,53301,53302],{"xmlns":27},[29,53303,53304,53312],{},[32,53305,53306],{},[1057,53307,53308,53310],{},[35,53309,4554],{},[35,53311,37],{},[45,53313,52918],{"encoding":47},[15,53315,53317],{"className":53316,"ariaHidden":53},[52],[15,53318,53320,53323],{"className":53319},[57],[15,53321],{"className":53322,"style":10192},[61],[15,53324,53326,53329],{"className":53325},[66],[15,53327,4554],{"className":53328},[66,67],[15,53330,53332],{"className":53331},[198],[15,53333,53335,53355],{"className":53334},[202,1090],[15,53336,53338,53352],{"className":53337},[206],[15,53339,53341],{"className":53340,"style":2198},[210],[15,53342,53343,53346],{"style":1584},[15,53344],{"className":53345,"style":218},[217],[15,53347,53349],{"className":53348},[222,223,224,225],[15,53350,37],{"className":53351},[66,67,225],[15,53353,1114],{"className":53354},[1113],[15,53356,53358],{"className":53357},[206],[15,53359,53361],{"className":53360,"style":1121},[210],[15,53362],{}," 收敛，再次使用比值判别法：",[15,53365,53367],{"className":53366,"translate":19},[294],[15,53368,53370,53492],{"className":53369,"translate":19},[18],[15,53371,53373],{"className":53372},[23],[25,53374,53375],{"xmlns":27,"display":303},[29,53376,53377,53489],{},[32,53378,53379,53385,53387,53411,53417,53419,53421,53433,53435,53477],{},[1057,53380,53381,53383],{},[35,53382,4554],{},[35,53384,37],{},[39,53386,129],{},[6465,53388,53389,53395],{},[32,53390,53391,53393],{},[35,53392,37],{},[39,53394,40764],{"stretchy":119},[131,53396,53397,53399],{},[35,53398,37],{},[32,53400,53401,53403,53405],{},[35,53402,37],{},[39,53404,1245],{},[6465,53406,53407,53409],{},[1209,53408,1211],{},[1209,53410,1263],{},[131,53412,53413,53415],{},[35,53414,48378],{},[35,53416,37],{},[39,53418,335],{"separator":53},[71,53420],{"width":9786},[1057,53422,53423,53425],{},[35,53424,4554],{},[32,53426,53427,53429,53431],{},[35,53428,37],{},[39,53430,1245],{},[1209,53432,1211],{},[39,53434,129],{},[6465,53436,53437,53451],{},[32,53438,53439,53441,53443,53445,53447,53449],{},[39,53440,120],{"stretchy":119},[35,53442,37],{},[39,53444,1245],{},[1209,53446,1211],{},[39,53448,126],{"stretchy":119},[39,53450,40764],{"stretchy":119},[32,53452,53453,53455,53457,53459,53461],{},[39,53454,120],{"stretchy":119},[35,53456,37],{},[39,53458,1245],{},[1209,53460,1211],{},[131,53462,53463,53465],{},[39,53464,126],{"stretchy":119},[32,53466,53467,53469,53471],{},[35,53468,37],{},[39,53470,1245],{},[6465,53472,53473,53475],{},[1209,53474,8673],{},[1209,53476,1263],{},[131,53478,53479,53481],{},[35,53480,48378],{},[32,53482,53483,53485,53487],{},[35,53484,37],{},[39,53486,1245],{},[1209,53488,1211],{},[45,53490,53491],{"encoding":47},"a_n = \\frac{n!}{n^{n + \\frac{1}{2}}} e^n, \\quad a_{n+1} = \\frac{(n+1)!}{(n+1)^{n + \\frac{3}{2}}} e^{n+1}",[15,53493,53495,53550,53817],{"className":53494,"ariaHidden":53},[52],[15,53496,53498,53501,53541,53544,53547],{"className":53497},[57],[15,53499],{"className":53500,"style":10192},[61],[15,53502,53504,53507],{"className":53503},[66],[15,53505,4554],{"className":53506},[66,67],[15,53508,53510],{"className":53509},[198],[15,53511,53513,53533],{"className":53512},[202,1090],[15,53514,53516,53530],{"className":53515},[206],[15,53517,53519],{"className":53518,"style":2198},[210],[15,53520,53521,53524],{"style":1584},[15,53522],{"className":53523,"style":218},[217],[15,53525,53527],{"className":53526},[222,223,224,225],[15,53528,37],{"className":53529},[66,67,225],[15,53531,1114],{"className":53532},[1113],[15,53534,53536],{"className":53535},[206],[15,53537,53539],{"className":53538,"style":1121},[210],[15,53540],{},[15,53542],{"className":53543,"style":173},[71],[15,53545,129],{"className":53546},[177],[15,53548],{"className":53549,"style":173},[71],[15,53551,53553,53556,53721,53750,53753,53756,53759,53808,53811,53814],{"className":53552},[57],[15,53554],{"className":53555,"style":53092},[61],[15,53557,53559,53562,53718],{"className":53558},[66],[15,53560],{"className":53561},[162,23412],[15,53563,53565],{"className":53564},[6465],[15,53566,53568,53710],{"className":53567},[202,1090],[15,53569,53571,53707],{"className":53570},[206],[15,53572,53574,53685,53693],{"className":53573,"style":27430},[210],[15,53575,53576,53579],{"style":53113},[15,53577],{"className":53578,"style":1368},[217],[15,53580,53582],{"className":53581},[66],[15,53583,53585,53588],{"className":53584},[66],[15,53586,37],{"className":53587},[66,67],[15,53589,53591],{"className":53590},[198],[15,53592,53594],{"className":53593},[202],[15,53595,53597],{"className":53596},[206],[15,53598,53600],{"className":53599,"style":53138},[210],[15,53601,53602,53605],{"style":53141},[15,53603],{"className":53604,"style":1368},[217],[15,53606,53608],{"className":53607},[222,223,224,225],[15,53609,53611,53614,53617],{"className":53610},[66,225],[15,53612,37],{"className":53613},[66,67,225],[15,53615,1245],{"className":53616},[76,225],[15,53618,53620,53623,53682],{"className":53619},[66,225],[15,53621],{"className":53622},[162,23412,222,3677,33576],[15,53624,53626],{"className":53625},[6465],[15,53627,53629,53674],{"className":53628},[202,1090],[15,53630,53632,53671],{"className":53631},[206],[15,53633,53635,53649,53657],{"className":53634,"style":49016},[210],[15,53636,53637,53640],{"style":33592},[15,53638],{"className":53639,"style":1368},[217],[15,53641,53643],{"className":53642},[222,3677,1508,225],[15,53644,53646],{"className":53645},[66,225],[15,53647,1263],{"className":53648},[66,225],[15,53650,53651,53654],{"style":33607},[15,53652],{"className":53653,"style":1368},[217],[15,53655],{"className":53656,"style":33614},[23447,225],[15,53658,53659,53662],{"style":33617},[15,53660],{"className":53661,"style":1368},[217],[15,53663,53665],{"className":53664},[222,3677,1508,225],[15,53666,53668],{"className":53667},[66,225],[15,53669,1211],{"className":53670},[66,225],[15,53672,1114],{"className":53673},[1113],[15,53675,53677],{"className":53676},[206],[15,53678,53680],{"className":53679,"style":33639},[210],[15,53681],{},[15,53683],{"className":53684},[169,23412,222,3677,33576],[15,53686,53687,53690],{"style":23440},[15,53688],{"className":53689,"style":1368},[217],[15,53691],{"className":53692,"style":4438},[23447],[15,53694,53695,53698],{"style":23450},[15,53696],{"className":53697,"style":1368},[217],[15,53699,53701,53704],{"className":53700},[66],[15,53702,37],{"className":53703},[66,67],[15,53705,40764],{"className":53706},[169],[15,53708,1114],{"className":53709},[1113],[15,53711,53713],{"className":53712},[206],[15,53714,53716],{"className":53715,"style":53256},[210],[15,53717],{},[15,53719],{"className":53720},[169,23412],[15,53722,53724,53727],{"className":53723},[66],[15,53725,48378],{"className":53726},[66,67],[15,53728,53730],{"className":53729},[198],[15,53731,53733],{"className":53732},[202],[15,53734,53736],{"className":53735},[206],[15,53737,53739],{"className":53738,"style":51746},[210],[15,53740,53741,53744],{"style":397},[15,53742],{"className":53743,"style":218},[217],[15,53745,53747],{"className":53746},[222,223,224,225],[15,53748,37],{"className":53749},[66,67,225],[15,53751,335],{"className":53752},[422],[15,53754],{"className":53755,"style":10044},[71],[15,53757],{"className":53758,"style":697},[71],[15,53760,53762,53765],{"className":53761},[66],[15,53763,4554],{"className":53764},[66,67],[15,53766,53768],{"className":53767},[198],[15,53769,53771,53800],{"className":53770},[202,1090],[15,53772,53774,53797],{"className":53773},[206],[15,53775,53777],{"className":53776,"style":2139},[210],[15,53778,53779,53782],{"style":1584},[15,53780],{"className":53781,"style":218},[217],[15,53783,53785],{"className":53784},[222,223,224,225],[15,53786,53788,53791,53794],{"className":53787},[66,225],[15,53789,37],{"className":53790},[66,67,2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4099,54100,54102,54104],{},[35,54101,37],{},[39,54103,1245],{},[1209,54105,1211],{},[1057,54107,54108,54110],{},[35,54109,4554],{},[35,54111,37],{},[39,54113,129],{},[6465,54115,54116,54118],{},[35,54117,48378],{},[131,54119,54120,54136],{},[32,54121,54122,54124,54126,54128,54134],{},[39,54123,120],{"fence":53},[1209,54125,1211],{},[39,54127,1245],{},[6465,54129,54130,54132],{},[1209,54131,1211],{},[35,54133,37],{},[39,54135,126],{"fence":53},[32,54137,54138,54140,54142],{},[35,54139,37],{},[39,54141,1245],{},[6465,54143,54144,54146],{},[1209,54145,1211],{},[1209,54147,1263],{},[45,54149,54150],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}}}",[15,54152,54154,54370],{"className":54153,"ariaHidden":53},[52],[15,54155,54157,54160,54213,54216,54361,54364,54367],{"className":54156},[57],[15,54158],{"className":54159,"style":52418},[61],[15,54161,54163],{"className":54162},[1339,1351],[15,54164,54166,54205],{"className":54165},[202,1090],[15,54167,54169,54202],{"className":54168},[206],[15,54170,54172,54192],{"className":54171,"style":840},[210],[15,54173,54174,54177],{"style":48819},[15,54175],{"className":54176,"style":1368},[217],[15,54178,54180],{"className":54179},[222,223,224,225],[15,54181,54183,54186,54189],{"className":54182},[66,225],[15,54184,37],{"className":54185},[66,67,225],[15,54187,449],{"className":54188},[177,225],[15,54190,7615],{"className":54191},[66,225],[15,54193,54194,54197],{"style":1386},[15,54195],{"className":54196,"style":1368},[217],[15,54198,54199],{},[15,54200,43663],{"className":54201},[1339],[15,54203,1114],{"className":54204},[1113],[15,54206,54208],{"className":54207},[206],[15,54209,54211],{"className":54210,"style":48857},[210],[15,54212],{},[15,54214],{"className":54215,"style":697},[71],[15,54217,54219,54222,54358],{"className":54218},[66],[15,54220],{"className":54221},[162,23412],[15,54223,54225],{"className":54224},[6465],[15,54226,54228,54350],{"className":54227},[202,1090],[15,54229,54231,54347],{"className":54230},[206],[15,54232,54234,54282,54290],{"className":54233,"style":39128},[210],[15,54235,54236,54239],{"style":23428},[15,54237],{"className":54238,"style":1368},[217],[15,54240,54242],{"className":54241},[66],[15,54243,54245,54248],{"className":54244},[66],[15,54246,4554],{"className":54247},[66,67],[15,54249,54251],{"className":54250},[198],[15,54252,54254,54274],{"className":54253},[202,1090],[15,54255,54257,54271],{"className":54256},[206],[15,54258,54260],{"className":54259,"style":2198},[210],[15,54261,54262,54265],{"style":1584},[15,54263],{"className":54264,"style":218},[217],[15,54266,54268],{"className":54267},[222,223,224,225],[15,54269,37],{"className":54270},[66,67,225],[15,54272,1114],{"className":54273},[1113],[15,54275,54277],{"className":54276},[206],[15,54278,54280],{"className":54279,"style":1121},[210],[15,54281],{},[15,54283,54284,54287],{"style":23440},[15,54285],{"className":54286,"style":1368},[217],[15,54288],{"className":54289,"style":4438},[23447],[15,54291,54292,54295],{"style":23450},[15,54293],{"className":54294,"style":1368},[217],[15,54296,54298],{"className":54297},[66],[15,54299,54301,54304],{"className":54300},[66],[15,54302,4554],{"className":54303},[66,67],[15,54305,54307],{"className":54306},[198],[15,54308,54310,54339],{"className":54309},[202,1090],[15,54311,54313,54336],{"className":54312},[206],[15,54314,54316],{"className":54315,"style":2139},[210],[15,54317,54318,54321],{"style":1584},[15,54319],{"className":54320,"style":218},[217],[15,54322,54324],{"className":54323},[222,223,224,225],[15,54325,54327,54330,54333],{"className":54326},[66,225],[15,54328,37],{"className":54329},[66,67,225],[15,54331,1245],{"className":54332},[76,225],[15,54334,1211],{"className":54335},[66,225],[15,54337,1114],{"className":54338},[1113],[15,54340,54342],{"className":54341},[206],[15,54343,54345],{"className":54344,"style":52136},[210],[15,54346],{},[15,54348,1114],{"className":54349},[1113],[15,54351,54353],{"className":54352},[206],[15,54354,54356],{"className":54355,"style":52615},[210],[15,54357],{},[15,54359],{"className":54360},[169,23412],[15,54362],{"className":54363,"style":173},[71],[15,54365,129],{"className":54366},[177],[15,54368],{"className":54369,"style":173},[71],[15,54371,54373,54377],{"className":54372},[57],[15,54374],{"className":54375,"style":54376},[61],"height:2.5415em;vertical-align:-1.4339em;",[15,54378,54380,54383,54634],{"className":54379},[66],[15,54381],{"className":54382},[162,23412],[15,54384,54386],{"className":54385},[6465],[15,54387,54389,54625],{"className":54388},[202,1090],[15,54390,54392,54622],{"className":54391},[206],[15,54393,54395,54601,54610],{"className":54394,"style":39128},[210],[15,54396,54398,54402],{"style":54397},"top:-2.11em;",[15,54399],{"className":54400,"style":54401},[217],"height:3.1939em;",[15,54403,54405],{"className":54404},[66],[15,54406,54408,54503],{"className":54407},[1413],[15,54409,54411,54417,54420,54423,54426,54429,54497],{"className":54410},[1413],[15,54412,54414],{"className":54413,"style":1418},[162,1417],[15,54415,120],{"className":54416},[1422,1508],[15,54418,1211],{"className":54419},[66],[15,54421],{"className":54422,"style":72},[71],[15,54424,1245],{"className":54425},[76],[15,54427],{"className":54428,"style":72},[71],[15,54430,54432,54435,54494],{"className":54431},[66],[15,54433],{"className":54434},[162,23412],[15,54436,54438],{"className":54437},[6465],[15,54439,54441,54486],{"className":54440},[202,1090],[15,54442,54444,54483],{"className":54443},[206],[15,54445,54447,54461,54469],{"className":54446,"style":30868},[210],[15,54448,54449,54452],{"style":30871},[15,54450],{"className":54451,"style":1368},[217],[15,54453,54455],{"className":54454},[222,223,224,225],[15,54456,54458],{"className":54457},[66,225],[15,54459,37],{"className":54460},[66,67,225],[15,54462,54463,54466],{"style":23440},[15,54464],{"className":54465,"style":1368},[217],[15,54467],{"className":54468,"style":4438},[23447],[15,54470,54471,54474],{"style":30894},[15,54472],{"className":54473,"style":1368},[217],[15,54475,54477],{"className":54476},[222,223,224,225],[15,54478,54480],{"className":54479},[66,225],[15,54481,1211],{"className":54482},[66,225],[15,54484,1114],{"className":54485},[1113],[15,54487,54489],{"className":54488},[206],[15,54490,54492],{"className":54491,"style":30916},[210],[15,54493],{},[15,54495],{"className":54496},[169,23412],[15,54498,54500],{"className":54499,"style":1418},[169,1417],[15,54501,126],{"className":54502},[1422,1508],[15,54504,54506],{"className":54505},[198],[15,54507,54509],{"className":54508},[202],[15,54510,54512],{"className":54511},[206],[15,54513,54516],{"className":54514,"style":54515},[210],"height:1.1939em;",[15,54517,54518,54521],{"style":51213},[15,54519],{"className":54520,"style":1368},[217],[15,54522,54524],{"className":54523},[222,223,224,225],[15,54525,54527,54530,54533],{"className":54526},[66,225],[15,54528,37],{"className":54529},[66,67,225],[15,54531,1245],{"className":54532},[76,225],[15,54534,54536,54539,54598],{"className":54535},[66,225],[15,54537],{"className":54538},[162,23412,222,3677,33576],[15,54540,54542],{"className":54541},[6465],[15,54543,54545,54590],{"className":54544},[202,1090],[15,54546,54548,54587],{"className":54547},[206],[15,54549,54551,54565,54573],{"className":54550,"style":49016},[210],[15,54552,54553,54556],{"style":33592},[15,54554],{"className":54555,"style":1368},[217],[15,54557,54559],{"className":54558},[222,3677,1508,225],[15,54560,54562],{"className":54561},[66,225],[15,54563,1263],{"className":54564},[66,225],[15,54566,54567,54570],{"style":33607},[15,54568],{"className":54569,"style":1368},[217],[15,54571],{"className":54572,"style":33614},[23447,225],[15,54574,54575,54578],{"style":33617},[15,54576],{"className":54577,"style":1368},[217],[15,54579,54581],{"className":54580},[222,3677,1508,225],[15,54582,54584],{"className":54583},[66,225],[15,54585,1211],{"className":54586},[66,225],[15,54588,1114],{"className":54589},[1113],[15,54591,54593],{"className":54592},[206],[15,54594,54596],{"className":54595,"style":33639},[210],[15,54597],{},[15,54599],{"className":54600},[169,23412,222,3677,33576],[15,54602,54604,54607],{"style":54603},"top:-3.4239em;",[15,54605],{"className":54606,"style":54401},[217],[15,54608],{"className":54609,"style":4438},[23447],[15,54611,54613,54616],{"style":54612},"top:-3.8709em;",[15,54614],{"className":54615,"style":54401},[217],[15,54617,54619],{"className":54618},[66],[15,54620,48378],{"className":54621},[66,67],[15,54623,1114],{"className":54624},[1113],[15,54626,54628],{"className":54627},[206],[15,54629,54632],{"className":54630,"style":54631},[210],"height:1.4339em;",[15,54633],{},[15,54635],{"className":54636},[169,23412],[11,54638,54639],{},"将其与下式比较：",[15,54641,54643],{"className":54642,"translate":19},[294],[15,54644,54646,54730],{"className":54645,"translate":19},[18],[15,54647,54649],{"className":54648},[23],[25,54650,54651],{"xmlns":27,"display":303},[29,54652,54653,54727],{},[32,54654,54655,54685,54687],{},[131,54656,54657,54673],{},[32,54658,54659,54661,54663,54665,54671],{},[39,54660,120],{"fence":53},[1209,54662,1211],{},[39,54664,1245],{},[6465,54666,54667,54669],{},[1209,54668,1211],{},[35,54670,37],{},[39,54672,126],{"fence":53},[32,54674,54675,54677,54679],{},[35,54676,37],{},[39,54678,1245],{},[6465,54680,54681,54683],{},[1209,54682,1211],{},[1209,54684,1263],{},[39,54686,129],{},[131,54688,54689,54691],{},[35,54690,48378],{},[32,54692,54693,54695,54697,54699,54705,54707,54709,54711],{},[39,54694,120],{"stretchy":119},[35,54696,37],{},[39,54698,1245],{},[6465,54700,54701,54703],{},[1209,54702,1211],{},[1209,54704,1263],{},[39,54706,126],{"stretchy":119},[35,54708,49239],{},[39,54710,1170],{},[32,54712,54713,54715,54717,54719,54725],{},[39,54714,120],{"fence":53},[1209,54716,1211],{},[39,54718,1245],{},[6465,54720,54721,54723],{},[1209,54722,1211],{},[35,54724,37],{},[39,54726,126],{"fence":53},[45,54728,54729],{"encoding":47},"\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}} = e^{(n + \\frac{1}{2}) \\ln \\left(1 + \\frac{1}{n}\\right)}",[15,54731,54733,54937],{"className":54732,"ariaHidden":53},[52],[15,54734,54736,54739,54928,54931,54934],{"className":54735},[57],[15,54737],{"className":54738,"style":48804},[61],[15,54740,54742,54831],{"className":54741},[1413],[15,54743,54745,54751,54754,54757,54760,54763,54825],{"className":54744},[1413],[15,54746,54748],{"className":54747,"style":1418},[162,1417],[15,54749,120],{"className":54750},[1422,224],[15,54752,1211],{"className":54753},[66],[15,54755],{"className":54756,"style":72},[71],[15,54758,1245],{"className":54759},[76],[15,54761],{"className":54762,"style":72},[71],[15,54764,54766,54769,54822],{"className":54765},[66],[15,54767],{"className":54768},[162,23412],[15,54770,54772],{"className":54771},[6465],[15,54773,54775,54814],{"className":54774},[202,1090],[15,54776,54778,54811],{"className":54777},[206],[15,54779,54781,54792,54800],{"className":54780,"style":23425},[210],[15,54782,54783,54786],{"style":23428},[15,54784],{"className":54785,"style":1368},[217],[15,54787,54789],{"className":54788},[66],[15,54790,37],{"className":54791},[66,67],[15,54793,54794,54797],{"style":23440},[15,54795],{"className":54796,"style":1368},[217],[15,54798],{"className":54799,"style":4438},[23447],[15,54801,54802,54805],{"style":23450},[15,54803],{"className":54804,"style":1368},[217],[15,54806,54808],{"className":54807},[66],[15,54809,1211],{"className":54810},[66],[15,54812,1114],{"className":54813},[1113],[15,54815,54817],{"className":54816},[206],[15,54818,54820],{"className":54819,"style":23469},[210],[15,54821],{},[15,54823],{"className":54824},[169,23412],[15,54826,54828],{"className":54827,"style":1418},[169,1417],[15,54829,126],{"className":54830},[1422,224],[15,54832,54834],{"className":54833},[198],[15,54835,54837],{"className":54836},[202],[15,54838,54840],{"className":54839},[206],[15,54841,54843],{"className":54842,"style":48985},[210],[15,54844,54845,54848],{"style":48988},[15,54846],{"className":54847,"style":1368},[217],[15,54849,54851],{"className":54850},[222,223,224,225],[15,54852,54854,54857,54860],{"className":54853},[66,225],[15,54855,37],{"className":54856},[66,67,225],[15,54858,1245],{"className":54859},[76,225],[15,54861,54863,54866,54925],{"className":54862},[66,225],[15,54864],{"className":54865},[162,23412,222,3677,33576],[15,54867,54869],{"className":54868},[6465],[15,54870,54872,54917],{"className":54871},[202,1090],[15,54873,54875,54914],{"className":54874},[206],[15,54876,54878,54892,54900],{"className":54877,"style":49016},[210],[15,54879,54880,54883],{"style":33592},[15,54881],{"className":54882,"style":1368},[217],[15,54884,54886],{"className":54885},[222,3677,1508,225],[15,54887,54889],{"className":54888},[66,225],[15,54890,1263],{"className":54891},[66,225],[15,54893,54894,54897],{"style":33607},[15,54895],{"className":54896,"style":1368},[217],[15,54898],{"className":54899,"style":33614},[23447,225],[15,54901,54902,54905],{"style":33617},[15,54903],{"className":54904,"style":1368},[217],[15,54906,54908],{"className":54907},[222,3677,1508,225],[15,54909,54911],{"className":54910},[66,225],[15,54912,1211],{"className":54913},[66,225],[15,54915,1114],{"className":54916},[1113],[15,54918,54920],{"className":54919},[206],[15,54921,54923],{"className":54922,"style":33639},[210],[15,54924],{},[15,54926],{"className":54927},[169,23412,222,3677,33576],[15,54929],{"className":54930,"style":173},[71],[15,54932,129],{"className":54933},[177],[15,54935],{"className":54936,"style":173},[71],[15,54938,54940,54943],{"className":54939},[57],[15,54941],{"className":54942,"style":49620},[61],[15,54944,54946,54949],{"className":54945},[66],[15,54947,48378],{"className":54948},[66,67],[15,54950,54952],{"className":54951},[198],[15,54953,54955],{"className":54954},[202],[15,54956,54958],{"className":54957},[206],[15,54959,54961],{"className":54960,"style":49620},[210],[15,54962,54963,54966],{"style":49623},[15,54964],{"className":54965,"style":1368},[217],[15,54967,54969],{"className":54968},[222,223,224,225],[15,54970,54972,54975,54978,54981,55049,55052,55055,55064],{"className":54971},[66,225],[15,54973,120],{"className":54974},[162,225],[15,54976,37],{"className":54977},[66,67,225],[15,54979,1245],{"className":54980},[76,225],[15,54982,54984,54987,55046],{"className":54983},[66,225],[15,54985],{"className":54986},[162,23412,222,3677,33576],[15,54988,54990],{"className":54989},[6465],[15,54991,54993,55038],{"className":54992},[202,1090],[15,54994,54996,55035],{"className":54995},[206],[15,54997,54999,55013,55021],{"className":54998,"style":49016},[210],[15,55000,55001,55004],{"style":33592},[15,55002],{"className":55003,"style":1368},[217],[15,55005,55007],{"className":55006},[222,3677,1508,225],[15,55008,55010],{"className":55009},[66,225],[15,55011,1263],{"className":55012},[66,225],[15,55014,55015,55018],{"style":33607},[15,55016],{"className":55017,"style":1368},[217],[15,55019],{"className":55020,"style":33614},[23447,225],[15,55022,55023,55026],{"style":33617},[15,55024],{"className":55025,"style":1368},[217],[15,55027,55029],{"className":55028},[222,3677,1508,225],[15,55030,55032],{"className":55031},[66,225],[15,55033,1211],{"className":55034},[66,225],[15,55036,1114],{"className":55037},[1113],[15,55039,55041],{"className":55040},[206],[15,55042,55044],{"className":55043,"style":33639},[210],[15,55045],{},[15,55047],{"className":55048},[169,23412,222,3677,33576],[15,55050,126],{"className":55051},[169,225],[15,55053],{"className":55054,"style":23363},[71,225],[15,55056,55058,55061],{"className":55057},[1339,225],[15,55059,49710],{"className":55060},[225],[15,55062,37],{"className":55063},[225],[15,55065,55067,55073,55076,55079,55147],{"className":55066},[1413,225],[15,55068,55070],{"className":55069,"style":49720},[162,222,3677,33576,225,1417],[15,55071,120],{"className":55072},[225],[15,55074,1211],{"className":55075},[66,225],[15,55077,1245],{"className":55078},[76,225],[15,55080,55082,55085,55144],{"className":55081},[66,225],[15,55083],{"className":55084},[162,23412,222,3677,33576],[15,55086,55088],{"className":55087},[6465],[15,55089,55091,55136],{"className":55090},[202,1090],[15,55092,55094,55133],{"className":55093},[206],[15,55095,55097,55111,55119],{"className":55096,"style":49016},[210],[15,55098,55099,55102],{"style":33592},[15,55100],{"className":55101,"style":1368},[217],[15,55103,55105],{"className":55104},[222,3677,1508,225],[15,55106,55108],{"className":55107},[66,225],[15,55109,37],{"className":55110},[66,67,225],[15,55112,55113,55116],{"style":33607},[15,55114],{"className":55115,"style":1368},[217],[15,55117],{"className":55118,"style":33614},[23447,225],[15,55120,55121,55124],{"style":33617},[15,55122],{"className":55123,"style":1368},[217],[15,55125,55127],{"className":55126},[222,3677,1508,225],[15,55128,55130],{"className":55129},[66,225],[15,55131,1211],{"className":55132},[66,225],[15,55134,1114],{"className":55135},[1113],[15,55137,55139],{"className":55138},[206],[15,55140,55142],{"className":55141,"style":33639},[210],[15,55143],{},[15,55145],{"className":55146},[169,23412,222,3677,33576],[15,55148,55150],{"className":55149,"style":49720},[169,222,3677,33576,225,1417],[15,55151,126],{"className":55152},[225],[11,55154,55155,55156,1124],{},"以及 ",[15,55157,55159,55172],{"className":55158,"translate":19},[18],[15,55160,55162],{"className":55161},[23],[25,55163,55164],{"xmlns":27},[29,55165,55166,55170],{},[32,55167,55168],{},[35,55169,48378],{},[45,55171,48378],{"encoding":47},[15,55173,55175],{"className":55174,"ariaHidden":53},[52],[15,55176,55178,55181],{"className":55177},[57],[15,55179],{"className":55180,"style":86},[61],[15,55182,48378],{"className":55183},[66,67],[11,55185,55186],{},"因此，比较：",[15,55188,55190],{"className":55189,"translate":19},[294],[15,55191,55193,55239],{"className":55192,"translate":19},[18],[15,55194,55196],{"className":55195},[23],[25,55197,55198],{"xmlns":27,"display":303},[29,55199,55200,55236],{},[32,55201,55202,55204,55206,55208,55214,55216,55218,55220],{},[39,55203,120],{"stretchy":119},[35,55205,37],{},[39,55207,1245],{},[6465,55209,55210,55212],{},[1209,55211,1211],{},[1209,55213,1263],{},[39,55215,126],{"stretchy":119},[35,55217,49239],{},[39,55219,1170],{},[32,55221,55222,55224,55226,55228,55234],{},[39,55223,120],{"fence":53},[1209,55225,1211],{},[39,55227,1245],{},[6465,55229,55230,55232],{},[1209,55231,1211],{},[35,55233,37],{},[39,55235,126],{"fence":53},[45,55237,55238],{"encoding":47},"(n + \\frac{1}{2}) \\ln \\left(1 + \\frac{1}{n}\\right)",[15,55240,55242,55263],{"className":55241,"ariaHidden":53},[52],[15,55243,55245,55248,55251,55254,55257,55260],{"className":55244},[57],[15,55246],{"className":55247,"style":154},[61],[15,55249,120],{"className":55250},[162],[15,55252,37],{"className":55253},[66,67],[15,55255],{"className":55256,"style":72},[71],[15,55258,1245],{"className":55259},[76],[15,55261],{"className":55262,"style":72},[71],[15,55264,55266,55269,55331,55334,55337,55340,55343],{"className":55265},[57],[15,55267],{"className":55268,"style":43968},[61],[15,55270,55272,55275,55328],{"className":55271},[66],[15,55273],{"className":55274},[162,23412],[15,55276,55278],{"className":55277},[6465],[15,55279,55281,55320],{"className":55280},[202,1090],[15,55282,55284,55317],{"className":55283},[206],[15,55285,55287,55298,55306],{"className":55286,"style":23425},[210],[15,55288,55289,55292],{"style":23428},[15,55290],{"className":55291,"style":1368},[217],[15,55293,55295],{"className":55294},[66],[15,55296,1263],{"className":55297},[66],[15,55299,55300,55303],{"style":23440},[15,55301],{"className":55302,"style":1368},[217],[15,55304],{"className":55305,"style":4438},[23447],[15,55307,55308,55311],{"style":23450},[15,55309],{"className":55310,"style":1368},[217],[15,55312,55314],{"className":55313},[66],[15,55315,1211],{"className":55316},[66],[15,55318,1114],{"className":55319},[1113],[15,55321,55323],{"className":55322},[206],[15,55324,55326],{"className":55325,"style":23469},[210],[15,55327],{},[15,55329],{"className":55330},[169,23412],[15,55332,126],{"className":55333},[169],[15,55335],{"className":55336,"style":697},[71],[15,55338,49239],{"className":55339},[1339],[15,55341],{"className":55342,"style":697},[71],[15,55344,55346,55352,55355,55358,55361,55364,55426],{"className":55345},[1413],[15,55347,55349],{"className":55348,"style":1418},[162,1417],[15,55350,120],{"className":55351},[1422,224],[15,55353,1211],{"className":55354},[66],[15,55356],{"className":55357,"style":72},[71],[15,55359,1245],{"className":55360},[76],[15,55362],{"className":55363,"style":72},[71],[15,55365,55367,55370,55423],{"className":55366},[66],[15,55368],{"className":55369},[162,23412],[15,55371,55373],{"className":55372},[6465],[15,55374,55376,55415],{"className":55375},[202,1090],[15,55377,55379,55412],{"className":55378},[206],[15,55380,55382,55393,55401],{"className":55381,"style":23425},[210],[15,55383,55384,55387],{"style":23428},[15,55385],{"className":55386,"style":1368},[217],[15,55388,55390],{"className":55389},[66],[15,55391,37],{"className":55392},[66,67],[15,55394,55395,55398],{"style":23440},[15,55396],{"className":55397,"style":1368},[217],[15,55399],{"className":55400,"style":4438},[23447],[15,55402,55403,55406],{"style":23450},[15,55404],{"className":55405,"style":1368},[217],[15,55407,55409],{"className":55408},[66],[15,55410,1211],{"className":55411},[66],[15,55413,1114],{"className":55414},[1113],[15,55416,55418],{"className":55417},[206],[15,55419,55421],{"className":55420,"style":23469},[210],[15,55422],{},[15,55424],{"className":55425},[169,23412],[15,55427,55429],{"className":55428,"style":1418},[169,1417],[15,55430,126],{"className":55431},[1422,224],[11,55433,55434],{},"与 1。",[11,55436,55437,55438,48361,55558,37639],{},"具体来说，比较 ",[15,55439,55441,55469],{"className":55440,"translate":19},[18],[15,55442,55444],{"className":55443},[23],[25,55445,55446],{"xmlns":27},[29,55447,55448,55466],{},[32,55449,55450,55452,55454],{},[35,55451,49239],{},[39,55453,1170],{},[6465,55455,55456,55464],{},[32,55457,55458,55460,55462],{},[35,55459,37],{},[39,55461,1245],{},[1209,55463,1211],{},[35,55465,37],{},[45,55467,55468],{"encoding":47},"\\ln \\frac{n+1}{n}",[15,55470,55472],{"className":55471,"ariaHidden":53},[52],[15,55473,55475,55478,55481,55484],{"className":55474},[57],[15,55476],{"className":55477,"style":30849},[61],[15,55479,49239],{"className":55480},[1339],[15,55482],{"className":55483,"style":697},[71],[15,55485,55487,55490,55555],{"className":55486},[66],[15,55488],{"className":55489},[162,23412],[15,55491,55493],{"className":55492},[6465],[15,55494,55496,55547],{"className":55495},[202,1090],[15,55497,55499,55544],{"className":55498},[206],[15,55500,55502,55516,55524],{"className":55501,"style":30868},[210],[15,55503,55504,55507],{"style":30871},[15,55505],{"className":55506,"style":1368},[217],[15,55508,55510],{"className":55509},[222,223,224,225],[15,55511,55513],{"className":55512},[66,225],[15,55514,37],{"className":55515},[66,67,225],[15,55517,55518,55521],{"style":23440},[15,55519],{"className":55520,"style":1368},[217],[15,55522],{"className":55523,"style":4438},[23447],[15,55525,55526,55529],{"style":30894},[15,55527],{"className":55528,"style":1368},[217],[15,55530,55532],{"className":55531},[222,223,224,225],[15,55533,55535,55538,55541],{"className":55534},[66,225],[15,55536,37],{"className":55537},[66,67,225],[15,55539,1245],{"className":55540},[76,225],[15,55542,1211],{"className":55543},[66,225],[15,55545,1114],{"className":55546},[1113],[15,55548,55550],{"className":55549},[206],[15,55551,55553],{"className":55552,"style":30916},[210],[15,55554],{},[15,55556],{"className":55557},[169,23412],[15,55559,55561,55589],{"className":55560,"translate":19},[18],[15,55562,55564],{"className":55563},[23],[25,55565,55566],{"xmlns":27},[29,55567,55568,55586],{},[32,55569,55570],{},[6465,55571,55572,55574],{},[1209,55573,1211],{},[32,55575,55576,55578,55580],{},[35,55577,37],{},[39,55579,1245],{},[6465,55581,55582,55584],{},[1209,55583,1211],{},[1209,55585,1263],{},[45,55587,55588],{"encoding":47},"\\frac{1}{n + \\frac{1}{2}}",[15,55590,55592],{"className":55591,"ariaHidden":53},[52],[15,55593,55595,55599],{"className":55594},[57],[15,55596],{"className":55597,"style":55598},[61],"height:1.4869em;vertical-align:-0.6418em;",[15,55600,55602,55605,55737],{"className":55601},[66],[15,55603],{"className":55604},[162,23412],[15,55606,55608],{"className":55607},[6465],[15,55609,55611,55728],{"className":55610},[202,1090],[15,55612,55614,55725],{"className":55613},[206],[15,55615,55617,55703,55711],{"className":55616,"style":30868},[210],[15,55618,55620,55623],{"style":55619},"top:-2.599em;",[15,55621],{"className":55622,"style":1368},[217],[15,55624,55626],{"className":55625},[222,223,224,225],[15,55627,55629,55632,55635],{"className":55628},[66,225],[15,55630,37],{"className":55631},[66,67,225],[15,55633,1245],{"className":55634},[76,225],[15,55636,55638,55641,55700],{"className":55637},[66,225],[15,55639],{"className":55640},[162,23412,222,3677,33576],[15,55642,55644],{"className":55643},[6465],[15,55645,55647,55692],{"className":55646},[202,1090],[15,55648,55650,55689],{"className":55649},[206],[15,55651,55653,55667,55675],{"className":55652,"style":49016},[210],[15,55654,55655,55658],{"style":33592},[15,55656],{"className":55657,"style":1368},[217],[15,55659,55661],{"className":55660},[222,3677,1508,225],[15,55662,55664],{"className":55663},[66,225],[15,55665,1263],{"className":55666},[66,225],[15,55668,55669,55672],{"style":33607},[15,55670],{"className":55671,"style":1368},[217],[15,55673],{"className":55674,"style":33614},[23447,225],[15,55676,55677,55680],{"style":33617},[15,55678],{"className":55679,"style":1368},[217],[15,55681,55683],{"className":55682},[222,3677,1508,225],[15,55684,55686],{"className":55685},[66,225],[15,55687,1211],{"className":55688},[66,225],[15,55690,1114],{"className":55691},[1113],[15,55693,55695],{"className":55694},[206],[15,55696,55698],{"className":55697,"style":33639},[210],[15,55699],{},[15,55701],{"className":55702},[169,23412,222,3677,33576],[15,55704,55705,55708],{"style":23440},[15,55706],{"className":55707,"style":1368},[217],[15,55709],{"className":55710,"style":4438},[23447],[15,55712,55713,55716],{"style":30894},[15,55714],{"className":55715,"style":1368},[217],[15,55717,55719],{"className":55718},[222,223,224,225],[15,55720,55722],{"className":55721},[66,225],[15,55723,1211],{"className":55724},[66,225],[15,55726,1114],{"className":55727},[1113],[15,55729,55731],{"className":55730},[206],[15,55732,55735],{"className":55733,"style":55734},[210],"height:0.6418em;",[15,55736],{},[15,55738],{"className":55739},[169,23412],[11,55741,55742],{},"由于：",[15,55744,55746],{"className":55745,"translate":19},[294],[15,55747,55749,55803],{"className":55748,"translate":19},[18],[15,55750,55752],{"className":55751},[23],[25,55753,55754],{"xmlns":27,"display":303},[29,55755,55756,55800],{},[32,55757,55758,55770,55772,55774,55776,55792,55794],{},[6465,55759,55760,55762],{},[1209,55761,1211],{},[32,55763,55764,55766,55768],{},[35,55765,37],{},[39,55767,1245],{},[1209,55769,1211],{},[39,55771,25767],{},[35,55773,49239],{},[39,55775,1170],{},[32,55777,55778,55780,55782,55784,55790],{},[39,55779,120],{"fence":53},[1209,55781,1211],{},[39,55783,1245],{},[6465,55785,55786,55788],{},[1209,55787,1211],{},[35,55789,37],{},[39,55791,126],{"fence":53},[39,55793,25767],{},[6465,55795,55796,55798],{},[1209,55797,1211],{},[35,55799,37],{},[45,55801,55802],{"encoding":47},"\\frac{1}{n + 1} \u003C \\ln \\left(1 + \\frac{1}{n}\\right) \u003C \\frac{1}{n}",[15,55804,55806,55895,56005],{"className":55805,"ariaHidden":53},[52],[15,55807,55809,55812,55886,55889,55892],{"className":55808},[57],[15,55810],{"className":55811,"style":44665},[61],[15,55813,55815,55818,55883],{"className":55814},[66],[15,55816],{"className":55817},[162,23412],[15,55819,55821],{"className":55820},[6465],[15,55822,55824,55875],{"className":55823},[202,1090],[15,55825,55827,55872],{"className":55826},[206],[15,55828,55830,55853,55861],{"className":55829,"style":23425},[210],[15,55831,55832,55835],{"style":23428},[15,55833],{"className":55834,"style":1368},[217],[15,55836,55838,55841,55844,55847,55850],{"className":55837},[66],[15,55839,37],{"className":55840},[66,67],[15,55842],{"className":55843,"style":72},[71],[15,55845,1245],{"className":55846},[76],[15,55848],{"className":55849,"style":72},[71],[15,55851,1211],{"className":55852},[66],[15,55854,55855,55858],{"style":23440},[15,55856],{"className":55857,"style":1368},[217],[15,55859],{"className":55860,"style":4438},[23447],[15,55862,55863,55866],{"style":23450},[15,55864],{"className":55865,"style":1368},[217],[15,55867,55869],{"className":55868},[66],[15,55870,1211],{"className":55871},[66],[15,55873,1114],{"className":55874},[1113],[15,55876,55878],{"className":55877},[206],[15,55879,55881],{"className":55880,"style":38565},[210],[15,55882],{},[15,55884],{"className":55885},[169,23412],[15,55887],{"className":55888,"style":173},[71],[15,55890,25767],{"className":55891},[177],[15,55893],{"className":55894,"style":173},[71],[15,55896,55898,55901,55904,55907,55996,55999,56002],{"className":55897},[57],[15,55899],{"className":55900,"style":43968},[61],[15,55902,49239],{"className":55903},[1339],[15,55905],{"className":55906,"style":697},[71],[15,55908,55910,55916,55919,55922,55925,55928,55990],{"className":55909},[1413],[15,55911,55913],{"className":55912,"style":1418},[162,1417],[15,55914,120],{"className":55915},[1422,224],[15,55917,1211],{"className":55918},[66],[15,55920],{"className":55921,"style":72},[71],[15,55923,1245],{"className":55924},[76],[15,55926],{"className":55927,"style":72},[71],[15,55929,55931,55934,55987],{"className":55930},[66],[15,55932],{"className":55933},[162,23412],[15,55935,55937],{"className":55936},[6465],[15,55938,55940,55979],{"className":55939},[202,1090],[15,55941,55943,55976],{"className":55942},[206],[15,55944,55946,55957,55965],{"className":55945,"style":23425},[210],[15,55947,55948,55951],{"style":23428},[15,55949],{"className":55950,"style":1368},[217],[15,55952,55954],{"className":55953},[66],[15,55955,37],{"className":55956},[66,67],[15,55958,55959,55962],{"style":23440},[15,55960],{"className":55961,"style":1368},[217],[15,55963],{"className":55964,"style":4438},[23447],[15,55966,55967,55970],{"style":23450},[15,55968],{"className":55969,"style":1368},[217],[15,55971,55973],{"className":55972},[66],[15,55974,1211],{"className":55975},[66],[15,55977,1114],{"className":55978},[1113],[15,55980,55982],{"className":55981},[206],[15,55983,55985],{"className":55984,"style":23469},[210],[15,55986],{},[15,55988],{"className":55989},[169,23412],[15,55991,55993],{"className":55992,"style":1418},[169,1417],[15,55994,126],{"className":55995},[1422,224],[15,55997],{"className":55998,"style":173},[71],[15,56000,25767],{"className":56001},[177],[15,56003],{"className":56004,"style":173},[71],[15,56006,56008,56011],{"className":56007},[57],[15,56009],{"className":56010,"style":49082},[61],[15,56012,56014,56017,56070],{"className":56013},[66],[15,56015],{"className":56016},[162,23412],[15,56018,56020],{"className":56019},[6465],[15,56021,56023,56062],{"className":56022},[202,1090],[15,56024,56026,56059],{"className":56025},[206],[15,56027,56029,56040,56048],{"className":56028,"style":23425},[210],[15,56030,56031,56034],{"style":23428},[15,56032],{"className":56033,"style":1368},[217],[15,56035,56037],{"className":56036},[66],[15,56038,37],{"className":56039},[66,67],[15,56041,56042,56045],{"style":23440},[15,56043],{"className":56044,"style":1368},[217],[15,56046],{"className":56047,"style":4438},[23447],[15,56049,56050,56053],{"style":23450},[15,56051],{"className":56052,"style":1368},[217],[15,56054,56056],{"className":56055},[66],[15,56057,1211],{"className":56058},[66],[15,56060,1114],{"className":56061},[1113],[15,56063,56065],{"className":56064},[206],[15,56066,56068],{"className":56067,"style":23469},[210],[15,56069],{},[15,56071],{"className":56072},[169,23412],[11,56074,56075,56076,56166,56167,56251],{},"注意：右侧由不等式 ",[15,56077,56079,56109],{"className":56078,"translate":19},[18],[15,56080,56082],{"className":56081},[23],[25,56083,56084],{"xmlns":27},[29,56085,56086,56106],{},[32,56087,56088,56090,56092,56094,56096,56098,56100,56102,56104],{},[35,56089,49239],{},[39,56091,1170],{},[39,56093,120],{"stretchy":119},[1209,56095,1211],{},[39,56097,1245],{},[35,56099,123],{},[39,56101,126],{"stretchy":119},[39,56103,25767],{},[35,56105,123],{},[45,56107,56108],{"encoding":47},"\\ln(1 + x) \u003C x",[15,56110,56112,56136,56157],{"className":56111,"ariaHidden":53},[52],[15,56113,56115,56118,56121,56124,56127,56130,56133],{"className":56114},[57],[15,56116],{"className":56117,"style":154},[61],[15,56119,49239],{"className":56120},[1339],[15,56122,120],{"className":56123},[162],[15,56125,1211],{"className":56126},[66],[15,56128],{"className":56129,"style":72},[71],[15,56131,1245],{"className":56132},[76],[15,56134],{"className":56135,"style":72},[71],[15,56137,56139,56142,56145,56148,56151,56154],{"className":56138},[57],[15,56140],{"className":56141,"style":154},[61],[15,56143,123],{"className":56144},[66,67],[15,56146,126],{"className":56147},[169],[15,56149],{"className":56150,"style":173},[71],[15,56152,25767],{"className":56153},[177],[15,56155],{"className":56156,"style":173},[71],[15,56158,56160,56163],{"className":56159},[57],[15,56161],{"className":56162,"style":86},[61],[15,56164,123],{"className":56165},[66,67],"（其中 ",[15,56168,56170,56188],{"className":56169,"translate":19},[18],[15,56171,56173],{"className":56172},[23],[25,56174,56175],{"xmlns":27},[29,56176,56177,56185],{},[32,56178,56179,56181,56183],{},[35,56180,123],{},[39,56182,13014],{"mathvariant":138},[1209,56184,1804],{},[45,56186,56187],{"encoding":47},"x \\neq 0",[15,56189,56191,56242],{"className":56190,"ariaHidden":53},[52],[15,56192,56194,56197,56200,56203,56239],{"className":56193},[57],[15,56195],{"className":56196,"style":3406},[61],[15,56198,123],{"className":56199},[66,67],[15,56201],{"className":56202,"style":173},[71],[15,56204,56206,56233,56236],{"className":56205},[177],[15,56207,56209],{"className":56208},[177],[15,56210,56212],{"className":56211},[66,13081],[15,56213,56215],{"className":56214},[13085],[15,56216,56218,56221,56230],{"className":56217},[13089],[15,56219],{"className":56220,"style":3406},[61],[15,56222,56224],{"className":56223},[13096],[15,56225,56227],{"className":56226},[66],[15,56228,13103],{"className":56229},[177],[15,56231],{"className":56232},[13107],[15,56234],{"className":56235},[71,13111],[15,56237,129],{"className":56238},[177],[15,56240],{"className":56241,"style":173},[71],[15,56243,56245,56248],{"className":56244},[57],[15,56246],{"className":56247,"style":1836},[61],[15,56249,1804],{"className":56250},[66],"）得出。",[11,56253,56254,56255,56522,56523,56633],{},"左侧由 ",[15,56256,56258,56310],{"className":56257,"translate":19},[18],[15,56259,56261],{"className":56260},[23],[25,56262,56263],{"xmlns":27},[29,56264,56265,56307],{},[32,56266,56267,56269,56271,56283,56285,56287,56289],{},[1209,56268,1211],{},[39,56270,6600],{},[6465,56272,56273,56275],{},[35,56274,37],{},[32,56276,56277,56279,56281],{},[35,56278,37],{},[39,56280,1245],{},[1209,56282,1211],{},[39,56284,25767],{},[35,56286,49239],{},[39,56288,1170],{},[32,56290,56291,56293,56305],{},[39,56292,120],{"fence":53},[6465,56294,56295,56303],{},[32,56296,56297,56299,56301],{},[35,56298,37],{},[39,56300,1245],{},[1209,56302,1211],{},[35,56304,37],{},[39,56306,126],{"fence":53},[45,56308,56309],{"encoding":47},"1 - \\frac{n}{n+1} \u003C \\ln \\left(\\frac{n+1}{n}\\right)",[15,56311,56313,56331,56421],{"className":56312,"ariaHidden":53},[52],[15,56314,56316,56319,56322,56325,56328],{"className":56315},[57],[15,56317],{"className":56318,"style":30463},[61],[15,56320,1211],{"className":56321},[66],[15,56323],{"className":56324,"style":72},[71],[15,56326,6600],{"className":56327},[76],[15,56329],{"className":56330,"style":72},[71],[15,56332,56334,56338,56412,56415,56418],{"className":56333},[57],[15,56335],{"className":56336,"style":56337},[61],"height:1.0987em;vertical-align:-0.4033em;",[15,56339,56341,56344,56409],{"className":56340},[66],[15,56342],{"className":56343},[162,23412],[15,56345,56347],{"className":56346},[6465],[15,56348,56350,56401],{"className":56349},[202,1090],[15,56351,56353,56398],{"className":56352},[206],[15,56354,56356,56376,56384],{"className":56355,"style":41843},[210],[15,56357,56358,56361],{"style":30871},[15,56359],{"className":56360,"style":1368},[217],[15,56362,56364],{"className":56363},[222,223,224,225],[15,56365,56367,56370,56373],{"className":56366},[66,225],[15,56368,37],{"className":56369},[66,67,225],[15,56371,1245],{"className":56372},[76,225],[15,56374,1211],{"className":56375},[66,225],[15,56377,56378,56381],{"style":23440},[15,56379],{"className":56380,"style":1368},[217],[15,56382],{"className":56383,"style":4438},[23447],[15,56385,56386,56389],{"style":30894},[15,56387],{"className":56388,"style":1368},[217],[15,56390,56392],{"className":56391},[222,223,224,225],[15,56393,56395],{"className":56394},[66,225],[15,56396,37],{"className":56397},[66,67,225],[15,56399,1114],{"className":56400},[1113],[15,56402,56404],{"className":56403},[206],[15,56405,56407],{"className":56406,"style":45071},[210],[15,56408],{},[15,56410],{"className":56411},[169,23412],[15,56413],{"className":56414,"style":173},[71],[15,56416,25767],{"className":56417},[177],[15,56419],{"className":56420,"style":173},[71],[15,56422,56424,56427,56430,56433],{"className":56423},[57],[15,56425],{"className":56426,"style":4398},[61],[15,56428,49239],{"className":56429},[1339],[15,56431],{"className":56432,"style":697},[71],[15,56434,56436,56442,56516],{"className":56435},[1413],[15,56437,56439],{"className":56438,"style":1418},[162,1417],[15,56440,120],{"className":56441},[1422,1508],[15,56443,56445,56448,56513],{"className":56444},[66],[15,56446],{"className":56447},[162,23412],[15,56449,56451],{"className":56450},[6465],[15,56452,56454,56505],{"className":56453},[202,1090],[15,56455,56457,56502],{"className":56456},[206],[15,56458,56460,56474,56482],{"className":56459,"style":30868},[210],[15,56461,56462,56465],{"style":30871},[15,56463],{"className":56464,"style":1368},[217],[15,56466,56468],{"className":56467},[222,223,224,225],[15,56469,56471],{"className":56470},[66,225],[15,56472,37],{"className":56473},[66,67,225],[15,56475,56476,56479],{"style":23440},[15,56477],{"className":56478,"style":1368},[217],[15,56480],{"className":56481,"style":4438},[23447],[15,56483,56484,56487],{"style":30894},[15,56485],{"className":56486,"style":1368},[217],[15,56488,56490],{"className":56489},[222,223,224,225],[15,56491,56493,56496,56499],{"className":56492},[66,225],[15,56494,37],{"className":56495},[66,67,225],[15,56497,1245],{"className":56498},[76,225],[15,56500,1211],{"className":56501},[66,225],[15,56503,1114],{"className":56504},[1113],[15,56506,56508],{"className":56507},[206],[15,56509,56511],{"className":56510,"style":30916},[210],[15,56512],{},[15,56514],{"className":56515},[169,23412],[15,56517,56519],{"className":56518,"style":1418},[169,1417],[15,56520,126],{"className":56521},[1422,1508]," 得出，其中 ",[15,56524,56526,56550],{"className":56525,"translate":19},[18],[15,56527,56529],{"className":56528},[23],[25,56530,56531],{"xmlns":27},[29,56532,56533,56547],{},[32,56534,56535],{},[6465,56536,56537,56539],{},[35,56538,37],{},[32,56540,56541,56543,56545],{},[35,56542,37],{},[39,56544,1245],{},[1209,56546,1211],{},[45,56548,56549],{"encoding":47},"\\frac{n}{n+1}",[15,56551,56553],{"className":56552,"ariaHidden":53},[52],[15,56554,56556,56559],{"className":56555},[57],[15,56557],{"className":56558,"style":56337},[61],[15,56560,56562,56565,56630],{"className":56561},[66],[15,56563],{"className":56564},[162,23412],[15,56566,56568],{"className":56567},[6465],[15,56569,56571,56622],{"className":56570},[202,1090],[15,56572,56574,56619],{"className":56573},[206],[15,56575,56577,56597,56605],{"className":56576,"style":41843},[210],[15,56578,56579,56582],{"style":30871},[15,56580],{"className":56581,"style":1368},[217],[15,56583,56585],{"className":56584},[222,223,224,225],[15,56586,56588,56591,56594],{"className":56587},[66,225],[15,56589,37],{"className":56590},[66,67,225],[15,56592,1245],{"className":56593},[76,225],[15,56595,1211],{"className":56596},[66,225],[15,56598,56599,56602],{"style":23440},[15,56600],{"className":56601,"style":1368},[217],[15,56603],{"className":56604,"style":4438},[23447],[15,56606,56607,56610],{"style":30894},[15,56608],{"className":56609,"style":1368},[217],[15,56611,56613],{"className":56612},[222,223,224,225],[15,56614,56616],{"className":56615},[66,225],[15,56617,37],{"className":56618},[66,67,225],[15,56620,1114],{"className":56621},[1113],[15,56623,56625],{"className":56624},[206],[15,56626,56628],{"className":56627,"style":45071},[210],[15,56629],{},[15,56631],{"className":56632},[169,23412]," 被视为一个整体。这是高中知识。",[11,56635,11612,56636,56734],{},[15,56637,56639,56657],{"className":56638,"translate":19},[18],[15,56640,56642],{"className":56641},[23],[25,56643,56644],{"xmlns":27},[29,56645,56646,56654],{},[32,56647,56648],{},[6465,56649,56650,56652],{},[1209,56651,1211],{},[35,56653,123],{},[45,56655,56656],{"encoding":47},"\\frac{1}{x}",[15,56658,56660],{"className":56659,"ariaHidden":53},[52],[15,56661,56663,56666],{"className":56662},[57],[15,56664],{"className":56665,"style":30849},[61],[15,56667,56669,56672,56731],{"className":56668},[66],[15,56670],{"className":56671},[162,23412],[15,56673,56675],{"className":56674},[6465],[15,56676,56678,56723],{"className":56677},[202,1090],[15,56679,56681,56720],{"className":56680},[206],[15,56682,56684,56698,56706],{"className":56683,"style":30868},[210],[15,56685,56686,56689],{"style":30871},[15,56687],{"className":56688,"style":1368},[217],[15,56690,56692],{"className":56691},[222,223,224,225],[15,56693,56695],{"className":56694},[66,225],[15,56696,123],{"className":56697},[66,67,225],[15,56699,56700,56703],{"style":23440},[15,56701],{"className":56702,"style":1368},[217],[15,56704],{"className":56705,"style":4438},[23447],[15,56707,56708,56711],{"style":30894},[15,56709],{"className":56710,"style":1368},[217],[15,56712,56714],{"className":56713},[222,223,224,225],[15,56715,56717],{"className":56716},[66,225],[15,56718,1211],{"className":56719},[66,225],[15,56721,1114],{"className":56722},[1113],[15,56724,56726],{"className":56725},[206],[15,56727,56729],{"className":56728,"style":30916},[210],[15,56730],{},[15,56732],{"className":56733},[169,23412]," 是凹函数：",[15,56736,56738],{"className":56737,"translate":19},[294],[15,56739,56741,56815],{"className":56740,"translate":19},[18],[15,56742,56744],{"className":56743},[23],[25,56745,56746],{"xmlns":27,"display":303},[29,56747,56748,56812],{},[32,56749,56750,56764,56770,56772,56774,56776,56778,56780,56782,56794,56796],{},[1255,56751,56752,56754,56756],{},[39,56753,29727],{},[35,56755,37],{},[32,56757,56758,56760,56762],{},[35,56759,37],{},[39,56761,1245],{},[1209,56763,1211],{},[6465,56765,56766,56768],{},[1209,56767,1211],{},[35,56769,123],{},[1274,56771,2462],{},[35,56773,458],{},[35,56775,123],{},[39,56777,129],{},[35,56779,49239],{},[39,56781,1170],{},[6465,56783,56784,56792],{},[32,56785,56786,56788,56790],{},[35,56787,37],{},[39,56789,1245],{},[1209,56791,1211],{},[35,56793,37],{},[39,56795,9278],{},[6465,56797,56798,56800],{},[1209,56799,1211],{},[32,56801,56802,56804,56806],{},[35,56803,37],{},[39,56805,1245],{},[6465,56807,56808,56810],{},[1209,56809,1211],{},[1209,56811,1263],{},[45,56813,56814],{"encoding":47},"\\int_{n}^{n+1} \\frac{1}{x} \\, dx = \\ln \\frac{n+1}{n} > \\frac{1}{n + \\frac{1}{2}}",[15,56816,56818,56970,57065],{"className":56817,"ariaHidden":53},[52],[15,56819,56821,56824,56887,56890,56952,56955,56958,56961,56964,56967],{"className":56820},[57],[15,56822],{"className":56823,"style":29778},[61],[15,56825,56827,56830],{"className":56826},[1339],[15,56828,29727],{"className":56829,"style":29791},[1339,1470,1471],[15,56831,56833],{"className":56832},[198],[15,56834,56836,56879],{"className":56835},[202,1090],[15,56837,56839,56876],{"className":56838},[206],[15,56840,56842,56856],{"className":56841,"style":29804},[210],[15,56843,56844,56847],{"style":29807},[15,56845],{"className":56846,"style":218},[217],[15,56848,56850],{"className":56849},[222,223,224,225],[15,56851,56853],{"className":56852},[66,225],[15,56854,37],{"className":56855},[66,67,225],[15,56857,56858,56861],{"style":29819},[15,56859],{"className":56860,"style":218},[217],[15,56862,56864],{"className":56863},[222,223,224,225],[15,56865,56867,56870,56873],{"className":56866},[66,225],[15,56868,37],{"className":56869},[66,67,225],[15,56871,1245],{"className":56872},[76,225],[15,56874,1211],{"className":56875},[66,225],[15,56877,1114],{"className":56878},[1113],[15,56880,56882],{"className":56881},[206],[15,56883,56885],{"className":56884,"style":29838},[210],[15,56886],{},[15,56888],{"className":56889,"style":697},[71],[15,56891,56893,56896,56949],{"className":56892},[66],[15,56894],{"className":56895},[162,23412],[15,56897,56899],{"className":56898},[6465],[15,56900,56902,56941],{"className":56901},[202,1090],[15,56903,56905,56938],{"className":56904},[206],[15,56906,56908,56919,56927],{"className":56907,"style":23425},[210],[15,56909,56910,56913],{"style":23428},[15,56911],{"className":56912,"style":1368},[217],[15,56914,56916],{"className":56915},[66],[15,56917,123],{"className":56918},[66,67],[15,56920,56921,56924],{"style":23440},[15,56922],{"className":56923,"style":1368},[217],[15,56925],{"className":56926,"style":4438},[23447],[15,56928,56929,56932],{"style":23450},[15,56930],{"className":56931,"style":1368},[217],[15,56933,56935],{"className":56934},[66],[15,56936,1211],{"className":56937},[66],[15,56939,1114],{"className":56940},[1113],[15,56942,56944],{"className":56943},[206],[15,56945,56947],{"className":56946,"style":23469},[210],[15,56948],{},[15,56950],{"className":56951},[169,23412],[15,56953],{"className":56954,"style":697},[71],[15,56956,458],{"className":56957},[66,67],[15,56959,123],{"className":56960},[66,67],[15,56962],{"className":56963,"style":173},[71],[15,56965,129],{"className":56966},[177],[15,56968],{"className":56969,"style":173},[71],[15,56971,56973,56976,56979,56982,57056,57059,57062],{"className":56972},[57],[15,56974],{"className":56975,"style":49082},[61],[15,56977,49239],{"className":56978},[1339],[15,56980],{"className":56981,"style":697},[71],[15,56983,56985,56988,57053],{"className":56984},[66],[15,56986],{"className":56987},[162,23412],[15,56989,56991],{"className":56990},[6465],[15,56992,56994,57045],{"className":56993},[202,1090],[15,56995,56997,57042],{"className":56996},[206],[15,56998,57000,57011,57019],{"className":56999,"style":23425},[210],[15,57001,57002,57005],{"style":23428},[15,57003],{"className":57004,"style":1368},[217],[15,57006,57008],{"className":57007},[66],[15,57009,37],{"className":57010},[66,67],[15,57012,57013,57016],{"style":23440},[15,57014],{"className":57015,"style":1368},[217],[15,57017],{"className":57018,"style":4438},[23447],[15,57020,57021,57024],{"style":23450},[15,57022],{"className":57023,"style":1368},[217],[15,57025,57027,57030,57033,57036,57039],{"className":57026},[66],[15,57028,37],{"className":57029},[66,67],[15,57031],{"className":57032,"style":72},[71],[15,57034,1245],{"className":57035},[76],[15,57037],{"className":57038,"style":72},[71],[15,57040,1211],{"className":57041},[66],[15,57043,1114],{"className":57044},[1113],[15,57046,57048],{"className":57047},[206],[15,57049,57051],{"className":57050,"style":23469},[210],[15,57052],{},[15,57054],{"className":57055},[169,23412],[15,57057],{"className":57058,"style":173},[71],[15,57060,9278],{"className":57061},[177],[15,57063],{"className":57064,"style":173},[71],[15,57066,57068,57072],{"className":57067},[57],[15,57069],{"className":57070,"style":57071},[61],"height:2.4015em;vertical-align:-1.0801em;",[15,57073,57075,57078,57210],{"className":57074},[66],[15,57076],{"className":57077},[162,23412],[15,57079,57081],{"className":57080},[6465],[15,57082,57084,57201],{"className":57083},[202,1090],[15,57085,57087,57198],{"className":57086},[206],[15,57088,57090,57179,57187],{"className":57089,"style":23425},[210],[15,57091,57093,57096],{"style":57092},"top:-2.2649em;",[15,57094],{"className":57095,"style":1368},[217],[15,57097,57099,57102,57105,57108,57111],{"className":57098},[66],[15,57100,37],{"className":57101},[66,67],[15,57103],{"className":57104,"style":72},[71],[15,57106,1245],{"className":57107},[76],[15,57109],{"className":57110,"style":72},[71],[15,57112,57114,57117,57176],{"className":57113},[66],[15,57115],{"className":57116},[162,23412],[15,57118,57120],{"className":57119},[6465],[15,57121,57123,57168],{"className":57122},[202,1090],[15,57124,57126,57165],{"className":57125},[206],[15,57127,57129,57143,57151],{"className":57128,"style":30868},[210],[15,57130,57131,57134],{"style":30871},[15,57132],{"className":57133,"style":1368},[217],[15,57135,57137],{"className":57136},[222,223,224,225],[15,57138,57140],{"className":57139},[66,225],[15,57141,1263],{"className":57142},[66,225],[15,57144,57145,57148],{"style":23440},[15,57146],{"className":57147,"style":1368},[217],[15,57149],{"className":57150,"style":4438},[23447],[15,57152,57153,57156],{"style":30894},[15,57154],{"className":57155,"style":1368},[217],[15,57157,57159],{"className":57158},[222,223,224,225],[15,57160,57162],{"className":57161},[66,225],[15,57163,1211],{"className":57164},[66,225],[15,57166,1114],{"className":57167},[1113],[15,57169,57171],{"className":57170},[206],[15,57172,57174],{"className":57173,"style":30916},[210],[15,57175],{},[15,57177],{"className":57178},[169,23412],[15,57180,57181,57184],{"style":23440},[15,57182],{"className":57183,"style":1368},[217],[15,57185],{"className":57186,"style":4438},[23447],[15,57188,57189,57192],{"style":23450},[15,57190],{"className":57191,"style":1368},[217],[15,57193,57195],{"className":57194},[66],[15,57196,1211],{"className":57197},[66],[15,57199,1114],{"className":57200},[1113],[15,57202,57204],{"className":57203},[206],[15,57205,57208],{"className":57206,"style":57207},[210],"height:1.0801em;",[15,57209],{},[15,57211],{"className":57212},[169,23412],[11,57214,35904],{},[15,57216,57218],{"className":57217,"translate":19},[294],[15,57219,57221,57309],{"className":57220,"translate":19},[18],[15,57222,57224],{"className":57223},[23],[25,57225,57226],{"xmlns":27,"display":303},[29,57227,57228,57306],{},[32,57229,57230,57246,57266,57268,57302,57304],{},[1172,57231,57232,57238],{},[32,57233,57234,57236],{},[35,57235,43663],{},[39,57237,1170],{},[32,57239,57240,57242,57244],{},[35,57241,37],{},[39,57243,449],{},[35,57245,7615],{"mathvariant":138},[6465,57247,57248,57260],{},[1057,57249,57250,57252],{},[35,57251,4554],{},[32,57253,57254,57256,57258],{},[35,57255,37],{},[39,57257,1245],{},[1209,57259,1211],{},[1057,57261,57262,57264],{},[35,57263,4554],{},[35,57265,37],{},[39,57267,129],{},[6465,57269,57270,57272],{},[35,57271,48378],{},[131,57273,57274,57290],{},[32,57275,57276,57278,57280,57282,57288],{},[39,57277,120],{"fence":53},[1209,57279,1211],{},[39,57281,1245],{},[6465,57283,57284,57286],{},[1209,57285,1211],{},[35,57287,37],{},[39,57289,126],{"fence":53},[32,57291,57292,57294,57296],{},[35,57293,37],{},[39,57295,1245],{},[6465,57297,57298,57300],{},[1209,57299,1211],{},[1209,57301,1263],{},[39,57303,25767],{},[1209,57305,1211],{},[45,57307,57308],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}}} \u003C 1",[15,57310,57312,57528,57797],{"className":57311,"ariaHidden":53},[52],[15,57313,57315,57318,57371,57374,57519,57522,57525],{"className":57314},[57],[15,57316],{"className":57317,"style":52418},[61],[15,57319,57321],{"className":57320},[1339,1351],[15,57322,57324,57363],{"className":57323},[202,1090],[15,57325,57327,57360],{"className":57326},[206],[15,57328,57330,57350],{"className":57329,"style":840},[210],[15,57331,57332,57335],{"style":48819},[15,57333],{"className":57334,"style":1368},[217],[15,57336,57338],{"className":57337},[222,223,224,225],[15,57339,57341,57344,57347],{"className":57340},[66,225],[15,57342,37],{"className":57343},[66,67,225],[15,57345,449],{"className":57346},[177,225],[15,57348,7615],{"className":57349},[66,225],[15,57351,57352,57355],{"style":1386},[15,57353],{"className":57354,"style":1368},[217],[15,57356,57357],{},[15,57358,43663],{"className":57359},[1339],[15,57361,1114],{"className":57362},[1113],[15,57364,57366],{"className":57365},[206],[15,57367,57369],{"className":57368,"style":48857},[210],[15,57370],{},[15,57372],{"className":57373,"style":697},[71],[15,57375,57377,57380,57516],{"className":57376},[66],[15,57378],{"className":57379},[162,23412],[15,57381,57383],{"className":57382},[6465],[15,57384,57386,57508],{"className":57385},[202,1090],[15,57387,57389,57505],{"className":57388},[206],[15,57390,57392,57440,57448],{"className":57391,"style":39128},[210],[15,57393,57394,57397],{"style":23428},[15,57395],{"className":57396,"style":1368},[217],[15,57398,57400],{"className":57399},[66],[15,57401,57403,57406],{"className":57402},[66],[15,57404,4554],{"className":57405},[66,67],[15,57407,57409],{"className":57408},[198],[15,57410,57412,57432],{"className":57411},[202,1090],[15,57413,57415,57429],{"className":57414},[206],[15,57416,57418],{"className":57417,"style":2198},[210],[15,57419,57420,57423],{"style":1584},[15,57421],{"className":57422,"style":218},[217],[15,57424,57426],{"className":57425},[222,223,224,225],[15,57427,37],{"className":57428},[66,67,225],[15,57430,1114],{"className":57431},[1113],[15,57433,57435],{"className":57434},[206],[15,57436,57438],{"className":57437,"style":1121},[210],[15,57439],{},[15,57441,57442,57445],{"style":23440},[15,57443],{"className":57444,"style":1368},[217],[15,57446],{"className":57447,"style":4438},[23447],[15,57449,57450,57453],{"style":23450},[15,57451],{"className":57452,"style":1368},[217],[15,57454,57456],{"className":57455},[66],[15,57457,57459,57462],{"className":57458},[66],[15,57460,4554],{"className":57461},[66,67],[15,57463,57465],{"className":57464},[198],[15,57466,57468,57497],{"className":57467},[202,1090],[15,57469,57471,57494],{"className":57470},[206],[15,57472,57474],{"className":57473,"style":2139},[210],[15,57475,57476,57479],{"style":1584},[15,57477],{"className":57478,"style":218},[217],[15,57480,57482],{"className":57481},[222,223,224,225],[15,57483,57485,57488,57491],{"className":57484},[66,225],[15,57486,37],{"className":57487},[66,67,225],[15,57489,1245],{"className":57490},[76,225],[15,57492,1211],{"className":57493},[66,225],[15,57495,1114],{"className":57496},[1113],[15,57498,57500],{"className":57499},[206],[15,57501,57503],{"className":57502,"style":52136},[210],[15,57504],{},[15,57506,1114],{"className":57507},[1113],[15,57509,57511],{"className":57510},[206],[15,57512,57514],{"className":57513,"style":52615},[210],[15,57515],{},[15,57517],{"className":57518},[169,23412],[15,57520],{"className":57521,"style":173},[71],[15,57523,129],{"className":57524},[177],[15,57526],{"className":57527,"style":173},[71],[15,57529,57531,57534,57788,57791,57794],{"className":57530},[57],[15,57532],{"className":57533,"style":54376},[61],[15,57535,57537,57540,57785],{"className":57536},[66],[15,57538],{"className":57539},[162,23412],[15,57541,57543],{"className":57542},[6465],[15,57544,57546,57777],{"className":57545},[202,1090],[15,57547,57549,57774],{"className":57548},[206],[15,57550,57552,57755,57763],{"className":57551,"style":39128},[210],[15,57553,57554,57557],{"style":54397},[15,57555],{"className":57556,"style":54401},[217],[15,57558,57560],{"className":57559},[66],[15,57561,57563,57658],{"className":57562},[1413],[15,57564,57566,57572,57575,57578,57581,57584,57652],{"className":57565},[1413],[15,57567,57569],{"className":57568,"style":1418},[162,1417],[15,57570,120],{"className":57571},[1422,1508],[15,57573,1211],{"className":57574},[66],[15,57576],{"className":57577,"style":72},[71],[15,57579,1245],{"className":57580},[76],[15,57582],{"className":57583,"style":72},[71],[15,57585,57587,57590,57649],{"className":57586},[66],[15,57588],{"className":57589},[162,23412],[15,57591,57593],{"className":57592},[6465],[15,57594,57596,57641],{"className":57595},[202,1090],[15,57597,57599,57638],{"className":57598},[206],[15,57600,57602,57616,57624],{"className":57601,"style":30868},[210],[15,57603,57604,57607],{"style":30871},[15,57605],{"className":57606,"style":1368},[217],[15,57608,57610],{"className":57609},[222,223,224,225],[15,57611,57613],{"className":57612},[66,225],[15,57614,37],{"className":57615},[66,67,225],[15,57617,57618,57621],{"style":23440},[15,57619],{"className":57620,"style":1368},[217],[15,57622],{"className":57623,"style":4438},[23447],[15,57625,57626,57629],{"style":30894},[15,57627],{"className":57628,"style":1368},[217],[15,57630,57632],{"className":57631},[222,223,224,225],[15,57633,57635],{"className":57634},[66,225],[15,57636,1211],{"className":57637},[66,225],[15,57639,1114],{"className":57640},[1113],[15,57642,57644],{"className":57643},[206],[15,57645,57647],{"className":57646,"style":30916},[210],[15,57648],{},[15,57650],{"className":57651},[169,23412],[15,57653,57655],{"className":57654,"style":1418},[169,1417],[15,57656,126],{"className":57657},[1422,1508],[15,57659,57661],{"className":57660},[198],[15,57662,57664],{"className":57663},[202],[15,57665,57667],{"className":57666},[206],[15,57668,57670],{"className":57669,"style":54515},[210],[15,57671,57672,57675],{"style":51213},[15,57673],{"className":57674,"style":1368},[217],[15,57676,57678],{"className":57677},[222,223,224,225],[15,57679,57681,57684,57687],{"className":57680},[66,225],[15,57682,37],{"className":57683},[66,67,225],[15,57685,1245],{"className":57686},[76,225],[15,57688,57690,57693,57752],{"className":57689},[66,225],[15,57691],{"className":57692},[162,23412,222,3677,33576],[15,57694,57696],{"className":57695},[6465],[15,57697,57699,57744],{"className":57698},[202,1090],[15,57700,57702,57741],{"className":57701},[206],[15,57703,57705,57719,57727],{"className":57704,"style":49016},[210],[15,57706,57707,57710],{"style":33592},[15,57708],{"className":57709,"style":1368},[217],[15,57711,57713],{"className":57712},[222,3677,1508,225],[15,57714,57716],{"className":57715},[66,225],[15,57717,1263],{"className":57718},[66,225],[15,57720,57721,57724],{"style":33607},[15,57722],{"className":57723,"style":1368},[217],[15,57725],{"className":57726,"style":33614},[23447,225],[15,57728,57729,57732],{"style":33617},[15,57730],{"className":57731,"style":1368},[217],[15,57733,57735],{"className":57734},[222,3677,1508,225],[15,57736,57738],{"className":57737},[66,225],[15,57739,1211],{"className":57740},[66,225],[15,57742,1114],{"className":57743},[1113],[15,57745,57747],{"className":57746},[206],[15,57748,57750],{"className":57749,"style":33639},[210],[15,57751],{},[15,57753],{"className":57754},[169,23412,222,3677,33576],[15,57756,57757,57760],{"style":54603},[15,57758],{"className":57759,"style":54401},[217],[15,57761],{"className":57762,"style":4438},[23447],[15,57764,57765,57768],{"style":54612},[15,57766],{"className":57767,"style":54401},[217],[15,57769,57771],{"className":57770},[66],[15,57772,48378],{"className":57773},[66,67],[15,57775,1114],{"className":57776},[1113],[15,57778,57780],{"className":57779},[206],[15,57781,57783],{"className":57782,"style":54631},[210],[15,57784],{},[15,57786],{"className":57787},[169,23412],[15,57789],{"className":57790,"style":173},[71],[15,57792,25767],{"className":57793},[177],[15,57795],{"className":57796,"style":173},[71],[15,57798,57800,57803],{"className":57799},[57],[15,57801],{"className":57802,"style":1836},[61],[15,57804,1211],{"className":57805},[66],[11,57807,57808,57809,57878,57879,57971],{},"于是 ",[15,57810,57812,57829],{"className":57811,"translate":19},[18],[15,57813,57815],{"className":57814},[23],[25,57816,57817],{"xmlns":27},[29,57818,57819,57827],{},[32,57820,57821],{},[1057,57822,57823,57825],{},[35,57824,4554],{},[35,57826,37],{},[45,57828,52918],{"encoding":47},[15,57830,57832],{"className":57831,"ariaHidden":53},[52],[15,57833,57835,57838],{"className":57834},[57],[15,57836],{"className":57837,"style":10192},[61],[15,57839,57841,57844],{"className":57840},[66],[15,57842,4554],{"className":57843},[66,67],[15,57845,57847],{"className":57846},[198],[15,57848,57850,57870],{"className":57849},[202,1090],[15,57851,57853,57867],{"className":57852},[206],[15,57854,57856],{"className":57855,"style":2198},[210],[15,57857,57858,57861],{"style":1584},[15,57859],{"className":57860,"style":218},[217],[15,57862,57864],{"className":57863},[222,223,224,225],[15,57865,37],{"className":57866},[66,67,225],[15,57868,1114],{"className":57869},[1113],[15,57871,57873],{"className":57872},[206],[15,57874,57876],{"className":57875,"style":1121},[210],[15,57877],{}," 单调递减且 ",[15,57880,57882,57904],{"className":57881,"translate":19},[18],[15,57883,57885],{"className":57884},[23],[25,57886,57887],{"xmlns":27},[29,57888,57889,57901],{},[32,57890,57891,57897,57899],{},[1057,57892,57893,57895],{},[35,57894,4554],{},[35,57896,37],{},[39,57898,9278],{},[1209,57900,1804],{},[45,57902,57903],{"encoding":47},"a_n > 0",[15,57905,57907,57962],{"className":57906,"ariaHidden":53},[52],[15,57908,57910,57913,57953,57956,57959],{"className":57909},[57],[15,57911],{"className":57912,"style":26344},[61],[15,57914,57916,57919],{"className":57915},[66],[15,57917,4554],{"className":57918},[66,67],[15,57920,57922],{"className":57921},[198],[15,57923,57925,57945],{"className":57924},[202,1090],[15,57926,57928,57942],{"className":57927},[206],[15,57929,57931],{"className":57930,"style":2198},[210],[15,57932,57933,57936],{"style":1584},[15,57934],{"className":57935,"style":218},[217],[15,57937,57939],{"className":57938},[222,223,224,225],[15,57940,37],{"className":57941},[66,67,225],[15,57943,1114],{"className":57944},[1113],[15,57946,57948],{"className":57947},[206],[15,57949,57951],{"className":57950,"style":1121},[210],[15,57952],{},[15,57954],{"className":57955,"style":173},[71],[15,57957,9278],{"className":57958},[177],[15,57960],{"className":57961,"style":173},[71],[15,57963,57965,57968],{"className":57964},[57],[15,57966],{"className":57967,"style":1836},[61],[15,57969,1804],{"className":57970},[66],"，因此它必然收敛，即存在极限。",[11,57973,57974],{},"设：",[15,57976,57978],{"className":57977,"translate":19},[294],[15,57979,57981,58043],{"className":57980,"translate":19},[18],[15,57982,57984],{"className":57983},[23],[25,57985,57986],{"xmlns":27,"display":303},[29,57987,57988,58040],{},[32,57989,57990,58006,58030,58036,58038],{},[1172,57991,57992,57998],{},[32,57993,57994,57996],{},[35,57995,43663],{},[39,57997,1170],{},[32,57999,58000,58002,58004],{},[35,58001,37],{},[39,58003,449],{},[35,58005,7615],{"mathvariant":138},[6465,58007,58008,58014],{},[32,58009,58010,58012],{},[35,58011,37],{},[39,58013,40764],{"stretchy":119},[131,58015,58016,58018],{},[35,58017,37],{},[32,58019,58020,58022,58024],{},[35,58021,37],{},[39,58023,1245],{},[6465,58025,58026,58028],{},[1209,58027,1211],{},[1209,58029,1263],{},[131,58031,58032,58034],{},[35,58033,48378],{},[35,58035,37],{},[39,58037,129],{},[35,58039,4554],{},[45,58041,58042],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{n!}{n^{n + \\frac{1}{2}}} e^n = a",[15,58044,58046,58311],{"className":58045,"ariaHidden":53},[52],[15,58047,58049,58052,58105,58108,58273,58302,58305,58308],{"className":58048},[57],[15,58050],{"className":58051,"style":53092},[61],[15,58053,58055],{"className":58054},[1339,1351],[15,58056,58058,58097],{"className":58057},[202,1090],[15,58059,58061,58094],{"className":58060},[206],[15,58062,58064,58084],{"className":58063,"style":840},[210],[15,58065,58066,58069],{"style":48819},[15,58067],{"className":58068,"style":1368},[217],[15,58070,58072],{"className":58071},[222,223,224,225],[15,58073,58075,58078,58081],{"className":58074},[66,225],[15,58076,37],{"className":58077},[66,67,225],[15,58079,449],{"className":58080},[177,225],[15,58082,7615],{"className":58083},[66,225],[15,58085,58086,58089],{"style":1386},[15,58087],{"className":58088,"style":1368},[217],[15,58090,58091],{},[15,58092,43663],{"className":58093},[1339],[15,58095,1114],{"className":58096},[1113],[15,58098,58100],{"className":58099},[206],[15,58101,58103],{"className":58102,"style":48857},[210],[15,58104],{},[15,58106],{"className":58107,"style":697},[71],[15,58109,58111,58114,58270],{"className":58110},[66],[15,58112],{"className":58113},[162,23412],[15,58115,58117],{"className":58116},[6465],[15,58118,58120,58262],{"className":58119},[202,1090],[15,58121,58123,58259],{"className":58122},[206],[15,58124,58126,58237,58245],{"className":58125,"style":27430},[210],[15,58127,58128,58131],{"style":53113},[15,58129],{"className":58130,"style":1368},[217],[15,58132,58134],{"className":58133},[66],[15,58135,58137,58140],{"className":58136},[66],[15,58138,37],{"className":58139},[66,67],[15,58141,58143],{"className":58142},[198],[15,58144,58146],{"className":58145},[202],[15,58147,58149],{"className":58148},[206],[15,58150,58152],{"className":58151,"style":53138},[210],[15,58153,58154,58157],{"style":53141},[15,58155],{"className":58156,"style":1368},[217],[15,58158,58160],{"className":58159},[222,223,224,225],[15,58161,58163,58166,58169],{"className":58162},[66,225],[15,58164,37],{"className":58165},[66,67,225],[15,58167,1245],{"className":58168},[76,225],[15,58170,58172,58175,58234],{"className":58171},[66,225],[15,58173],{"className":58174},[162,23412,222,3677,33576],[15,58176,58178],{"className":58177},[6465],[15,58179,58181,58226],{"className":58180},[202,1090],[15,58182,58184,58223],{"className":58183},[206],[15,58185,58187,58201,58209],{"className":58186,"style":49016},[210],[15,58188,58189,58192],{"style":33592},[15,58190],{"className":58191,"style":1368},[217],[15,58193,58195],{"className":58194},[222,3677,1508,225],[15,58196,58198],{"className":58197},[66,225],[15,58199,1263],{"className":58200},[66,225],[15,58202,58203,58206],{"style":33607},[15,58204],{"className":58205,"style":1368},[217],[15,58207],{"className":58208,"style":33614},[23447,225],[15,58210,58211,58214],{"style":33617},[15,58212],{"className":58213,"style":1368},[217],[15,58215,58217],{"className":58216},[222,3677,1508,225],[15,58218,58220],{"className":58219},[66,225],[15,58221,1211],{"className":58222},[66,225],[15,58224,1114],{"className":58225},[1113],[15,58227,58229],{"className":58228},[206],[15,58230,58232],{"className":58231,"style":33639},[210],[15,58233],{},[15,58235],{"className":58236},[169,23412,222,3677,33576],[15,58238,58239,58242],{"style":23440},[15,58240],{"className":58241,"style":1368},[217],[15,58243],{"className":58244,"style":4438},[23447],[15,58246,58247,58250],{"style":23450},[15,58248],{"className":58249,"style":1368},[217],[15,58251,58253,58256],{"className":58252},[66],[15,58254,37],{"className":58255},[66,67],[15,58257,40764],{"className":58258},[169],[15,58260,1114],{"className":58261},[1113],[15,58263,58265],{"className":58264},[206],[15,58266,58268],{"className":58267,"style":53256},[210],[15,58269],{},[15,58271],{"className":58272},[169,23412],[15,58274,58276,58279],{"className":58275},[66],[15,58277,48378],{"className":58278},[66,67],[15,58280,58282],{"className":58281},[198],[15,58283,58285],{"className":58284},[202],[15,58286,58288],{"className":58287},[206],[15,58289,58291],{"className":58290,"style":51746},[210],[15,58292,58293,58296],{"style":397},[15,58294],{"className":58295,"style":218},[217],[15,58297,58299],{"className":58298},[222,223,224,225],[15,58300,37],{"className":58301},[66,67,225],[15,58303],{"className":58304,"style":173},[71],[15,58306,129],{"className":58307},[177],[15,58309],{"className":58310,"style":173},[71],[15,58312,58314,58317],{"className":58313},[57],[15,58315],{"className":58316,"style":86},[61],[15,58318,4554],{"className":58319},[66,67],[11,58321,58322],{},"接下来我们需要求出这个极限。",[11,58324,58325],{},"由前面的沃利斯公式：",[15,58327,58329],{"className":58328,"translate":19},[294],[15,58330,58332,58421],{"className":58331,"translate":19},[18],[15,58333,58335],{"className":58334},[23],[25,58336,58337],{"xmlns":27,"display":303},[29,58338,58339,58419],{},[32,58340,58341,58357,58411,58413],{},[1172,58342,58343,58349],{},[32,58344,58345,58347],{},[35,58346,43663],{},[39,58348,1170],{},[32,58350,58351,58353,58355],{},[35,58352,621],{},[39,58354,449],{},[35,58356,7615],{"mathvariant":138},[6465,58358,58359,58377],{},[32,58360,58361,58363,58365,58367,58369,58371],{},[39,58362,120],{"stretchy":119},[1209,58364,1263],{},[35,58366,621],{},[39,58368,126],{"stretchy":119},[39,58370,40764],{"stretchy":119},[131,58372,58373,58375],{},[39,58374,40764],{"stretchy":119},[1209,58376,1263],{},[32,58378,58379,58381,58383,58385,58387,58389,58391,58393,58395,58397,58399,58401,58403,58405,58407,58409],{},[39,58380,120],{"stretchy":119},[1209,58382,1263],{},[35,58384,621],{},[39,58386,1245],{},[1209,58388,1211],{},[39,58390,126],{"stretchy":119},[39,58392,40764],{"stretchy":119},[39,58394,40764],{"stretchy":119},[39,58396,120],{"stretchy":119},[1209,58398,1263],{},[35,58400,621],{},[39,58402,6600],{},[1209,58404,1211],{},[39,58406,126],{"stretchy":119},[39,58408,40764],{"stretchy":119},[39,58410,40764],{"stretchy":119},[39,58412,129],{},[6465,58414,58415,58417],{},[35,58416,33489],{},[1209,58418,1263],{},[45,58420,47971],{"encoding":47},[15,58422,58424,58640],{"className":58423,"ariaHidden":53},[52],[15,58425,58427,58430,58483,58486,58631,58634,58637],{"className":58426},[57],[15,58428],{"className":58429,"style":45939},[61],[15,58431,58433],{"className":58432},[1339,1351],[15,58434,58436,58475],{"className":58435},[202,1090],[15,58437,58439,58472],{"className":58438},[206],[15,58440,58442,58462],{"className":58441,"style":840},[210],[15,58443,58444,58447],{"style":1364},[15,58445],{"className":58446,"style":1368},[217],[15,58448,58450],{"className":58449},[222,223,224,225],[15,58451,58453,58456,58459],{"className":58452},[66,225],[15,58454,621],{"className":58455,"style":684},[66,67,225],[15,58457,449],{"className":58458},[177,225],[15,58460,7615],{"className":58461},[66,225],[15,58463,58464,58467],{"style":1386},[15,58465],{"className":58466,"style":1368},[217],[15,58468,58469],{},[15,58470,43663],{"className":58471},[1339],[15,58473,1114],{"className":58474},[1113],[15,58476,58478],{"className":58477},[206],[15,58479,58481],{"className":58480,"style":44020},[210],[15,58482],{},[15,58484],{"className":58485,"style":697},[71],[15,58487,58489,58492,58628],{"className":58488},[66],[15,58490],{"className":58491},[162,23412],[15,58493,58495],{"className":58494},[6465],[15,58496,58498,58620],{"className":58497},[202,1090],[15,58499,58501,58617],{"className":58500},[206],[15,58502,58504,58560,58568],{"className":58503,"style":45958},[210],[15,58505,58506,58509],{"style":23428},[15,58507],{"className":58508,"style":1368},[217],[15,58510,58512,58515,58518,58521,58524,58527,58530,58533,58536,58539,58542,58545,58548,58551,58554,58557],{"className":58511},[66],[15,58513,120],{"className":58514},[162],[15,58516,1263],{"className":58517},[66],[15,58519,621],{"className":58520,"style":684},[66,67],[15,58522],{"className":58523,"style":72},[71],[15,58525,1245],{"className":58526},[76],[15,58528],{"className":58529,"style":72},[71],[15,58531,1211],{"className":58532},[66],[15,58534,41227],{"className":58535},[169],[15,58537,120],{"className":58538},[162],[15,58540,1263],{"className":58541},[66],[15,58543,621],{"className":58544,"style":684},[66,67],[15,58546],{"className":58547,"style":72},[71],[15,58549,6600],{"className":58550},[76],[15,58552],{"className":58553,"style":72},[71],[15,58555,1211],{"className":58556},[66],[15,58558,41227],{"className":58559},[169],[15,58561,58562,58565],{"style":23440},[15,58563],{"className":58564,"style":1368},[217],[15,58566],{"className":58567,"style":4438},[23447],[15,58569,58570,58573],{"style":23450},[15,58571],{"className":58572,"style":1368},[217],[15,58574,58576,58579,58582,58585,58588],{"className":58575},[66],[15,58577,120],{"className":58578},[162],[15,58580,1263],{"className":58581},[66],[15,58583,621],{"className":58584,"style":684},[66,67],[15,58586,46043],{"className":58587},[169],[15,58589,58591,58594],{"className":58590},[169],[15,58592,40764],{"className":58593},[169],[15,58595,58597],{"className":58596},[198],[15,58598,58600],{"className":58599},[202],[15,58601,58603],{"className":58602},[206],[15,58604,58606],{"className":58605,"style":6738},[210],[15,58607,58608,58611],{"style":213},[15,58609],{"className":58610,"style":218},[217],[15,58612,58614],{"className":58613},[222,223,224,225],[15,58615,1263],{"className":58616},[66,225],[15,58618,1114],{"className":58619},[1113],[15,58621,58623],{"className":58622},[206],[15,58624,58626],{"className":58625,"style":41277},[210],[15,58627],{},[15,58629],{"className":58630},[169,23412],[15,58632],{"className":58633,"style":173},[71],[15,58635,129],{"className":58636},[177],[15,58638],{"className":58639,"style":173},[71],[15,58641,58643,58646],{"className":58642},[57],[15,58644],{"className":58645,"style":39109},[61],[15,58647,58649,58652,58705],{"className":58648},[66],[15,58650],{"className":58651},[162,23412],[15,58653,58655],{"className":58654},[6465],[15,58656,58658,58697],{"className":58657},[202,1090],[15,58659,58661,58694],{"className":58660},[206],[15,58662,58664,58675,58683],{"className":58663,"style":39128},[210],[15,58665,58666,58669],{"style":23428},[15,58667],{"className":58668,"style":1368},[217],[15,58670,58672],{"className":58671},[66],[15,58673,1263],{"className":58674},[66],[15,58676,58677,58680],{"style":23440},[15,58678],{"className":58679,"style":1368},[217],[15,58681],{"className":58682,"style":4438},[23447],[15,58684,58685,58688],{"style":23450},[15,58686],{"className":58687,"style":1368},[217],[15,58689,58691],{"className":58690},[66],[15,58692,33489],{"className":58693,"style":1515},[66,67],[15,58695,1114],{"className":58696},[1113],[15,58698,58700],{"className":58699},[206],[15,58701,58703],{"className":58702,"style":23469},[210],[15,58704],{},[15,58706],{"className":58707},[169,23412],[15,58709,58711],{"className":58710,"translate":19},[294],[15,58712,58714,58772],{"className":58713,"translate":19},[18],[15,58715,58717],{"className":58716},[23],[25,58718,58719],{"xmlns":27,"display":303},[29,58720,58721,58769],{},[32,58722,58723,58725,58727,58729,58731,58733,58735,58737,58739,58741,58743,58745,58747,58749,58751,58753,58755,58757,58759,58765,58767],{},[39,58724,120],{"stretchy":119},[1209,58726,1263],{},[35,58728,37],{},[39,58730,126],{"stretchy":119},[39,58732,40764],{"stretchy":119},[39,58734,40764],{"stretchy":119},[39,58736,129],{},[1209,58738,1263],{},[35,58740,37],{},[39,58742,120],{"stretchy":119},[1209,58744,1263],{},[35,58746,37],{},[39,58748,6600],{},[1209,58750,1263],{},[39,58752,126],{"stretchy":119},[39,58754,37795],{},[1209,58756,1263],{},[39,58758,129],{},[131,58760,58761,58763],{},[1209,58762,1263],{},[35,58764,37],{},[35,58766,37],{},[39,58768,40764],{"stretchy":119},[45,58770,58771],{"encoding":47},"(2n)!! = 2n (2n-2) \\cdots 2 = 2^n n!",[15,58773,58775,58802,58832,58865],{"className":58774,"ariaHidden":53},[52],[15,58776,58778,58781,58784,58787,58790,58793,58796,58799],{"className":58777},[57],[15,58779],{"className":58780,"style":154},[61],[15,58782,120],{"className":58783},[162],[15,58785,1263],{"className":58786},[66],[15,58788,37],{"className":58789},[66,67],[15,58791,41227],{"className":58792},[169],[15,58794],{"className":58795,"style":173},[71],[15,58797,129],{"className":58798},[177],[15,58800],{"className":58801,"style":173},[71],[15,58803,58805,58808,58811,58814,58817,58820,58823,58826,58829],{"className":58804},[57],[15,58806],{"className":58807,"style":154},[61],[15,58809,1263],{"className":58810},[66],[15,58812,37],{"className":58813},[66,67],[15,58815,120],{"className":58816},[162],[15,58818,1263],{"className":58819},[66],[15,58821,37],{"className":58822},[66,67],[15,58824],{"className":58825,"style":72},[71],[15,58827,6600],{"className":58828},[76],[15,58830],{"className":58831,"style":72},[71],[15,58833,58835,58838,58841,58844,58847,58850,58853,58856,58859,58862],{"className":58834},[57],[15,58836],{"className":58837,"style":154},[61],[15,58839,1263],{"className":58840},[66],[15,58842,126],{"className":58843},[169],[15,58845],{"className":58846,"style":697},[71],[15,58848,37795],{"className":58849},[1413],[15,58851],{"className":58852,"style":697},[71],[15,58854,1263],{"className":58855},[66],[15,58857],{"className":58858,"style":173},[71],[15,58860,129],{"className":58861},[177],[15,58863],{"className":58864,"style":173},[71],[15,58866,58868,58871,58900,58903],{"className":58867},[57],[15,58869],{"className":58870,"style":51746},[61],[15,58872,58874,58877],{"className":58873},[66],[15,58875,1263],{"className":58876},[66],[15,58878,58880],{"className":58879},[198],[15,58881,58883],{"className":58882},[202],[15,58884,58886],{"className":58885},[206],[15,58887,58889],{"className":58888,"style":51746},[210],[15,58890,58891,58894],{"style":397},[15,58892],{"className":58893,"style":218},[217],[15,58895,58897],{"className":58896},[222,223,224,225],[15,58898,37],{"className":58899},[66,67,225],[15,58901,37],{"className":58902},[66,67],[15,58904,40764],{"className":58905},[169],[15,58907,58909],{"className":58908,"translate":19},[294],[15,58910,58912,58998],{"className":58911,"translate":19},[18],[15,58913,58915],{"className":58914},[23],[25,58916,58917],{"xmlns":27,"display":303},[29,58918,58919,58995],{},[32,58920,58921,58923,58925,58927,58929,58931,58933,58935,58937,58939,58967,58969],{},[39,58922,120],{"stretchy":119},[1209,58924,1263],{},[35,58926,37],{},[39,58928,6600],{},[1209,58930,1211],{},[39,58932,126],{"stretchy":119},[39,58934,40764],{"stretchy":119},[39,58936,40764],{"stretchy":119},[39,58938,129],{},[6465,58940,58941,58953],{},[32,58942,58943,58945,58947,58949,58951],{},[39,58944,120],{"stretchy":119},[1209,58946,1263],{},[35,58948,37],{},[39,58950,126],{"stretchy":119},[39,58952,40764],{"stretchy":119},[32,58954,58955,58957,58959,58961,58963,58965],{},[39,58956,120],{"stretchy":119},[1209,58958,1263],{},[35,58960,37],{},[39,58962,126],{"stretchy":119},[39,58964,40764],{"stretchy":119},[39,58966,40764],{"stretchy":119},[39,58968,129],{},[6465,58970,58971,58983],{},[32,58972,58973,58975,58977,58979,58981],{},[39,58974,120],{"stretchy":119},[1209,58976,1263],{},[35,58978,37],{},[39,58980,126],{"stretchy":119},[39,58982,40764],{"stretchy":119},[32,58984,58985,58991,58993],{},[131,58986,58987,58989],{},[1209,58988,1263],{},[35,58990,37],{},[35,58992,37],{},[39,58994,40764],{"stretchy":119},[45,58996,58997],{"encoding":47},"(2n-1)!! = \\frac{(2n)!}{(2n)!!} = \\frac{(2n)!}{2^n n!}",[15,58999,59001,59025,59046,59141],{"className":59000,"ariaHidden":53},[52],[15,59002,59004,59007,59010,59013,59016,59019,59022],{"className":59003},[57],[15,59005],{"className":59006,"style":154},[61],[15,59008,120],{"className":59009},[162],[15,59011,1263],{"className":59012},[66],[15,59014,37],{"className":59015},[66,67],[15,59017],{"className":59018,"style":72},[71],[15,59020,6600],{"className":59021},[76],[15,59023],{"className":59024,"style":72},[71],[15,59026,59028,59031,59034,59037,59040,59043],{"className":59027},[57],[15,59029],{"className":59030,"style":154},[61],[15,59032,1211],{"className":59033},[66],[15,59035,41227],{"className":59036},[169],[15,59038],{"className":59039,"style":173},[71],[15,59041,129],{"className":59042},[177],[15,59044],{"className":59045,"style":173},[71],[15,59047,59049,59052,59132,59135,59138],{"className":59048},[57],[15,59050],{"className":59051,"style":41187},[61],[15,59053,59055,59058,59129],{"className":59054},[66],[15,59056],{"className":59057},[162,23412],[15,59059,59061],{"className":59060},[6465],[15,59062,59064,59121],{"className":59063},[202,1090],[15,59065,59067,59118],{"className":59066},[206],[15,59068,59070,59090,59098],{"className":59069,"style":41206},[210],[15,59071,59072,59075],{"style":23428},[15,59073],{"className":59074,"style":1368},[217],[15,59076,59078,59081,59084,59087],{"className":59077},[66],[15,59079,120],{"className":59080},[162],[15,59082,1263],{"className":59083},[66],[15,59085,37],{"className":59086},[66,67],[15,59088,41227],{"className":59089},[169],[15,59091,59092,59095],{"style":23440},[15,59093],{"className":59094,"style":1368},[217],[15,59096],{"className":59097,"style":4438},[23447],[15,59099,59100,59103],{"style":23450},[15,59101],{"className":59102,"style":1368},[217],[15,59104,59106,59109,59112,59115],{"className":59105},[66],[15,59107,120],{"className":59108},[162],[15,59110,1263],{"className":59111},[66],[15,59113,37],{"className":59114},[66,67],[15,59116,46043],{"className":59117},[169],[15,59119,1114],{"className":59120},[1113],[15,59122,59124],{"className":59123},[206],[15,59125,59127],{"className":59126,"style":41277},[210],[15,59128],{},[15,59130],{"className":59131},[169,23412],[15,59133],{"className":59134,"style":173},[71],[15,59136,129],{"className":59137},[177],[15,59139],{"className":59140,"style":173},[71],[15,59142,59144,59148],{"className":59143},[57],[15,59145],{"className":59146,"style":59147},[61],"height:2.113em;vertical-align:-0.686em;",[15,59149,59151,59154,59248],{"className":59150},[66],[15,59152],{"className":59153},[162,23412],[15,59155,59157],{"className":59156},[6465],[15,59158,59160,59240],{"className":59159},[202,1090],[15,59161,59163,59237],{"className":59162},[206],[15,59164,59166,59209,59217],{"className":59165,"style":41206},[210],[15,59167,59168,59171],{"style":23428},[15,59169],{"className":59170,"style":1368},[217],[15,59172,59174,59203,59206],{"className":59173},[66],[15,59175,59177,59180],{"className":59176},[66],[15,59178,1263],{"className":59179},[66],[15,59181,59183],{"className":59182},[198],[15,59184,59186],{"className":59185},[202],[15,59187,59189],{"className":59188},[206],[15,59190,59192],{"className":59191,"style":48919},[210],[15,59193,59194,59197],{"style":46250},[15,59195],{"className":59196,"style":218},[217],[15,59198,59200],{"className":59199},[222,223,224,225],[15,59201,37],{"className":59202},[66,67,225],[15,59204,37],{"className":59205},[66,67],[15,59207,40764],{"className":59208},[169],[15,59210,59211,59214],{"style":23440},[15,59212],{"className":59213,"style":1368},[217],[15,59215],{"className":59216,"style":4438},[23447],[15,59218,59219,59222],{"style":23450},[15,59220],{"className":59221,"style":1368},[217],[15,59223,59225,59228,59231,59234],{"className":59224},[66],[15,59226,120],{"className":59227},[162],[15,59229,1263],{"className":59230},[66],[15,59232,37],{"className":59233},[66,67],[15,59235,46043],{"className":59236},[169],[15,59238,1114],{"className":59239},[1113],[15,59241,59243],{"className":59242},[206],[15,59244,59246],{"className":59245,"style":23469},[210],[15,59247],{},[15,59249],{"className":59250},[169,23412],[15,59252,59254],{"className":59253,"translate":19},[294],[15,59255,59257,59369],{"className":59256,"translate":19},[18],[15,59258,59260],{"className":59259},[23],[25,59261,59262],{"xmlns":27,"display":303},[29,59263,59264,59366],{},[32,59265,59266,59268,59270,59272,59274,59276,59278,59280,59282,59284,59320,59322],{},[39,59267,120],{"stretchy":119},[1209,59269,1263],{},[35,59271,37],{},[39,59273,1245],{},[1209,59275,1211],{},[39,59277,126],{"stretchy":119},[39,59279,40764],{"stretchy":119},[39,59281,40764],{"stretchy":119},[39,59283,129],{},[6465,59285,59286,59302],{},[32,59287,59288,59290,59292,59294,59296,59298,59300],{},[39,59289,120],{"stretchy":119},[1209,59291,1263],{},[35,59293,37],{},[39,59295,1245],{},[1209,59297,1263],{},[39,59299,126],{"stretchy":119},[39,59301,40764],{"stretchy":119},[32,59303,59304,59306,59308,59310,59312,59314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= \\frac{(2n+2)!}{(2n+2)!!} = \\frac{(2n+2)!}{2^{n+1} (n+1)!}",[15,59370,59372,59396,59417,59536],{"className":59371,"ariaHidden":53},[52],[15,59373,59375,59378,59381,59384,59387,59390,59393],{"className":59374},[57],[15,59376],{"className":59377,"style":154},[61],[15,59379,120],{"className":59380},[162],[15,59382,1263],{"className":59383},[66],[15,59385,37],{"className":59386},[66,67],[15,59388],{"className":59389,"style":72},[71],[15,59391,1245],{"className":59392},[76],[15,59394],{"className":59395,"style":72},[71],[15,59397,59399,59402,59405,59408,59411,59414],{"className":59398},[57],[15,59400],{"className":59401,"style":154},[61],[15,59403,1211],{"className":59404},[66],[15,59406,41227],{"className":59407},[169],[15,59409],{"className":59410,"style":173},[71],[15,59412,129],{"className":59413},[177],[15,59415],{"className":59416,"style":173},[71],[15,59418,59420,59423,59527,59530,59533],{"className":59419},[57],[15,59421],{"className":59422,"style":41187},[61],[15,59424,59426,59429,59524],{"className":59425},[66],[15,59427],{"className":59428},[162,23412],[15,59430,59432],{"className":59431},[6465],[15,59433,59435,59516],{"className":59434},[202,1090],[15,59436,59438,59513],{"className":59437},[206],[15,59439,59441,59473,59481],{"className":59440,"style":41206},[210],[15,59442,59443,59446],{"style":23428},[15,59444],{"className":59445,"style":1368},[217],[15,59447,59449,59452,59455,59458,59461,59464,59467,59470],{"className":59448},[66],[15,59450,120],{"className":59451},[162],[15,59453,1263],{"className":59454},[66],[15,59456,37],{"className":59457},[66,67],[15,59459],{"className":59460,"style":72},[71],[15,59462,1245],{"className":59463},[76],[15,59465],{"className":59466,"style":72},[71],[15,59468,1263],{"className":59469},[66],[15,59471,41227],{"className":59472},[169],[15,59474,59475,59478],{"style":23440},[15,59476],{"className":59477,"style":1368},[217],[15,59479],{"className":59480,"style":4438},[23447],[15,59482,59483,59486],{"style":23450},[15,59484],{"className":59485,"style":1368},[217],[15,59487,59489,59492,59495,59498,59501,59504,59507,59510],{"className":59488},[66],[15,59490,120],{"className":59491},[162],[15,59493,1263],{"className":59494},[66],[15,59496,37],{"className":59497},[66,67],[15,59499],{"className":59500,"style":72},[71],[15,59502,1245],{"className":59503},[76],[15,59505],{"className":59506,"style":72},[71],[15,59508,1263],{"className":59509},[66],[15,59511,46043],{"className":59512},[169],[15,59514,1114],{"className":59515},[1113],[15,59517,59519],{"className":59518},[206],[15,59520,59522],{"className":59521,"style":41277},[210],[15,59523],{},[15,59525],{"className":59526},[169,23412],[15,59528],{"className":59529,"style":173},[71],[15,59531,129],{"className":59532},[177],[15,59534],{"className":59535,"style":173},[71],[15,59537,59539,59542],{"className":59538},[57],[15,59540],{"className":59541,"style":41187},[61],[15,59543,59545,59548,59678],{"className":59544},[66],[15,59546],{"className":59547},[162,23412],[15,59549,59551],{"className":59550},[6465],[15,59552,59554,59670],{"className":59553},[202,1090],[15,59555,59557,59667],{"className":59556},[206],[15,59558,59560,59627,59635],{"className":59559,"style":41206},[210],[15,59561,59562,59565],{"style":23428},[15,59563],{"className":59564,"style":1368},[217],[15,59566,59568,59606,59609,59612,59615,59618,59621,59624],{"className":59567},[66],[15,59569,59571,59574],{"className":59570},[66],[15,59572,1263],{"className":59573},[66],[15,59575,59577],{"className":59576},[198],[15,59578,59580],{"className":59579},[202],[15,59581,59583],{"className":59582},[206],[15,59584,59586],{"className":59585,"style":46247},[210],[15,59587,59588,59591],{"style":46250},[15,59589],{"className":59590,"style":218},[217],[15,59592,59594],{"className":59593},[222,223,224,225],[15,59595,59597,59600,59603],{"className":59596},[66,225],[15,59598,37],{"className":59599},[66,67,225],[15,59601,1245],{"className":59602},[76,225],[15,59604,1211],{"className":59605},[66,225],[15,59607,120],{"className":59608},[162],[15,59610,37],{"className":59611},[66,67],[15,59613],{"className":59614,"style":72},[71],[15,59616,1245],{"className":59617},[76],[15,59619],{"className":59620,"style":72},[71],[15,59622,1211],{"className":59623},[66],[15,59625,46043],{"className":59626},[169],[15,59628,59629,59632],{"style":23440},[15,59630],{"className":59631,"style":1368},[217],[15,59633],{"className":59634,"style":4438},[23447],[15,59636,59637,59640],{"style":23450},[15,59638],{"className":59639,"style":1368},[217],[15,59641,59643,59646,59649,59652,59655,59658,59661,59664],{"className":59642},[66],[15,59644,120],{"className":59645},[162],[15,59647,1263],{"className":59648},[66],[15,59650,37],{"className":59651},[66,67],[15,59653],{"className":59654,"style":72},[71],[15,59656,1245],{"className":59657},[76],[15,59659],{"className":59660,"style":72},[71],[15,59662,1263],{"className":59663},[66],[15,59665,46043],{"className":59666},[169],[15,59668,1114],{"className":59669},[1113],[15,59671,59673],{"className":59672},[206],[15,59674,59676],{"className":59675,"style":41277},[210],[15,59677],{},[15,59679],{"className":59680},[169,23412],[11,59682,59683],{},"于是沃利斯公式变为：",[15,59685,59687],{"className":59686,"translate":19},[294],[15,59688,59690,59787],{"className":59689,"translate":19},[18],[15,59691,59693],{"className":59692},[23],[25,59694,59695],{"xmlns":27,"display":303},[29,59696,59697,59784],{},[32,59698,59699,59715,59776,59778],{},[1172,59700,59701,59707],{},[32,59702,59703,59705],{},[35,59704,43663],{},[39,59706,1170],{},[32,59708,59709,59711,59713],{},[35,59710,37],{},[39,59712,449],{},[35,59714,7615],{"mathvariant":138},[6465,59716,59717,59750],{},[32,59718,59719,59726,59728,59730,59732,59738,59740,59742,59744,59746,59748],{},[131,59720,59721,59724],{},[1209,59722,59723],{},"16",[35,59725,37],{},[39,59727,120],{"stretchy":119},[35,59729,37],{},[39,59731,40764],{"stretchy":119},[131,59733,59734,59736],{},[39,59735,126],{"stretchy":119},[1209,59737,15850],{},[39,59739,120],{"stretchy":119},[1209,59741,1263],{},[35,59743,37],{},[39,59745,1245],{},[1209,59747,1263],{},[39,59749,126],{"stretchy":119},[32,59751,59752,59754,59756,59758,59760,59762,59764,59766,59768,59770,59772,59774],{},[39,59753,120],{"stretchy":119},[1209,59755,1263],{},[35,59757,37],{},[39,59759,126],{"stretchy":119},[39,59761,40764],{"stretchy":119},[39,59763,120],{"stretchy":119},[1209,59765,1263],{},[35,59767,37],{},[39,59769,1245],{},[1209,59771,1263],{},[39,59773,126],{"stretchy":119},[39,59775,40764],{"stretchy":119},[39,59777,129],{},[6465,59779,59780,59782],{},[35,59781,33489],{},[1209,59783,1263],{},[45,59785,59786],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{16^n (n!)^4 (2n+2)}{(2n)!(2n+2)!} = \\frac{\\pi}{2}",[15,59788,59790,60048],{"className":59789,"ariaHidden":53},[52],[15,59791,59793,59796,59849,59852,60039,60042,60045],{"className":59792},[57],[15,59794],{"className":59795,"style":45939},[61],[15,59797,59799],{"className":59798},[1339,1351],[15,59800,59802,59841],{"className":59801},[202,1090],[15,59803,59805,59838],{"className":59804},[206],[15,59806,59808,59828],{"className":59807,"style":840},[210],[15,59809,59810,59813],{"style":48819},[15,59811],{"className":59812,"style":1368},[217],[15,59814,59816],{"className":59815},[222,223,224,225],[15,59817,59819,59822,59825],{"className":59818},[66,225],[15,59820,37],{"className":59821},[66,67,225],[15,59823,449],{"className":59824},[177,225],[15,59826,7615],{"className":59827},[66,225],[15,59829,59830,59833],{"style":1386},[15,59831],{"className":59832,"style":1368},[217],[15,59834,59835],{},[15,59836,43663],{"className":59837},[1339],[15,59839,1114],{"className":59840},[1113],[15,59842,59844],{"className":59843},[206],[15,59845,59847],{"className":59846,"style":48857},[210],[15,59848],{},[15,59850],{"className":59851,"style":697},[71],[15,59853,59855,59858,60036],{"className":59854},[66],[15,59856],{"className":59857},[162,23412],[15,59859,59861],{"className":59860},[6465],[15,59862,59864,60028],{"className":59863},[202,1090],[15,59865,59867,60025],{"className":59866},[206],[15,59868,59870,59914,59922],{"className":59869,"style":45958},[210],[15,59871,59872,59875],{"style":23428},[15,59873],{"className":59874,"style":1368},[217],[15,59876,59878,59881,59884,59887,59890,59893,59896,59899,59902,59905,59908,59911],{"className":59877},[66],[15,59879,120],{"className":59880},[162],[15,59882,1263],{"className":59883},[66],[15,59885,37],{"className":59886},[66,67],[15,59888,46043],{"className":59889},[169],[15,59891,120],{"className":59892},[162],[15,59894,1263],{"className":59895},[66],[15,59897,37],{"className":59898},[66,67],[15,59900],{"className":59901,"style":72},[71],[15,59903,1245],{"className":59904},[76],[15,59906],{"className":59907,"style":72},[71],[15,59909,1263],{"className":59910},[66],[15,59912,46043],{"className":59913},[169],[15,59915,59916,59919],{"style":23440},[15,59917],{"className":59918,"style":1368},[217],[15,59920],{"className":59921,"style":4438},[23447],[15,59923,59924,59927],{"style":23450},[15,59925],{"className":59926,"style":1368},[217],[15,59928,59930,59933,59963,59966,59969,59972,60001,60004,60007,60010,60013,60016,60019,60022],{"className":59929},[66],[15,59931,1211],{"className":59932},[66],[15,59934,59936,59940],{"className":59935},[66],[15,59937,59939],{"className":59938},[66],"6",[15,59941,59943],{"className":59942},[198],[15,59944,59946],{"className":59945},[202],[15,59947,59949],{"className":59948},[206],[15,59950,59952],{"className":59951,"style":6436},[210],[15,59953,59954,59957],{"style":213},[15,59955],{"className":59956,"style":218},[217],[15,59958,59960],{"className":59959},[222,223,224,225],[15,59961,37],{"className":59962},[66,67,225],[15,59964,120],{"className":59965},[162],[15,59967,37],{"className":59968},[66,67],[15,59970,40764],{"className":59971},[169],[15,59973,59975,59978],{"className":59974},[169],[15,59976,126],{"className":59977},[169],[15,59979,59981],{"className":59980},[198],[15,59982,59984],{"className":59983},[202],[15,59985,59987],{"className":59986},[206],[15,59988,59990],{"className":59989,"style":6738},[210],[15,59991,59992,59995],{"style":213},[15,59993],{"className":59994,"style":218},[217],[15,59996,59998],{"className":59997},[222,223,224,225],[15,59999,15850],{"className":60000},[66,225],[15,60002,120],{"className":60003},[162],[15,60005,1263],{"className":60006},[66],[15,60008,37],{"className":60009},[66,67],[15,60011],{"className":60012,"style":72},[71],[15,60014,1245],{"className":60015},[76],[15,60017],{"className":60018,"style":72},[71],[15,60020,1263],{"className":60021},[66],[15,60023,126],{"className":60024},[169],[15,60026,1114],{"className":60027},[1113],[15,60029,60031],{"className":60030},[206],[15,60032,60034],{"className":60033,"style":41277},[210],[15,60035],{},[15,60037],{"className":60038},[169,23412],[15,60040],{"className":60041,"style":173},[71],[15,60043,129],{"className":60044},[177],[15,60046],{"className":60047,"style":173},[71],[15,60049,60051,60054],{"className":60050},[57],[15,60052],{"className":60053,"style":39109},[61],[15,60055,60057,60060,60113],{"className":60056},[66],[15,60058],{"className":60059},[162,23412],[15,60061,60063],{"className":60062},[6465],[15,60064,60066,60105],{"className":60065},[202,1090],[15,60067,60069,60102],{"className":60068},[206],[15,60070,60072,60083,60091],{"className":60071,"style":39128},[210],[15,60073,60074,60077],{"style":23428},[15,60075],{"className":60076,"style":1368},[217],[15,60078,60080],{"className":60079},[66],[15,60081,1263],{"className":60082},[66],[15,60084,60085,60088],{"style":23440},[15,60086],{"className":60087,"style":1368},[217],[15,60089],{"className":60090,"style":4438},[23447],[15,60092,60093,60096],{"style":23450},[15,60094],{"className":60095,"style":1368},[217],[15,60097,60099],{"className":60098},[66],[15,60100,33489],{"className":60101,"style":1515},[66,67],[15,60103,1114],{"className":60104},[1113],[15,60106,60108],{"className":60107},[206],[15,60109,60111],{"className":60110,"style":23469},[210],[15,60112],{},[15,60114],{"className":60115},[169,23412],[11,60117,60118],{},"利用：",[15,60120,60122],{"className":60121,"translate":19},[294],[15,60123,60125,60171],{"className":60124,"translate":19},[18],[15,60126,60128],{"className":60127},[23],[25,60129,60130],{"xmlns":27,"display":303},[29,60131,60132,60168],{},[32,60133,60134,60136,60138,60140,60142,60152],{},[35,60135,37],{},[39,60137,40764],{"stretchy":119},[39,60139,28386],{},[35,60141,4554],{},[131,60143,60144,60146],{},[35,60145,48378],{},[32,60147,60148,60150],{},[39,60149,6600],{},[35,60151,37],{},[131,60153,60154,60156],{},[35,60155,37],{},[32,60157,60158,60160,60162],{},[35,60159,37],{},[39,60161,1245],{},[6465,60163,60164,60166],{},[1209,60165,1211],{},[1209,60167,1263],{},[45,60169,60170],{"encoding":47},"n! \\sim a e^{-n} n^{n + \\frac{1}{2}}",[15,60172,60174,60195],{"className":60173,"ariaHidden":53},[52],[15,60175,60177,60180,60183,60186,60189,60192],{"className":60176},[57],[15,60178],{"className":60179,"style":840},[61],[15,60181,37],{"className":60182},[66,67],[15,60184,40764],{"className":60185},[169],[15,60187],{"className":60188,"style":173},[71],[15,60190,28386],{"className":60191},[177],[15,60193],{"className":60194,"style":173},[71],[15,60196,60198,60202,60205,60241],{"className":60197},[57],[15,60199],{"className":60200,"style":60201},[61],"height:1.004em;",[15,60203,4554],{"className":60204},[66,67],[15,60206,60208,60211],{"className":60207},[66],[15,60209,48378],{"className":60210},[66,67],[15,60212,60214],{"className":60213},[198],[15,60215,60217],{"className":60216},[202],[15,60218,60220],{"className":60219},[206],[15,60221,60224],{"className":60222,"style":60223},[210],"height:0.8213em;",[15,60225,60226,60229],{"style":397},[15,60227],{"className":60228,"style":218},[217],[15,60230,60232],{"className":60231},[222,223,224,225],[15,60233,60235,60238],{"className":60234},[66,225],[15,60236,6600],{"className":60237},[66,225],[15,60239,37],{"className":60240},[66,67,225],[15,60242,60244,60247],{"className":60243},[66],[15,60245,37],{"className":60246},[66,67],[15,60248,60250],{"className":60249},[198],[15,60251,60253],{"className":60252},[202],[15,60254,60256],{"className":60255},[206],[15,60257,60259],{"className":60258,"style":60201},[210],[15,60260,60261,60264],{"style":49623},[15,60262],{"className":60263,"style":1368},[217],[15,60265,60267],{"className":60266},[222,223,224,225],[15,60268,60270,60273,60276],{"className":60269},[66,225],[15,60271,37],{"className":60272},[66,67,225],[15,60274,1245],{"className":60275},[76,225],[15,60277,60279,60282,60341],{"className":60278},[66,225],[15,60280],{"className":60281},[162,23412,222,3677,33576],[15,60283,60285],{"className":60284},[6465],[15,60286,60288,60333],{"className":60287},[202,1090],[15,60289,60291,60330],{"className":60290},[206],[15,60292,60294,60308,60316],{"className":60293,"style":49016},[210],[15,60295,60296,60299],{"style":33592},[15,60297],{"className":60298,"style":1368},[217],[15,60300,60302],{"className":60301},[222,3677,1508,225],[15,60303,60305],{"className":60304},[66,225],[15,60306,1263],{"className":60307},[66,225],[15,60309,60310,60313],{"style":33607},[15,60311],{"className":60312,"style":1368},[217],[15,60314],{"className":60315,"style":33614},[23447,225],[15,60317,60318,60321],{"style":33617},[15,60319],{"className":60320,"style":1368},[217],[15,60322,60324],{"className":60323},[222,3677,1508,225],[15,60325,60327],{"className":60326},[66,225],[15,60328,1211],{"className":60329},[66,225],[15,60331,1114],{"className":60332},[1113],[15,60334,60336],{"className":60335},[206],[15,60337,60339],{"className":60338,"style":33639},[210],[15,60340],{},[15,60342],{"className":60343},[169,23412,222,3677,33576],[11,60345,60346],{},"代入：",[15,60348,60350],{"className":60349,"translate":19},[294],[15,60351,60353,60443],{"className":60352,"translate":19},[18],[15,60354,60356],{"className":60355},[23],[25,60357,60358],{"xmlns":27,"display":303},[29,60359,60360,60440],{},[32,60361,60362,60378,60432,60434],{},[1172,60363,60364,60370],{},[32,60365,60366,60368],{},[35,60367,43663],{},[39,60369,1170],{},[32,60371,60372,60374,60376],{},[35,60373,37],{},[39,60375,449],{},[35,60377,7615],{"mathvariant":138},[6465,60379,60380,60394],{},[32,60381,60382,60388],{},[131,60383,60384,60386],{},[35,60385,4554],{},[1209,60387,1263],{},[131,60389,60390,60392],{},[35,60391,48378],{},[1209,60393,1263],{},[32,60395,60396,60398],{},[1209,60397,15850],{},[131,60399,60400,60418],{},[32,60401,60402,60404,60416],{},[39,60403,120],{"fence":53},[6465,60405,60406,60414],{},[32,60407,60408,60410,60412],{},[35,60409,37],{},[39,60411,1245],{},[1209,60413,1211],{},[35,60415,37],{},[39,60417,126],{"fence":53},[32,60419,60420,60422,60424,60426],{},[1209,60421,1263],{},[35,60423,37],{},[39,60425,1245],{},[6465,60427,60428,60430],{},[1209,60429,8673],{},[1209,60431,1263],{},[39,60433,129],{},[6465,60435,60436,60438],{},[35,60437,33489],{},[1209,60439,1263],{},[45,60441,60442],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{a^2 e^2}{4 \\left(\\frac{n+1}{n}\\right)^{2n + \\frac{3}{2}}} = \\frac{\\pi}{2}",[15,60444,60446,60830],{"className":60445,"ariaHidden":53},[52],[15,60447,60449,60453,60506,60509,60821,60824,60827],{"className":60448},[57],[15,60450],{"className":60451,"style":60452},[61],"height:2.925em;vertical-align:-1.4339em;",[15,60454,60456],{"className":60455},[1339,1351],[15,60457,60459,60498],{"className":60458},[202,1090],[15,60460,60462,60495],{"className":60461},[206],[15,60463,60465,60485],{"className":60464,"style":840},[210],[15,60466,60467,60470],{"style":48819},[15,60468],{"className":60469,"style":1368},[217],[15,60471,60473],{"className":60472},[222,223,224,225],[15,60474,60476,60479,60482],{"className":60475},[66,225],[15,60477,37],{"className":60478},[66,67,225],[15,60480,449],{"className":60481},[177,225],[15,60483,7615],{"className":60484},[66,225],[15,60486,60487,60490],{"style":1386},[15,60488],{"className":60489,"style":1368},[217],[15,60491,60492],{},[15,60493,43663],{"className":60494},[1339],[15,60496,1114],{"className":60497},[1113],[15,60499,60501],{"className":60500},[206],[15,60502,60504],{"className":60503,"style":48857},[210],[15,60505],{},[15,60507],{"className":60508,"style":697},[71],[15,60510,60512,60515,60818],{"className":60511},[66],[15,60513],{"className":60514},[162,23412],[15,60516,60518],{"className":60517},[6465],[15,60519,60521,60810],{"className":60520},[202,1090],[15,60522,60524,60807],{"className":60523},[206],[15,60525,60527,60733,60741],{"className":60526,"style":45958},[210],[15,60528,60529,60532],{"style":54397},[15,60530],{"className":60531,"style":54401},[217],[15,60533,60535,60538,60541],{"className":60534},[66],[15,60536,15850],{"className":60537},[66],[15,60539],{"className":60540,"style":697},[71],[15,60542,60544,60633],{"className":60543},[1413],[15,60545,60547,60553,60627],{"className":60546},[1413],[15,60548,60550],{"className":60549,"style":1418},[162,1417],[15,60551,120],{"className":60552},[1422,1508],[15,60554,60556,60559,60624],{"className":60555},[66],[15,60557],{"className":60558},[162,23412],[15,60560,60562],{"className":60561},[6465],[15,60563,60565,60616],{"className":60564},[202,1090],[15,60566,60568,60613],{"className":60567},[206],[15,60569,60571,60585,60593],{"className":60570,"style":30868},[210],[15,60572,60573,60576],{"style":30871},[15,60574],{"className":60575,"style":1368},[217],[15,60577,60579],{"className":60578},[222,223,224,225],[15,60580,60582],{"className":60581},[66,225],[15,60583,37],{"className":60584},[66,67,225],[15,60586,60587,60590],{"style":23440},[15,60588],{"className":60589,"style":1368},[217],[15,60591],{"className":60592,"style":4438},[23447],[15,60594,60595,60598],{"style":30894},[15,60596],{"className":60597,"style":1368},[217],[15,60599,60601],{"className":60600},[222,223,224,225],[15,60602,60604,60607,60610],{"className":60603},[66,225],[15,60605,37],{"className":60606},[66,67,225],[15,60608,1245],{"className":60609},[76,225],[15,60611,1211],{"className":60612},[66,225],[15,60614,1114],{"className":60615},[1113],[15,60617,60619],{"className":60618},[206],[15,60620,60622],{"className":60621,"style":30916},[210],[15,60623],{},[15,60625],{"className":60626},[169,23412],[15,60628,60630],{"className":60629,"style":1418},[169,1417],[15,60631,126],{"className":60632},[1422,1508],[15,60634,60636],{"className":60635},[198],[15,60637,60639],{"className":60638},[202],[15,60640,60642],{"className":60641},[206],[15,60643,60645],{"className":60644,"style":54515},[210],[15,60646,60647,60650],{"style":51213},[15,60648],{"className":60649,"style":1368},[217],[15,60651,60653],{"className":60652},[222,223,224,225],[15,60654,60656,60659,60662,60665],{"className":60655},[66,225],[15,60657,1263],{"className":60658},[66,225],[15,60660,37],{"className":60661},[66,67,225],[15,60663,1245],{"className":60664},[76,225],[15,60666,60668,60671,60730],{"className":60667},[66,225],[15,60669],{"className":60670},[162,23412,222,3677,33576],[15,60672,60674],{"className":60673},[6465],[15,60675,60677,60722],{"className":60676},[202,1090],[15,60678,60680,60719],{"className":60679},[206],[15,60681,60683,60697,60705],{"className":60682,"style":49016},[210],[15,60684,60685,60688],{"style":33592},[15,60686],{"className":60687,"style":1368},[217],[15,60689,60691],{"className":60690},[222,3677,1508,225],[15,60692,60694],{"className":60693},[66,225],[15,60695,1263],{"className":60696},[66,225],[15,60698,60699,60702],{"style":33607},[15,60700],{"className":60701,"style":1368},[217],[15,60703],{"className":60704,"style":33614},[23447,225],[15,60706,60707,60710],{"style":33617},[15,60708],{"className":60709,"style":1368},[217],[15,60711,60713],{"className":60712},[222,3677,1508,225],[15,60714,60716],{"className":60715},[66,225],[15,60717,8673],{"className":60718},[66,225],[15,60720,1114],{"className":60721},[1113],[15,60723,60725],{"className":60724},[206],[15,60726,60728],{"className":60727,"style":33639},[210],[15,60729],{},[15,60731],{"className":60732},[169,23412,222,3677,33576],[15,60734,60735,60738],{"style":54603},[15,60736],{"className":60737,"style":54401},[217],[15,60739],{"className":60740,"style":4438},[23447],[15,60742,60743,60746],{"style":54612},[15,60744],{"className":60745,"style":54401},[217],[15,60747,60749,60778],{"className":60748},[66],[15,60750,60752,60755],{"className":60751},[66],[15,60753,4554],{"className":60754},[66,67],[15,60756,60758],{"className":60757},[198],[15,60759,60761],{"className":60760},[202],[15,60762,60764],{"className":60763},[206],[15,60765,60767],{"className":60766,"style":6738},[210],[15,60768,60769,60772],{"style":213},[15,60770],{"className":60771,"style":218},[217],[15,60773,60775],{"className":60774},[222,223,224,225],[15,60776,1263],{"className":60777},[66,225],[15,60779,60781,60784],{"className":60780},[66],[15,60782,48378],{"className":60783},[66,67],[15,60785,60787],{"className":60786},[198],[15,60788,60790],{"className":60789},[202],[15,60791,60793],{"className":60792},[206],[15,60794,60796],{"className":60795,"style":6738},[210],[15,60797,60798,60801],{"style":213},[15,60799],{"className":60800,"style":218},[217],[15,60802,60804],{"className":60803},[222,223,224,225],[15,60805,1263],{"className":60806},[66,225],[15,60808,1114],{"className":60809},[1113],[15,60811,60813],{"className":60812},[206],[15,60814,60816],{"className":60815,"style":54631},[210],[15,60817],{},[15,60819],{"className":60820},[169,23412],[15,60822],{"className":60823,"style":173},[71],[15,60825,129],{"className":60826},[177],[15,60828],{"className":60829,"style":173},[71],[15,60831,60833,60836],{"className":60832},[57],[15,60834],{"className":60835,"style":39109},[61],[15,60837,60839,60842,60895],{"className":60838},[66],[15,60840],{"className":60841},[162,23412],[15,60843,60845],{"className":60844},[6465],[15,60846,60848,60887],{"className":60847},[202,1090],[15,60849,60851,60884],{"className":60850},[206],[15,60852,60854,60865,60873],{"className":60853,"style":39128},[210],[15,60855,60856,60859],{"style":23428},[15,60857],{"className":60858,"style":1368},[217],[15,60860,60862],{"className":60861},[66],[15,60863,1263],{"className":60864},[66],[15,60866,60867,60870],{"style":23440},[15,60868],{"className":60869,"style":1368},[217],[15,60871],{"className":60872,"style":4438},[23447],[15,60874,60875,60878],{"style":23450},[15,60876],{"className":60877,"style":1368},[217],[15,60879,60881],{"className":60880},[66],[15,60882,33489],{"className":60883,"style":1515},[66,67],[15,60885,1114],{"className":60886},[1113],[15,60888,60890],{"className":60889},[206],[15,60891,60893],{"className":60892,"style":23469},[210],[15,60894],{},[15,60896],{"className":60897},[169,23412],[11,60899,60900],{},"解得：",[15,60902,60904],{"className":60903,"translate":19},[294],[15,60905,60907,60931],{"className":60906,"translate":19},[18],[15,60908,60910],{"className":60909},[23],[25,60911,60912],{"xmlns":27,"display":303},[29,60913,60914,60928],{},[32,60915,60916,60918,60920],{},[35,60917,4554],{},[39,60919,129],{},[48504,60921,60922],{},[32,60923,60924,60926],{},[1209,60925,1263],{},[35,60927,33489],{},[45,60929,60930],{"encoding":47},"a = \\sqrt{2 \\pi}",[15,60932,60934,60952],{"className":60933,"ariaHidden":53},[52],[15,60935,60937,60940,60943,60946,60949],{"className":60936},[57],[15,60938],{"className":60939,"style":86},[61],[15,60941,4554],{"className":60942},[66,67],[15,60944],{"className":60945,"style":173},[71],[15,60947,129],{"className":60948},[177],[15,60950],{"className":60951,"style":173},[71],[15,60953,60955,60959],{"className":60954},[57],[15,60956],{"className":60957,"style":60958},[61],"height:1.04em;vertical-align:-0.0839em;",[15,60960,60962],{"className":60961},[66,48546],[15,60963,60965,61003],{"className":60964},[202,1090],[15,60966,60968,61000],{"className":60967},[206],[15,60969,60972,60987],{"className":60970,"style":60971},[210],"height:0.9561em;",[15,60973,60975,60978],{"className":60974,"style":1386},[48560],[15,60976],{"className":60977,"style":1368},[217],[15,60979,60981,60984],{"className":60980,"style":48567},[66],[15,60982,1263],{"className":60983},[66],[15,60985,33489],{"className":60986,"style":1515},[66,67],[15,60988,60990,60993],{"style":60989},"top:-2.9161em;",[15,60991],{"className":60992,"style":1368},[217],[15,60994,60996],{"className":60995,"style":48587},[48586],[34823,60997,60998],{"xmlns":34825,"width":48590,"height":48591,"viewBox":48592,"preserveAspectRatio":48593},[34829,60999],{"d":48596},[15,61001,1114],{"className":61002},[1113],[15,61004,61006],{"className":61005},[206],[15,61007,61010],{"className":61008,"style":61009},[210],"height:0.0839em;",[15,61011],{},[11,61013,61014,61015,61043],{},"因此，当 ",[15,61016,61018,61031],{"className":61017,"translate":19},[18],[15,61019,61021],{"className":61020},[23],[25,61022,61023],{"xmlns":27},[29,61024,61025,61029],{},[32,61026,61027],{},[35,61028,37],{},[45,61030,37],{"encoding":47},[15,61032,61034],{"className":61033,"ariaHidden":53},[52],[15,61035,61037,61040],{"className":61036},[57],[15,61038],{"className":61039,"style":86},[61],[15,61041,37],{"className":61042},[66,67]," 趋近无穷大时：",[15,61045,61047],{"className":61046,"translate":19},[294],[15,61048,61050,61102],{"className":61049,"translate":19},[18],[15,61051,61053],{"className":61052},[23],[25,61054,61055],{"xmlns":27,"display":303},[29,61056,61057,61099],{},[32,61058,61059,61061,61063,61065,61073,61083],{},[35,61060,37],{},[39,61062,40764],{"stretchy":119},[39,61064,28386],{},[48504,61066,61067],{},[32,61068,61069,61071],{},[1209,61070,1263],{},[35,61072,33489],{},[131,61074,61075,61077],{},[35,61076,48378],{},[32,61078,61079,61081],{},[39,61080,6600],{},[35,61082,37],{},[131,61084,61085,61087],{},[35,61086,37],{},[32,61088,61089,61091,61093],{},[35,61090,37],{},[39,61092,1245],{},[6465,61094,61095,61097],{},[1209,61096,1211],{},[1209,61098,1263],{},[45,61100,61101],{"encoding":47},"n! \\sim \\sqrt{2 \\pi} e^{-n} n^{n + \\frac{1}{2}}",[15,61103,61105,61126],{"className":61104,"ariaHidden":53},[52],[15,61106,61108,61111,61114,61117,61120,61123],{"className":61107},[57],[15,61109],{"className":61110,"style":840},[61],[15,61112,37],{"className":61113},[66,67],[15,61115,40764],{"className":61116},[169],[15,61118],{"className":61119,"style":173},[71],[15,61121,28386],{"className":61122},[177],[15,61124],{"className":61125,"style":173},[71],[15,61127,61129,61133,61183,61218],{"className":61128},[57],[15,61130],{"className":61131,"style":61132},[61],"height:1.0879em;vertical-align:-0.0839em;",[15,61134,61136],{"className":61135},[66,48546],[15,61137,61139,61175],{"className":61138},[202,1090],[15,61140,61142,61172],{"className":61141},[206],[15,61143,61145,61160],{"className":61144,"style":60971},[210],[15,61146,61148,61151],{"className":61147,"style":1386},[48560],[15,61149],{"className":61150,"style":1368},[217],[15,61152,61154,61157],{"className":61153,"style":48567},[66],[15,61155,1263],{"className":61156},[66],[15,61158,33489],{"className":61159,"style":1515},[66,67],[15,61161,61162,61165],{"style":60989},[15,61163],{"className":61164,"style":1368},[217],[15,61166,61168],{"className":61167,"style":48587},[48586],[34823,61169,61170],{"xmlns":34825,"width":48590,"height":48591,"viewBox":48592,"preserveAspectRatio":48593},[34829,61171],{"d":48596},[15,61173,1114],{"className":61174},[1113],[15,61176,61178],{"className":61177},[206],[15,61179,61181],{"className":61180,"style":61009},[210],[15,61182],{},[15,61184,61186,61189],{"className":61185},[66],[15,61187,48378],{"className":61188},[66,67],[15,61190,61192],{"className":61191},[198],[15,61193,61195],{"className":61194},[202],[15,61196,61198],{"className":61197},[206],[15,61199,61201],{"className":61200,"style":60223},[210],[15,61202,61203,61206],{"style":397},[15,61204],{"className":61205,"style":218},[217],[15,61207,61209],{"className":61208},[222,223,224,225],[15,61210,61212,61215],{"className":61211},[66,225],[15,61213,6600],{"className":61214},[66,225],[15,61216,37],{"className":61217},[66,67,225],[15,61219,61221,61224],{"className":61220},[66],[15,61222,37],{"className":61223},[66,67],[15,61225,61227],{"className":61226},[198],[15,61228,61230],{"className":61229},[202],[15,61231,61233],{"className":61232},[206],[15,61234,61236],{"className":61235,"style":60201},[210],[15,61237,61238,61241],{"style":49623},[15,61239],{"className":61240,"style":1368},[217],[15,61242,61244],{"className":61243},[222,223,224,225],[15,61245,61247,61250,61253],{"className":61246},[66,225],[15,61248,37],{"className":61249},[66,67,225],[15,61251,1245],{"className":61252},[76,225],[15,61254,61256,61259,61318],{"className":61255},[66,225],[15,61257],{"className":61258},[162,23412,222,3677,33576],[15,61260,61262],{"className":61261},[6465],[15,61263,61265,61310],{"className":61264},[202,1090],[15,61266,61268,61307],{"className":61267},[206],[15,61269,61271,61285,61293],{"className":61270,"style":49016},[210],[15,61272,61273,61276],{"style":33592},[15,61274],{"className":61275,"style":1368},[217],[15,61277,61279],{"className":61278},[222,3677,1508,225],[15,61280,61282],{"className":61281},[66,225],[15,61283,1263],{"className":61284},[66,225],[15,61286,61287,61290],{"style":33607},[15,61288],{"className":61289,"style":1368},[217],[15,61291],{"className":61292,"style":33614},[23447,225],[15,61294,61295,61298],{"style":33617},[15,61296],{"className":61297,"style":1368},[217],[15,61299,61301],{"className":61300},[222,3677,1508,225],[15,61302,61304],{"className":61303},[66,225],[15,61305,1211],{"className":61306},[66,225],[15,61308,1114],{"className":61309},[1113],[15,61311,61313],{"className":61312},[206],[15,61314,61316],{"className":61315,"style":33639},[210],[15,61317],{},[15,61319],{"className":61320},[169,23412,222,3677,33576],[11,61322,61323],{},"或者等价地：",[15,61325,61327],{"className":61326,"translate":19},[294],[15,61328,61330,61374],{"className":61329,"translate":19},[18],[15,61331,61333],{"className":61332},[23],[25,61334,61335],{"xmlns":27,"display":303},[29,61336,61337,61371],{},[32,61338,61339,61341,61343,61345,61355],{},[35,61340,37],{},[39,61342,40764],{"stretchy":119},[39,61344,28386],{},[48504,61346,61347],{},[32,61348,61349,61351,61353],{},[1209,61350,1263],{},[35,61352,33489],{},[35,61354,37],{},[131,61356,61357,61369],{},[32,61358,61359,61361,61367],{},[39,61360,120],{"fence":53},[6465,61362,61363,61365],{},[35,61364,37],{},[35,61366,48378],{},[39,61368,126],{"fence":53},[35,61370,37],{},[45,61372,61373],{"encoding":47},"n! \\sim \\sqrt{2 \\pi n} \\left(\\frac{n}{e}\\right)^n",[15,61375,61377,61398],{"className":61376,"ariaHidden":53},[52],[15,61378,61380,61383,61386,61389,61392,61395],{"className":61379},[57],[15,61381],{"className":61382,"style":840},[61],[15,61384,37],{"className":61385},[66,67],[15,61387,40764],{"className":61388},[169],[15,61390],{"className":61391,"style":173},[71],[15,61393,28386],{"className":61394},[177],[15,61396],{"className":61397,"style":173},[71],[15,61399,61401,61405,61458,61461],{"className":61400},[57],[15,61402],{"className":61403,"style":61404},[61],"height:1.8903em;vertical-align:-0.686em;",[15,61406,61408],{"className":61407},[66,48546],[15,61409,61411,61450],{"className":61410},[202,1090],[15,61412,61414,61447],{"className":61413},[206],[15,61415,61417,61435],{"className":61416,"style":60971},[210],[15,61418,61420,61423],{"className":61419,"style":1386},[48560],[15,61421],{"className":61422,"style":1368},[217],[15,61424,61426,61429,61432],{"className":61425,"style":48567},[66],[15,61427,1263],{"className":61428},[66],[15,61430,33489],{"className":61431,"style":1515},[66,67],[15,61433,37],{"className":61434},[66,67],[15,61436,61437,61440],{"style":60989},[15,61438],{"className":61439,"style":1368},[217],[15,61441,61443],{"className":61442,"style":48587},[48586],[34823,61444,61445],{"xmlns":34825,"width":48590,"height":48591,"viewBox":48592,"preserveAspectRatio":48593},[34829,61446],{"d":48596},[15,61448,1114],{"className":61449},[1113],[15,61451,61453],{"className":61452},[206],[15,61454,61456],{"className":61455,"style":61009},[210],[15,61457],{},[15,61459],{"className":61460,"style":697},[71],[15,61462,61464,61541],{"className":61463},[1413],[15,61465,61467,61473,61535],{"className":61466},[1413],[15,61468,61470],{"className":61469,"style":1418},[162,1417],[15,61471,120],{"className":61472},[1422,4675],[15,61474,61476,61479,61532],{"className":61475},[66],[15,61477],{"className":61478},[162,23412],[15,61480,61482],{"className":61481},[6465],[15,61483,61485,61524],{"className":61484},[202,1090],[15,61486,61488,61521],{"className":61487},[206],[15,61489,61491,61502,61510],{"className":61490,"style":39128},[210],[15,61492,61493,61496],{"style":23428},[15,61494],{"className":61495,"style":1368},[217],[15,61497,61499],{"className":61498},[66],[15,61500,48378],{"className":61501},[66,67],[15,61503,61504,61507],{"style":23440},[15,61505],{"className":61506,"style":1368},[217],[15,61508],{"className":61509,"style":4438},[23447],[15,61511,61512,61515],{"style":23450},[15,61513],{"className":61514,"style":1368},[217],[15,61516,61518],{"className":61517},[66],[15,61519,37],{"className":61520},[66,67],[15,61522,1114],{"className":61523},[1113],[15,61525,61527],{"className":61526},[206],[15,61528,61530],{"className":61529,"style":23469},[210],[15,61531],{},[15,61533],{"className":61534},[169,23412],[15,61536,61538],{"className":61537,"style":1418},[169,1417],[15,61539,126],{"className":61540},[1422,4675],[15,61542,61544],{"className":61543},[198],[15,61545,61547],{"className":61546},[202],[15,61548,61550],{"className":61549},[206],[15,61551,61554],{"className":61552,"style":61553},[210],"height:1.2043em;",[15,61555,61556,61559],{"style":51213},[15,61557],{"className":61558,"style":218},[217],[15,61560,61562],{"className":61561},[222,223,224,225],[15,61563,37],{"className":61564},[66,67,225],{"title":33407,"searchDepth":33408,"depth":33408,"links":61566},[61567,61568],{"id":33460,"depth":33408,"text":33460},{"id":33455,"depth":33408,"text":33455},{},"\u002Fblog\u002F2025\u002F2025-09-18-the-stirling-formula",{"title":33455,"description":33407},"blog\u002F2025\u002F2025-09-18-the-stirling-formula","斯特林公式揭示了当 n 趋向无穷大时 n!、nⁿ 与 eⁿ 之间的关系。",[61575],"mathematics","2025-09-18T00:00:00+08:00","kCcJ1796DlLa-Cf3vQ5ANr8rCBs2ieEkP55C1qxB2oo",{"id":61579,"title":61580,"body":61581,"description":61585,"draft":33440,"enableCodeHighlight":33440,"enableCodeHightlight":33440,"enableComment":33441,"enableD2":33441,"enableKatex":33441,"extension":33442,"image":33407,"important":33440,"location":33443,"meta":121799,"navigation":33441,"ogImage":33443,"path":121800,"seo":121801,"stem":121802,"summary":121803,"tags":121804,"time":121806,"__hash__":121807},"blog\u002Fblog\u002F2025\u002F2025-09-02-bei-ye-si-jue-ce.md","贝叶斯决策理论",{"type":8,"value":61582,"toc":121741},[61583,61586,61593,61828,61835,61842,61845,61852,61856,61859,62194,62205,62341,62348,62563,62624,62929,62961,63545,63548,63723,63932,63936,63940,63943,64205,64408,64411,64643,64691,64695,64815,65190,65193,65499,65502,65510,65585,65744,65791,65823,65827,65904,66196,66257,66465,66468,66721,66869,66872,67085,67088,67685,67773,67776,68116,68148,68152,68155,68158,68165,68169,68173,68243,68687,69039,69045,69392,69627,69744,69747,70228,70322,70326,70329,70488,70491,70794,70854,71085,71218,71699,71877,71881,72026,72542,72660,72823,72915,72919,73263,73539,73571,73914,73975,73978,74014,74210,74561,74672,75012,75083,75090,75093,75421,75674,75889,75892,75896,75899,76253,76346,76349,76541,76544,76789,76792,76799,77131,77470,77684,77743,77945,78183,78540,78545,79427,79643,79646,79650,79653,79885,79889,79893,80091,80416,80419,81234,81237,81695,81874,82623,82944,82947,83319,83322,83356,83359,83687,83690,84605,84641,85039,85224,85713,85818,85821,86218,86221,86750,87097,87457,87463,87467,87470,88123,88126,88553,88777,89291,89298,89301,89604,89922,89994,89998,90221,90368,90372,90375,90681,90827,92030,92033,92037,92041,92318,92321,92784,92846,93316,93745,93843,94012,94125,94241,94587,94708,94712,95046,95555,95675,95939,95942,96615,97105,97515,97518,97521,97525,97528,98107,98110,98253,98659,98662,99097,99103,99107,99165,99168,99303,99364,99367,99371,99374,99813,100634,100637,100640,100644,100648,100709,101477,101480,101592,101897,101900,101904,101973,101976,102620,102623,102626,102630,102633,102666,102669,102672,102676,102683,103072,103160,103543,103549,103652,103656,103663,103879,103885,104712,105092,105463,105570,105901,105903,106103,106198,106204,106208,106212,106488,107135,107138,107259,107262,107266,107371,107607,107613,107880,107969,107973,108019,108957,109018,109083,109086,109090,109167,109446,109493,110045,110052,110351,110592,110821,110882,110886,110985,110988,111321,111324,111327,111331,111504,111507,111510,111514,111517,111905,111982,111985,112237,112240,112244,112342,112487,112519,112555,112558,112564,112567,112571,112575,112705,112892,112895,113335,113544,113667,114126,114186,114190,114587,114792,114794,115594,115667,116256,116536,117220,117223,117227,117333,117336,118091,118325,118328,118670,118907,119281,119546,119549,119676,119679,119683,119877,120023,120280,120283,120541,120703,120706,120709,121724,121727,121738],[11,61584,61585],{},"机器学习里，「贝叶斯」这个词至少承担了三项不同的工作，而它们经常被混为一谈。",[11,61587,61588,61589,61592],{},"第一项是",[2040,61590,61591],{},"贝叶斯公式","。它是一条关于条件概率的恒等式，本身没有争议，也不要求任何哲学立场：",[15,61594,61596],{"className":61595,"translate":19},[294],[15,61597,61599,61664],{"className":61598,"translate":19},[18],[15,61600,61602],{"className":61601},[23],[25,61603,61604],{"xmlns":27,"display":303},[29,61605,61606,61661],{},[32,61607,61608,61610,61612,61615,61617,61619,61621,61623,61659],{},[35,61609,11],{},[39,61611,120],{"stretchy":119},[35,61613,61614],{},"θ",[39,61616,34425],{},[35,61618,32453],{},[39,61620,126],{"stretchy":119},[39,61622,129],{},[6465,61624,61625,61649],{},[32,61626,61627,61629,61631,61633,61635,61637,61639,61641,61643,61645,61647],{},[35,61628,11],{},[39,61630,120],{"stretchy":119},[35,61632,32453],{},[39,61634,34425],{},[35,61636,61614],{},[39,61638,126],{"stretchy":119},[1274,61640,2462],{},[35,61642,11],{},[39,61644,120],{"stretchy":119},[35,61646,61614],{},[39,61648,126],{"stretchy":119},[32,61650,61651,61653,61655,61657],{},[35,61652,11],{},[39,61654,120],{"stretchy":119},[35,61656,32453],{},[39,61658,126],{"stretchy":119},[35,61660,3571],{"mathvariant":138},[45,61662,61663],{"encoding":47},"p(\\theta\\mid D)=\\frac{p(D\\mid\\theta)\\,p(\\theta)}{p(D)}.",[15,61665,61667,61691,61712],{"className":61666,"ariaHidden":53},[52],[15,61668,61670,61673,61676,61679,61682,61685,61688],{"className":61669},[57],[15,61671],{"className":61672,"style":154},[61],[15,61674,11],{"className":61675},[66,67],[15,61677,120],{"className":61678},[162],[15,61680,61614],{"className":61681,"style":23020},[66,67],[15,61683],{"className":61684,"style":173},[71],[15,61686,34425],{"className":61687},[177],[15,61689],{"className":61690,"style":173},[71],[15,61692,61694,61697,61700,61703,61706,61709],{"className":61693},[57],[15,61695],{"className":61696,"style":154},[61],[15,61698,32453],{"className":61699,"style":23020},[66,67],[15,61701,126],{"className":61702},[169],[15,61704],{"className":61705,"style":173},[71],[15,61707,129],{"className":61708},[177],[15,61710],{"className":61711,"style":173},[71],[15,61713,61715,61718,61825],{"className":61714},[57],[15,61716],{"className":61717,"style":41187},[61],[15,61719,61721,61724,61822],{"className":61720},[66],[15,61722],{"className":61723},[162,23412],[15,61725,61727],{"className":61726},[6465],[15,61728,61730,61814],{"className":61729},[202,1090],[15,61731,61733,61811],{"className":61732},[206],[15,61734,61736,61756,61764],{"className":61735,"style":41206},[210],[15,61737,61738,61741],{"style":23428},[15,61739],{"className":61740,"style":1368},[217],[15,61742,61744,61747,61750,61753],{"className":61743},[66],[15,61745,11],{"className":61746},[66,67],[15,61748,120],{"className":61749},[162],[15,61751,32453],{"className":61752,"style":23020},[66,67],[15,61754,126],{"className":61755},[169],[15,61757,61758,61761],{"style":23440},[15,61759],{"className":61760,"style":1368},[217],[15,61762],{"className":61763,"style":4438},[23447],[15,61765,61766,61769],{"style":23450},[15,61767],{"className":61768,"style":1368},[217],[15,61770,61772,61775,61778,61781,61784,61787,61790,61793,61796,61799,61802,61805,61808],{"className":61771},[66],[15,61773,11],{"className":61774},[66,67],[15,61776,120],{"className":61777},[162],[15,61779,32453],{"className":61780,"style":23020},[66,67],[15,61782],{"className":61783,"style":173},[71],[15,61785,34425],{"className":61786},[177],[15,61788],{"className":61789,"style":173},[71],[15,61791,61614],{"className":61792,"style":23020},[66,67],[15,61794,126],{"className":61795},[169],[15,61797],{"className":61798,"style":697},[71],[15,61800,11],{"className":61801},[66,67],[15,61803,120],{"className":61804},[162],[15,61806,61614],{"className":61807,"style":23020},[66,67],[15,61809,126],{"className":61810},[169],[15,61812,1114],{"className":61813},[1113],[15,61815,61817],{"className":61816},[206],[15,61818,61820],{"className":61819,"style":41277},[210],[15,61821],{},[15,61823],{"className":61824},[169,23412],[15,61826,3571],{"className":61827},[66],[11,61829,61830,61831,61834],{},"第二项是",[2040,61832,61833],{},"贝叶斯决策","。它问的不是「参数是多少」，而是「在还不确定的情况下，应该采取什么行动」。给定后验分布与一个损失函数，最优行动就是使后验期望损失最小的那个。",[11,61836,61837,61838,61841],{},"第三项是",[2040,61839,61840],{},"贝叶斯学习","。它把模型参数也看成随机变量，用后验分布而不是点估计来描述知识状态；预测时对参数做积分，而不是代入一个「最佳值」。",[11,61843,61844],{},"三者可以独立使用，也可以串成一条链：先验与似然给出后验，后验与损失给出行动，行动的结果又成为新的数据。本文沿着这条链走一遍：先补齐概率论的语言，再建立决策论的骨架，最后看这套骨架如何解释机器学习里的分类器、正则化、错误率、泛化界与不确定性。",[11,61846,61847,61848,61851],{},"这条链的终点是一个看起来过于理想化的量，即",[2040,61849,61850],{},"贝叶斯风险","。它假设真实分布已知，因而在实践中永远算不出来。但正是这个「算不出来的量」充当了学习理论的坐标原点：它告诉我们，在给定问题里错误率的下限在哪里，模型的哪些部分在逼近这个下限，哪些部分只是在拟合噪声。",[95,61853,61855],{"id":61854},"_1-一个决策问题的解剖","1. 一个决策问题的解剖",[11,61857,61858],{},"把所有监督学习问题写成同一个形式，只需要四样东西：",[16131,61860,61861,61921,61978,62036],{},[3192,61862,61863,61866,61867,61920],{},[2040,61864,61865],{},"状态"," ",[15,61868,61870,61889],{"className":61869,"translate":19},[18],[15,61871,61873],{"className":61872},[23],[25,61874,61875],{"xmlns":27},[29,61876,61877,61886],{},[32,61878,61879,61882,61884],{},[35,61880,61881],{},"Y",[39,61883,1187],{},[35,61885,61881],{"mathvariant":445},[45,61887,61888],{"encoding":47},"Y\\in\\mathcal Y",[15,61890,61892,61910],{"className":61891,"ariaHidden":53},[52],[15,61893,61895,61898,61901,61904,61907],{"className":61894},[57],[15,61896],{"className":61897,"style":14335},[61],[15,61899,61881],{"className":61900,"style":72},[66,67],[15,61902],{"className":61903,"style":173},[71],[15,61905,1187],{"className":61906},[177],[15,61908],{"className":61909,"style":173},[71],[15,61911,61913,61917],{"className":61912},[57],[15,61914],{"className":61915,"style":61916},[61],"height:0.7805em;vertical-align:-0.0972em;",[15,61918,61881],{"className":61919,"style":28447},[66,494],"：真实类别、待预测的实数值，或环境里我们关心但看不见的量；",[3192,61922,61923,61866,61926,61977],{},[2040,61924,61925],{},"观测",[15,61927,61929,61947],{"className":61928,"translate":19},[18],[15,61930,61932],{"className":61931},[23],[25,61933,61934],{"xmlns":27},[29,61935,61936,61944],{},[32,61937,61938,61940,61942],{},[35,61939,446],{},[39,61941,1187],{},[35,61943,446],{"mathvariant":445},[45,61945,61946],{"encoding":47},"X\\in\\mathcal X",[15,61948,61950,61968],{"className":61949,"ariaHidden":53},[52],[15,61951,61953,61956,61959,61962,61965],{"className":61952},[57],[15,61954],{"className":61955,"style":14335},[61],[15,61957,446],{"className":61958,"style":16492},[66,67],[15,61960],{"className":61961,"style":173},[71],[15,61963,1187],{"className":61964},[177],[15,61966],{"className":61967,"style":173},[71],[15,61969,61971,61974],{"className":61970},[57],[15,61972],{"className":61973,"style":490},[61],[15,61975,446],{"className":61976,"style":495},[66,494],"：我们拿到的特征；",[3192,61979,61980,61866,61983,62035],{},[2040,61981,61982],{},"行动",[15,61984,61986,62005],{"className":61985,"translate":19},[18],[15,61987,61989],{"className":61988},[23],[25,61990,61991],{"xmlns":27},[29,61992,61993,62002],{},[32,61994,61995,61997,61999],{},[35,61996,4554],{},[39,61998,1187],{},[35,62000,62001],{"mathvariant":445},"A",[45,62003,62004],{"encoding":47},"a\\in\\mathcal A",[15,62006,62008,62026],{"className":62007,"ariaHidden":53},[52],[15,62009,62011,62014,62017,62020,62023],{"className":62010},[57],[15,62012],{"className":62013,"style":3225},[61],[15,62015,4554],{"className":62016},[66,67],[15,62018],{"className":62019,"style":173},[71],[15,62021,1187],{"className":62022},[177],[15,62024],{"className":62025,"style":173},[71],[15,62027,62029,62032],{"className":62028},[57],[15,62030],{"className":62031,"style":490},[61],[15,62033,62001],{"className":62034},[66,494],"：模型输出的预测、决策或控制量；",[3192,62037,62038,61866,62040,62135,62136,62164,62165,62193],{},[2040,62039,30588],{},[15,62041,62043,62069],{"className":62042,"translate":19},[18],[15,62044,62046],{"className":62045},[23],[25,62047,62048],{"xmlns":27},[29,62049,62050,62066],{},[32,62051,62052,62054,62056,62058,62060,62062,62064],{},[35,62053,1216],{},[39,62055,442],{},[35,62057,61881],{"mathvariant":445},[39,62059,41],{},[35,62061,62001],{"mathvariant":445},[39,62063,449],{},[35,62065,455],{"mathvariant":454},[45,62067,62068],{"encoding":47},"L:\\mathcal Y\\times\\mathcal A\\to\\mathbb R",[15,62070,62072,62090,62108,62126],{"className":62071,"ariaHidden":53},[52],[15,62073,62075,62078,62081,62084,62087],{"className":62074},[57],[15,62076],{"className":62077,"style":490},[61],[15,62079,1216],{"className":62080},[66,67],[15,62082],{"className":62083,"style":173},[71],[15,62085,442],{"className":62086},[177],[15,62088],{"className":62089,"style":173},[71],[15,62091,62093,62096,62099,62102,62105],{"className":62092},[57],[15,62094],{"className":62095,"style":61916},[61],[15,62097,61881],{"className":62098,"style":28447},[66,494],[15,62100],{"className":62101,"style":72},[71],[15,62103,41],{"className":62104},[76],[15,62106],{"className":62107,"style":72},[71],[15,62109,62111,62114,62117,62120,62123],{"className":62110},[57],[15,62112],{"className":62113,"style":490},[61],[15,62115,62001],{"className":62116},[66,494],[15,62118],{"className":62119,"style":173},[71],[15,62121,449],{"className":62122},[177],[15,62124],{"className":62125,"style":173},[71],[15,62127,62129,62132],{"className":62128},[57],[15,62130],{"className":62131,"style":2034},[61],[15,62133,455],{"className":62134},[66,517],"：当真实状态为 ",[15,62137,62139,62152],{"className":62138,"translate":19},[18],[15,62140,62142],{"className":62141},[23],[25,62143,62144],{"xmlns":27},[29,62145,62146,62150],{},[32,62147,62148],{},[35,62149,1224],{},[45,62151,1224],{"encoding":47},[15,62153,62155],{"className":62154,"ariaHidden":53},[52],[15,62156,62158,62161],{"className":62157},[57],[15,62159],{"className":62160,"style":471},[61],[15,62162,1224],{"className":62163,"style":1515},[66,67],"、采取行动 ",[15,62166,62168,62181],{"className":62167,"translate":19},[18],[15,62169,62171],{"className":62170},[23],[25,62172,62173],{"xmlns":27},[29,62174,62175,62179],{},[32,62176,62177],{},[35,62178,4554],{},[45,62180,4554],{"encoding":47},[15,62182,62184],{"className":62183,"ariaHidden":53},[52],[15,62185,62187,62190],{"className":62186},[57],[15,62188],{"className":62189,"style":86},[61],[15,62191,4554],{"className":62192},[66,67]," 时付出的代价。",[11,62195,62196,62197,62200,62201,62204],{},"四者合起来称为一个",[2040,62198,62199],{},"决策问题","。一个",[2040,62202,62203],{},"决策规则","（decision rule）是一个映射",[15,62206,62208],{"className":62207,"translate":19},[294],[15,62209,62211,62251],{"className":62210,"translate":19},[18],[15,62212,62214],{"className":62213},[23],[25,62215,62216],{"xmlns":27,"display":303},[29,62217,62218,62248],{},[32,62219,62220,62222,62224,62226,62228,62230,62232,62234,62236,62238,62240,62242,62244,62246],{},[35,62221,3883],{},[39,62223,442],{},[35,62225,446],{"mathvariant":445},[39,62227,449],{},[35,62229,62001],{"mathvariant":445},[39,62231,335],{"separator":53},[71,62233],{"width":3902},[35,62235,123],{},[39,62237,15989],{},[35,62239,3883],{},[39,62241,120],{"stretchy":119},[35,62243,123],{},[39,62245,126],{"stretchy":119},[39,62247,335],{"separator":53},[45,62249,62250],{"encoding":47},"\\delta:\\mathcal X\\to\\mathcal A,\\qquad x\\mapsto \\delta(x),",[15,62252,62254,62272,62290,62320],{"className":62253,"ariaHidden":53},[52],[15,62255,62257,62260,62263,62266,62269],{"className":62256},[57],[15,62258],{"className":62259,"style":840},[61],[15,62261,3883],{"className":62262,"style":3944},[66,67],[15,62264],{"className":62265,"style":173},[71],[15,62267,442],{"className":62268},[177],[15,62270],{"className":62271,"style":173},[71],[15,62273,62275,62278,62281,62284,62287],{"className":62274},[57],[15,62276],{"className":62277,"style":490},[61],[15,62279,446],{"className":62280,"style":495},[66,494],[15,62282],{"className":62283,"style":173},[71],[15,62285,449],{"className":62286},[177],[15,62288],{"className":62289,"style":173},[71],[15,62291,62293,62296,62299,62302,62305,62308,62311,62314,62317],{"className":62292},[57],[15,62294],{"className":62295,"style":17374},[61],[15,62297,62001],{"className":62298},[66,494],[15,62300,335],{"className":62301},[422],[15,62303],{"className":62304,"style":4059},[71],[15,62306],{"className":62307,"style":697},[71],[15,62309,123],{"className":62310},[66,67],[15,62312],{"className":62313,"style":173},[71],[15,62315,15989],{"className":62316},[177],[15,62318],{"className":62319,"style":173},[71],[15,62321,62323,62326,62329,62332,62335,62338],{"className":62322},[57],[15,62324],{"className":62325,"style":154},[61],[15,62327,3883],{"className":62328,"style":3944},[66,67],[15,62330,120],{"className":62331},[162],[15,62333,123],{"className":62334},[66,67],[15,62336,126],{"className":62337},[169],[15,62339,335],{"className":62340},[422],[11,62342,62343,62344,62347],{},"它的",[2040,62345,62346],{},"风险","（risk）定义为",[15,62349,62351],{"className":62350,"translate":19},[294],[15,62352,62354,62421],{"className":62353,"translate":19},[18],[15,62355,62357],{"className":62356},[23],[25,62358,62359],{"xmlns":27,"display":303},[29,62360,62361,62418],{},[32,62362,62363,62365,62367,62369,62371,62373,62394,62396,62398,62400,62402,62404,62406,62408,62410,62412,62414,62416],{},[35,62364,455],{},[39,62366,120],{"stretchy":119},[35,62368,3883],{},[39,62370,126],{"stretchy":119},[39,62372,129],{},[1057,62374,62375,62378],{},[35,62376,62377],{"mathvariant":454},"E",[32,62379,62380,62382,62384,62386,62388,62390,62392],{},[39,62381,120],{"stretchy":119},[35,62383,446],{},[39,62385,335],{"separator":53},[35,62387,61881],{},[39,62389,126],{"stretchy":119},[39,62391,28386],{},[35,62393,28394],{},[39,62395,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,62397,1216],{},[39,62399,120],{"stretchy":119},[35,62401,61881],{},[39,62403,335],{"separator":53},[35,62405,3883],{},[39,62407,120],{"stretchy":119},[35,62409,446],{},[39,62411,126],{"stretchy":119},[39,62413,126],{"stretchy":119},[39,62415,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,62417,3571],{"mathvariant":138},[45,62419,62420],{"encoding":47},"R(\\delta)=\\mathbb E_{(X,Y)\\sim P}\\big[L(Y,\\delta(X))\\big].",[15,62422,62424,62452],{"className":62423,"ariaHidden":53},[52],[15,62425,62427,62430,62434,62437,62440,62443,62446,62449],{"className":62426},[57],[15,62428],{"className":62429,"style":154},[61],[15,62431,455],{"className":62432,"style":62433},[66,67],"margin-right:0.0077em;",[15,62435,120],{"className":62436},[162],[15,62438,3883],{"className":62439,"style":3944},[66,67],[15,62441,126],{"className":62442},[169],[15,62444],{"className":62445,"style":173},[71],[15,62447,129],{"className":62448},[177],[15,62450],{"className":62451,"style":173},[71],[15,62453,62455,62458,62521,62527,62530,62533,62536,62539,62542,62545,62548,62551,62554,62560],{"className":62454},[57],[15,62456],{"className":62457,"style":12286},[61],[15,62459,62461,62464],{"className":62460},[66],[15,62462,62377],{"className":62463},[66,517],[15,62465,62467],{"className":62466},[198],[15,62468,62470,62513],{"className":62469},[202,1090],[15,62471,62473,62510],{"className":62472},[206],[15,62474,62476],{"className":62475,"style":3666},[210],[15,62477,62479,62482],{"style":62478},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[15,62480],{"className":62481,"style":218},[217],[15,62483,62485],{"className":62484},[222,223,224,225],[15,62486,62488,62491,62494,62497,62500,62503,62506],{"className":62487},[66,225],[15,62489,120],{"className":62490},[162,225],[15,62492,446],{"className":62493,"style":16492},[66,67,225],[15,62495,335],{"className":62496},[422,225],[15,62498,61881],{"className":62499,"style":72},[66,67,225],[15,62501,126],{"className":62502},[169,225],[15,62504,28386],{"className":62505},[177,225],[15,62507,28394],{"className":62508,"style":62509},[66,67,225],"margin-right:0.1389em;",[15,62511,1114],{"className":62512},[1113],[15,62514,62516],{"className":62515},[206],[15,62517,62519],{"className":62518,"style":9875},[210],[15,62520],{},[15,62522,62524],{"className":62523},[66],[15,62525,1194],{"className":62526},[1422,1508],[15,62528,1216],{"className":62529},[66,67],[15,62531,120],{"className":62532},[162],[15,62534,61881],{"className":62535,"style":72},[66,67],[15,62537,335],{"className":62538},[422],[15,62540],{"className":62541,"style":697},[71],[15,62543,3883],{"className":62544,"style":3944},[66,67],[15,62546,120],{"className":62547},[162],[15,62549,446],{"className":62550,"style":16492},[66,67],[15,62552,18766],{"className":62553},[169],[15,62555,62557],{"className":62556},[66],[15,62558,1266],{"className":62559},[1422,1508],[15,62561,3571],{"className":62562},[66],[11,62564,62565,62566,62623],{},"这里期望是对联合分布 ",[15,62567,62569,62593],{"className":62568,"translate":19},[18],[15,62570,62572],{"className":62571},[23],[25,62573,62574],{"xmlns":27},[29,62575,62576,62590],{},[32,62577,62578,62580,62582,62584,62586,62588],{},[35,62579,28394],{},[39,62581,120],{"stretchy":119},[35,62583,446],{},[39,62585,335],{"separator":53},[35,62587,61881],{},[39,62589,126],{"stretchy":119},[45,62591,62592],{"encoding":47},"P(X,Y)",[15,62594,62596],{"className":62595,"ariaHidden":53},[52],[15,62597,62599,62602,62605,62608,62611,62614,62617,62620],{"className":62598},[57],[15,62600],{"className":62601,"style":154},[61],[15,62603,28394],{"className":62604,"style":62509},[66,67],[15,62606,120],{"className":62607},[162],[15,62609,446],{"className":62610,"style":16492},[66,67],[15,62612,335],{"className":62613},[422],[15,62615],{"className":62616,"style":697},[71],[15,62618,61881],{"className":62619,"style":72},[66,67],[15,62621,126],{"className":62622},[169]," 取的。这个分布在实践中未知，于是我们退而求其次：用样本来估计它，或者直接在一个受限的函数类里最小化经验风险。这个「退而求其次」的过程，就是统计学习理论。",[11,62625,62626,62627,62655,62656,62684,62685,62713,62714,62742,62743,62771,62772,62800,62801,62829,62830,1124],{},"同一套语言可以覆盖很多看似不同的任务。医疗诊断中，",[15,62628,62630,62643],{"className":62629,"translate":19},[18],[15,62631,62633],{"className":62632},[23],[25,62634,62635],{"xmlns":27},[29,62636,62637,62641],{},[32,62638,62639],{},[35,62640,61881],{},[45,62642,61881],{"encoding":47},[15,62644,62646],{"className":62645,"ariaHidden":53},[52],[15,62647,62649,62652],{"className":62648},[57],[15,62650],{"className":62651,"style":490},[61],[15,62653,61881],{"className":62654,"style":72},[66,67]," 是是否患病，",[15,62657,62659,62672],{"className":62658,"translate":19},[18],[15,62660,62662],{"className":62661},[23],[25,62663,62664],{"xmlns":27},[29,62665,62666,62670],{},[32,62667,62668],{},[35,62669,4554],{},[45,62671,4554],{"encoding":47},[15,62673,62675],{"className":62674,"ariaHidden":53},[52],[15,62676,62678,62681],{"className":62677},[57],[15,62679],{"className":62680,"style":86},[61],[15,62682,4554],{"className":62683},[66,67]," 是是否治疗，损失把漏诊与误诊的代价写成两个不同的数；垃圾邮件过滤中，",[15,62686,62688,62701],{"className":62687,"translate":19},[18],[15,62689,62691],{"className":62690},[23],[25,62692,62693],{"xmlns":27},[29,62694,62695,62699],{},[32,62696,62697],{},[35,62698,61881],{},[45,62700,61881],{"encoding":47},[15,62702,62704],{"className":62703,"ariaHidden":53},[52],[15,62705,62707,62710],{"className":62706},[57],[15,62708],{"className":62709,"style":490},[61],[15,62711,61881],{"className":62712,"style":72},[66,67]," 是邮件类别，",[15,62715,62717,62730],{"className":62716,"translate":19},[18],[15,62718,62720],{"className":62719},[23],[25,62721,62722],{"xmlns":27},[29,62723,62724,62728],{},[32,62725,62726],{},[35,62727,4554],{},[45,62729,4554],{"encoding":47},[15,62731,62733],{"className":62732,"ariaHidden":53},[52],[15,62734,62736,62739],{"className":62735},[57],[15,62737],{"className":62738,"style":86},[61],[15,62740,4554],{"className":62741},[66,67]," 是放行或拦截；回归中，",[15,62744,62746,62759],{"className":62745,"translate":19},[18],[15,62747,62749],{"className":62748},[23],[25,62750,62751],{"xmlns":27},[29,62752,62753,62757],{},[32,62754,62755],{},[35,62756,4554],{},[45,62758,4554],{"encoding":47},[15,62760,62762],{"className":62761,"ariaHidden":53},[52],[15,62763,62765,62768],{"className":62764},[57],[15,62766],{"className":62767,"style":86},[61],[15,62769,4554],{"className":62770},[66,67]," 是一个实数，损失通常是平方误差；强化学习中，",[15,62773,62775,62788],{"className":62774,"translate":19},[18],[15,62776,62778],{"className":62777},[23],[25,62779,62780],{"xmlns":27},[29,62781,62782,62786],{},[32,62783,62784],{},[35,62785,446],{},[45,62787,446],{"encoding":47},[15,62789,62791],{"className":62790,"ariaHidden":53},[52],[15,62792,62794,62797],{"className":62793},[57],[15,62795],{"className":62796,"style":490},[61],[15,62798,446],{"className":62799,"style":16492},[66,67]," 是历史状态，",[15,62802,62804,62817],{"className":62803,"translate":19},[18],[15,62805,62807],{"className":62806},[23],[25,62808,62809],{"xmlns":27},[29,62810,62811,62815],{},[32,62812,62813],{},[35,62814,4554],{},[45,62816,4554],{"encoding":47},[15,62818,62820],{"className":62819,"ariaHidden":53},[52],[15,62821,62823,62826],{"className":62822},[57],[15,62824],{"className":62825,"style":86},[61],[15,62827,4554],{"className":62828},[66,67]," 是动作，损失是负回报。它们的共同点是：",[2040,62831,62832,62833,62899,62900,62928],{},"不确定性由 ",[15,62834,62836,62860],{"className":62835,"translate":19},[18],[15,62837,62839],{"className":62838},[23],[25,62840,62841],{"xmlns":27},[29,62842,62843,62857],{},[32,62844,62845,62847,62849,62851,62853,62855],{},[35,62846,28394],{},[39,62848,120],{"stretchy":119},[35,62850,61881],{},[39,62852,34425],{},[35,62854,446],{},[39,62856,126],{"stretchy":119},[45,62858,62859],{"encoding":47},"P(Y\\mid X)",[15,62861,62863,62887],{"className":62862,"ariaHidden":53},[52],[15,62864,62866,62869,62872,62875,62878,62881,62884],{"className":62865},[57],[15,62867],{"className":62868,"style":154},[61],[15,62870,28394],{"className":62871,"style":62509},[66,67],[15,62873,120],{"className":62874},[162],[15,62876,61881],{"className":62877,"style":72},[66,67],[15,62879],{"className":62880,"style":173},[71],[15,62882,34425],{"className":62883},[177],[15,62885],{"className":62886,"style":173},[71],[15,62888,62890,62893,62896],{"className":62889},[57],[15,62891],{"className":62892,"style":154},[61],[15,62894,446],{"className":62895,"style":16492},[66,67],[15,62897,126],{"className":62898},[169]," 描述，偏好由 ",[15,62901,62903,62916],{"className":62902,"translate":19},[18],[15,62904,62906],{"className":62905},[23],[25,62907,62908],{"xmlns":27},[29,62909,62910,62914],{},[32,62911,62912],{},[35,62913,1216],{},[45,62915,1216],{"encoding":47},[15,62917,62919],{"className":62918,"ariaHidden":53},[52],[15,62920,62922,62925],{"className":62921},[57],[15,62923],{"className":62924,"style":490},[61],[15,62926,1216],{"className":62927},[66,67]," 描述，决策则是两者的交汇",[11,62930,62931,62932,62960],{},"把联合风险按 ",[15,62933,62935,62948],{"className":62934,"translate":19},[18],[15,62936,62938],{"className":62937},[23],[25,62939,62940],{"xmlns":27},[29,62941,62942,62946],{},[32,62943,62944],{},[35,62945,446],{},[45,62947,446],{"encoding":47},[15,62949,62951],{"className":62950,"ariaHidden":53},[52],[15,62952,62954,62957],{"className":62953},[57],[15,62955],{"className":62956,"style":490},[61],[15,62958,446],{"className":62959,"style":16492},[66,67]," 分解，可以得到一个更有用的形式：",[15,62962,62964],{"className":62963,"translate":19},[294],[15,62965,62967,63127],{"className":62966,"translate":19},[18],[15,62968,62970],{"className":62969},[23],[25,62971,62972],{"xmlns":27,"display":303},[29,62973,62974,63124],{},[1142,62975,62978,63075],{"rowspacing":62976,"columnalign":62977,"columnspacing":634},"0.25em","right left",[1146,62979,62980,62995],{},[1149,62981,62982],{},[62983,62984,62985],"mstyle",{"scriptlevel":1804,"displaystyle":53},[32,62986,62987,62989,62991,62993],{},[35,62988,455],{},[39,62990,120],{"stretchy":119},[35,62992,3883],{},[39,62994,126],{"stretchy":119},[1149,62996,62997],{},[62983,62998,62999],{"scriptlevel":1804,"displaystyle":53},[32,63000,63001,63003,63005,63011,63061,63063,63065,63067,63069,63071,63073],{},[32,63002],{},[39,63004,129],{},[1057,63006,63007,63009],{},[39,63008,29727],{},[35,63010,446],{"mathvariant":445},[32,63012,63013,63015,63021,63023,63025,63027,63029,63031,63033,63035,63037,63039,63041,63043,63045,63047,63049,63051,63053,63055,63057,63059],{},[39,63014,120],{"fence":53},[1057,63016,63017,63019],{},[39,63018,29727],{},[35,63020,61881],{"mathvariant":445},[35,63022,1216],{},[39,63024,120],{"stretchy":119},[35,63026,1224],{},[39,63028,335],{"separator":53},[35,63030,3883],{},[39,63032,120],{"stretchy":119},[35,63034,123],{},[39,63036,126],{"stretchy":119},[39,63038,126],{"stretchy":119},[1274,63040,2462],{},[35,63042,11],{},[39,63044,120],{"stretchy":119},[35,63046,1224],{},[39,63048,34425],{},[35,63050,123],{},[39,63052,126],{"stretchy":119},[1274,63054,2462],{},[35,63056,458],{},[35,63058,1224],{},[39,63060,126],{"fence":53},[35,63062,11],{},[39,63064,120],{"stretchy":119},[35,63066,123],{},[39,63068,126],{"stretchy":119},[1274,63070,2462],{},[35,63072,458],{},[35,63074,123],{},[1146,63076,63077,63083],{},[1149,63078,63079],{},[62983,63080,63081],{"scriptlevel":1804,"displaystyle":53},[32,63082],{},[1149,63084,63085],{},[62983,63086,63087],{"scriptlevel":1804,"displaystyle":53},[32,63088,63089,63091,63093,63099,63101,63104,63106,63108,63110,63112,63114,63116,63118,63120,63122],{},[32,63090],{},[39,63092,129],{},[1057,63094,63095,63097],{},[35,63096,62377],{"mathvariant":454},[35,63098,446],{},[39,63100,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,63102,63103],{},"r",[39,63105,120],{"stretchy":119},[35,63107,3883],{},[39,63109,120],{"stretchy":119},[35,63111,446],{},[39,63113,126],{"stretchy":119},[39,63115,335],{"separator":53},[35,63117,446],{},[39,63119,126],{"stretchy":119},[39,63121,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,63123,335],{"separator":53},[45,63125,63126],{"encoding":47},"\\begin{aligned}\nR(\\delta)\n&=\\int_{\\mathcal X}\\left(\\int_{\\mathcal Y}L(y,\\delta(x))\\,p(y\\mid x)\\,dy\\right)p(x)\\,dx\\\\\n&=\\mathbb E_X\\big[r(\\delta(X),X)\\big],\n\\end{aligned}",[15,63128,63130],{"className":63129,"ariaHidden":53},[52],[15,63131,63133,63137],{"className":63132},[57],[15,63134],{"className":63135,"style":63136},[61],"height:3.94em;vertical-align:-1.72em;",[15,63138,63140],{"className":63139},[66],[15,63141,63143,63200],{"className":63142},[1142],[15,63144,63147],{"className":63145},[63146],"col-align-r",[15,63148,63150,63191],{"className":63149},[202,1090],[15,63151,63153,63188],{"className":63152},[206],[15,63154,63157,63179],{"className":63155,"style":63156},[210],"height:2.22em;",[15,63158,63160,63164],{"style":63159},"top:-4.22em;",[15,63161],{"className":63162,"style":63163},[217],"height:3.45em;",[15,63165,63167,63170,63173,63176],{"className":63166},[66],[15,63168,455],{"className":63169,"style":62433},[66,67],[15,63171,120],{"className":63172},[162],[15,63174,3883],{"className":63175,"style":3944},[66,67],[15,63177,126],{"className":63178},[169],[15,63180,63182,63185],{"style":63181},"top:-2.09em;",[15,63183],{"className":63184,"style":63163},[217],[15,63186],{"className":63187},[66],[15,63189,1114],{"className":63190},[1113],[15,63192,63194],{"className":63193},[206],[15,63195,63198],{"className":63196,"style":63197},[210],"height:1.72em;",[15,63199],{},[15,63201,63204],{"className":63202},[63203],"col-align-l",[15,63205,63207,63537],{"className":63206},[202,1090],[15,63208,63210,63534],{"className":63209},[206],[15,63211,63213,63429],{"className":63212,"style":63156},[210],[15,63214,63215,63218],{"style":63159},[15,63216],{"className":63217,"style":63163},[217],[15,63219,63221,63224,63227,63230,63233,63277,63280,63405,63408,63411,63414,63417,63420,63423,63426],{"className":63220},[66],[15,63222],{"className":63223},[66],[15,63225],{"className":63226,"style":173},[71],[15,63228,129],{"className":63229},[177],[15,63231],{"className":63232,"style":173},[71],[15,63234,63236,63239],{"className":63235},[1339],[15,63237,29727],{"className":63238,"style":29791},[1339,1470,1471],[15,63240,63242],{"className":63241},[198],[15,63243,63245,63269],{"className":63244},[202,1090],[15,63246,63248,63266],{"className":63247},[206],[15,63249,63252],{"className":63250,"style":63251},[210],"height:-0.4336em;",[15,63253,63254,63257],{"style":29807},[15,63255],{"className":63256,"style":218},[217],[15,63258,63260],{"className":63259},[222,223,224,225],[15,63261,63263],{"className":63262},[66,225],[15,63264,446],{"className":63265,"style":495},[66,494,225],[15,63267,1114],{"className":63268},[1113],[15,63270,63272],{"className":63271},[206],[15,63273,63275],{"className":63274,"style":29838},[210],[15,63276],{},[15,63278],{"className":63279,"style":697},[71],[15,63281,63283,63289,63333,63336,63339,63342,63345,63348,63351,63354,63357,63360,63363,63366,63369,63372,63375,63378,63381,63384,63387,63390,63393,63396,63399],{"className":63282},[1413],[15,63284,63286],{"className":63285,"style":1418},[162,1417],[15,63287,120],{"className":63288},[1422,224],[15,63290,63292,63295],{"className":63291},[1339],[15,63293,29727],{"className":63294,"style":29791},[1339,1470,1471],[15,63296,63298],{"className":63297},[198],[15,63299,63301,63324],{"className":63300},[202,1090],[15,63302,63304,63321],{"className":63303},[206],[15,63305,63307],{"className":63306,"style":63251},[210],[15,63308,63309,63312],{"style":29807},[15,63310],{"className":63311,"style":218},[217],[15,63313,63315],{"className":63314},[222,223,224,225],[15,63316,63318],{"className":63317},[66,225],[15,63319,61881],{"className":63320,"style":28447},[66,494,225],[15,63322,1114],{"className":63323},[1113],[15,63325,63327],{"className":63326},[206],[15,63328,63331],{"className":63329,"style":63330},[210],"height:0.98em;",[15,63332],{},[15,63334],{"className":63335,"style":697},[71],[15,63337,1216],{"className":63338},[66,67],[15,63340,120],{"className":63341},[162],[15,63343,1224],{"className":63344,"style":1515},[66,67],[15,63346,335],{"className":63347},[422],[15,63349],{"className":63350,"style":697},[71],[15,63352,3883],{"className":63353,"style":3944},[66,67],[15,63355,120],{"className":63356},[162],[15,63358,123],{"className":63359},[66,67],[15,63361,18766],{"className":63362},[169],[15,63364],{"className":63365,"style":697},[71],[15,63367,11],{"className":63368},[66,67],[15,63370,120],{"className":63371},[162],[15,63373,1224],{"className":63374,"style":1515},[66,67],[15,63376],{"className":63377,"style":173},[71],[15,63379,34425],{"className":63380},[177],[15,63382],{"className":63383,"style":173},[71],[15,63385,123],{"className":63386},[66,67],[15,63388,126],{"className":63389},[169],[15,63391],{"className":63392,"style":697},[71],[15,63394,458],{"className":63395},[66,67],[15,63397,1224],{"className":63398,"style":1515},[66,67],[15,63400,63402],{"className":63401,"style":1418},[169,1417],[15,63403,126],{"className":63404},[1422,224],[15,63406],{"className":63407,"style":697},[71],[15,63409,11],{"className":63410},[66,67],[15,63412,120],{"className":63413},[162],[15,63415,123],{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E\\big[L(Y,a)\\mid X=x\\big]",[15,63607,63609,63645,63690,63708],{"className":63608,"ariaHidden":53},[52],[15,63610,63612,63615,63618,63621,63624,63627,63630,63633,63636,63639,63642],{"className":63611},[57],[15,63613],{"className":63614,"style":154},[61],[15,63616,63103],{"className":63617,"style":23020},[66,67],[15,63619,120],{"className":63620},[162],[15,63622,4554],{"className":63623},[66,67],[15,63625,335],{"className":63626},[422],[15,63628],{"className":63629,"style":697},[71],[15,63631,123],{"className":63632},[66,67],[15,63634,126],{"className":63635},[169],[15,63637],{"className":63638,"style":173},[71],[15,63640,129],{"className":63641},[177],[15,63643],{"className":63644,"style":173},[71],[15,63646,63648,63651,63654,63660,63663,63666,63669,63672,63675,63678,63681,63684,63687],{"className":63647},[57],[15,63649],{"className":63650,"style":4398},[61],[15,63652,62377],{"className":63653},[66,517],[15,63655,63657],{"className":63656},[66],[15,63658,1194],{"className":63659},[1422,1508],[15,63661,1216],{"className":63662},[66,67],[15,63664,120],{"className":63665},[162],[15,63667,61881],{"className":63668,"style":72},[66,67],[15,63670,335],{"className":63671},[422],[15,63673],{"className":63674,"style":697},[71],[15,63676,4554],{"className":63677},[66,67],[15,63679,126],{"className":63680},[169],[15,63682],{"className":63683,"style":173},[71],[15,63685,34425],{"className":63686},[177],[15,63688],{"className":63689,"style":173},[71],[15,63691,63693,63696,63699,63702,63705],{"className":63692},[57],[15,63694],{"className":63695,"style":490},[61],[15,63697,446],{"className":63698,"style":16492},[66,67],[15,63700],{"className":63701,"style":173},[71],[15,63703,129],{"className":63704},[177],[15,63706],{"className":63707,"style":173},[71],[15,63709,63711,63714,63717],{"className":63710},[57],[15,63712],{"className":63713,"style":4398},[61],[15,63715,123],{"className":63716},[66,67],[15,63718,63720],{"className":63719},[66],[15,63721,1266],{"className":63722},[1422,1508],[11,63724,63725,63726,63754,63755,3766,63783,63786,63787,63815,63816,63844,63845,63902,63903,63931],{},"称为在 ",[15,63727,63729,63742],{"className":63728,"translate":19},[18],[15,63730,63732],{"className":63731},[23],[25,63733,63734],{"xmlns":27},[29,63735,63736,63740],{},[32,63737,63738],{},[35,63739,123],{},[45,63741,123],{"encoding":47},[15,63743,63745],{"className":63744,"ariaHidden":53},[52],[15,63746,63748,63751],{"className":63747},[57],[15,63749],{"className":63750,"style":86},[61],[15,63752,123],{"className":63753},[66,67]," 处采取行动 ",[15,63756,63758,63771],{"className":63757,"translate":19},[18],[15,63759,63761],{"className":63760},[23],[25,63762,63763],{"xmlns":27},[29,63764,63765,63769],{},[32,63766,63767],{},[35,63768,4554],{},[45,63770,4554],{"encoding":47},[15,63772,63774],{"className":63773,"ariaHidden":53},[52],[15,63775,63777,63780],{"className":63776},[57],[15,63778],{"className":63779,"style":86},[61],[15,63781,4554],{"className":63782},[66,67],[2040,63784,63785],{},"条件风险","。这个分解是整个理论的第一块基石：联合风险被拆成了「每个观测点上的局部风险」再对 ",[15,63788,63790,63803],{"className":63789,"translate":19},[18],[15,63791,63793],{"className":63792},[23],[25,63794,63795],{"xmlns":27},[29,63796,63797,63801],{},[32,63798,63799],{},[35,63800,446],{},[45,63802,446],{"encoding":47},[15,63804,63806],{"className":63805,"ariaHidden":53},[52],[15,63807,63809,63812],{"className":63808},[57],[15,63810],{"className":63811,"style":490},[61],[15,63813,446],{"className":63814,"style":16492},[66,67]," 平均。于是，只要在每个 ",[15,63817,63819,63832],{"className":63818,"translate":19},[18],[15,63820,63822],{"className":63821},[23],[25,63823,63824],{"xmlns":27},[29,63825,63826,63830],{},[32,63827,63828],{},[35,63829,123],{},[45,63831,123],{"encoding":47},[15,63833,63835],{"className":63834,"ariaHidden":53},[52],[15,63836,63838,63841],{"className":63837},[57],[15,63839],{"className":63840,"style":86},[61],[15,63842,123],{"className":63843},[66,67]," 上把 ",[15,63846,63848,63872],{"className":63847,"translate":19},[18],[15,63849,63851],{"className":63850},[23],[25,63852,63853],{"xmlns":27},[29,63854,63855,63869],{},[32,63856,63857,63859,63861,63863,63865,63867],{},[35,63858,63103],{},[39,63860,120],{"stretchy":119},[35,63862,4554],{},[39,63864,335],{"separator":53},[35,63866,123],{},[39,63868,126],{"stretchy":119},[45,63870,63871],{"encoding":47},"r(a,x)",[15,63873,63875],{"className":63874,"ariaHidden":53},[52],[15,63876,63878,63881,63884,63887,63890,63893,63896,63899],{"className":63877},[57],[15,63879],{"className":63880,"style":154},[61],[15,63882,63103],{"className":63883,"style":23020},[66,67],[15,63885,120],{"className":63886},[162],[15,63888,4554],{"className":63889},[66,67],[15,63891,335],{"className":63892},[422],[15,63894],{"className":63895,"style":697},[71],[15,63897,123],{"className":63898},[66,67],[15,63900,126],{"className":63901},[169]," 压到最低，全局风险就自动最低，不同 ",[15,63904,63906,63919],{"className":63905,"translate":19},[18],[15,63907,63909],{"className":63908},[23],[25,63910,63911],{"xmlns":27},[29,63912,63913,63917],{},[32,63914,63915],{},[35,63916,123],{},[45,63918,123],{"encoding":47},[15,63920,63922],{"className":63921,"ariaHidden":53},[52],[15,63923,63925,63928],{"className":63924},[57],[15,63926],{"className":63927,"style":86},[61],[15,63929,123],{"className":63930},[66,67]," 之间没有耦合。",[95,63933,63935],{"id":63934},"_2-概率的语言","2. 概率的语言",[1904,63937,63939],{"id":63938},"_21-条件概率信息如何改变概率","2.1 条件概率：信息如何改变概率",[11,63941,63942],{},"条件概率的定义是",[15,63944,63946],{"className":63945,"translate":19},[294],[15,63947,63949,64019],{"className":63948,"translate":19},[18],[15,63950,63952],{"className":63951},[23],[25,63953,63954],{"xmlns":27,"display":303},[29,63955,63956,64016],{},[32,63957,63958,63960,63962,63964,63966,63969,63971,63973,64000,64002,64004,64006,64008,64010,64012,64014],{},[35,63959,28394],{},[39,63961,120],{"stretchy":119},[35,63963,62001],{},[39,63965,34425],{},[35,63967,63968],{},"B",[39,63970,126],{"stretchy":119},[39,63972,129],{},[6465,63974,63975,63990],{},[32,63976,63977,63979,63981,63983,63986,63988],{},[35,63978,28394],{},[39,63980,120],{"stretchy":119},[35,63982,62001],{},[39,63984,63985],{},"∩",[35,63987,63968],{},[39,63989,126],{"stretchy":119},[32,63991,63992,63994,63996,63998],{},[35,63993,28394],{},[39,63995,120],{"stretchy":119},[35,63997,63968],{},[39,63999,126],{"stretchy":119},[39,64001,335],{"separator":53},[71,64003],{"width":3902},[35,64005,28394],{},[39,64007,120],{"stretchy":119},[35,64009,63968],{},[39,64011,126],{"stretchy":119},[39,64013,9278],{},[1209,64015,2487],{},[45,64017,64018],{"encoding":47},"P(A\\mid B)=\\frac{P(A\\cap B)}{P(B)},\\qquad P(B)>0.",[15,64020,64022,64046,64068,64196],{"className":64021,"ariaHidden":53},[52],[15,64023,64025,64028,64031,64034,64037,64040,64043],{"className":64024},[57],[15,64026],{"className":64027,"style":154},[61],[15,64029,28394],{"className":64030,"style":62509},[66,67],[15,64032,120],{"className":64033},[162],[15,64035,62001],{"className":64036},[66,67],[15,64038],{"className":64039,"style":173},[71],[15,64041,34425],{"className":64042},[177],[15,64044],{"className":64045,"style":173},[71],[15,64047,64049,64052,64056,64059,64062,64065],{"className":64048},[57],[15,64050],{"className":64051,"style":154},[61],[15,64053,63968],{"className":64054,"style":64055},[66,67],"margin-right:0.0502em;",[15,64057,126],{"className":64058},[169],[15,64060],{"className":64061,"style":173},[71],[15,64063,129],{"className":64064},[177],[15,64066],{"className":64067,"style":173},[71],[15,64069,64071,64074,64166,64169,64172,64175,64178,64181,64184,64187,64190,64193],{"className":64070},[57],[15,64072],{"className":64073,"style":41187},[61],[15,64075,64077,64080,64163],{"className":64076},[66],[15,64078],{"className":64079},[162,23412],[15,64081,64083],{"className":64082},[6465],[15,64084,64086,64155],{"className":64085},[202,1090],[15,64087,64089,64152],{"className":64088},[206],[15,64090,64092,64112,64120],{"className":64091,"style":41206},[210],[15,64093,64094,64097],{"style":23428},[15,64095],{"className":64096,"style":1368},[217],[15,64098,64100,64103,64106,64109],{"className":64099},[66],[15,64101,28394],{"className":64102,"style":62509},[66,67],[15,64104,120],{"className":64105},[162],[15,64107,63968],{"className":64108,"style":64055},[66,67],[15,64110,126],{"className":64111},[169],[15,64113,64114,64117],{"style":23440},[15,64115],{"className":64116,"style":1368},[217],[15,64118],{"className":64119,"style":4438},[23447],[15,64121,64122,64125],{"style":23450},[15,64123],{"className":64124,"style":1368},[217],[15,64126,64128,64131,64134,64137,64140,64143,64146,64149],{"className":64127},[66],[15,64129,28394],{"className":64130,"style":62509},[66,67],[15,64132,120],{"className":64133},[162],[15,64135,62001],{"className":64136},[66,67],[15,64138],{"className":64139,"style":72},[71],[15,64141,63985],{"className":64142},[76],[15,64144],{"className":64145,"style":72},[71],[15,64147,63968],{"className":64148,"style":64055},[66,67],[15,64150,126],{"className":64151},[169],[15,64153,1114],{"className":64154},[1113],[15,64156,64158],{"className":64157},[206],[15,64159,64161],{"className":64160,"style":41277},[210],[15,64162],{},[15,64164],{"className":64165},[169,23412],[15,64167,335],{"className":64168},[422],[15,64170],{"className":64171,"style":4059},[71],[15,64173],{"className":64174,"style":697},[71],[15,64176,28394],{"className":64177,"style":62509},[66,67],[15,64179,120],{"className":64180},[162],[15,64182,63968],{"className":64183,"style":64055},[66,67],[15,64185,126],{"className":64186},[169],[15,64188],{"className":64189,"style":173},[71],[15,64191,9278],{"className":64192},[177],[15,64194],{"className":64195,"style":173},[71],[15,64197,64199,64202],{"className":64198},[57],[15,64200],{"className":64201,"style":1836},[61],[15,64203,2487],{"className":64204},[66],[11,64206,64207,64208,64236,64237,64265,64266,64407],{},"它不是在「重新定义」概率，而是在缩小样本空间：一旦知道 ",[15,64209,64211,64224],{"className":64210,"translate":19},[18],[15,64212,64214],{"className":64213},[23],[25,64215,64216],{"xmlns":27},[29,64217,64218,64222],{},[32,64219,64220],{},[35,64221,63968],{},[45,64223,63968],{"encoding":47},[15,64225,64227],{"className":64226,"ariaHidden":53},[52],[15,64228,64230,64233],{"className":64229},[57],[15,64231],{"className":64232,"style":490},[61],[15,64234,63968],{"className":64235,"style":64055},[66,67]," 发生，所有不含 ",[15,64238,64240,64253],{"className":64239,"translate":19},[18],[15,64241,64243],{"className":64242},[23],[25,64244,64245],{"xmlns":27},[29,64246,64247,64251],{},[32,64248,64249],{},[35,64250,63968],{},[45,64252,63968],{"encoding":47},[15,64254,64256],{"className":64255,"ariaHidden":53},[52],[15,64257,64259,64262],{"className":64258},[57],[15,64260],{"className":64261,"style":490},[61],[15,64263,63968],{"className":64264,"style":64055},[66,67]," 的结果都被排除，原来的概率需要在剩下的世界里重新归一化。连续情形下，条件分布由条件密度 ",[15,64267,64269,64317],{"className":64268,"translate":19},[18],[15,64270,64272],{"className":64271},[23],[25,64273,64274],{"xmlns":27},[29,64275,64276,64314],{},[32,64277,64278,64280,64282,64284,64286,64288,64290,64292,64294,64296,64298,64300,64302,64304,64306,64308,64310,64312],{},[35,64279,11],{},[39,64281,120],{"stretchy":119},[35,64283,1224],{},[39,64285,34425],{},[35,64287,123],{},[39,64289,126],{"stretchy":119},[39,64291,129],{},[35,64293,11],{},[39,64295,120],{"stretchy":119},[35,64297,123],{},[39,64299,335],{"separator":53},[35,64301,1224],{},[39,64303,126],{"stretchy":119},[35,64305,6623],{"mathvariant":138},[35,64307,11],{},[39,64309,120],{"stretchy":119},[35,64311,123],{},[39,64313,126],{"stretchy":119},[45,64315,64316],{"encoding":47},"p(y\\mid x)=p(x,y)\u002Fp(x)",[15,64318,64320,64344,64365],{"className":64319,"ariaHidden":53},[52],[15,64321,64323,64326,64329,64332,64335,64338,64341],{"className":64322},[57],[15,64324],{"className":64325,"style":154},[61],[15,64327,11],{"className":64328},[66,67],[15,64330,120],{"className":64331},[162],[15,64333,1224],{"className":64334,"style":1515},[66,67],[15,64336],{"className":64337,"style":173},[71],[15,64339,34425],{"className":64340},[177],[15,64342],{"className":64343,"style":173},[71],[15,64345,64347,64350,64353,64356,64359,64362],{"className":64346},[57],[15,64348],{"className":64349,"style":154},[61],[15,64351,123],{"className":64352},[66,67],[15,64354,126],{"className":64355},[169],[15,64357],{"className":64358,"style":173},[71],[15,64360,129],{"className":64361},[177],[15,64363],{"className":64364,"style":173},[71],[15,64366,64368,64371,64374,64377,64380,64383,64386,64389,64392,64395,64398,64401,64404],{"className":64367},[57],[15,64369],{"className":64370,"style":154},[61],[15,64372,11],{"className":64373},[66,67],[15,64375,120],{"className":64376},[162],[15,64378,123],{"className":64379},[66,67],[15,64381,335],{"className":64382},[422],[15,64384],{"className":64385,"style":697},[71],[15,64387,1224],{"className":64388,"style":1515},[66,67],[15,64390,126],{"className":64391},[169],[15,64393,6623],{"className":64394},[66],[15,64396,11],{"className":64397},[66,67],[15,64399,120],{"className":64400},[162],[15,64402,123],{"className":64403},[66,67],[15,64405,126],{"className":64406},[169]," 给出；更一般地，它是测度论意义下的正则条件分布，本文不展开技术细节。",[11,64409,64410],{},"由条件概率的定义立刻得到乘法公式",[15,64412,64414],{"className":64413,"translate":19},[294],[15,64415,64417,64487],{"className":64416,"translate":19},[18],[15,64418,64420],{"className":64419},[23],[25,64421,64422],{"xmlns":27,"display":303},[29,64423,64424,64484],{},[32,64425,64426,64428,64430,64432,64434,64436,64438,64440,64442,64444,64446,64448,64450,64452,64454,64456,64458,64460,64462,64464,64466,64468,64470,64472,64474,64476,64478,64480,64482],{},[35,64427,28394],{},[39,64429,120],{"stretchy":119},[35,64431,62001],{},[39,64433,63985],{},[35,64435,63968],{},[39,64437,126],{"stretchy":119},[39,64439,129],{},[35,64441,28394],{},[39,64443,120],{"stretchy":119},[35,64445,62001],{},[39,64447,34425],{},[35,64449,63968],{},[39,64451,126],{"stretchy":119},[35,64453,28394],{},[39,64455,120],{"stretchy":119},[35,64457,63968],{},[39,64459,126],{"stretchy":119},[39,64461,129],{},[35,64463,28394],{},[39,64465,120],{"stretchy":119},[35,64467,63968],{},[39,64469,34425],{},[35,64471,62001],{},[39,64473,126],{"stretchy":119},[35,64475,28394],{},[39,64477,120],{"stretchy":119},[35,64479,62001],{},[39,64481,126],{"stretchy":119},[39,64483,335],{"separator":53},[45,64485,64486],{"encoding":47},"P(A\\cap B)=P(A\\mid B)P(B)=P(B\\mid A)P(A),",[15,64488,64490,64514,64535,64559,64592,64616],{"className":64489,"ariaHidden":53},[52],[15,64491,64493,64496,64499,64502,64505,64508,64511],{"className":64492},[57],[15,64494],{"className":64495,"style":154},[61],[15,64497,28394],{"className":64498,"style":62509},[66,67],[15,64500,120],{"className":64501},[162],[15,64503,62001],{"className":64504},[66,67],[15,64506],{"className":64507,"style":72},[71],[15,64509,63985],{"className":64510},[76],[15,64512],{"className":64513,"style":72},[71],[15,64515,64517,64520,64523,64526,64529,64532],{"className":64516},[57],[15,64518],{"className":64519,"style":154},[61],[15,64521,63968],{"className":64522,"style":64055},[66,67],[15,64524,126],{"className":64525},[169],[15,64527],{"className":64528,"style":173},[71],[15,64530,129],{"className":64531},[177],[15,64533],{"className":64534,"style":173},[71],[15,64536,64538,64541,64544,64547,64550,64553,64556],{"className":64537},[57],[15,64539],{"className":64540,"style":154},[61],[15,64542,28394],{"className":64543,"style":62509},[66,67],[15,64545,120],{"className":64546},[162],[15,64548,62001],{"className":64549},[66,67],[15,64551],{"className":64552,"style":173},[71],[15,64554,34425],{"className":64555},[177],[15,64557],{"className":64558,"style":173},[71],[15,64560,64562,64565,64568,64571,64574,64577,64580,64583,64586,64589],{"className":64561},[57],[15,64563],{"className":64564,"style":154},[61],[15,64566,63968],{"className":64567,"style":64055},[66,67],[15,64569,126],{"className":64570},[169],[15,64572,28394],{"className":64573,"style":62509},[66,67],[15,64575,120],{"className":64576},[162],[15,64578,63968],{"className":64579,"style":64055},[66,67],[15,64581,126],{"className":64582},[169],[15,64584],{"className":64585,"style":173},[71],[15,64587,129],{"className":64588},[177],[15,64590],{"className":64591,"style":173},[71],[15,64593,64595,64598,64601,64604,64607,64610,64613],{"className":64594},[57],[15,64596],{"className":64597,"style":154},[61],[15,64599,28394],{"className":64600,"style":62509},[66,67],[15,64602,120],{"className":64603},[162],[15,64605,63968],{"className":64606,"style":64055},[66,67],[15,64608],{"className":64609,"style":173},[71],[15,64611,34425],{"className":64612},[177],[15,64614],{"className":64615,"style":173},[71],[15,64617,64619,64622,64625,64628,64631,64634,64637,64640],{"className":64618},[57],[15,64620],{"className":64621,"style":154},[61],[15,64623,62001],{"className":64624},[66,67],[15,64626,126],{"className":64627},[169],[15,64629,28394],{"className":64630,"style":62509},[66,67],[15,64632,120],{"className":64633},[162],[15,64635,62001],{"className":64636},[66,67],[15,64638,126],{"className":64639},[169],[15,64641,335],{"className":64642},[422],[11,64644,64645,64646,64690],{},"两边同除以 ",[15,64647,64649,64669],{"className":64648,"translate":19},[18],[15,64650,64652],{"className":64651},[23],[25,64653,64654],{"xmlns":27},[29,64655,64656,64666],{},[32,64657,64658,64660,64662,64664],{},[35,64659,28394],{},[39,64661,120],{"stretchy":119},[35,64663,63968],{},[39,64665,126],{"stretchy":119},[45,64667,64668],{"encoding":47},"P(B)",[15,64670,64672],{"className":64671,"ariaHidden":53},[52],[15,64673,64675,64678,64681,64684,64687],{"className":64674},[57],[15,64676],{"className":64677,"style":154},[61],[15,64679,28394],{"className":64680,"style":62509},[66,67],[15,64682,120],{"className":64683},[162],[15,64685,63968],{"className":64686,"style":64055},[66,67],[15,64688,126],{"className":64689},[169],"，就是贝叶斯公式。",[1904,64692,64694],{"id":64693},"_22-贝叶斯公式","2.2 贝叶斯公式",[11,64696,64697,64698,64726,64727,64755,64756,64784,64785,64814],{},"把 ",[15,64699,64701,64714],{"className":64700,"translate":19},[18],[15,64702,64704],{"className":64703},[23],[25,64705,64706],{"xmlns":27},[29,64707,64708,64712],{},[32,64709,64710],{},[35,64711,63968],{},[45,64713,63968],{"encoding":47},[15,64715,64717],{"className":64716,"ariaHidden":53},[52],[15,64718,64720,64723],{"className":64719},[57],[15,64721],{"className":64722,"style":490},[61],[15,64724,63968],{"className":64725,"style":64055},[66,67]," 换成数据 ",[15,64728,64730,64743],{"className":64729,"translate":19},[18],[15,64731,64733],{"className":64732},[23],[25,64734,64735],{"xmlns":27},[29,64736,64737,64741],{},[32,64738,64739],{},[35,64740,32453],{},[45,64742,32453],{"encoding":47},[15,64744,64746],{"className":64745,"ariaHidden":53},[52],[15,64747,64749,64752],{"className":64748},[57],[15,64750],{"className":64751,"style":490},[61],[15,64753,32453],{"className":64754,"style":23020},[66,67],"，把 ",[15,64757,64759,64772],{"className":64758,"translate":19},[18],[15,64760,64762],{"className":64761},[23],[25,64763,64764],{"xmlns":27},[29,64765,64766,64770],{},[32,64767,64768],{},[35,64769,62001],{},[45,64771,62001],{"encoding":47},[15,64773,64775],{"className":64774,"ariaHidden":53},[52],[15,64776,64778,64781],{"className":64777},[57],[15,64779],{"className":64780,"style":490},[61],[15,64782,62001],{"className":64783},[66,67]," 换成参数或隐变量 ",[15,64786,64788,64802],{"className":64787,"translate":19},[18],[15,64789,64791],{"className":64790},[23],[25,64792,64793],{"xmlns":27},[29,64794,64795,64799],{},[32,64796,64797],{},[35,64798,61614],{},[45,64800,64801],{"encoding":47},"\\theta",[15,64803,64805],{"className":64804,"ariaHidden":53},[52],[15,64806,64808,64811],{"className":64807},[57],[15,64809],{"className":64810,"style":840},[61],[15,64812,61614],{"className":64813,"style":23020},[66,67],"，得到",[15,64816,64818],{"className":64817,"translate":19},[294],[15,64819,64821,64929],{"className":64820,"translate":19},[18],[15,64822,64824],{"className":64823},[23],[25,64825,64826],{"xmlns":27,"display":303},[29,64827,64828,64926],{},[32,64829,64830,64832,64834,64836,64838,64840,64842,64844,64880,64882,64884,64886,64888,64890,64892,64894,64896,64898,64900,64902,64904,64906,64908,64910,64912,64914,64916,64918,64920,64922,64924],{},[35,64831,11],{},[39,64833,120],{"stretchy":119},[35,64835,61614],{},[39,64837,34425],{},[35,64839,32453],{},[39,64841,126],{"stretchy":119},[39,64843,129],{},[6465,64845,64846,64870],{},[32,64847,64848,64850,64852,64854,64856,64858,64860,64862,64864,64866,64868],{},[35,64849,11],{},[39,64851,120],{"stretchy":119},[35,64853,32453],{},[39,64855,34425],{},[35,64857,61614],{},[39,64859,126],{"stretchy":119},[1274,64861,2462],{},[35,64863,11],{},[39,64865,120],{"stretchy":119},[35,64867,61614],{},[39,64869,126],{"stretchy":119},[32,64871,64872,64874,64876,64878],{},[35,64873,11],{},[39,64875,120],{"stretchy":119},[35,64877,32453],{},[39,64879,126],{"stretchy":119},[39,64881,335],{"separator":53},[71,64883],{"width":3902},[35,64885,11],{},[39,64887,120],{"stretchy":119},[35,64889,32453],{},[39,64891,126],{"stretchy":119},[39,64893,129],{},[39,64895,29727],{},[35,64897,11],{},[39,64899,120],{"stretchy":119},[35,64901,32453],{},[39,64903,34425],{},[35,64905,61614],{},[39,64907,126],{"stretchy":119},[1274,64909,2462],{},[35,64911,11],{},[39,64913,120],{"stretchy":119},[35,64915,61614],{},[39,64917,126],{"stretchy":119},[1274,64919,2462],{},[35,64921,458],{},[35,64923,61614],{},[35,64925,3571],{"mathvariant":138},[45,64927,64928],{"encoding":47},"p(\\theta\\mid D)=\\frac{p(D\\mid\\theta)\\,p(\\theta)}{p(D)},\n\\qquad\np(D)=\\int p(D\\mid\\theta)\\,p(\\theta)\\,d\\theta .",[15,64930,64932,64956,64977,65120,65151],{"className":64931,"ariaHidden":53},[52],[15,64933,64935,64938,64941,64944,64947,64950,64953],{"className":64934},[57],[15,64936],{"className":64937,"style":154},[61],[15,64939,11],{"className":64940},[66,67],[15,64942,120],{"className":64943},[162],[15,64945,61614],{"className":64946,"style":23020},[66,67],[15,64948],{"className":64949,"style":173},[71],[15,64951,34425],{"className":64952},[177],[15,64954],{"className":64955,"style":173},[71],[15,64957,64959,64962,64965,64968,64971,64974],{"className":64958},[57],[15,64960],{"className":64961,"style":154},[61],[15,64963,32453],{"className":64964,"style":23020},[66,67],[15,64966,126],{"className":64967},[169],[15,64969],{"className":64970,"style":173},[71],[15,64972,129],{"className":64973},[177],[15,64975],{"className":64976,"style":173},[71],[15,64978,64980,64983,65090,65093,65096,65099,65102,65105,65108,65111,65114,65117],{"className":64979},[57],[15,64981],{"className":64982,"style":41187},[61],[15,64984,64986,64989,65087],{"className":64985},[66],[15,64987],{"className":64988},[162,23412],[15,64990,64992],{"className":64991},[6465],[15,64993,64995,65079],{"className":64994},[202,1090],[15,64996,64998,65076],{"className":64997},[206],[15,64999,65001,65021,65029],{"className":65000,"style":41206},[210],[15,65002,65003,65006],{"style":23428},[15,65004],{"className":65005,"style":1368},[217],[15,65007,65009,65012,65015,65018],{"className":65008},[66],[15,65010,11],{"className":65011},[66,67],[15,65013,120],{"className":65014},[162],[15,65016,32453],{"className":65017,"style":23020},[66,67],[15,65019,126],{"className":65020},[169],[15,65022,65023,65026],{"style":23440},[15,65024],{"className":65025,"style":1368},[217],[15,65027],{"className":65028,"style":4438},[23447],[15,65030,65031,65034],{"style":23450},[15,65032],{"className":65033,"style":1368},[217],[15,65035,65037,65040,65043,65046,65049,65052,65055,65058,65061,65064,65067,65070,65073],{"className":65036},[66],[15,65038,11],{"className":65039},[66,67],[15,65041,120],{"className":65042},[162],[15,65044,32453],{"className":65045,"style":23020},[66,67],[15,65047],{"className":65048,"style":173},[71],[15,65050,34425],{"className":65051},[177],[15,65053],{"className":65054,"style":173},[71],[15,65056,61614],{"className":65057,"style":23020},[66,67],[15,65059,126],{"className":65060},[169],[15,65062],{"className":65063,"style":697},[71],[15,65065,11],{"className":65066},[66,67],[15,65068,120],{"className":65069},[162],[15,65071,61614],{"className":65072,"style":23020},[66,67],[15,65074,126],{"className":65075},[169],[15,65077,1114],{"className":65078},[1113],[15,65080,65082],{"className":65081},[206],[15,65083,65085],{"className":65084,"style":41277},[210],[15,65086],{},[15,65088],{"className":65089},[169,23412],[15,65091,335],{"className":65092},[422],[15,65094],{"className":65095,"style":4059},[71],[15,65097],{"className":65098,"style":697},[71],[15,65100,11],{"className":65101},[66,67],[15,65103,120],{"className":65104},[162],[15,65106,32453],{"className":65107,"style":23020},[66,67],[15,65109,126],{"className":65110},[169],[15,65112],{"className":65113,"style":173},[71],[15,65115,129],{"className":65116},[177],[15,65118],{"className":65119,"style":173},[71],[15,65121,65123,65127,65130,65133,65136,65139,65142,65145,65148],{"className":65122},[57],[15,65124],{"className":65125,"style":65126},[61],"height:2.2222em;vertical-align:-0.8622em;",[15,65128,29727],{"className":65129,"style":29791},[1339,1470,1471],[15,65131],{"className":65132,"style":697},[71],[15,65134,11],{"className":65135},[66,67],[15,65137,120],{"className":65138},[162],[15,65140,32453],{"className":65141,"style":23020},[66,67],[15,65143],{"className":65144,"style":173},[71],[15,65146,34425],{"className":65147},[177],[15,65149],{"className":65150,"style":173},[71],[15,65152,65154,65157,65160,65163,65166,65169,65172,65175,65178,65181,65184,65187],{"className":65153},[57],[15,65155],{"className":65156,"style":154},[61],[15,65158,61614],{"className":65159,"style":23020},[66,67],[15,65161,126],{"className":65162},[169],[15,65164],{"className":65165,"style":697},[71],[15,65167,11],{"className":65168},[66,67],[15,65170,120],{"className":65171},[162],[15,65173,61614],{"className":65174,"style":23020},[66,67],[15,65176,126],{"className":65177},[169],[15,65179],{"className":65180,"style":697},[71],[15,65182,458],{"className":65183},[66,67],[15,65185,61614],{"className":65186,"style":23020},[66,67],[15,65188,3571],{"className":65189},[66],[11,65191,65192],{},"公式的四个部分各有名字：",[6150,65194,65195,65208],{},[6153,65196,65197],{},[6156,65198,65199,65202,65205],{},[6159,65200,65201],{},"名称",[6159,65203,65204],{},"记号",[6159,65206,65207],{},"含义",[6169,65209,65210,65264,65340,65423],{},[6156,65211,65212,65215,65261],{},[6174,65213,65214],{},"先验",[6174,65216,65217],{},[15,65218,65220,65240],{"className":65219,"translate":19},[18],[15,65221,65223],{"className":65222},[23],[25,65224,65225],{"xmlns":27},[29,65226,65227,65237],{},[32,65228,65229,65231,65233,65235],{},[35,65230,11],{},[39,65232,120],{"stretchy":119},[35,65234,61614],{},[39,65236,126],{"stretchy":119},[45,65238,65239],{"encoding":47},"p(\\theta)",[15,65241,65243],{"className":65242,"ariaHidden":53},[52],[15,65244,65246,65249,65252,65255,65258],{"className":65245},[57],[15,65247],{"className":65248,"style":154},[61],[15,65250,11],{"className":65251},[66,67],[15,65253,120],{"className":65254},[162],[15,65256,61614],{"className":65257,"style":23020},[66,67],[15,65259,126],{"className":65260},[169],[6174,65262,65263],{},"看到数据前的信念",[6156,65265,65266,65269,65337],{},[6174,65267,65268],{},"似然",[6174,65270,65271],{},[15,65272,65274,65298],{"className":65273,"translate":19},[18],[15,65275,65277],{"className":65276},[23],[25,65278,65279],{"xmlns":27},[29,65280,65281,65295],{},[32,65282,65283,65285,65287,65289,65291,65293],{},[35,65284,11],{},[39,65286,120],{"stretchy":119},[35,65288,32453],{},[39,65290,34425],{},[35,65292,61614],{},[39,65294,126],{"stretchy":119},[45,65296,65297],{"encoding":47},"p(D\\mid\\theta)",[15,65299,65301,65325],{"className":65300,"ariaHidden":53},[52],[15,65302,65304,65307,65310,65313,65316,65319,65322],{"className":65303},[57],[15,65305],{"className":65306,"style":154},[61],[15,65308,11],{"className":65309},[66,67],[15,65311,120],{"className":65312},[162],[15,65314,32453],{"className":65315,"style":23020},[66,67],[15,65317],{"className":65318,"style":173},[71],[15,65320,34425],{"className":65321},[177],[15,65323],{"className":65324,"style":173},[71],[15,65326,65328,65331,65334],{"className":65327},[57],[15,65329],{"className":65330,"style":154},[61],[15,65332,61614],{"className":65333,"style":23020},[66,67],[15,65335,126],{"className":65336},[169],[6174,65338,65339],{},"给定参数时数据的概率",[6156,65341,65342,65345,65391],{},[6174,65343,65344],{},"证据",[6174,65346,65347],{},[15,65348,65350,65370],{"className":65349,"translate":19},[18],[15,65351,65353],{"className":65352},[23],[25,65354,65355],{"xmlns":27},[29,65356,65357,65367],{},[32,65358,65359,65361,65363,65365],{},[35,65360,11],{},[39,65362,120],{"stretchy":119},[35,65364,32453],{},[39,65366,126],{"stretchy":119},[45,65368,65369],{"encoding":47},"p(D)",[15,65371,65373],{"className":65372,"ariaHidden":53},[52],[15,65374,65376,65379,65382,65385,65388],{"className":65375},[57],[15,65377],{"className":65378,"style":154},[61],[15,65380,11],{"className":65381},[66,67],[15,65383,120],{"className":65384},[162],[15,65386,32453],{"className":65387,"style":23020},[66,67],[15,65389,126],{"className":65390},[169],[6174,65392,65393,65394,65422],{},"与 ",[15,65395,65397,65410],{"className":65396,"translate":19},[18],[15,65398,65400],{"className":65399},[23],[25,65401,65402],{"xmlns":27},[29,65403,65404,65408],{},[32,65405,65406],{},[35,65407,61614],{},[45,65409,64801],{"encoding":47},[15,65411,65413],{"className":65412,"ariaHidden":53},[52],[15,65414,65416,65419],{"className":65415},[57],[15,65417],{"className":65418,"style":840},[61],[15,65420,61614],{"className":65421,"style":23020},[66,67]," 无关的归一化常数，也是模型比较的关键",[6156,65424,65425,65428,65496],{},[6174,65426,65427],{},"后验",[6174,65429,65430],{},[15,65431,65433,65457],{"className":65432,"translate":19},[18],[15,65434,65436],{"className":65435},[23],[25,65437,65438],{"xmlns":27},[29,65439,65440,65454],{},[32,65441,65442,65444,65446,65448,65450,65452],{},[35,65443,11],{},[39,65445,120],{"stretchy":119},[35,65447,61614],{},[39,65449,34425],{},[35,65451,32453],{},[39,65453,126],{"stretchy":119},[45,65455,65456],{"encoding":47},"p(\\theta\\mid D)",[15,65458,65460,65484],{"className":65459,"ariaHidden":53},[52],[15,65461,65463,65466,65469,65472,65475,65478,65481],{"className":65462},[57],[15,65464],{"className":65465,"style":154},[61],[15,65467,11],{"className":65468},[66,67],[15,65470,120],{"className":65471},[162],[15,65473,61614],{"className":65474,"style":23020},[66,67],[15,65476],{"className":65477,"style":173},[71],[15,65479,34425],{"className":65480},[177],[15,65482],{"className":65483,"style":173},[71],[15,65485,65487,65490,65493],{"className":65486},[57],[15,65488],{"className":65489,"style":154},[61],[15,65491,32453],{"className":65492,"style":23020},[66,67],[15,65494,126],{"className":65495},[169],[6174,65497,65498],{},"看到数据后的信念",[11,65500,65501],{},"下面的图把这条链画全。注意「后验」不是终点：它只是决策的输入。",[13528,65503,65504],{},[13531,65505],{"height":65506,"width":65507,"style":13535,"data-d2-theme":13536,"alt":65508,"src":65509},257,720,"贝叶斯决策流程：先验与似然合成后验，后验与损失共同决定最优行动","data:image\u002Fsvg+xml;base64,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\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\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",[11,65511,65512,65513,48361,65556,65584],{},"公式右端的 ",[15,65514,65516,65535],{"className":65515,"translate":19},[18],[15,65517,65519],{"className":65518},[23],[25,65520,65521],{"xmlns":27},[29,65522,65523,65533],{},[32,65524,65525,65527,65529,65531],{},[35,65526,11],{},[39,65528,120],{"stretchy":119},[35,65530,32453],{},[39,65532,126],{"stretchy":119},[45,65534,65369],{"encoding":47},[15,65536,65538],{"className":65537,"ariaHidden":53},[52],[15,65539,65541,65544,65547,65550,65553],{"className":65540},[57],[15,65542],{"className":65543,"style":154},[61],[15,65545,11],{"className":65546},[66,67],[15,65548,120],{"className":65549},[162],[15,65551,32453],{"className":65552,"style":23020},[66,67],[15,65554,126],{"className":65555},[169],[15,65557,65559,65572],{"className":65558,"translate":19},[18],[15,65560,65562],{"className":65561},[23],[25,65563,65564],{"xmlns":27},[29,65565,65566,65570],{},[32,65567,65568],{},[35,65569,61614],{},[45,65571,64801],{"encoding":47},[15,65573,65575],{"className":65574,"ariaHidden":53},[52],[15,65576,65578,65581],{"className":65577},[57],[15,65579],{"className":65580,"style":840},[61],[15,65582,61614],{"className":65583,"style":23020},[66,67]," 无关，所以在求后验时常常被略去，写成",[15,65586,65588],{"className":65587,"translate":19},[294],[15,65589,65591,65642],{"className":65590,"translate":19},[18],[15,65592,65594],{"className":65593},[23],[25,65595,65596],{"xmlns":27,"display":303},[29,65597,65598,65639],{},[32,65599,65600,65602,65604,65606,65608,65610,65612,65615,65617,65619,65621,65623,65625,65627,65629,65631,65633,65635,65637],{},[35,65601,11],{},[39,65603,120],{"stretchy":119},[35,65605,61614],{},[39,65607,34425],{},[35,65609,32453],{},[39,65611,126],{"stretchy":119},[39,65613,65614],{},"∝",[35,65616,11],{},[39,65618,120],{"stretchy":119},[35,65620,32453],{},[39,65622,34425],{},[35,65624,61614],{},[39,65626,126],{"stretchy":119},[1274,65628,2462],{},[35,65630,11],{},[39,65632,120],{"stretchy":119},[35,65634,61614],{},[39,65636,126],{"stretchy":119},[35,65638,3571],{"mathvariant":138},[45,65640,65641],{"encoding":47},"p(\\theta\\mid D)\\propto p(D\\mid\\theta)\\,p(\\theta).",[15,65643,65645,65669,65690,65714],{"className":65644,"ariaHidden":53},[52],[15,65646,65648,65651,65654,65657,65660,65663,65666],{"className":65647},[57],[15,65649],{"className":65650,"style":154},[61],[15,65652,11],{"className":65653},[66,67],[15,65655,120],{"className":65656},[162],[15,65658,61614],{"className":65659,"style":23020},[66,67],[15,65661],{"className":65662,"style":173},[71],[15,65664,34425],{"className":65665},[177],[15,65667],{"className":65668,"style":173},[71],[15,65670,65672,65675,65678,65681,65684,65687],{"className":65671},[57],[15,65673],{"className":65674,"style":154},[61],[15,65676,32453],{"className":65677,"style":23020},[66,67],[15,65679,126],{"className":65680},[169],[15,65682],{"className":65683,"style":173},[71],[15,65685,65614],{"className":65686},[177],[15,65688],{"className":65689,"style":173},[71],[15,65691,65693,65696,65699,65702,65705,65708,65711],{"className":65692},[57],[15,65694],{"className":65695,"style":154},[61],[15,65697,11],{"className":65698},[66,67],[15,65700,120],{"className":65701},[162],[15,65703,32453],{"className":65704,"style":23020},[66,67],[15,65706],{"className":65707,"style":173},[71],[15,65709,34425],{"className":65710},[177],[15,65712],{"className":65713,"style":173},[71],[15,65715,65717,65720,65723,65726,65729,65732,65735,65738,65741],{"className":65716},[57],[15,65718],{"className":65719,"style":154},[61],[15,65721,61614],{"className":65722,"style":23020},[66,67],[15,65724,126],{"className":65725},[169],[15,65727],{"className":65728,"style":697},[71],[15,65730,11],{"className":65731},[66,67],[15,65733,120],{"className":65734},[162],[15,65736,61614],{"className":65737,"style":23020},[66,67],[15,65739,126],{"className":65740},[169],[15,65742,3571],{"className":65743},[66],[11,65745,65746,65747,65790],{},"但这个省略只在「给定模型」内部成立。一旦要比较不同模型，",[15,65748,65750,65769],{"className":65749,"translate":19},[18],[15,65751,65753],{"className":65752},[23],[25,65754,65755],{"xmlns":27},[29,65756,65757,65767],{},[32,65758,65759,65761,65763,65765],{},[35,65760,11],{},[39,65762,120],{"stretchy":119},[35,65764,32453],{},[39,65766,126],{"stretchy":119},[45,65768,65369],{"encoding":47},[15,65770,65772],{"className":65771,"ariaHidden":53},[52],[15,65773,65775,65778,65781,65784,65787],{"className":65774},[57],[15,65776],{"className":65777,"style":154},[61],[15,65779,11],{"className":65780},[66,67],[15,65782,120],{"className":65783},[162],[15,65785,32453],{"className":65786,"style":23020},[66,67],[15,65788,126],{"className":65789},[169]," 就是主角，第 6.5 节会回到这一点。",[11,65792,65793,65794,65822],{},"还要强调一点：贝叶斯公式本身只是概率论的一条定理，接受它不需要接受任何统计哲学。真正带有立场的是下一步——把 ",[15,65795,65797,65810],{"className":65796,"translate":19},[18],[15,65798,65800],{"className":65799},[23],[25,65801,65802],{"xmlns":27},[29,65803,65804,65808],{},[32,65805,65806],{},[35,65807,61614],{},[45,65809,64801],{"encoding":47},[15,65811,65813],{"className":65812,"ariaHidden":53},[52],[15,65814,65816,65819],{"className":65815},[57],[15,65817],{"className":65818,"style":840},[61],[15,65820,61614],{"className":65821,"style":23020},[66,67]," 看成随机变量，并为它指定先验。",[1904,65824,65826],{"id":65825},"_23-betabernoulli一个可以手算的更新","2.3 Beta–Bernoulli：一个可以手算的更新",[11,65828,1910,65829,65903],{},[15,65830,65832,65858],{"className":65831,"translate":19},[18],[15,65833,65835],{"className":65834},[23],[25,65836,65837],{"xmlns":27},[29,65838,65839,65855],{},[32,65840,65841,65843,65845,65847,65849,65851,65853],{},[35,65842,61614],{},[39,65844,1187],{},[39,65846,1194],{"stretchy":119},[1209,65848,1804],{},[39,65850,335],{"separator":53},[1209,65852,1211],{},[39,65854,1266],{"stretchy":119},[45,65856,65857],{"encoding":47},"\\theta\\in[0,1]",[15,65859,65861,65879],{"className":65860,"ariaHidden":53},[52],[15,65862,65864,65867,65870,65873,65876],{"className":65863},[57],[15,65865],{"className":65866,"style":9293},[61],[15,65868,61614],{"className":65869,"style":23020},[66,67],[15,65871],{"className":65872,"style":173},[71],[15,65874,1187],{"className":65875},[177],[15,65877],{"className":65878,"style":173},[71],[15,65880,65882,65885,65888,65891,65894,65897,65900],{"className":65881},[57],[15,65883],{"className":65884,"style":154},[61],[15,65886,1194],{"className":65887},[162],[15,65889,1804],{"className":65890},[66],[15,65892,335],{"className":65893},[422],[15,65895],{"className":65896,"style":697},[71],[15,65898,1211],{"className":65899},[66],[15,65901,1266],{"className":65902},[169]," 是一枚硬币正面向上的概率，先验取 Beta 分布",[15,65905,65907],{"className":65906,"translate":19},[294],[15,65908,65910,65984],{"className":65909,"translate":19},[18],[15,65911,65913],{"className":65912},[23],[25,65914,65915],{"xmlns":27,"display":303},[29,65916,65917,65981],{},[32,65918,65919,65921,65923,65925,65927,65929,65979],{},[35,65920,11],{},[39,65922,120],{"stretchy":119},[35,65924,61614],{},[39,65926,126],{"stretchy":119},[39,65928,129],{},[6465,65930,65931,65965],{},[32,65932,65933,65945,65947,65949,65951,65953],{},[131,65934,65935,65937],{},[35,65936,61614],{},[32,65938,65939,65941,65943],{},[35,65940,8385],{},[39,65942,6600],{},[1209,65944,1211],{},[39,65946,120],{"stretchy":119},[1209,65948,1211],{},[39,65950,6600],{},[35,65952,61614],{},[131,65954,65955,65957],{},[39,65956,126],{"stretchy":119},[32,65958,65959,65961,65963],{},[35,65960,17082],{},[39,65962,6600],{},[1209,65964,1211],{},[32,65966,65967,65969,65971,65973,65975,65977],{},[35,65968,63968],{},[39,65970,120],{"stretchy":119},[35,65972,8385],{},[39,65974,335],{"separator":53},[35,65976,17082],{},[39,65978,126],{"stretchy":119},[35,65980,3571],{"mathvariant":138},[45,65982,65983],{"encoding":47},"p(\\theta)=\\frac{\\theta^{\\alpha-1}(1-\\theta)^{\\beta-1}}{B(\\alpha,\\beta)}.",[15,65985,65987,66014],{"className":65986,"ariaHidden":53},[52],[15,65988,65990,65993,65996,65999,66002,66005,66008,66011],{"className":65989},[57],[15,65991],{"className":65992,"style":154},[61],[15,65994,11],{"className":65995},[66,67],[15,65997,120],{"className":65998},[162],[15,66000,61614],{"className":66001,"style":23020},[66,67],[15,66003,126],{"className":66004},[169],[15,66006],{"className":66007,"style":173},[71],[15,66009,129],{"className":66010},[177],[15,66012],{"className":66013,"style":173},[71],[15,66015,66017,66021,66193],{"className":66016},[57],[15,66018],{"className":66019,"style":66020},[61],"height:2.4621em;vertical-align:-0.936em;",[15,66022,66024,66027,66190],{"className":66023},[66],[15,66025],{"className":66026},[162,23412],[15,66028,66030],{"className":66029},[6465],[15,66031,66033,66182],{"className":66032},[202,1090],[15,66034,66036,66179],{"className":66035},[206],[15,66037,66040,66069,66077],{"className":66038,"style":66039},[210],"height:1.5261em;",[15,66041,66042,66045],{"style":23428},[15,66043],{"className":66044,"style":1368},[217],[15,66046,66048,66051,66054,66057,66060,66063,66066],{"className":66047},[66],[15,66049,63968],{"className":66050,"style":64055},[66,67],[15,66052,120],{"className":66053},[162],[15,66055,8385],{"className":66056,"style":8555},[66,67],[15,66058,335],{"className":66059},[422],[15,66061],{"className":66062,"style":697},[71],[15,66064,17082],{"className":66065,"style":17235},[66,67],[15,66067,126],{"className":66068},[169],[15,66070,66071,66074],{"style":23440},[15,66072],{"className":66073,"style":1368},[217],[15,66075],{"className":66076,"style":4438},[23447],[15,66078,66079,66082],{"style":23450},[15,66080],{"className":66081,"style":1368},[217],[15,66083,66085,66123,66126,66129,66132,66135,66138,66141],{"className":66084},[66],[15,66086,66088,66091],{"className":66087},[66],[15,66089,61614],{"className":66090,"style":23020},[66,67],[15,66092,66094],{"className":66093},[198],[15,66095,66097],{"className":66096},[202],[15,66098,66100],{"className":66099},[206],[15,66101,66103],{"className":66102,"style":6738},[210],[15,66104,66105,66108],{"style":213},[15,66106],{"className":66107,"style":218},[217],[15,66109,66111],{"className":66110},[222,223,224,225],[15,66112,66114,66117,66120],{"className":66113},[66,225],[15,66115,8385],{"className":66116,"style":8555},[66,67,225],[15,66118,6600],{"className":66119},[76,225],[15,66121,1211],{"className":66122},[66,225],[15,66124,120],{"className":66125},[162],[15,66127,1211],{"className":66128},[66],[15,66130],{"className":66131,"style":72},[71],[15,66133,6600],{"className":66134},[76],[15,66136],{"className":66137,"style":72},[71],[15,66139,61614],{"className":66140,"style":23020},[66,67],[15,66142,66144,66147],{"className":66143},[169],[15,66145,126],{"className":66146},[169],[15,66148,66150],{"className":66149},[198],[15,66151,66153],{"className":66152},[202],[15,66154,66156],{"className":66155},[206],[15,66157,66159],{"className":66158,"style":187},[210],[15,66160,66161,66164],{"style":213},[15,66162],{"className":66163,"style":218},[217],[15,66165,66167],{"className":66166},[222,223,224,225],[15,66168,66170,66173,66176],{"className":66169},[66,225],[15,66171,17082],{"className":66172,"style":17235},[66,67,225],[15,66174,6600],{"className":66175},[76,225],[15,66177,1211],{"className":66178},[66,225],[15,66180,1114],{"className":66181},[1113],[15,66183,66185],{"className":66184},[206],[15,66186,66188],{"className":66187,"style":41277},[210],[15,66189],{},[15,66191],{"className":66192},[169,23412],[15,66194,3571],{"className":66195},[66],[11,66197,66198,66199,66227,66228,66256],{},"观测到 ",[15,66200,66202,66215],{"className":66201,"translate":19},[18],[15,66203,66205],{"className":66204},[23],[25,66206,66207],{"xmlns":27},[29,66208,66209,66213],{},[32,66210,66211],{},[35,66212,37],{},[45,66214,37],{"encoding":47},[15,66216,66218],{"className":66217,"ariaHidden":53},[52],[15,66219,66221,66224],{"className":66220},[57],[15,66222],{"className":66223,"style":86},[61],[15,66225,37],{"className":66226},[66,67]," 次投掷中 ",[15,66229,66231,66244],{"className":66230,"translate":19},[18],[15,66232,66234],{"className":66233},[23],[25,66235,66236],{"xmlns":27},[29,66237,66238,66242],{},[32,66239,66240],{},[35,66241,621],{},[45,66243,621],{"encoding":47},[15,66245,66247],{"className":66246,"ariaHidden":53},[52],[15,66248,66250,66253],{"className":66249},[57],[15,66251],{"className":66252,"style":840},[61],[15,66254,621],{"className":66255,"style":684},[66,67]," 次正面，似然为",[15,66258,66260],{"className":66259,"translate":19},[294],[15,66261,66263,66317],{"className":66262,"translate":19},[18],[15,66264,66266],{"className":66265},[23],[25,66267,66268],{"xmlns":27,"display":303},[29,66269,66270,66314],{},[32,66271,66272,66274,66276,66278,66280,66282,66284,66286,66292,66294,66296,66298,66300,66312],{},[35,66273,11],{},[39,66275,120],{"stretchy":119},[35,66277,32453],{},[39,66279,34425],{},[35,66281,61614],{},[39,66283,126],{"stretchy":119},[39,66285,129],{},[131,66287,66288,66290],{},[35,66289,61614],{},[35,66291,621],{},[39,66293,120],{"stretchy":119},[1209,66295,1211],{},[39,66297,6600],{},[35,66299,61614],{},[131,66301,66302,66304],{},[39,66303,126],{"stretchy":119},[32,66305,66306,66308,66310],{},[35,66307,37],{},[39,66309,6600],{},[35,66311,621],{},[35,66313,3571],{"mathvariant":138},[45,66315,66316],{"encoding":47},"p(D\\mid\\theta)=\\theta^k(1-\\theta)^{n-k}.",[15,66318,66320,66344,66365,66415],{"className":66319,"ariaHidden":53},[52],[15,66321,66323,66326,66329,66332,66335,66338,66341],{"className":66322},[57],[15,66324],{"className":66325,"style":154},[61],[15,66327,11],{"className":66328},[66,67],[15,66330,120],{"className":66331},[162],[15,66333,32453],{"className":66334,"style":23020},[66,67],[15,66336],{"className":66337,"style":173},[71],[15,66339,34425],{"className":66340},[177],[15,66342],{"className":66343,"style":173},[71],[15,66345,66347,66350,66353,66356,66359,66362],{"className":66346},[57],[15,66348],{"className":66349,"style":154},[61],[15,66351,61614],{"className":66352,"style":23020},[66,67],[15,66354,126],{"className":66355},[169],[15,66357],{"className":66358,"style":173},[71],[15,66360,129],{"className":66361},[177],[15,66363],{"className":66364,"style":173},[71],[15,66366,66368,66371,66400,66403,66406,66409,66412],{"className":66367},[57],[15,66369],{"className":66370,"style":375},[61],[15,66372,66374,66377],{"className":66373},[66],[15,66375,61614],{"className":66376,"style":23020},[66,67],[15,66378,66380],{"className":66379},[198],[15,66381,66383],{"className":66382},[202],[15,66384,66386],{"className":66385},[206],[15,66387,66389],{"className":66388,"style":394},[210],[15,66390,66391,66394],{"style":397},[15,66392],{"className":66393,"style":218},[217],[15,66395,66397],{"className":66396},[222,223,224,225],[15,66398,621],{"className":66399,"style":684},[66,67,225],[15,66401,120],{"className":66402},[162],[15,66404,1211],{"className":66405},[66],[15,66407],{"className":66408,"style":72},[71],[15,66410,6600],{"className":66411},[76],[15,66413],{"className":66414,"style":72},[71],[15,66416,66418,66421,66424,66462],{"className":66417},[57],[15,66419],{"className":66420,"style":375},[61],[15,66422,61614],{"className":66423,"style":23020},[66,67],[15,66425,66427,66430],{"className":66426},[169],[15,66428,126],{"className":66429},[169],[15,66431,66433],{"className":66432},[198],[15,66434,66436],{"className":66435},[202],[15,66437,66439],{"className":66438},[206],[15,66440,66442],{"className":66441,"style":394},[210],[15,66443,66444,66447],{"style":397},[15,66445],{"className":66446,"style":218},[217],[15,66448,66450],{"className":66449},[222,223,224,225],[15,66451,66453,66456,66459],{"className":66452},[66,225],[15,66454,37],{"className":66455},[66,67,225],[15,66457,6600],{"className":66458},[76,225],[15,66460,621],{"className":66461,"style":684},[66,67,225],[15,66463,3571],{"className":66464},[66],[11,66466,66467],{},"两者相乘，后验的核为",[15,66469,66471],{"className":66470,"translate":19},[294],[15,66472,66474,66546],{"className":66473,"translate":19},[18],[15,66475,66477],{"className":66476},[23],[25,66478,66479],{"xmlns":27,"display":303},[29,66480,66481,66543],{},[32,66482,66483,66485,66487,66489,66491,66493,66495,66497,66513,66515,66517,66519,66521,66541],{},[35,66484,11],{},[39,66486,120],{"stretchy":119},[35,66488,61614],{},[39,66490,34425],{},[35,66492,32453],{},[39,66494,126],{"stretchy":119},[39,66496,65614],{},[131,66498,66499,66501],{},[35,66500,61614],{},[32,66502,66503,66505,66507,66509,66511],{},[35,66504,8385],{},[39,66506,1245],{},[35,66508,621],{},[39,66510,6600],{},[1209,66512,1211],{},[39,66514,120],{"stretchy":119},[1209,66516,1211],{},[39,66518,6600],{},[35,66520,61614],{},[131,66522,66523,66525],{},[39,66524,126],{"stretchy":119},[32,66526,66527,66529,66531,66533,66535,66537,66539],{},[35,66528,17082],{},[39,66530,1245],{},[35,66532,37],{},[39,66534,6600],{},[35,66536,621],{},[39,66538,6600],{},[1209,66540,1211],{},[39,66542,335],{"separator":53},[45,66544,66545],{"encoding":47},"p(\\theta\\mid D)\\propto \\theta^{\\alpha+k-1}(1-\\theta)^{\\beta+n-k-1},",[15,66547,66549,66573,66594,66659],{"className":66548,"ariaHidden":53},[52],[15,66550,66552,66555,66558,66561,66564,66567,66570],{"className":66551},[57],[15,66553],{"className":66554,"style":154},[61],[15,66556,11],{"className":66557},[66,67],[15,66559,120],{"className":66560},[162],[15,66562,61614],{"className":66563,"style":23020},[66,67],[15,66565],{"className":66566,"style":173},[71],[15,66568,34425],{"className":66569},[177],[15,66571],{"className":66572,"style":173},[71],[15,66574,66576,66579,66582,66585,66588,66591],{"className":66575},[57],[15,66577],{"className":66578,"style":154},[61],[15,66580,32453],{"className":66581,"style":23020},[66,67],[15,66583,126],{"className":66584},[169],[15,66586],{"className":66587,"style":173},[71],[15,66589,65614],{"className":66590},[177],[15,66592],{"className":66593,"style":173},[71],[15,66595,66597,66600,66644,66647,66650,66653,66656],{"className":66596},[57],[15,66598],{"className":66599,"style":375},[61],[15,66601,66603,66606],{"className":66602},[66],[15,66604,61614],{"className":66605,"style":23020},[66,67],[15,66607,66609],{"className":66608},[198],[15,66610,66612],{"className":66611},[202],[15,66613,66615],{"className":66614},[206],[15,66616,66618],{"className":66617,"style":394},[210],[15,66619,66620,66623],{"style":397},[15,66621],{"className":66622,"style":218},[217],[15,66624,66626],{"className":66625},[222,223,224,225],[15,66627,66629,66632,66635,66638,66641],{"className":66628},[66,225],[15,66630,8385],{"className":66631,"style":8555},[66,67,225],[15,66633,1245],{"className":66634},[76,225],[15,66636,621],{"className":66637,"style":684},[66,67,225],[15,66639,6600],{"className":66640},[76,225],[15,66642,1211],{"className":66643},[66,225],[15,66645,120],{"className":66646},[162],[15,66648,1211],{"className":66649},[66],[15,66651],{"className":66652,"style":72},[71],[15,66654,6600],{"className":66655},[76],[15,66657],{"className":66658,"style":72},[71],[15,66660,66662,66665,66668,66718],{"className":66661},[57],[15,66663],{"className":66664,"style":375},[61],[15,66666,61614],{"className":66667,"style":23020},[66,67],[15,66669,66671,66674],{"className":66670},[169],[15,66672,126],{"className":66673},[169],[15,66675,66677],{"className":66676},[198],[15,66678,66680],{"className":66679},[202],[15,66681,66683],{"className":66682},[206],[15,66684,66686],{"className":66685,"style":394},[210],[15,66687,66688,66691],{"style":397},[15,66689],{"className":66690,"style":218},[217],[15,66692,66694],{"className":66693},[222,223,224,225],[15,66695,66697,66700,66703,66706,66709,66712,66715],{"className":66696},[66,225],[15,66698,17082],{"className":66699,"style":17235},[66,67,225],[15,66701,1245],{"className":66702},[76,225],[15,66704,37],{"className":66705},[66,67,225],[15,66707,6600],{"className":66708},[76,225],[15,66710,621],{"className":66711,"style":684},[66,67,225],[15,66713,6600],{"className":66714},[76,225],[15,66716,1211],{"className":66717},[66,225],[15,66719,335],{"className":66720},[422],[11,66722,66723,66724,66864,66865,66868],{},"这正是 ",[15,66725,66727,66773],{"className":66726,"translate":19},[18],[15,66728,66730],{"className":66729},[23],[25,66731,66732],{"xmlns":27},[29,66733,66734,66770],{},[32,66735,66736,66746,66748,66750,66752,66754,66756,66758,66760,66762,66764,66766,66768],{},[32,66737,66738,66740,66742,66744],{},[35,66739,63968],{"mathvariant":138},[35,66741,48378],{"mathvariant":138},[35,66743,9207],{"mathvariant":138},[35,66745,4554],{"mathvariant":138},[39,66747,120],{"stretchy":119},[35,66749,8385],{},[39,66751,1245],{},[35,66753,621],{},[39,66755,335],{"separator":53},[1274,66757,24228],{},[35,66759,17082],{},[39,66761,1245],{},[35,66763,37],{},[39,66765,6600],{},[35,66767,621],{},[39,66769,126],{"stretchy":119},[45,66771,66772],{"encoding":47},"\\mathrm{Beta}(\\alpha+k,\\ \\beta+n-k)",[15,66774,66776,66804,66834,66852],{"className":66775,"ariaHidden":53},[52],[15,66777,66779,66782,66789,66792,66795,66798,66801],{"className":66778},[57],[15,66780],{"className":66781,"style":154},[61],[15,66783,66785],{"className":66784},[66],[15,66786,66788],{"className":66787},[66,4427],"Beta",[15,66790,120],{"className":66791},[162],[15,66793,8385],{"className":66794,"style":8555},[66,67],[15,66796],{"className":66797,"style":72},[71],[15,66799,1245],{"className":66800},[76],[15,66802],{"className":66803,"style":72},[71],[15,66805,66807,66810,66813,66816,66819,66822,66825,66828,66831],{"className":66806},[57],[15,66808],{"className":66809,"style":3406},[61],[15,66811,621],{"className":66812,"style":684},[66,67],[15,66814,335],{"className":66815},[422],[15,66817,24228],{"className":66818},[71],[15,66820],{"className":66821,"style":697},[71],[15,66823,17082],{"className":66824,"style":17235},[66,67],[15,66826],{"className":66827,"style":72},[71],[15,66829,1245],{"className":66830},[76],[15,66832],{"className":66833,"style":72},[71],[15,66835,66837,66840,66843,66846,66849],{"className":66836},[57],[15,66838],{"className":66839,"style":62},[61],[15,66841,37],{"className":66842},[66,67],[15,66844],{"className":66845,"style":72},[71],[15,66847,6600],{"className":66848},[76],[15,66850],{"className":66851,"style":72},[71],[15,66853,66855,66858,66861],{"className":66854},[57],[15,66856],{"className":66857,"style":154},[61],[15,66859,621],{"className":66860,"style":684},[66,67],[15,66862,126],{"className":66863},[169],"。归一化常数不用算，因为 Beta 分布的形式已经把它确定了。这就是",[2040,66866,66867],{},"共轭先验","的好处：后验与先验属于同一个分布族，更新只需要改两个参数。",[11,66870,66871],{},"后验均值是",[15,66873,66875],{"className":66874,"translate":19},[294],[15,66876,66878,66928],{"className":66877,"translate":19},[18],[15,66879,66881],{"className":66880},[23],[25,66882,66883],{"xmlns":27,"display":303},[29,66884,66885,66925],{},[32,66886,66887,66889,66891,66893,66895,66897,66899,66901,66923],{},[35,66888,62377],{"mathvariant":454},[39,66890,1194],{"stretchy":119},[35,66892,61614],{},[39,66894,34425],{},[35,66896,32453],{},[39,66898,1266],{"stretchy":119},[39,66900,129],{},[6465,66902,66903,66911],{},[32,66904,66905,66907,66909],{},[35,66906,8385],{},[39,66908,1245],{},[35,66910,621],{},[32,66912,66913,66915,66917,66919,66921],{},[35,66914,8385],{},[39,66916,1245],{},[35,66918,17082],{},[39,66920,1245],{},[35,66922,37],{},[39,66924,335],{"separator":53},[45,66926,66927],{"encoding":47},"\\mathbb E[\\theta\\mid D]=\\frac{\\alpha+k}{\\alpha+\\beta+n},",[15,66929,66931,66955,66976],{"className":66930,"ariaHidden":53},[52],[15,66932,66934,66937,66940,66943,66946,66949,66952],{"className":66933},[57],[15,66935],{"className":66936,"style":154},[61],[15,66938,62377],{"className":66939},[66,517],[15,66941,1194],{"className":66942},[162],[15,66944,61614],{"className":66945,"style":23020},[66,67],[15,66947],{"className":66948,"style":173},[71],[15,66950,34425],{"className":66951},[177],[15,66953],{"className":66954,"style":173},[71],[15,66956,66958,66961,66964,66967,66970,66973],{"className":66957},[57],[15,66959],{"className":66960,"style":154},[61],[15,66962,32453],{"className":66963,"style":23020},[66,67],[15,66965,1266],{"className":66966},[169],[15,66968],{"className":66969,"style":173},[71],[15,66971,129],{"className":66972},[177],[15,66974],{"className":66975,"style":173},[71],[15,66977,66979,66983,67082],{"className":66978},[57],[15,66980],{"className":66981,"style":66982},[61],"height:2.2519em;vertical-align:-0.8804em;",[15,66984,66986,66989,67079],{"className":66985},[66],[15,66987],{"className":66988},[162,23412],[15,66990,66992],{"className":66991},[6465],[15,66993,66995,67070],{"className":66994},[202,1090],[15,66996,66998,67067],{"className":66997},[206],[15,66999,67001,67036,67044],{"className":67000,"style":27430},[210],[15,67002,67003,67006],{"style":23428},[15,67004],{"className":67005,"style":1368},[217],[15,67007,67009,67012,67015,67018,67021,67024,67027,67030,67033],{"className":67008},[66],[15,67010,8385],{"className":67011,"style":8555},[66,67],[15,67013],{"className":67014,"style":72},[71],[15,67016,1245],{"className":67017},[76],[15,67019],{"className":67020,"style":72},[71],[15,67022,17082],{"className":67023,"style":17235},[66,67],[15,67025],{"className":67026,"style":72},[71],[15,67028,1245],{"className":67029},[76],[15,67031],{"className":67032,"style":72},[71],[15,67034,37],{"className":67035},[66,67],[15,67037,67038,67041],{"style":23440},[15,67039],{"className":67040,"style":1368},[217],[15,67042],{"className":67043,"style":4438},[23447],[15,67045,67046,67049],{"style":23450},[15,67047],{"className":67048,"style":1368},[217],[15,67050,67052,67055,67058,67061,67064],{"className":67051},[66],[15,67053,8385],{"className":67054,"style":8555},[66,67],[15,67056],{"className":67057,"style":72},[71],[15,67059,1245],{"className":67060},[76],[15,67062],{"className":67063,"style":72},[71],[15,67065,621],{"className":67066,"style":684},[66,67],[15,67068,1114],{"className":67069},[1113],[15,67071,67073],{"className":67072},[206],[15,67074,67077],{"className":67075,"style":67076},[210],"height:0.8804em;",[15,67078],{},[15,67080],{"className":67081},[169,23412],[15,67083,335],{"className":67084},[422],[11,67086,67087],{},"把它改写成加权平均：",[15,67089,67091],{"className":67090,"translate":19},[294],[15,67092,67094,67194],{"className":67093,"translate":19},[18],[15,67095,67097],{"className":67096},[23],[25,67098,67099],{"xmlns":27,"display":303},[29,67100,67101,67191],{},[32,67102,67103,67125,67127,67149,67151,67163,67165,67181,67183,67189],{},[6465,67104,67105,67113],{},[32,67106,67107,67109,67111],{},[35,67108,8385],{},[39,67110,1245],{},[35,67112,621],{},[32,67114,67115,67117,67119,67121,67123],{},[35,67116,8385],{},[39,67118,1245],{},[35,67120,17082],{},[39,67122,1245],{},[35,67124,37],{},[39,67126,129],{},[6465,67128,67129,67137],{},[32,67130,67131,67133,67135],{},[35,67132,8385],{},[39,67134,1245],{},[35,67136,17082],{},[32,67138,67139,67141,67143,67145,67147],{},[35,67140,8385],{},[39,67142,1245],{},[35,67144,17082],{},[39,67146,1245],{},[35,67148,37],{},[39,67150,3266],{},[6465,67152,67153,67155],{},[35,67154,8385],{},[32,67156,67157,67159,67161],{},[35,67158,8385],{},[39,67160,1245],{},[35,67162,17082],{},[39,67164,1245],{},[6465,67166,67167,67169],{},[35,67168,37],{},[32,67170,67171,67173,67175,67177,67179],{},[35,67172,8385],{},[39,67174,1245],{},[35,67176,17082],{},[39,67178,1245],{},[35,67180,37],{},[39,67182,3266],{},[6465,67184,67185,67187],{},[35,67186,621],{},[35,67188,37],{},[35,67190,3571],{"mathvariant":138},[45,67192,67193],{"encoding":47},"\\frac{\\alpha+k}{\\alpha+\\beta+n}\n=\n\\frac{\\alpha+\\beta}{\\alpha+\\beta+n}\\cdot\\frac{\\alpha}{\\alpha+\\beta}\n+\n\\frac{n}{\\alpha+\\beta+n}\\cdot\\frac{k}{n}.",[15,67195,67197,67310,67423,67513,67614],{"className":67196,"ariaHidden":53},[52],[15,67198,67200,67203,67301,67304,67307],{"className":67199},[57],[15,67201],{"className":67202,"style":66982},[61],[15,67204,67206,67209,67298],{"className":67205},[66],[15,67207],{"className":67208},[162,23412],[15,67210,67212],{"className":67211},[6465],[15,67213,67215,67290],{"className":67214},[202,1090],[15,67216,67218,67287],{"className":67217},[206],[15,67219,67221,67256,67264],{"className":67220,"style":27430},[210],[15,67222,67223,67226],{"style":23428},[15,67224],{"className":67225,"style":1368},[217],[15,67227,67229,67232,67235,67238,67241,67244,67247,67250,67253],{"className":67228},[66],[15,67230,8385],{"className":67231,"style":8555},[66,67],[15,67233],{"className":67234,"style":72},[71],[15,67236,1245],{"className":67237},[76],[15,67239],{"className":67240,"style":72},[71],[15,67242,17082],{"className":67243,"style":17235},[66,67],[15,67245],{"className":67246,"style":72},[71],[15,67248,1245],{"className":67249},[76],[15,67251],{"className":67252,"style":72},[71],[15,67254,37],{"className":67255},[66,67],[15,67257,67258,67261],{"style":23440},[15,67259],{"className":67260,"style":1368},[217],[15,67262],{"className":67263,"style":4438},[23447],[15,67265,67266,67269],{"style":23450},[15,67267],{"className":67268,"style":1368},[217],[15,67270,67272,67275,67278,67281,67284],{"className":67271},[66],[15,67273,8385],{"className":67274,"style":8555},[66,67],[15,67276],{"className":67277,"style":72},[71],[15,67279,1245],{"className":67280},[76],[15,67282],{"className":67283,"style":72},[71],[15,67285,621],{"className":67286,"style":684},[66,67],[15,67288,1114],{"className":67289},[1113],[15,67291,67293],{"className":67292},[206],[15,67294,67296],{"className":67295,"style":67076},[210],[15,67297],{},[15,67299],{"className":67300},[169,23412],[15,67302],{"className":67303,"style":173},[71],[15,67305,129],{"className":67306},[177],[15,67308],{"className":67309,"style":173},[71],[15,67311,67313,67316,67414,67417,67420],{"className":67312},[57],[15,67314],{"className":67315,"style":66982},[61],[15,67317,67319,67322,67411],{"className":67318},[66],[15,67320],{"className":67321},[162,23412],[15,67323,67325],{"className":67324},[6465],[15,67326,67328,67403],{"className":67327},[202,1090],[15,67329,67331,67400],{"className":67330},[206],[15,67332,67334,67369,67377],{"className":67333,"style":27430},[210],[15,67335,67336,67339],{"style":23428},[15,67337],{"className":67338,"style":1368},[217],[15,67340,67342,67345,67348,67351,67354,67357,67360,67363,67366],{"className":67341},[66],[15,67343,8385],{"className":67344,"style":8555},[66,67],[15,67346],{"className":67347,"style":72},[71],[15,67349,1245],{"className":67350},[76],[15,67352],{"className":67353,"style":72},[71],[15,67355,17082],{"className":67356,"style":17235},[66,67],[15,67358],{"className":67359,"style":72},[71],[15,67361,1245],{"className":67362},[76],[15,67364],{"className":67365,"style":72},[71],[15,67367,37],{"className":67368},[66,67],[15,67370,67371,67374],{"style":23440},[15,67372],{"className":67373,"style":1368},[217],[15,67375],{"className":67376,"style":4438},[23447],[15,67378,67379,67382],{"style":23450},[15,67380],{"className":67381,"style":1368},[217],[15,67383,67385,67388,67391,67394,67397],{"className":67384},[66],[15,67386,8385],{"className":67387,"style":8555},[66,67],[15,67389],{"className":67390,"style":72},[71],[15,67392,1245],{"className":67393},[76],[15,67395],{"className":67396,"style":72},[71],[15,67398,17082],{"className":67399,"style":17235},[66,67],[15,67401,1114],{"className":67402},[1113],[15,67404,67406],{"className":67405},[206],[15,67407,67409],{"className":67408,"style":67076},[210],[15,67410],{},[15,67412],{"className":67413},[169,23412],[15,67415],{"className":67416,"style":72},[71],[15,67418,3266],{"className":67419},[76],[15,67421],{"className":67422,"style":72},[71],[15,67424,67426,67430,67504,67507,67510],{"className":67425},[57],[15,67427],{"className":67428,"style":67429},[61],"height:1.988em;vertical-align:-0.8804em;",[15,67431,67433,67436,67501],{"className":67432},[66],[15,67434],{"className":67435},[162,23412],[15,67437,67439],{"className":67438},[6465],[15,67440,67442,67493],{"className":67441},[202,1090],[15,67443,67445,67490],{"className":67444},[206],[15,67446,67448,67471,67479],{"className":67447,"style":39128},[210],[15,67449,67450,67453],{"style":23428},[15,67451],{"className":67452,"style":1368},[217],[15,67454,67456,67459,67462,67465,67468],{"className":67455},[66],[15,67457,8385],{"className":67458,"style":8555},[66,67],[15,67460],{"className":67461,"style":72},[71],[15,67463,1245],{"className":67464},[76],[15,67466],{"className":67467,"style":72},[71],[15,67469,17082],{"className":67470,"style":17235},[66,67],[15,67472,67473,67476],{"style":23440},[15,67474],{"className":67475,"style":1368},[217],[15,67477],{"className":67478,"style":4438},[23447],[15,67480,67481,67484],{"style":23450},[15,67482],{"className":67483,"style":1368},[217],[15,67485,67487],{"className":67486},[66],[15,67488,8385],{"className":67489,"style":8555},[66,67],[15,67491,1114],{"className":67492},[1113],[15,67494,67496],{"className":67495},[206],[15,67497,67499],{"className":67498,"style":67076},[210],[15,67500],{},[15,67502],{"className":67503},[169,23412],[15,67505],{"className":67506,"style":72},[71],[15,67508,1245],{"className":67509},[76],[15,67511],{"className":67512,"style":72},[71],[15,67514,67516,67519,67605,67608,67611],{"className":67515},[57],[15,67517],{"className":67518,"style":67429},[61],[15,67520,67522,67525,67602],{"className":67521},[66],[15,67523],{"className":67524},[162,23412],[15,67526,67528],{"className"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44,67846,67848],{},[35,67841,8385],{},[39,67843,1245],{},[35,67845,17082],{},[39,67847,1245],{},[35,67849,37],{},[35,67851,3571],{"mathvariant":138},[45,67853,67854],{"encoding":47},"p(x_{\\text{new}}=1\\mid D)=\\mathbb E[\\theta\\mid D]=\\frac{\\alpha+k}{\\alpha+\\beta+n}.",[15,67856,67858,67925,67943,67964,67988,68009],{"className":67857,"ariaHidden":53},[52],[15,67859,67861,67864,67867,67870,67916,67919,67922],{"className":67860},[57],[15,67862],{"className":67863,"style":154},[61],[15,67865,11],{"className":67866},[66,67],[15,67868,120],{"className":67869},[162],[15,67871,67873,67876],{"className":67872},[66],[15,67874,123],{"className":67875},[66,67],[15,67877,67879],{"className":67878},[198],[15,67880,67882,67908],{"className":67881},[202,1090],[15,67883,67885,67905],{"className":67884},[206],[15,67886,67888],{"className":67887,"style":2198},[210],[15,67889,67890,67893],{"style":1584},[15,67891],{"className":67892,"style":218},[217],[15,67894,67896],{"className":67895},[222,223,224,225],[15,67897,67899],{"className":67898},[66,225],[15,67900,67902],{"className":67901},[66,1711,225],[15,67903,67801],{"className":67904},[66,225],[15,67906,1114],{"className":67907},[1113],[15,67909,67911],{"className":67910},[206],[15,67912,67914],{"className":67913,"style":1121},[210],[15,67915],{},[15,67917],{"className":67918,"style":173},[71],[15,67920,129],{"className":67921},[177],[15,67923],{"className":67924,"style":173},[71],[15,67926,67928,67931,67934,67937,67940],{"className":67927},[57],[15,67929],{"className":67930,"style":154},[61],[15,67932,1211],{"className":67933},[66],[15,67935],{"className":67936,"style":173},[71],[15,67938,34425],{"className":67939},[177],[15,67941],{"className":67942,"style":173},[71],[15,67944,67946,67949,67952,67955,67958,67961],{"className":67945},[57],[15,67947],{"className":67948,"style":154},[61],[15,67950,32453],{"className":67951,"style":23020},[66,67],[15,67953,126],{"className":67954},[169],[15,67956],{"className":67957,"style":173},[71],[15,67959,129],{"className":67960},[177],[15,67962],{"className":67963,"style":173},[71],[15,67965,67967,67970,67973,67976,67979,67982,67985],{"className":67966},[57],[15,67968],{"className":67969,"style":154},[61],[15,67971,62377],{"className":67972},[66,517],[15,67974,1194],{"className":67975},[162],[15,67977,61614],{"className":67978,"style":23020},[66,67],[15,67980],{"className":67981,"style":173},[71],[15,67983,34425],{"className":67984},[177],[15,67986],{"className":67987,"style":173},[71],[15,67989,67991,67994,67997,68000,68003,68006],{"className":67990},[57],[15,67992],{"className":67993,"style":154},[61],[15,67995,32453],{"className":67996,"style":23020},[66,67],[15,67998,1266],{"className":67999},[169],[15,68001],{"className":68002,"style":173},[71],[15,68004,129],{"className":68005},[177],[15,68007],{"className":68008,"style":173},[71],[15,68010,68012,68015,68113],{"className":68011},[57],[15,68013],{"className":68014,"style":66982},[61],[15,68016,68018,68021,68110],{"className":68017},[66],[15,68019],{"className":68020},[162,23412],[15,68022,68024],{"className":68023},[6465],[15,68025,68027,68102],{"className":68026},[202,1090],[15,68028,68030,68099],{"className":68029},[206],[15,68031,68033,68068,68076],{"className":68032,"style":27430},[210],[15,68034,68035,68038],{"style":23428},[15,68036],{"className":68037,"style":1368},[217],[15,68039,68041,68044,68047,68050,68053,68056,68059,68062,68065],{"className":68040},[66],[15,68042,8385],{"className":68043,"style":8555},[66,67],[15,68045],{"className":68046,"style":72},[71],[15,68048,1245],{"className":68049},[76],[15,68051],{"className":68052,"style":72},[71],[15,68054,17082],{"className":68055,"style":17235},[66,67],[15,68057],{"className":68058,"style":72},[71],[15,68060,1245],{"className":68061},[76],[15,68063],{"className":68064,"style":72},[71],[15,68066,37],{"className":68067},[66,67],[15,68069,68070,68073],{"style":23440},[15,68071],{"className":68072,"style":1368},[217],[15,68074],{"className":68075,"style":4438},[23447],[15,68077,68078,68081],{"style":23450},[15,68079],{"className":68080,"style":1368},[217],[15,68082,68084,68087,68090,68093,68096],{"className":68083},[66],[15,68085,8385],{"className":68086,"style":8555},[66,67],[15,68088],{"className":68089,"style":72},[71],[15,68091,1245],{"className":68092},[76],[15,68094],{"className":68095,"style":72},[71],[15,68097,621],{"className":68098,"style":684},[66,67],[15,68100,1114],{"className":68101},[1113],[15,68103,68105],{"className":68104},[206],[15,68106,68108],{"className":68107,"style":67076},[210],[15,68109],{},[15,68111],{"className":68112},[169,23412],[15,68114,3571],{"className":68115},[66],[11,68117,68118,68119,68147],{},"这里恰好等于后验均值，是因为伯努利概率是 ",[15,68120,68122,68135],{"className":68121,"translate":19},[18],[15,68123,68125],{"className":68124},[23],[25,68126,68127],{"xmlns":27},[29,68128,68129,68133],{},[32,68130,68131],{},[35,68132,61614],{},[45,68134,64801],{"encoding":47},[15,68136,68138],{"className":68137,"ariaHidden":53},[52],[15,68139,68141,68144],{"className":68140},[57],[15,68142],{"className":68143,"style":840},[61],[15,68145,61614],{"className":68146,"style":23020},[66,67]," 的线性函数。对一般模型，后验预测分布并不等于「把后验均值代进去」，这个区别在第 6 节会反复出现。",[1904,68149,68151],{"id":68150},"_24-先验是什么不是什么","2.4 先验是什么，不是什么",[11,68153,68154],{},"先验常被批评为「主观」。这个批评有一半是对的：先验确实是一个假设，而且不同先验会给出不同后验。但「主观」不等于「任意」。先验可以被数据稀释：在正则条件下，后验的渐近行为与先验无关（见 6.4）。先验也可以层级化：给超参数再加一层先验，让数据同时决定参数与超参数。先验还可以被检验：通过后验预测检验、敏感性分析，看结论是否依赖于某个特定选择。",[11,68156,68157],{},"更重要的是，非贝叶斯方法同样有假设。模型类、正则项、损失函数、优化算法，都是假设，只是它们不以「先验」的名字出现。贝叶斯的优点不是没有假设，而是把假设写在了明处，并且提供了一个用概率语言修改假设的机制。",[11,68159,68160,68161,68164],{},"第 7.8 节会给出一个更强的说法：在相当一般的条件下，所有",[2040,68162,68163],{},"可容许","（admissible）的决策规则都可以看成某个先验（或先验序列）下的贝叶斯规则。换句话说，贝叶斯规则不是一个「流派」，而是可容许决策规则的一般形式。这个结论把「为什么要用先验」从个人偏好提升到了决策论的结构性事实。",[95,68166,68168],{"id":68167},"_3-从后验到行动","3. 从后验到行动",[1904,68170,68172],{"id":68171},"_31-条件风险与逐点最优","3.1 条件风险与逐点最优",[11,68174,68175,68176,68242],{},"有了后验 ",[15,68177,68179,68203],{"className":68178,"translate":19},[18],[15,68180,68182],{"className":68181},[23],[25,68183,68184],{"xmlns":27},[29,68185,68186,68200],{},[32,68187,68188,68190,68192,68194,68196,68198],{},[35,68189,11],{},[39,68191,120],{"stretchy":119},[35,68193,1224],{},[39,68195,34425],{},[35,68197,123],{},[39,68199,126],{"stretchy":119},[45,68201,68202],{"encoding":47},"p(y\\mid x)",[15,68204,68206,68230],{"className":68205,"ariaHidden":53},[52],[15,68207,68209,68212,68215,68218,68221,68224,68227],{"className":68208},[57],[15,68210],{"className":68211,"style":154},[61],[15,68213,11],{"className":68214},[66,67],[15,68216,120],{"className":68217},[162],[15,68219,1224],{"className":68220,"style":1515},[66,67],[15,68222],{"className":68223,"style":173},[71],[15,68225,34425],{"className":68226},[177],[15,68228],{"className":68229,"style":173},[71],[15,68231,68233,68236,68239],{"className":68232},[57],[15,68234],{"className":68235,"style":154},[61],[15,68237,123],{"className":68238},[66,67],[15,68240,126],{"className":68241},[169],"，决策问题就变成了一个逐点的优化问题。",[3054,68244,68246,68309,68621],{"title":68245},"定义 3.1（贝叶斯决策规则）",[11,68247,68248,68249,68308],{},"若决策规则 ",[15,68250,68252,68270],{"className":68251,"translate":19},[18],[15,68253,68255],{"className":68254},[23],[25,68256,68257],{"xmlns":27},[29,68258,68259,68267],{},[32,68260,68261],{},[131,68262,68263,68265],{},[35,68264,3883],{},[39,68266,1162],{},[45,68268,68269],{"encoding":47},"\\delta^*",[15,68271,68273],{"className":68272,"ariaHidden":53},[52],[15,68274,68276,68279],{"className":68275},[57],[15,68277],{"className":68278,"style":840},[61],[15,68280,68282,68285],{"className":68281},[66],[15,68283,3883],{"className":68284,"style":3944},[66,67],[15,68286,68288],{"className":68287},[198],[15,68289,68291],{"className":68290},[202],[15,68292,68294],{"className":68293},[206],[15,68295,68297],{"className":68296,"style":16244},[210],[15,68298,68299,68302],{"style":213},[15,68300],{"className":68301,"style":218},[217],[15,68303,68305],{"className":68304},[222,223,224,225],[15,68306,1162],{"className":68307},[76,225]," 满足",[15,68310,68312],{"className":68311,"translate":19},[294],[15,68313,68315,68398],{"className":68314,"translate":19},[18],[15,68316,68318],{"className":68317},[23],[25,68319,68320],{"xmlns":27,"display":303},[29,68321,68322,68395],{},[32,68323,68324,68330,68332,68334,68336,68338,68340,68342,68358,68360,68362,68364,68366,68368,68370,68372,68374,68376,68378,68380,68382,68384,68386,68388,68391,68393],{},[131,68325,68326,68328],{},[35,68327,3883],{},[39,68329,1162],{},[39,68331,120],{"stretchy":119},[35,68333,123],{},[39,68335,126],{"stretchy":119},[39,68337,1187],{},[35,68339,1167],{},[39,68341,1170],{},[1172,68343,68344,68350],{},[32,68345,68346,68348],{},[35,68347,1178],{},[39,68349,1170],{},[32,68351,68352,68354,68356],{},[35,68353,4554],{},[39,68355,1187],{},[35,68357,62001],{"mathvariant":445},[1274,68359,24228],{},[35,68361,62377],{"mathvariant":454},[39,68363,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,68365,1216],{},[39,68367,120],{"stretchy":119},[35,68369,61881],{},[39,68371,335],{"separator":53},[35,68373,4554],{},[39,68375,126],{"stretchy":119},[39,68377,34425],{},[35,68379,446],{},[39,68381,129],{},[35,68383,123],{},[39,68385,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[71,68387],{"width":3902},[1274,68389,68390],{},"对几乎所有的 ",[35,68392,123],{},[39,68394,335],{"separator":53},[45,68396,68397],{"encoding":47},"\\delta^*(x)\\in\\arg\\min_{a\\in\\mathcal A}\\ \\mathbb E\\big[L(Y,a)\\mid X=x\\big]\n\\qquad \\text{对几乎所有的 }x,",[15,68399,68401,68454,68568,68586],{"className":68400,"ariaHidden":53},[52],[15,68402,68404,68407,68436,68439,68442,68445,68448,68451],{"className":68403},[57],[15,68405],{"className":68406,"style":154},[61],[15,68408,68410,68413],{"className":68409},[66],[15,68411,3883],{"className":68412,"style":3944},[66,67],[15,68414,68416],{"className":68415},[198],[15,68417,68419],{"className":68418},[202],[15,68420,68422],{"className":68421},[206],[15,68423,68425],{"className":68424,"style":1308},[210],[15,68426,68427,68430],{"style":397},[15,68428],{"className":68429,"style":218},[217],[15,68431,68433],{"className":68432},[222,223,224,225],[15,68434,1162],{"className":68435},[76,225],[15,68437,120],{"className":68438},[162],[15,68440,123],{"className":68441},[66,67],[15,68443,126],{"className":68444},[169],[15,68446],{"className":68447,"style":173},[71],[15,68449,1187],{"className":68450},[177],[15,68452],{"className":68453,"style":173},[71],[15,68455,68457,68461,68466,68469,68523,68526,68529,68532,68538,68541,68544,68547,68550,68553,68556,68559,68562,68565],{"className":68456},[57],[15,68458],{"className":68459,"style":68460},[61],"height:1.6217em;vertical-align:-0.7717em;",[15,68462,1340,68464],{"className":68463},[1339],[15,68465,1344],{"style":1343},[15,68467],{"className":68468,"style":697},[71],[15,68470,68472],{"className":68471},[1339,1351],[15,68473,68475,68514],{"className":68474},[202,1090],[15,68476,68478,68511],{"className":68477},[206],[15,68479,68481,68501],{"className":68480,"style":1361},[210],[15,68482,68483,68486],{"style":24349},[15,68484],{"className":68485,"style":1368},[217],[15,68487,68489],{"className":68488},[222,223,224,225],[15,68490,68492,68495,68498],{"className":68491},[66,225],[15,68493,4554],{"className":68494},[66,67,225],[15,68496,1187],{"className":68497},[177,225],[15,68499,62001],{"className":68500},[66,494,225],[15,68502,68503,68506],{"style":1386},[15,68504],{"className":68505,"style":1368},[217],[15,68507,68508],{},[15,68509,1178],{"className":68510},[1339],[15,68512,1114],{"className":68513},[1113],[15,68515,68517],{"className":68516},[206],[15,68518,68521],{"className":68519,"style":68520},[210],"height:0.7717em;",[15,68522],{},[15,68524,24228],{"className":68525},[71],[15,68527],{"className":68528,"style":697},[71],[15,68530,62377],{"className":68531},[66,517],[15,68533,68535],{"className":68534},[66],[15,68536,1194],{"className":68537},[1422,1508],[15,68539,1216],{"className":68540},[66,67],[15,68542,120],{"className":68543},[162],[15,68545,61881],{"className":68546,"style":72},[66,67],[15,68548,335],{"className":68549},[422],[15,68551],{"className":68552,"style":697},[71],[15,68554,4554],{"className":68555},[66,67],[15,68557,126],{"className":68558},[169],[15,68560],{"className":68561,"style":173},[71],[15,68563,34425],{"className":68564},[177],[15,68566],{"className":68567,"style":173},[71],[15,68569,68571,68574,68577,68580,68583],{"className":68570},[57],[15,68572],{"className":68573,"style":490},[61],[15,68575,446],{"className":68576,"style":16492},[66,67],[15,68578],{"className":68579,"style":173},[71],[15,68581,129],{"className":68582},[177],[15,68584],{"className":68585,"style":173},[71],[15,68587,68589,68592,68595,68601,68604,68615,68618],{"className":68588},[57],[15,68590],{"className":68591,"style":4398},[61],[15,68593,123],{"className":68594},[66,67],[15,68596,68598],{"className":68597},[66],[15,68599,1266],{"className":68600},[1422,1508],[15,68602],{"className":68603,"style":4059},[71],[15,68605,68607,68612],{"className":68606},[66,1711],[15,68608,68611],{"className":68609},[66,68610],"cjk_fallback","对几乎所有的",[15,68613,24228],{"className":68614},[66],[15,68616,123],{"className":68617},[66,67],[15,68619,335],{"className":68620},[422],[11,68622,68623,68624,68682,68683,68686],{},"则称 ",[15,68625,68627,68644],{"className":68626,"translate":19},[18],[15,68628,68630],{"className":68629},[23],[25,68631,68632],{"xmlns":27},[29,68633,68634,68642],{},[32,68635,68636],{},[131,68637,68638,68640],{},[35,68639,3883],{},[39,68641,1162],{},[45,68643,68269],{"encoding":47},[15,68645,68647],{"className":68646,"ariaHidden":53},[52],[15,68648,68650,68653],{"className":68649},[57],[15,68651],{"className":68652,"style":840},[61],[15,68654,68656,68659],{"className":68655},[66],[15,68657,3883],{"className":68658,"style":3944},[66,67],[15,68660,68662],{"className":68661},[198],[15,68663,68665],{"className":68664},[202],[15,68666,68668],{"className":68667},[206],[15,68669,68671],{"className":68670,"style":16244},[210],[15,68672,68673,68676],{"style":213},[15,68674],{"className":68675,"style":218},[217],[15,68677,68679],{"className":68678},[222,223,224,225],[15,68680,1162],{"className":68681},[76,225]," 为",[2040,68684,68685],{},"贝叶斯决策规则","（Bayes decision rule）。",[3054,68688,68690,68722],{"title":68689},"定理 3.1（逐点最优即全局最优）",[11,68691,68692,68693,2403],{},"对任意决策规则 ",[15,68694,68696,68710],{"className":68695,"translate":19},[18],[15,68697,68699],{"className":68698},[23],[25,68700,68701],{"xmlns":27},[29,68702,68703,68707],{},[32,68704,68705],{},[35,68706,3883],{},[45,68708,68709],{"encoding":47},"\\delta",[15,68711,68713],{"className":68712,"ariaHidden":53},[52],[15,68714,68716,68719],{"className":68715},[57],[15,68717],{"className":68718,"style":840},[61],[15,68720,3883],{"className":68721,"style":3944},[66,67],[15,68723,68725],{"className":68724,"translate":19},[294],[15,68726,68728,68808],{"className":68727,"translate":19},[18],[15,68729,68731],{"className":68730},[23],[25,68732,68733],{"xmlns":27,"display":303},[29,68734,68735,68805],{},[32,68736,68737,68739,68741,68743,68745,68747,68749,68751,68753,68755,68761,68763,68765,68771,68773,68789,68791,68793,68795,68797,68799,68801,68803],{},[35,68738,455],{},[39,68740,120],{"stretchy":119},[35,68742,3883],{},[39,68744,126],{"stretchy":119},[1274,68746,24228],{},[39,68748,1801],{},[1274,68750,24228],{},[35,68752,455],{},[39,68754,120],{"stretchy":119},[131,68756,68757,68759],{},[35,68758,3883],{},[39,68760,1162],{},[39,68762,126],{"stretchy":119},[39,68764,129],{},[1057,68766,68767,68769],{},[35,68768,62377],{"mathvariant":454},[35,68770,446],{},[39,68772,1194],{"fence":119,"stretchy":53,"minsize":4543,"maxsize":4543},[1172,68774,68775,68781],{},[32,68776,68777,68779],{},[35,68778,1178],{},[39,68780,1170],{},[32,68782,68783,68785,68787],{},[35,68784,4554],{},[39,68786,1187],{},[35,68788,62001],{"mathvariant":445},[35,68790,63103],{},[39,68792,120],{"stretchy":119},[35,68794,4554],{},[39,68796,335],{"separator":53},[35,68798,446],{},[39,68800,126],{"stretchy":119},[39,68802,1266],{"fence":119,"stretchy":53,"minsize":4543,"maxsize":4543},[35,68804,3571],{"mathvariant":138},[45,68806,68807],{"encoding":47},"R(\\delta)\\ \\ge\\ R(\\delta^*)=\\mathbb E_X\\Big[\\min_{a\\in\\mathcal A}r(a,X)\\Big].",[15,68809,68811,68844,68897],{"className":68810,"ariaHidden":53},[52],[15,68812,68814,68817,68820,68823,68826,68829,68832,68835,68838,68841],{"className":68813},[57],[15,68815],{"className":68816,"style":154},[61],[15,68818,455],{"className":68819,"style":62433},[66,67],[15,68821,120],{"className":68822},[162],[15,68824,3883],{"className":68825,"style":3944},[66,67],[15,68827,126],{"className":68828},[169],[15,68830,24228],{"className":68831},[71],[15,68833],{"className":68834,"style":173},[71],[15,68836,1801],{"className":68837},[177],[15,68839,24228],{"className":68840},[71],[15,68842],{"className":68843,"style":173},[71],[15,68845,68847,68850,68853,68856,68885,68888,68891,68894],{"className":68846},[57],[15,68848],{"className":68849,"style":154},[61],[15,68851,455],{"className":68852,"style":62433},[66,67],[15,68854,120],{"className":68855},[162],[15,68857,68859,68862],{"className":68858},[66],[15,68860,3883],{"className":68861,"style":3944},[66,67],[15,68863,68865],{"className":68864},[198],[15,68866,68868],{"className":68867},[202],[15,68869,68871],{"className":68870},[206],[15,68872,68874],{"className":68873,"style":1308},[210],[15,68875,68876,68879],{"style":397},[15,68877],{"className":68878,"style":218},[217],[15,68880,68882],{"className":68881},[222,223,224,225],[15,68883,1162],{"className":68884},[76,225],[15,68886,126],{"className":68887},[169],[15,68889],{"className":68890,"style":173},[71],[15,68892,129],{"className":68893},[177],[15,68895],{"className":68896,"style":173},[71],[15,68898,68900,68904,68944,68950,68953,69006,69009,69012,69015,69018,69021,69024,69027,69030,69036],{"className":68899},[57],[15,68901],{"className":68902,"style":68903},[61],"height:1.9217em;vertical-align:-0.7717em;",[15,68905,68907,68910],{"className":68906},[66],[15,68908,62377],{"className":68909},[66,517],[15,68911,68913],{"className":68912},[198],[15,68914,68916,68936],{"className":68915},[202,1090],[15,68917,68919,68933],{"className":68918},[206],[15,68920,68922],{"className":68921,"style":3636},[210],[15,68923,68924,68927],{"style":1584},[15,68925],{"className":68926,"style":218},[217],[15,68928,68930],{"className":68929},[222,223,224,225],[15,68931,446],{"className":68932,"style":16492},[66,67,225],[15,68934,1114],{"className":68935},[1113],[15,68937,68939],{"className":68938},[206],[15,68940,68942],{"className":68941,"style":1121},[210],[15,68943],{},[15,68945,68947],{"className":68946},[66],[15,68948,1194],{"className":68949},[1422,4675],[15,68951],{"className":68952,"style":697},[71],[15,68954,68956],{"className":68955},[1339,1351],[15,68957,68959,68998],{"className":68958},[202,1090],[15,68960,68962,68995],{"className":68961},[206],[15,68963,68965,68985],{"className":68964,"style":1361},[210],[15,68966,68967,68970],{"style":24349},[15,68968],{"className":68969,"style":1368},[217],[15,68971,68973],{"className":68972},[222,223,224,225],[15,68974,68976,68979,68982],{"className":68975},[66,225],[15,68977,4554],{"className":68978},[66,67,225],[15,68980,1187],{"className":68981},[177,225],[15,68983,62001],{"className":68984},[66,494,225],[15,68986,68987,68990],{"style":1386},[15,68988],{"className":68989,"style":1368},[217],[15,68991,68992],{},[15,68993,1178],{"className":68994},[1339],[15,68996,1114],{"className":68997},[1113],[15,68999,69001],{"className":69000},[206],[15,69002,69004],{"className":69003,"style":68520},[210],[15,69005],{},[15,69007],{"className":69008,"style":697},[71],[15,69010,63103],{"className":69011,"style":23020},[66,67],[15,69013,120],{"className":69014},[162],[15,69016,4554],{"className":69017},[66,67],[15,69019,335],{"className":69020},[422],[15,69022],{"className":69023,"style":697},[71],[15,69025,446],{"className":69026,"style":16492},[66,67],[15,69028,126],{"className":69029},[169],[15,69031,69033],{"className":69032},[66],[15,69034,1266],{"className":69035},[1422,4675],[15,69037,3571],{"className":69038},[66],[11,69040,69041,69044],{},[2040,69042,69043],{},"证明。"," 由塔性质，",[15,69046,69048],{"className":69047,"translate":19},[294],[15,69049,69051,69143],{"className":69050,"translate":19},[18],[15,69052,69054],{"className":69053},[23],[25,69055,69056],{"xmlns":27,"display":303},[29,69057,69058,69140],{},[32,69059,69060,69062,69064,69066,69068,69070,69076,69078,69080,69082,69084,69086,69088,69090,69092,69094,69096,69098,69100,69102,69104,69106,69108,69110,69116,69118,69120,69122,69124,69126,69128,69130,69132,69134,69136,69138],{},[35,69061,455],{},[39,69063,120],{"stretchy":119},[35,69065,3883],{},[39,69067,126],{"stretchy":119},[39,69069,129],{},[1057,69071,69072,69074],{},[35,69073,62377],{"mathvariant":454},[35,69075,446],{},[39,69077,1194],{"fence":119,"stretchy":53,"minsize":4543,"maxsize":4543},[35,69079,62377],{"mathvariant":454},[39,69081,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,69083,1216],{},[39,69085,120],{"stretchy":119},[35,69087,61881],{},[39,69089,335],{"separator":53},[35,69091,3883],{},[39,69093,120],{"stretchy":119},[35,69095,446],{},[39,69097,126],{"stretchy":119},[39,69099,126],{"stretchy":119},[39,69101,34425],{},[35,69103,446],{},[39,69105,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,69107,1266],{"fence":119,"stretchy":53,"minsize":4543,"maxsize":4543},[39,69109,129],{},[1057,69111,69112,69114],{},[35,69113,62377],{"mathvariant":454},[35,69115,446],{},[39,69117,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,69119,63103],{},[39,69121,120],{"stretchy":119},[35,69123,3883],{},[39,69125,120],{"stretchy":119},[35,69127,446],{},[39,69129,126],{"stretchy":119},[39,69131,335],{"separator":53},[35,69133,446],{},[39,69135,126],{"stretchy":119},[39,69137,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,69139,3571],{"mathvariant":138},[45,69141,69142],{"encoding":47},"R(\\delta)=\\mathbb E_X\\Big[\\mathbb E\\big[L(Y,\\delta(X))\\mid X\\big]\\Big]\n=\\mathbb E_X\\big[r(\\delta(X),X)\\big].",[15,69144,69146,69173,69271,69301],{"className":69145,"ariaHidden":53},[52],[15,69147,69149,69152,69155,69158,69161,69164,69167,69170],{"className":69148},[57],[15,69150],{"className":69151,"style":154},[61],[15,69153,455],{"className":69154,"style":62433},[66,67],[15,69156,120],{"className":69157},[162],[15,69159,3883],{"className":69160,"style":3944},[66,67],[15,69162,126],{"className":69163},[169],[15,69165],{"className":69166,"style":173},[71],[15,69168,129],{"className":69169},[177],[15,69171],{"className":69172,"style":173},[71],[15,69174,69176,69180,69220,69226,69229,69235,69238,69241,69244,69247,69250,69253,69256,69259,69262,69265,69268],{"className":69175},[57],[15,69177],{"className":69178,"style":69179},[61],"height:1.8em;vertical-align:-0.65em;",[15,69181,69183,69186],{"className":69182},[66],[15,69184,62377],{"className":69185},[66,517],[15,69187,69189],{"className":69188},[198],[15,69190,69192,69212],{"className":69191},[202,1090],[15,69193,69195,69209],{"className":69194},[206],[15,69196,69198],{"className":69197,"style":3636},[210],[15,69199,69200,69203],{"style":1584},[15,69201],{"className":69202,"style":218},[217],[15,69204,69206],{"className":69205},[222,223,224,225],[15,69207,446],{"className":69208,"style":16492},[66,67,225],[15,69210,1114],{"className":69211},[1113],[15,69213,69215],{"className":69214},[206],[15,69216,69218],{"className":69217,"style":1121},[210],[15,69219],{},[15,69221,69223],{"className":69222},[66],[15,69224,1194],{"className":69225},[1422,4675],[15,69227,62377],{"className":69228},[66,517],[15,69230,69232],{"className":69231},[66],[15,69233,1194],{"className":69234},[1422,1508],[15,69236,1216],{"className":69237},[66,67],[15,69239,120],{"className":69240},[162],[15,69242,61881],{"className":69243,"style":72},[66,67],[15,69245,335],{"className":69246},[422],[15,69248],{"className":69249,"style":697},[71],[15,69251,3883],{"className":69252,"style":3944},[66,67],[15,69254,120],{"className":69255},[162],[15,69257,446],{"className":69258,"style":16492},[66,67],[15,69260,18766],{"className":69261},[169],[15,69263],{"className":69264,"style":173},[71],[15,69266,34425],{"className":69267},[177],[15,69269],{"className":69270,"style":173},[71],[15,69272,69274,69277,69280,69286,69292,69295,69298],{"className":69273},[57],[15,69275],{"className":69276,"style":69179},[61],[15,69278,446],{"className":69279,"style":16492},[66,67],[15,69281,69283],{"className":69282},[66],[15,69284,1266],{"className":69285},[1422,1508],[15,69287,69289],{"className":69288},[66],[15,69290,1266],{"className":69291},[1422,4675],[15,69293],{"className":69294,"style":173},[71],[15,69296,129],{"className":69297},[177],[15,69299],{"className":69300,"style":173},[71],[15,69302,69304,69307,69347,69353,69356,69359,69362,69365,69368,69371,69374,69377,69380,69383,69389],{"className":69303},[57],[15,69305],{"className":69306,"style":4398},[61],[15,69308,69310,69313],{"className":69309},[66],[15,69311,62377],{"className":69312},[66,517],[15,69314,69316],{"className":69315},[198],[15,69317,69319,69339],{"className":69318},[202,1090],[15,69320,69322,69336],{"className":69321},[206],[15,69323,69325],{"className":69324,"style":3636},[210],[15,69326,69327,69330],{"style":1584},[15,69328],{"className":69329,"style":218},[217],[15,69331,69333],{"className":69332},[222,223,224,225],[15,69334,446],{"className":69335,"style":16492},[66,67,225],[15,69337,1114],{"className":69338},[1113],[15,69340,69342],{"className":69341},[206],[15,69343,69345],{"className":69344,"style":1121},[210],[15,69346],{},[15,69348,69350],{"className":69349},[66],[15,69351,1194],{"className":69352},[1422,1508],[15,69354,63103],{"className":69355,"style":23020},[66,67],[15,69357,120],{"className":69358},[162],[15,69360,3883],{"className":69361,"style":3944},[66,67],[15,69363,120],{"className":69364},[162],[15,69366,446],{"className":69367,"style":16492},[66,67],[15,69369,126],{"className":69370},[169],[15,69372,335],{"className":69373},[422],[15,69375],{"className":69376,"style":697},[71],[15,69378,446],{"className":69379,"style":16492},[66,67],[15,69381,126],{"className":69382},[169],[15,69384,69386],{"className":69385},[66],[15,69387,1266],{"className":69388},[1422,1508],[15,69390,3571],{"className":69391},[66],[11,69393,3194,69394,3247,69422,69598,69599],{},[15,69395,69397,69410],{"className":69396,"translate":19},[18],[15,69398,69400],{"className":69399},[23],[25,69401,69402],{"xmlns":27},[29,69403,69404,69408],{},[32,69405,69406],{},[35,69407,123],{},[45,69409,123],{"encoding":47},[15,69411,69413],{"className":69412,"ariaHidden":53},[52],[15,69414,69416,69419],{"className":69415},[57],[15,69417],{"className":69418,"style":86},[61],[15,69420,123],{"className":69421},[66,67],[15,69423,69425,69479],{"className":69424,"translate":19},[18],[15,69426,69428],{"className":69427},[23],[25,69429,69430],{"xmlns":27},[29,69431,69432,69476],{},[32,69433,69434,69436,69438,69440,69442,69444,69446,69448,69450,69452,69454,69464,69466,69468,69470,69472,69474],{},[35,69435,63103],{},[39,69437,120],{"stretchy":119},[35,69439,3883],{},[39,69441,120],{"stretchy":119},[35,69443,123],{},[39,69445,126],{"stretchy":119},[39,69447,335],{"separator":53},[35,69449,123],{},[39,69451,126],{"stretchy":119},[39,69453,1801],{},[1057,69455,69456,69462],{},[32,69457,69458,69460],{},[35,69459,1178],{},[39,69461,1170],{},[35,69463,4554],{},[35,69465,63103],{},[39,69467,120],{"stretchy":119},[35,69469,4554],{},[39,69471,335],{"separator":53},[35,69473,123],{},[39,69475,126],{"stretchy":119},[45,69477,69478],{"encoding":47},"r(\\delta(x),x)\\ge \\min_a r(a,x)",[15,69480,69482,69527],{"className":69481,"ariaHidden":53},[52],[15,69483,69485,69488,69491,69494,69497,69500,69503,69506,69509,69512,69515,69518,69521,69524],{"className":69484},[57],[15,69486],{"className":69487,"style":154},[61],[15,69489,63103],{"className":69490,"style":23020},[66,67],[15,69492,120],{"className":69493},[162],[15,69495,3883],{"className":69496,"style":3944},[66,67],[15,69498,120],{"className":69499},[162],[15,69501,123],{"className":69502},[66,67],[15,69504,126],{"className":69505},[169],[15,69507,335],{"className":69508},[422],[15,69510],{"className":69511,"style":697},[71],[15,69513,123],{"className":69514},[66,67],[15,69516,126],{"className":69517},[169],[15,69519],{"className":69520,"style":173},[71],[15,69522,1801],{"className":69523},[177],[15,69525],{"className":69526,"style":173},[71],[15,69528,69530,69533,69574,69577,69580,69583,69586,69589,69592,69595],{"className":69529},[57],[15,69531],{"className":69532,"style":154},[61],[15,69534,69536,69539],{"className":69535},[1339],[15,69537,1178],{"className":69538},[1339],[15,69540,69542],{"className":69541},[198],[15,69543,69545,69566],{"className":69544},[202,1090],[15,69546,69548,69563],{"className":69547},[206],[15,69549,69551],{"className":69550,"style":2198},[210],[15,69552,69554,69557],{"style":69553},"top:-2.55em;margin-right:0.05em;",[15,69555],{"className":69556,"style":218},[217],[15,69558,69560],{"className":69559},[222,223,224,225],[15,69561,4554],{"className":69562},[66,67,225],[15,69564,1114],{"className":69565},[1113],[15,69567,69569],{"className":69568},[206],[15,69570,69572],{"className":69571,"style":1121},[210],[15,69573],{},[15,69575],{"className":69576,"style":697},[71],[15,69578,63103],{"className":69579,"style":23020},[66,67],[15,69581,120],{"className":69582},[162],[15,69584,4554],{"className":69585},[66,67],[15,69587,335],{"className":69588},[422],[15,69590],{"className":69591,"style":697},[71],[15,69593,123],{"className":69594},[66,67],[15,69596,126],{"className":69597},[169],"，因此逐点取期望即得结论。",[15,69600,69602,69615],{"className":69601,"translate":19},[18],[15,69603,69605],{"className":69604},[23],[25,69606,69607],{"xmlns":27},[29,69608,69609,69613],{},[32,69610,69611],{},[35,69612,13509],{"mathvariant":138},[45,69614,13512],{"encoding":47},[15,69616,69618],{"className":69617,"ariaHidden":53},[52],[15,69619,69621,69624],{"className":69620},[57],[15,69622],{"className":69623,"style":13522},[61],[15,69625,13509],{"className":69626},[66,13526],[11,69628,69629,69630,69661,69662,69698,69699,69743],{},"证明只用了两件事：期望的塔性质，以及「逐点最小 ≤ 先对点取最小再平均」这样一个显而易见的事实。它不要求损失凸、不要求后验有密度、也不要求行动空间是有限的。这种一般性是贝叶斯决策理论最漂亮的地方之一。严格地说，若逐点下确界不可达，把 ",[15,69631,69633,69649],{"className":69632,"translate":19},[18],[15,69634,69636],{"className":69635},[23],[25,69637,69638],{"xmlns":27},[29,69639,69640,69646],{},[32,69641,69642,69644],{},[35,69643,1178],{},[39,69645,1170],{},[45,69647,69648],{"encoding":47},"\\min",[15,69650,69652],{"className":69651,"ariaHidden":53},[52],[15,69653,69655,69658],{"className":69654},[57],[15,69656],{"className":69657,"style":1361},[61],[15,69659,1178],{"className":69660},[1339]," 换成 ",[15,69663,69665,69682],{"className":69664,"translate":19},[18],[15,69666,69668],{"className":69667},[23],[25,69669,69670],{"xmlns":27},[29,69671,69672,69679],{},[32,69673,69674,69677],{},[35,69675,69676],{},"inf",[39,69678,1170],{},[45,69680,69681],{"encoding":47},"\\inf",[15,69683,69685],{"className":69684,"ariaHidden":53},[52],[15,69686,69688,69691],{"className":69687},[57],[15,69689],{"className":69690,"style":840},[61],[15,69692,69694,69695],{"className":69693},[1339],"in",[15,69696,116],{"style":69697},"margin-right:0.0778em;","、把 ",[15,69700,69702,69722],{"className":69701,"translate":19},[18],[15,69703,69705],{"className":69704},[23],[25,69706,69707],{"xmlns":27},[29,69708,69709,69719],{},[32,69710,69711,69713,69715,69717],{},[35,69712,1167],{},[39,69714,1170],{},[35,69716,1178],{},[39,69718,1170],{},[45,69720,69721],{"encoding":47},"\\arg\\min",[15,69723,69725],{"className":69724,"ariaHidden":53},[52],[15,69726,69728,69732,69737,69740],{"className":69727},[57],[15,69729],{"className":69730,"style":69731},[61],"height:0.8623em;vertical-align:-0.1944em;",[15,69733,1340,69735],{"className":69734},[1339],[15,69736,1344],{"style":1343},[15,69738],{"className":69739,"style":697},[71],[15,69741,1178],{"className":69742},[1339]," 换成「取一列逼近的行动」即可，证明不变。",[11,69745,69746],{},"把上式改写成差值，可以得到一个更有解释力的恒等式：",[15,69748,69750],{"className":69749,"translate":19},[294],[15,69751,69753,69852],{"className":69752,"translate":19},[18],[15,69754,69756],{"className":69755},[23],[25,69757,69758],{"xmlns":27,"display":303},[29,69759,69760,69849],{},[32,69761,69762,69764,69766,69768,69770,69772,69778,69780,69786,69788,69845,69847],{},[35,69763,455],{},[39,69765,120],{"stretchy":119},[35,69767,3883],{},[39,69769,126],{"stretchy":119},[39,69771,6600],{},[131,69773,69774,69776],{},[35,69775,455],{},[39,69777,1162],{},[39,69779,129],{},[1057,69781,69782,69784],{},[35,69783,62377],{"mathvariant":454},[35,69785,446],{},[39,69787,1194],{"fence":119,"stretchy":53,"minsize":4543,"maxsize":4543},[1172,69789,69790,69839],{},[1172,69791,69792,69836],{},[32,69793,69794,69796,69798,69800,69802,69804,69806,69808,69810,69812,69814,69824,69826,69828,69830,69832,69834],{},[35,69795,63103],{},[39,69797,120],{"stretchy":119},[35,69799,3883],{},[39,69801,120],{"stretchy":119},[35,69803,446],{},[39,69805,126],{"stretchy":119},[39,69807,335],{"separator":53},[35,69809,446],{},[39,69811,126],{"stretchy":119},[39,69813,6600],{},[1172,69815,69816,69822],{},[32,69817,69818,69820],{},[35,69819,1178],{},[39,69821,1170],{},[35,69823,4554],{},[35,69825,63103],{},[39,69827,120],{"stretchy":119},[35,69829,4554],{},[39,69831,335],{"separator":53},[35,69833,446],{},[39,69835,126],{"stretchy":119},[39,69837,69838],{"stretchy":53},"⏟",[32,69840,69841,69843],{},[39,69842,1801],{},[1209,69844,1804],{},[39,69846,1266],{"fence":119,"stretchy":53,"minsize":4543,"maxsize":4543},[35,69848,3571],{"mathvariant":138},[45,69850,69851],{"encoding":47},"R(\\delta)-R^*\n=\\mathbb E_X\\Big[\\underbrace{r(\\delta(X),X)-\\min_{a}r(a,X)}_{\\ge 0}\\Big].",[15,69853,69855,69882,69926],{"className":69854,"ariaHidden":53},[52],[15,69856,69858,69861,69864,69867,69870,69873,69876,69879],{"className":69857},[57],[15,69859],{"className":69860,"style":154},[61],[15,69862,455],{"className":69863,"style":62433},[66,67],[15,69865,120],{"className":69866},[162],[15,69868,3883],{"className":69869,"style":3944},[66,67],[15,69871,126],{"className":69872},[169],[15,69874],{"className":69875,"style":72},[71],[15,69877,6600],{"className":69878},[76],[15,69880],{"className":69881,"style":72},[71],[15,69883,69885,69888,69917,69920,69923],{"className":69884},[57],[15,69886],{"className":69887,"style":1308},[61],[15,69889,69891,69894],{"className":69890},[66],[15,69892,455],{"className":69893,"style":62433},[66,67],[15,69895,69897],{"className":69896},[198],[15,69898,69900],{"className":69899},[202],[15,69901,69903],{"className":69902},[206],[15,69904,69906],{"className":69905,"style":1308},[210],[15,69907,69908,69911],{"style":397},[15,69909],{"className":69910,"style":218},[217],[15,69912,69914],{"className":69913},[222,223,224,225],[15,69915,1162],{"className":69916},[76,225],[15,69918],{"className":69919,"style":173},[71],[15,69921,129],{"className":69922},[177],[15,69924],{"className":69925,"style":173},[71],[15,69927,69929,69933,69973,69979,69982,70216,70219,70225],{"className":69928},[57],[15,69930],{"className":69931,"style":69932},[61],"height:3.2443em;vertical-align:-2.0943em;",[15,69934,69936,69939],{"className":69935},[66],[15,69937,62377],{"className":69938},[66,517],[15,69940,69942],{"className":69941},[198],[15,69943,69945,69965],{"className":69944},[202,1090],[15,69946,69948,69962],{"className":69947},[206],[15,69949,69951],{"className":69950,"style":3636},[210],[15,69952,69953,69956],{"style":1584},[15,69954],{"className":69955,"style":218},[217],[15,69957,69959],{"className":69958},[222,223,224,225],[15,69960,446],{"className":69961,"style":16492},[66,67,225],[15,69963,1114],{"className":69964},[1113],[15,69966,69968],{"className":69967},[206],[15,69969,69971],{"className":69970,"style":1121},[210],[15,69972],{},[15,69974,69976],{"className":69975},[66],[15,69977,1194],{"className":69978},[1422,4675],[15,69980],{"className":69981,"style":697},[71],[15,69983,69985],{"className":69984},[1413,1172],[15,69986,69988,70207],{"className":69987},[202,1090],[15,69989,69991,70204],{"className":69990},[206],[15,69992,69995,70013],{"className":69993,"style":69994},[210],"height:0.75em;",[15,69996,69998,70001],{"style":69997},"top:-1.0009em;",[15,69999],{"className":70000,"style":1368},[217],[15,70002,70004],{"className":70003},[222,223,224,225],[15,70005,70007,70010],{"className":70006},[66,225],[15,70008,1801],{"className":70009},[177,225],[15,70011,1804],{"className":70012},[66,225],[15,70014,70015,70018],{"style":1386},[15,70016],{"className":70017,"style":1368},[217],[15,70019,70021],{"className":70020},[1413,1172],[15,70022,70024,70195],{"className":70023},[202,1090],[15,70025,70027,70192],{"className":70026},[206],[15,70028,70030,70074],{"className":70029,"style":69994},[210],[15,70031,70034,70037],{"className":70032,"style":70033},[48560],"top:-1.652em;",[15,70035],{"className":70036,"style":1368},[217],[15,70038,70042,70054,70064],{"className":70039,"style":70041},[70040],"stretchy","height:0.548em;min-width:1.6em;",[15,70043,70047],{"className":70044,"style":70046},[70045],"brace-left","height:0.548em;",[34823,70048,70051],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":48593},"0.548em","0 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20-1h17z",[15,70075,70076,70079],{"style":1386},[15,70077],{"className":70078,"style":1368},[217],[15,70080,70082,70085,70088,70091,70094,70097,70100,70103,70106,70109,70112,70115,70118,70121,70168,70171,70174,70177,70180,70183,70186,70189],{"className":70081},[66],[15,70083,63103],{"className":70084,"style":23020},[66,67],[15,70086,120],{"className":70087},[162],[15,70089,3883],{"className":70090,"style":3944},[66,67],[15,70092,120],{"className":70093},[162],[15,70095,446],{"className":70096,"style":16492},[66,67],[15,70098,126],{"className":70099},[169],[15,70101,335],{"className":70102},[422],[15,70104],{"className":70105,"style":697},[71],[15,70107,446],{"className":70108,"style":16492},[66,67],[15,70110,126],{"className":70111},[169],[15,70113],{"className":70114,"style":72},[71],[15,70116,6600],{"className":70117},[76],[15,70119],{"className":70120,"style":72},[71],[15,70122,70124],{"className":70123},[1339,1351],[15,70125,70127,70160],{"className":70126},[202,1090],[15,70128,70130,70157],{"className":70129},[206],[15,70131,70133,70147],{"className":70132,"style":1361},[210],[15,70134,70135,70138],{"style":48819},[15,70136],{"className":70137,"style":1368},[217],[15,70139,70141],{"className":70140},[222,223,224,225],[15,70142,70144],{"className":70143},[66,225],[15,70145,4554],{"className":70146},[66,67,225],[15,70148,70149,70152],{"style":1386},[15,70150],{"className":70151,"style":1368},[217],[15,70153,70154],{},[15,70155,1178],{"className":70156},[1339],[15,70158,1114],{"className":70159},[1113],[15,70161,70163],{"className":70162},[206],[15,70164,70166],{"className":70165,"style":48857},[210],[15,70167],{},[15,70169],{"className":70170,"style":697},[71],[15,70172,63103],{"className":70173,"style":23020},[66,67],[15,70175,120],{"className":70176},[162],[15,70178,4554],{"className":70179},[66,67],[15,70181,335],{"className":70182},[422],[15,70184],{"className":70185,"style":697},[71],[15,70187,446],{"className":70188,"style":16492},[66,67],[15,70190,126],{"className":70191},[169],[15,70193,1114],{"className":70194},[1113],[15,70196,70198],{"className":70197},[206],[15,70199,70202],{"className":70200,"style":70201},[210],"height:1.348em;",[15,70203],{},[15,70205,1114],{"className":70206},[1113],[15,70208,70210],{"className":70209},[206],[15,70211,70214],{"className":70212,"style":70213},[210],"height:2.0943em;",[15,70215],{},[15,70217],{"className":70218,"style":697},[71],[15,70220,70222],{"className":70221},[66],[15,70223,1266],{"className":70224},[1422,4675],[15,70226,3571],{"className":70227},[66],[11,70229,70230,70231,70259,70260,70263,70264,70292,70293,70321],{},"右边每一项都是非负的，称为在 ",[15,70232,70234,70247],{"className":70233,"translate":19},[18],[15,70235,70237],{"className":70236},[23],[25,70238,70239],{"xmlns":27},[29,70240,70241,70245],{},[32,70242,70243],{},[35,70244,123],{},[45,70246,123],{"encoding":47},[15,70248,70250],{"className":70249,"ariaHidden":53},[52],[15,70251,70253,70256],{"className":70252},[57],[15,70254],{"className":70255,"style":86},[61],[15,70257,123],{"className":70258},[66,67]," 处的",[2040,70261,70262],{},"逐点遗憾","（pointwise regret）。它说明：一个决策规则的次优性完全来自它在某些 ",[15,70265,70267,70280],{"className":70266,"translate":19},[18],[15,70268,70270],{"className":70269},[23],[25,70271,70272],{"xmlns":27},[29,70273,70274,70278],{},[32,70275,70276],{},[35,70277,123],{},[45,70279,123],{"encoding":47},[15,70281,70283],{"className":70282,"ariaHidden":53},[52],[15,70284,70286,70289],{"className":70285},[57],[15,70287],{"className":70288,"style":86},[61],[15,70290,123],{"className":70291},[66,67]," 上没有选到局部最优行动。不同 ",[15,70294,70296,70309],{"className":70295,"translate":19},[18],[15,70297,70299],{"className":70298},[23],[25,70300,70301],{"xmlns":27},[29,70302,70303,70307],{},[32,70304,70305],{},[35,70306,123],{},[45,70308,123],{"encoding":47},[15,70310,70312],{"className":70311,"ariaHidden":53},[52],[15,70313,70315,70318],{"className":70314},[57],[15,70316],{"className":70317,"style":86},[61],[15,70319,123],{"className":70320},[66,67]," 之间的错误不会相互抵消，也不会相互放大。",[1904,70323,70325],{"id":70324},"_32-0-1-损失最大后验","3.2 0-1 损失：最大后验",[11,70327,70328],{},"分类问题最常用的损失是 0-1 损失：",[15,70330,70332],{"className":70331,"translate":19},[294],[15,70333,70335,70376],{"className":70334,"translate":19},[18],[15,70336,70338],{"className":70337},[23],[25,70339,70340],{"xmlns":27,"display":303},[29,70341,70342,70373],{},[32,70343,70344,70346,70348,70350,70352,70354,70356,70358,70361,70363,70365,70367,70369,70371],{},[35,70345,1216],{},[39,70347,120],{"stretchy":119},[35,70349,1224],{},[39,70351,335],{"separator":53},[35,70353,4554],{},[39,70355,126],{"stretchy":119},[39,70357,129],{},[1209,70359,1211],{"mathvariant":70360},"bold",[39,70362,4305],{"stretchy":119},[35,70364,1224],{},[39,70366,13014],{"mathvariant":138},[35,70368,4554],{},[39,70370,4328],{"stretchy":119},[35,70372,3571],{"mathvariant":138},[45,70374,70375],{"encoding":47},"L(y,a)=\\mathbf 1\\{y\\ne a\\}.",[15,70377,70379,70415,70473],{"className":70378,"ariaHidden":53},[52],[15,70380,70382,70385,70388,70391,70394,70397,70400,70403,70406,70409,70412],{"className":70381},[57],[15,70383],{"className":70384,"style":154},[61],[15,70386,1216],{"className":70387},[66,67],[15,70389,120],{"className":70390},[162],[15,70392,1224],{"className":70393,"style":1515},[66,67],[15,70395,335],{"className":70396},[422],[15,70398],{"className":70399,"style":697},[71],[15,70401,4554],{"className":70402},[66,67],[15,70404,126],{"className":70405},[169],[15,70407],{"className":70408,"style":173},[71],[15,70410,129],{"className":70411},[177],[15,70413],{"className":70414,"style":173},[71],[15,70416,70418,70421,70425,70428,70431,70434,70470],{"className":70417},[57],[15,70419],{"className":70420,"style":154},[61],[15,70422,1211],{"className":70423},[66,70424],"mathbf",[15,70426,4305],{"className":70427},[162],[15,70429,1224],{"className":70430,"style":1515},[66,67],[15,70432],{"className":70433,"style":173},[71],[15,70435,70437,70464,70467],{"className":70436},[177],[15,70438,70440],{"className":70439},[177],[15,70441,70443],{"className":70442},[66,13081],[15,70444,70446],{"className":70445},[13085],[15,70447,70449,70452,70461],{"className":70448},[13089],[15,70450],{"className":70451,"style":3406},[61],[15,70453,70455],{"className":70454},[13096],[15,70456,70458],{"className":70457},[66],[15,70459,13103],{"className":70460},[177],[15,70462],{"className":70463},[13107],[15,70465],{"className":70466},[71,13111],[15,70468,129],{"className":70469},[177],[15,70471],{"className":70472,"style":173},[71],[15,70474,70476,70479,70482,70485],{"className":70475},[57],[15,70477],{"className":70478,"style":154},[61],[15,70480,4554],{"className":70481},[66,67],[15,70483,4328],{"className":70484},[169],[15,70486,3571],{"className":70487},[66],[11,70489,70490],{},"此时条件风险为",[15,70492,70494],{"className":70493,"translate":19},[294],[15,70495,70497,70569],{"className":70496,"translate":19},[18],[15,70498,70500],{"className":70499},[23],[25,70501,70502],{"xmlns":27,"display":303},[29,70503,70504,70566],{},[32,70505,70506,70508,70510,70512,70514,70516,70518,70520,70522,70524,70526,70528,70530,70532,70534,70536,70538,70540,70542,70544,70546,70548,70550,70552,70554,70556,70558,70560,70562,70564],{},[35,70507,63103],{},[39,70509,120],{"stretchy":119},[35,70511,4554],{},[39,70513,335],{"separator":53},[35,70515,123],{},[39,70517,126],{"stretchy":119},[39,70519,129],{},[35,70521,62377],{"mathvariant":454},[39,70523,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,70525,1211],{"mathvariant":70360},[39,70527,4305],{"stretchy":119},[35,70529,61881],{},[39,70531,13014],{"mathvariant":138},[35,70533,4554],{},[39,70535,4328],{"stretchy":119},[39,70537,34425],{},[35,70539,446],{},[39,70541,129],{},[35,70543,123],{},[39,70545,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,70547,129],{},[1209,70549,1211],{},[39,70551,6600],{},[35,70553,11],{},[39,70555,120],{"stretchy":119},[35,70557,4554],{},[39,70559,34425],{},[35,70561,123],{},[39,70563,126],{"stretchy":119},[39,70565,335],{"separator":53},[45,70567,70568],{"encoding":47},"r(a,x)=\\mathbb E\\big[\\mathbf 1\\{Y\\ne a\\}\\mid X=x\\big]=1-p(a\\mid x),",[15,70570,70572,70608,70674,70695,70713,70737,70755,70779],{"className":70571,"ariaHidden":53},[52],[15,70573,70575,70578,70581,70584,70587,70590,70593,70596,70599,70602,70605],{"className":70574},[57],[15,70576],{"className":70577,"style":154},[61],[15,70579,63103],{"className":70580,"style":23020},[66,67],[15,70582,120],{"className":70583},[162],[15,70585,4554],{"className":70586},[66,67],[15,70588,335],{"className":70589},[422],[15,70591],{"className":70592,"style":697},[71],[15,70594,123],{"className":70595},[66,67],[15,70597,126],{"className":70598},[169],[15,70600],{"className":70601,"style":173},[71],[15,70603,129],{"className":70604},[177],[15,70606],{"className":70607,"style":173},[71],[15,70609,70611,70614,70617,70623,70626,70629,70632,70635,70671],{"className":70610},[57],[15,70612],{"className":70613,"style":4398},[61],[15,70615,62377],{"className":70616},[66,517],[15,70618,70620],{"className":70619},[66],[15,70621,1194],{"className":70622},[1422,1508],[15,70624,1211],{"className":70625},[66,70424],[15,70627,4305],{"className":70628},[162],[15,70630,61881],{"className":70631,"style":72},[66,67],[15,70633],{"className":70634,"style":173},[71],[15,70636,70638,70665,70668],{"className":70637},[177],[15,70639,70641],{"className":70640},[177],[15,70642,70644],{"className":70643},[66,13081],[15,70645,70647],{"className":70646},[13085],[15,70648,70650,70653,70662],{"className":70649},[13089],[15,70651],{"className":70652,"style":3406},[61],[15,70654,70656],{"className":70655},[13096],[15,70657,70659],{"className":70658},[66],[15,70660,13103],{"className":70661},[177],[15,70663],{"className":70664},[13107],[15,70666],{"className":70667},[71,13111],[15,70669,129],{"className":70670},[177],[15,70672],{"className":70673,"style":173},[71],[15,70675,70677,70680,70683,70686,70689,70692],{"className":70676},[57],[15,70678],{"className":70679,"style":154},[61],[15,70681,4554],{"className":70682},[66,67],[15,70684,4328],{"className":70685},[169],[15,70687],{"className":70688,"style":173},[71],[15,70690,34425],{"className":70691},[177],[15,70693],{"className":70694,"style":173},[71],[15,70696,70698,70701,70704,70707,70710],{"className":70697},[57],[15,70699],{"className":70700,"style":490},[61],[15,70702,446],{"className":70703,"style":16492},[66,67],[15,70705],{"className":70706,"style":173},[71],[15,70708,129],{"className":70709},[177],[15,70711],{"className":70712,"style":173},[71],[15,70714,70716,70719,70722,70728,70731,70734],{"className":70715},[57],[15,70717],{"className":70718,"style":4398},[61],[15,70720,123],{"className":70721},[66,67],[15,70723,70725],{"className":70724},[66],[15,70726,1266],{"className":70727},[1422,1508],[15,70729],{"className":70730,"style":173},[71],[15,70732,129],{"className":70733},[177],[15,70735],{"className":70736,"style":173},[71],[15,70738,70740,70743,70746,70749,70752],{"className":70739},[57],[15,70741],{"className":70742,"style":30463},[61],[15,70744,1211],{"className":70745},[66],[15,70747],{"className":70748,"style":72},[71],[15,70750,6600],{"className":70751},[76],[15,70753],{"className":70754,"style":72},[71],[15,70756,70758,70761,70764,70767,70770,70773,70776],{"className":70757},[57],[15,70759],{"className":70760,"style":154},[61],[15,70762,11],{"className":70763},[66,67],[15,70765,120],{"className":70766},[162],[15,70768,4554],{"className":70769},[66,67],[15,70771],{"className":70772,"style":173},[71],[15,70774,34425],{"className":70775},[177],[15,70777],{"className":70778,"style":173},[71],[15,70780,70782,70785,70788,70791],{"className":70781},[57],[15,70783],{"className":70784,"style":154},[61],[15,70786,123],{"className":70787},[66,67],[15,70789,126],{"className":70790},[169],[15,70792,335],{"className":70793},[422],[11,70795,70796,70797,70853],{},"最小化 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等价于最大化后验概率：",[15,70855,70857],{"className":70856,"translate":19},[294],[15,70858,70860,70922],{"className":70859,"translate":19},[18],[15,70861,70863],{"className":70862},[23],[25,70864,70865],{"xmlns":27,"display":303},[29,70866,70867,70919],{},[32,70868,70869,70875,70877,70879,70881,70883,70885,70887,70903,70905,70907,70909,70911,70913,70915,70917],{},[131,70870,70871,70873],{},[35,70872,3883],{},[39,70874,1162],{},[39,70876,120],{"stretchy":119},[35,70878,123],{},[39,70880,126],{"stretchy":119},[39,70882,129],{},[35,70884,1167],{},[39,70886,1170],{},[1172,70888,70889,70895],{},[32,70890,70891,70893],{},[35,70892,23268],{},[39,70894,1170],{},[32,70896,70897,70899,70901],{},[35,70898,1224],{},[39,70900,1187],{},[35,70902,61881],{"mathvariant":445},[1274,70904,24228],{},[35,70906,11],{},[39,70908,120],{"stretchy":119},[35,70910,1224],{},[39,70912,34425],{},[35,70914,123],{},[39,70916,126],{"stretchy":119},[35,70918,3571],{"mathvariant":138},[45,70920,70921],{"encoding":47},"\\delta^*(x)=\\arg\\max_{y\\in\\mathcal Y}\\ p(y\\mid x).",[15,70923,70925,70978,71070],{"className":70924,"ariaHidden":53},[52],[15,70926,70928,70931,70960,70963,70966,70969,70972,70975],{"className":70927},[57],[15,70929],{"className":70930,"style":154},[61],[15,70932,70934,70937],{"className":70933},[66],[15,70935,3883],{"className":70936,"style":3944},[66,67],[15,70938,70940],{"className":70939},[198],[15,70941,70943],{"className":70942},[202],[15,70944,70946],{"className":70945},[206],[15,70947,70949],{"className":70948,"style":1308},[210],[15,70950,70951,70954],{"style":397},[15,70952],{"className":70953,"style":218},[217],[15,70955,70957],{"className":70956},[222,223,224,225],[15,70958,1162],{"className":70959},[76,225],[15,70961,120],{"className":70962},[162],[15,70964,123],{"className":70965},[66,67],[15,70967,126],{"className":70968},[169],[15,70970],{"className":70971,"style":173},[71],[15,70973,129],{"className":70974},[177],[15,70976],{"className":70977,"style":173},[71],[15,70979,70981,70985,70990,70993,71046,71049,71052,71055,71058,71061,71064,71067],{"className":70980},[57],[15,70982],{"className":70983,"style":70984},[61],"height:1.6304em;vertical-align:-0.8804em;",[15,70986,1340,70988],{"className":70987},[1339],[15,70989,1344],{"style":1343},[15,70991],{"className":70992,"style":697},[71],[15,70994,70996],{"className":70995},[1339,1351],[15,70997,70999,71038],{"className":70998},[202,1090],[15,71000,71002,71035],{"className":71001},[206],[15,71003,71005,71025],{"className":71004,"style":86},[210],[15,71006,71007,71010],{"style":24349},[15,71008],{"className":71009,"style":1368},[217],[15,71011,71013],{"className":71012},[222,223,224,225],[15,71014,71016,71019,71022],{"className":71015},[66,225],[15,71017,1224],{"className":71018,"style":1515},[66,67,225],[15,71020,1187],{"className":71021},[177,225],[15,71023,61881],{"className":71024,"style":28447},[66,494,225],[15,71026,71027,71030],{"style":1386},[15,71028],{"className":71029,"style":1368},[217],[15,71031,71032],{},[15,71033,23268],{"className":71034},[1339],[15,71036,1114],{"className":71037},[1113],[15,71039,71041],{"className":71040},[206],[15,71042,71044],{"className":71043,"style":67076},[210],[15,71045],{},[15,71047,24228],{"className":71048},[71],[15,71050],{"className":71051,"style":697},[71],[15,71053,11],{"className":71054},[66,67],[15,71056,120],{"className":71057},[162],[15,71059,1224],{"className":71060,"style":1515},[66,67],[15,71062],{"className":71063,"style":173},[71],[15,71065,34425],{"className":71066},[177],[15,71068],{"className":71069,"style":173},[71],[15,71071,71073,71076,71079,71082],{"className":71072},[57],[15,71074],{"className":71075,"style":154},[61],[15,71077,123],{"className":71078},[66,67],[15,71080,126],{"className":71081},[169],[15,71083,3571],{"className":71084},[66],[11,71086,5902,71087,71090,71091,71217],{},[2040,71088,71089],{},"最大后验决策","（maximum a posteriori decision）。二分类时，若 ",[15,71092,71094,71133],{"className":71093,"translate":19},[18],[15,71095,71097],{"className":71096},[23],[25,71098,71099],{"xmlns":27},[29,71100,71101,71130],{},[32,71102,71103,71106,71108,71110,71112,71114,71116,71118,71120,71122,71124,71126,71128],{},[35,71104,71105],{},"η",[39,71107,120],{"stretchy":119},[35,71109,123],{},[39,71111,126],{"stretchy":119},[39,71113,129],{},[35,71115,11],{},[39,71117,120],{"stretchy":119},[35,71119,61881],{},[39,71121,129],{},[1209,71123,1211],{},[39,71125,34425],{},[35,71127,123],{},[39,71129,126],{"stretchy":119},[45,71131,71132],{"encoding":47},"\\eta(x)=p(Y=1\\mid x)",[15,71134,71136,71163,71187,71205],{"className":71135,"ariaHidden":53},[52],[15,71137,71139,71142,71145,71148,71151,71154,71157,71160],{"className":71138},[57],[15,71140],{"className":71141,"style":154},[61],[15,71143,71105],{"className":71144,"style":1515},[66,67],[15,71146,120],{"className":71147},[162],[15,71149,123],{"className":71150},[66,67],[15,71152,126],{"className":71153},[169],[15,71155],{"className":71156,"style":173},[71],[15,71158,129],{"className":71159},[177],[15,71161],{"className":71162,"style":173},[71],[15,71164,71166,71169,71172,71175,71178,71181,71184],{"className":71165},[57],[15,71167],{"className":71168,"style":154},[61],[15,71170,11],{"className":71171},[66,67],[15,71173,120],{"className":71174},[162],[15,71176,61881],{"className":71177,"style":72},[66,67],[15,71179],{"className":71180,"style":173},[71],[15,71182,129],{"className":71183},[177],[15,71185],{"className":71186,"style":173},[71],[15,71188,71190,71193,71196,71199,71202],{"className":71189},[57],[15,71191],{"className":71192,"style":154},[61],[15,71194,1211],{"className":71195},[66],[15,71197],{"className":71198,"style":173},[71],[15,71200,34425],{"className":71201},[177],[15,71203],{"className":71204,"style":173},[71],[15,71206,71208,71211,71214],{"className":71207},[57],[15,71209],{"className":71210,"style":154},[61],[15,71212,123],{"className":71213},[66,67],[15,71215,126],{"className":71216},[169],"，规则简化为",[15,71219,71221],{"className":71220,"translate":19},[294],[15,71222,71224,71334],{"className":71223,"translate":19},[18],[15,71225,71227],{"className":71226},[23],[25,71228,71229],{"xmlns":27,"display":303},[29,71230,71231,71331],{},[32,71232,71233,71239,71241,71243,71245,71247],{},[131,71234,71235,71237],{},[35,71236,3883],{},[39,71238,1162],{},[39,71240,120],{"stretchy":119},[35,71242,123],{},[39,71244,126],{"stretchy":119},[39,71246,129],{},[32,71248,71249,71251],{},[39,71250,4305],{"fence":53},[1142,71252,71255,71293],{"rowspacing":71253,"columnalign":71254,"columnspacing":9786},"0.36em","left left",[1146,71256,71257,71267],{},[1149,71258,71259],{},[62983,71260,71261],{"scriptlevel":1804,"displaystyle":119},[32,71262,71263,71265],{},[1209,71264,1211],{},[39,71266,335],{"separator":53},[1149,71268,71269],{},[62983,71270,71271],{"scriptlevel":1804,"displaystyle":119},[32,71272,71273,71275,71277,71279,71281,71283,71291],{},[35,71274,71105],{},[39,71276,120],{"stretchy":119},[35,71278,123],{},[39,71280,126],{"stretchy":119},[39,71282,1801],{},[62983,71284,71285],{"scriptlevel":1804,"displaystyle":119},[6465,71286,71287,71289],{},[1209,71288,1211],{},[1209,71290,1263],{},[39,71292,335],{"separator":53},[1146,71294,71295,71305],{},[1149,71296,71297],{},[62983,71298,71299],{"scriptlevel":1804,"displaystyle":119},[32,71300,71301,71303],{},[1209,71302,1804],{},[39,71304,335],{"separator":53},[1149,71306,71307],{},[62983,71308,71309],{"scriptlevel":1804,"displaystyle":119},[32,71310,71311,71313,71315,71317,71319,71321,71329],{},[35,71312,71105],{},[39,71314,120],{"stretchy":119},[35,71316,123],{},[39,71318,126],{"stretchy":119},[39,71320,25767],{},[62983,71322,71323],{"scriptlevel":1804,"displaystyle":119},[6465,71324,71325,71327],{},[1209,71326,1211],{},[1209,71328,1263],{},[39,71330,335],{"separator":53},[45,71332,71333],{"encoding":47},"\\delta^*(x)=\n\\begin{cases}\n1, & \\eta(x)\\ge \\tfrac12,\\\\[2pt]\n0, & 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",[15,71703,71705,71741],{"className":71704,"translate":19},[18],[15,71706,71708],{"className":71707},[23],[25,71709,71710],{"xmlns":27},[29,71711,71712,71738],{},[32,71713,71714,71716,71718,71720,71722,71724,71726,71728,71730,71732,71734,71736],{},[39,71715,4305],{"stretchy":119},[35,71717,123],{},[39,71719,442],{},[35,71721,71105],{},[39,71723,120],{"stretchy":119},[35,71725,123],{},[39,71727,126],{"stretchy":119},[39,71729,129],{},[1209,71731,1211],{},[35,71733,6623],{"mathvariant":138},[1209,71735,1263],{},[39,71737,4328],{"stretchy":119},[45,71739,71740],{"encoding":47},"\\{x:\\eta(x)=1\u002F2\\}",[15,71742,71744,71765,71792],{"className":71743,"ariaHidden":53},[52],[15,71745,71747,71750,71753,71756,71759,71762],{"className":71746},[57],[15,71748],{"className":71749,"style":154},[61],[15,71751,4305],{"className":71752},[162],[15,71754,123],{"className":71755},[66,67],[15,71757],{"className":71758,"style":173},[71],[15,71760,442],{"className":71761},[177],[15,71763],{"className":71764,"style":173},[71],[15,71766,71768,71771,71774,71777,71780,71783,71786,71789],{"className":71767},[57],[15,71769],{"className":71770,"style":154},[61],[15,71772,71105],{"className":71773,"style":1515},[66,67],[15,71775,120],{"className":71776},[162],[15,71778,123],{"className":71779},[66,67],[15,71781,126],{"className":71782},[169],[15,71784],{"className":71785,"style":173},[71],[15,71787,129],{"className":71788},[177],[15,71790],{"className":71791,"style":173},[71],[15,71793,71795,71798,71802],{"className":71794},[57],[15,71796],{"className":71797,"style":154},[61],[15,71799,71801],{"className":71800},[66],"1\u002F2",[15,71803,4328],{"className":71804},[169],"。当 ",[15,71807,71809,71837],{"className":71808,"translate":19},[18],[15,71810,71812],{"className":71811},[23],[25,71813,71814],{"xmlns":27},[29,71815,71816,71834],{},[32,71817,71818,71820,71822,71824,71826,71828,71830,71832],{},[35,71819,71105],{},[39,71821,120],{"stretchy":119},[35,71823,123],{},[39,71825,126],{"stretchy":119},[39,71827,129],{},[1209,71829,1211],{},[35,71831,6623],{"mathvariant":138},[1209,71833,1263],{},[45,71835,71836],{"encoding":47},"\\eta(x)=1\u002F2",[15,71838,71840,71867],{"className":71839,"ariaHidden":53},[52],[15,71841,71843,71846,71849,71852,71855,71858,71861,71864],{"className":71842},[57],[15,71844],{"className":71845,"style":154},[61],[15,71847,71105],{"className":71848,"style":1515},[66,67],[15,71850,120],{"className":71851},[162],[15,71853,123],{"className":71854},[66,67],[15,71856,126],{"className":71857},[169],[15,71859],{"className":71860,"style":173},[71],[15,71862,129],{"className":71863},[177],[15,71865],{"className":71866,"style":173},[71],[15,71868,71870,71873],{"className":71869},[57],[15,71871],{"className":71872,"style":154},[61],[15,71874,71801],{"className":71875},[66]," 时两种行动同样好，实践中用固定的打破平局规则即可。",[1904,71878,71880],{"id":71879},"_33-平方损失后验均值","3.3 平方损失：后验均值",[11,71882,71883,71884,72025],{},"回归问题常用平方损失 ",[15,71885,71887,71927],{"className":71886,"translate":19},[18],[15,71888,71890],{"className":71889},[23],[25,71891,71892],{"xmlns":27},[29,71893,71894,71924],{},[32,71895,71896,71898,71900,71902,71904,71906,71908,71910,71912,71914,71916,71918],{},[35,71897,1216],{},[39,71899,120],{"stretchy":119},[35,71901,1224],{},[39,71903,335],{"separator":53},[35,71905,4554],{},[39,71907,126],{"stretchy":119},[39,71909,129],{},[39,71911,120],{"stretchy":119},[35,71913,1224],{},[39,71915,6600],{},[35,71917,4554],{},[131,71919,71920,71922],{},[39,71921,126],{"stretchy":119},[1209,71923,1263],{},[45,71925,71926],{"encoding":47},"L(y,a)=(y-a)^2",[15,71928,71930,71966,71987],{"className":71929,"ariaHidden":53},[52],[15,71931,71933,71936,71939,71942,71945,71948,71951,71954,71957,71960,71963],{"className":71932},[57],[15,71934],{"className":71935,"style":154},[61],[15,71937,1216],{"className":71938},[66,67],[15,71940,120],{"className":71941},[162],[15,71943,1224],{"className":71944,"style":1515},[66,67],[15,71946,335],{"className":71947},[422],[15,71949],{"className":71950,"style":697},[71],[15,71952,4554],{"className":71953},[66,67],[15,71955,126],{"className":71956},[169],[15,71958],{"className":71959,"style":173},[71],[15,71961,129],{"className":71962},[177],[15,71964],{"className":71965,"style":173},[71],[15,71967,71969,71972,71975,71978,71981,71984],{"className":71968},[57],[15,71970],{"className":71971,"style":154},[61],[15,71973,120],{"className":71974},[162],[15,71976,1224],{"className":71977,"style":1515},[66,67],[15,71979],{"className":71980,"style":72},[71],[15,71982,6600],{"className":71983},[76],[15,71985],{"className":71986,"style":72},[71],[15,71988,71990,71993,71996],{"className":71989},[57],[15,71991],{"className":71992,"style":9149},[61],[15,71994,4554],{"className":71995},[66,67],[15,71997,71999,72002],{"className":71998},[169],[15,72000,126],{"className":72001},[169],[15,72003,72005],{"className":72004},[198],[15,72006,72008],{"className":72007},[202],[15,72009,72011],{"className":72010},[206],[15,72012,72014],{"className":72013,"style":6738},[210],[15,72015,72016,72019],{"style":213},[15,72017],{"className":72018,"style":218},[217],[15,72020,72022],{"className":72021},[222,223,224,225],[15,72023,1263],{"className":72024},[66,225],"。此时",[15,72027,72029],{"className":72028,"translate":19},[294],[15,72030,72032,72174],{"className":72031,"translate":19},[18],[15,72033,72035],{"className":72034},[23],[25,72036,72037],{"xmlns":27,"display":303},[29,72038,72039,72171],{},[1142,72040,72041,72099],{"rowspacing":62976,"columnalign":62977,"columnspacing":634},[1146,72042,72043,72061],{},[1149,72044,72045],{},[62983,72046,72047],{"scriptlevel":1804,"displaystyle":53},[32,72048,72049,72051,72053,72055,72057,72059],{},[35,72050,63103],{},[39,72052,120],{"stretchy":119},[35,72054,4554],{},[39,72056,335],{"separator":53},[35,72058,123],{},[39,72060,126],{"stretchy":119},[1149,72062,72063],{},[62983,72064,72065],{"scriptlevel":1804,"displaystyle":53},[32,72066,72067,72069,72071,72073,72075,72077,72079,72081,72083,72089,72091,72093,72095,72097],{},[32,72068],{},[39,72070,129],{},[35,72072,62377],{"mathvariant":454},[39,72074,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,72076,120],{"stretchy":119},[35,72078,61881],{},[39,72080,6600],{},[35,72082,4554],{},[131,72084,72085,72087],{},[39,72086,126],{"stretchy":119},[1209,72088,1263],{},[39,72090,34425],{},[35,72092,446],{},[39,72094,129],{},[35,72096,123],{},[39,72098,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1146,72100,72101,72107],{},[1149,72102,72103],{},[62983,72104,72105],{"scriptlevel":1804,"displaystyle":53},[32,72106],{},[1149,72108,72109],{},[62983,72110,72111],{"scriptlevel":1804,"displaystyle":53},[32,72112,72113,72115,72117,72125,72127,72129,72131,72133,72135,72137,72139,72141,72143,72145,72147,72149,72151,72153,72155,72157,72159,72161,72163,72169],{},[32,72114],{},[39,72116,129],{},[32,72118,72119,72121,72123],{},[35,72120,31727],{"mathvariant":138},[35,72122,4554],{"mathvariant":138},[35,72124,63103],{"mathvariant":138},[39,72126,120],{"stretchy":119},[35,72128,61881],{},[39,72130,34425],{},[35,72132,446],{},[39,72134,129],{},[35,72136,123],{},[39,72138,126],{"stretchy":119},[39,72140,1245],{},[39,72142,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,72144,4554],{},[39,72146,6600],{},[35,72148,62377],{"mathvariant":454},[39,72150,1194],{"stretchy":119},[35,72152,61881],{},[39,72154,34425],{},[35,72156,446],{},[39,72158,129],{},[35,72160,123],{},[39,72162,1266],{"stretchy":119},[131,72164,72165,72167],{},[39,72166,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,72168,1263],{},[39,72170,335],{"separator":53},[45,72172,72173],{"encoding":47},"\\begin{aligned}\nr(a,x)\n&=\\mathbb E\\big[(Y-a)^2\\mid X=x\\big]\\\\\n&=\\mathrm{Var}(Y\\mid X=x)+\\big(a-\\mathbb E[Y\\mid X=x]\\big)^2,\n\\end{aligned}",[15,72175,72177],{"className":72176,"ariaHidden":53},[52],[15,72178,72180,72184],{"className":72179},[57],[15,72181],{"className":72182,"style":72183},[61],"height:2.9381em;vertical-align:-1.2191em;",[15,72185,72187],{"className":72186},[66],[15,72188,72190,72255],{"className":72189},[1142],[15,72191,72193],{"className":72192},[63146],[15,72194,72196,72246],{"className":72195},[202,1090],[15,72197,72199,72243],{"className":72198},[206],[15,72200,72203,72234],{"className":72201,"style":72202},[210],"height:1.7191em;",[15,72204,72206,72210],{"style":72205},"top:-3.909em;",[15,72207],{"className":72208,"style":72209},[217],"height:3.054em;",[15,72211,72213,72216,72219,72222,72225,72228,72231],{"className":72212},[66],[15,72214,63103],{"className":72215,"style":23020},[66,67],[15,72217,120],{"className":72218},[162],[15,72220,4554],{"className":72221},[66,67],[15,72223,335],{"className":72224},[422],[15,72226],{"className":72227,"style":697},[71],[15,72229,123],{"className":72230},[66,67],[15,72232,126],{"className":72233},[169],[15,72235,72237,72240],{"style":72236},"top:-2.1949em;",[15,72238],{"className":72239,"style":72209},[217],[15,72241],{"className":72242},[66],[15,72244,1114],{"className":72245},[1113],[15,72247,72249],{"className":72248},[206],[15,72250,72253],{"className":72251,"style":72252},[210],"height:1.2191em;",[15,72254],{},[15,72256,72258],{"className":72257},[63203],[15,72259,72261,72534],{"className":72260},[202,1090],[15,72262,72264,72531],{"className":72263},[206],[15,72265,72267,72373],{"className":72266,"style":72202},[210],[15,72268,72269,72272],{"style":72205},[15,72270],{"className":72271,"style":72209},[217],[15,72273,72275,72278,72281,72284,72287,72290,72296,72299,72302,72305,72308,72311,72314,72343,72346,72349,72352,72355,72358,72361,72364,72367],{"className":72274},[66],[15,72276],{"className":72277},[66],[15,72279],{"className":72280,"style":173},[71],[15,72282,129],{"className":72283},[177],[15,72285],{"className":72286,"style":173},[71],[15,72288,62377],{"className":72289},[66,517],[15,72291,72293],{"className":72292},[66],[15,72294,1194],{"className":72295},[1422,1508],[15,72297,120],{"className":72298},[162],[15,72300,61881],{"className":72301,"style":72},[66,67],[15,72303],{"className":72304,"style":72},[71],[15,72306,6600],{"className":72307},[76],[15,72309],{"className":72310,"style":72},[71],[15,72312,4554],{"className":72313},[66,67],[15,72315,72317,72320],{"className":72316},[169],[15,72318,126],{"className":72319},[169],[15,72321,72323],{"className":72322},[198],[15,72324,72326],{"className":72325},[202],[15,72327,72329],{"className":72328},[206],[15,72330,72332],{"className":72331,"style":1650},[210],[15,72333,72334,72337],{"style":397},[15,72335],{"className":72336,"style":218},[217],[15,72338,72340],{"className":72339},[222,223,224,225],[15,72341,1263],{"className":72342},[66,225],[15,72344],{"className":72345,"style":173},[71],[15,72347,34425],{"className":72348},[177],[15,72350],{"className":72351,"style":173},[71],[15,72353,446],{"className":72354,"style":16492},[66,67],[15,72356],{"className":72357,"style":173},[71],[15,72359,129],{"className":72360},[177],[15,72362],{"className":72363,"style":173},[71],[15,72365,123],{"className":72366},[66,67],[15,72368,72370],{"className":72369},[66],[15,72371,1266],{"className":72372},[1422,1508],[15,72374,72375,72378],{"style":72236},[15,72376],{"className":72377,"style":72209},[217],[15,72379,72381,72384,72387,72390,72393,72400,72403,72406,72409,72412,72415,72418,72421,72424,72427,72430,72433,72436,72439,72442,72448,72451,72454,72457,72460,72463,72466,72469,72472,72475,72478,72481,72484,72487,72490,72493,72496,72528],{"className":72380},[66],[15,72382],{"className":72383},[66],[15,72385],{"className":72386,"style":173},[71],[15,72388,129],{"className":72389},[177],[15,72391],{"className":72392,"style":173},[71],[15,72394,72396],{"className":72395},[66],[15,72397,72399],{"className":72398},[66,4427],"Var",[15,72401,120],{"className":72402},[162],[15,72404,61881],{"className":72405,"style":72},[66,67],[15,72407],{"className":72408,"style":173},[71],[15,72410,34425],{"className":72411},[177],[15,72413],{"className":72414,"style":173},[71],[15,72416,446],{"className":72417,"style":16492},[66,67],[15,72419],{"className":72420,"style":173},[71],[15,72422,129],{"className":72423},[177],[15,72425],{"className":72426,"style":173},[71],[15,72428,123],{"className":72429},[66,67],[15,72431,126],{"className":72432},[169],[15,72434],{"className":72435,"style":72},[71],[15,72437,1245],{"className":72438},[76],[15,72440],{"className":72441,"style":72},[71],[15,72443,72445],{"className":72444},[66],[15,72446,120],{"className":72447},[1422,1508],[15,72449,4554],{"className":72450},[66,67],[15,72452],{"className":72453,"style":72},[71],[15,72455,6600],{"className":72456},[76],[15,72458],{"className":72459,"style":72},[71],[15,72461,62377],{"className":72462},[66,517],[15,72464,1194],{"className":72465},[162],[15,72467,61881],{"className":72468,"style":72},[66,67],[15,72470],{"className":72471,"style":173},[71],[15,72473,34425],{"className":72474},[177],[15,72476],{"className":72477,"style":173},[71],[15,72479,446],{"className":72480,"style":16492},[66,67],[15,72482],{"className":72483,"style":173},[71],[15,72485,129],{"className":72486},[177],[15,72488],{"className":72489,"style":173},[71],[15,72491,123],{"className":72492},[66,67],[15,72494,1266],{"className":72495},[169],[15,72497,72499,72505],{"className":72498},[66],[15,72500,72502],{"className":72501},[66],[15,72503,126],{"className":72504},[1422,1508],[15,72506,72508],{"className":72507},[198],[15,72509,72511],{"className":72510},[202],[15,72512,72514],{"className":72513},[206],[15,72515,72517],{"className":72516,"style":21266},[210],[15,72518,72519,72522],{"style":21269},[15,72520],{"className":72521,"style":218},[217],[15,72523,72525],{"className":72524},[222,223,224,225],[15,72526,1263],{"className":72527},[66,225],[15,72529,335],{"className":72530},[422],[15,72532,1114],{"className":72533},[1113],[15,72535,72537],{"className":72536},[206],[15,72538,72540],{"className":72539,"style":72252},[210],[15,72541],{},[11,72543,72544,72545,72655,72656,72659],{},"第二项在 ",[15,72546,72548,72580],{"className":72547,"translate":19},[18],[15,72549,72551],{"className":72550},[23],[25,72552,72553],{"xmlns":27},[29,72554,72555,72577],{},[32,72556,72557,72559,72561,72563,72565,72567,72569,72571,72573,72575],{},[35,72558,4554],{},[39,72560,129],{},[35,72562,62377],{"mathvariant":454},[39,72564,1194],{"stretchy":119},[35,72566,61881],{},[39,72568,34425],{},[35,72570,446],{},[39,72572,129],{},[35,72574,123],{},[39,72576,1266],{"stretchy":119},[45,72578,72579],{"encoding":47},"a=\\mathbb E[Y\\mid X=x]",[15,72581,72583,72601,72625,72643],{"className":72582,"ariaHidden":53},[52],[15,72584,72586,72589,72592,72595,72598],{"className":72585},[57],[15,72587],{"className":72588,"style":86},[61],[15,72590,4554],{"className":72591},[66,67],[15,72593],{"className":72594,"style":173},[71],[15,72596,129],{"className":72597},[177],[15,72599],{"className":72600,"style":173},[71],[15,72602,72604,72607,72610,72613,72616,72619,72622],{"className":72603},[57],[15,72605],{"className":72606,"style":154},[61],[15,72608,62377],{"className":72609},[66,517],[15,72611,1194],{"className":72612},[162],[15,72614,61881],{"className":72615,"style":72},[66,67],[15,72617],{"className":72618,"style":173},[71],[15,72620,34425],{"className":72621},[177],[15,72623],{"className":72624,"style":173},[71],[15,72626,72628,72631,72634,72637,72640],{"className":72627},[57],[15,72629],{"className":72630,"style":490},[61],[15,72632,446],{"className":72633,"style":16492},[66,67],[15,72635],{"className":72636,"style":173},[71],[15,72638,129],{"className":72639},[177],[15,72641],{"className":72642,"style":173},[71],[15,72644,72646,72649,72652],{"className":72645},[57],[15,72647],{"className":72648,"style":154},[61],[15,72650,123],{"className":72651},[66,67],[15,72653,1266],{"className":72654},[169]," 时取到最小值。因此平方损失下的贝叶斯行动是",[2040,72657,72658],{},"后验均值","，也就是回归函数",[15,72661,72663],{"className":72662,"translate":19},[294],[15,72664,72666,72710],{"className":72665,"translate":19},[18],[15,72667,72669],{"className":72668},[23],[25,72670,72671],{"xmlns":27,"display":303},[29,72672,72673,72707],{},[32,72674,72675,72681,72683,72685,72687,72689,72691,72693,72695,72697,72699,72701,72703,72705],{},[131,72676,72677,72679],{},[35,72678,116],{},[39,72680,1162],{},[39,72682,120],{"stretchy":119},[35,72684,123],{},[39,72686,126],{"stretchy":119},[39,72688,129],{},[35,72690,62377],{"mathvariant":454},[39,72692,1194],{"stretchy":119},[35,72694,61881],{},[39,72696,34425],{},[35,72698,446],{},[39,72700,129],{},[35,72702,123],{},[39,72704,1266],{"stretchy":119},[35,72706,3571],{"mathvariant":138},[45,72708,72709],{"encoding":47},"f^*(x)=\\mathbb E[Y\\mid X=x].",[15,72711,72713,72766,72790,72808],{"className":72712,"ariaHidden":53},[52],[15,72714,72716,72719,72748,72751,72754,72757,72760,72763],{"className":72715},[57],[15,72717],{"className":72718,"style":154},[61],[15,72720,72722,72725],{"className":72721},[66],[15,72723,116],{"className":72724,"style":158},[66,67],[15,72726,72728],{"className":72727},[198],[15,72729,72731],{"className":72730},[202],[15,72732,72734],{"className":72733},[206],[15,72735,72737],{"className":72736,"style":1308},[210],[15,72738,72739,72742],{"style":397},[15,72740],{"className":72741,"style":218},[217],[15,72743,72745],{"className":72744},[222,223,224,225],[15,72746,1162],{"className":72747},[76,225],[15,72749,120],{"className":72750},[162],[15,72752,123],{"className":72753},[66,67],[15,72755,126],{"className":72756},[169],[15,72758],{"className":72759,"style":173},[71],[15,72761,129],{"className":72762},[177],[15,72764],{"className":72765,"style":173},[71],[15,72767,72769,72772,72775,72778,72781,72784,72787],{"className":72768},[57],[15,72770],{"className":72771,"style":154},[61],[15,72773,62377],{"className":72774},[66,517],[15,72776,1194],{"className":72777},[162],[15,72779,61881],{"className":72780,"style":72},[66,67],[15,72782],{"className":72783,"style":173},[71],[15,72785,34425],{"className":72786},[177],[15,72788],{"className":72789,"style":173},[71],[15,72791,72793,72796,72799,72802,72805],{"className":72792},[57],[15,72794],{"className":72795,"style":490},[61],[15,72797,446],{"className":72798,"style":16492},[66,67],[15,72800],{"className":72801,"style":173},[71],[15,72803,129],{"className":72804},[177],[15,72806],{"className":72807,"style":173},[71],[15,72809,72811,72814,72817,72820],{"className":72810},[57],[15,72812],{"className":72813,"style":154},[61],[15,72815,123],{"className":72816},[66,67],[15,72818,1266],{"className":72819},[169],[15,72821,3571],{"className":72822},[66],[11,72824,72825,72826,72854,72855,72914],{},"这个结论还可以从另一个角度理解：后验均值是 ",[15,72827,72829,72842],{"className":72828,"translate":19},[18],[15,72830,72832],{"className":72831},[23],[25,72833,72834],{"xmlns":27},[29,72835,72836,72840],{},[32,72837,72838],{},[35,72839,61881],{},[45,72841,61881],{"encoding":47},[15,72843,72845],{"className":72844,"ariaHidden":53},[52],[15,72846,72848,72851],{"className":72847},[57],[15,72849],{"className":72850,"style":490},[61],[15,72852,61881],{"className":72853,"style":72},[66,67]," 在 ",[15,72856,72858,72876],{"className":72857,"translate":19},[18],[15,72859,72861],{"className":72860},[23],[25,72862,72863],{"xmlns":27},[29,72864,72865,72873],{},[32,72866,72867],{},[131,72868,72869,72871],{},[35,72870,1216],{},[1209,72872,1263],{},[45,72874,72875],{"encoding":47},"L^2",[15,72877,72879],{"className":72878,"ariaHidden":53},[52],[15,72880,72882,72885],{"className":72881},[57],[15,72883],{"className":72884,"style":6738},[61],[15,72886,72888,72891],{"className":72887},[66],[15,72889,1216],{"className":72890},[66,67],[15,72892,72894],{"className":72893},[198],[15,72895,72897],{"className":72896},[202],[15,72898,72900],{"className":72899},[206],[15,72901,72903],{"className":72902,"style":6738},[210],[15,72904,72905,72908],{"style":213},[15,72906],{"className":72907,"style":218},[217],[15,72909,72911],{"className":72910},[222,223,224,225],[15,72912,1263],{"className":72913},[66,225]," 意义下最好的预测，任何其他预测都会多出一项方差。",[1904,72916,72918],{"id":72917},"_34-绝对损失后验中位数","3.4 绝对损失：后验中位数",[11,72920,72921,72922,73033,73034,73204,73205,73233,73234,73262],{},"若损失为 ",[15,72923,72925,72961],{"className":72924,"translate":19},[18],[15,72926,72928],{"className":72927},[23],[25,72929,72930],{"xmlns":27},[29,72931,72932,72958],{},[32,72933,72934,72936,72938,72940,72942,72944,72946,72948,72950,72952,72954,72956],{},[35,72935,1216],{},[39,72937,120],{"stretchy":119},[35,72939,1224],{},[39,72941,335],{"separator":53},[35,72943,4554],{},[39,72945,126],{"stretchy":119},[39,72947,129],{},[35,72949,34425],{"mathvariant":138},[35,72951,1224],{},[39,72953,6600],{},[35,72955,4554],{},[35,72957,34425],{"mathvariant":138},[45,72959,72960],{"encoding":47},"L(y,a)=|y-a|",[15,72962,72964,73000,73021],{"className":72963,"ariaHidden":53},[52],[15,72965,72967,72970,72973,72976,72979,72982,72985,72988,72991,72994,72997],{"className":72966},[57],[15,72968],{"className":72969,"style":154},[61],[15,72971,1216],{"className":72972},[66,67],[15,72974,120],{"className":72975},[162],[15,72977,1224],{"className":72978,"style":1515},[66,67],[15,72980,335],{"className":72981},[422],[15,72983],{"className":72984,"style":697},[71],[15,72986,4554],{"className":72987},[66,67],[15,72989,126],{"className":72990},[169],[15,72992],{"className":72993,"style":173},[71],[15,72995,129],{"className":72996},[177],[15,72998],{"className":72999,"style":173},[71],[15,73001,73003,73006,73009,73012,73015,73018],{"className":73002},[57],[15,73004],{"className":73005,"style":154},[61],[15,73007,34425],{"className":73008},[66],[15,73010,1224],{"className":73011,"style":1515},[66,67],[15,73013],{"className":73014,"style":72},[71],[15,73016,6600],{"className":73017},[76],[15,73019],{"className":73020,"style":72},[71],[15,73022,73024,73027,73030],{"className":73023},[57],[15,73025],{"className":73026,"style":154},[61],[15,73028,4554],{"className":73029},[66,67],[15,73031,34425],{"className":73032},[66],"，条件风险为 ",[15,73035,73037,73087],{"className":73036,"translate":19},[18],[15,73038,73040],{"className":73039},[23],[25,73041,73042],{"xmlns":27},[29,73043,73044,73084],{},[32,73045,73046,73048,73050,73052,73054,73056,73058,73060,73062,73064,73066,73068,73070,73072,73074,73076,73078,73080,73082],{},[35,73047,63103],{},[39,73049,120],{"stretchy":119},[35,73051,4554],{},[39,73053,335],{"separator":53},[35,73055,123],{},[39,73057,126],{"stretchy":119},[39,73059,129],{},[35,73061,62377],{"mathvariant":454},[39,73063,1194],{"stretchy":119},[35,73065,34425],{"mathvariant":138},[35,73067,61881],{},[39,73069,6600],{},[35,73071,4554],{},[35,73073,34425],{"mathvariant":138},[39,73075,34425],{},[35,73077,446],{},[39,73079,129],{},[35,73081,123],{},[39,73083,1266],{"stretchy":119},[45,73085,73086],{"encoding":47},"r(a,x)=\\mathbb E[|Y-a|\\mid X=x]",[15,73088,73090,73126,73153,73174,73192],{"className":73089,"ariaHidden":53},[52],[15,73091,73093,73096,73099,73102,73105,73108,73111,73114,73117,73120,73123],{"className":73092},[57],[15,73094],{"className":73095,"style":154},[61],[15,73097,63103],{"className":73098,"style":23020},[66,67],[15,73100,120],{"className":73101},[162],[15,73103,4554],{"className":73104},[66,67],[15,73106,335],{"className":73107},[422],[15,73109],{"className":73110,"style":697},[71],[15,73112,123],{"className":73113},[66,67],[15,73115,126],{"className":73116},[169],[15,73118],{"className":73119,"style":173},[71],[15,73121,129],{"className":73122},[177],[15,73124],{"className":73125,"style":173},[71],[15,73127,73129,73132,73135,73138,73141,73144,73147,73150],{"className":73128},[57],[15,73130],{"className":73131,"style":154},[61],[15,73133,62377],{"className":73134},[66,517],[15,73136,1194],{"className":73137},[162],[15,73139,34425],{"className":73140},[66],[15,73142,61881],{"className":73143,"style":72},[66,67],[15,73145],{"className":73146,"style":72},[71],[15,73148,6600],{"className":73149},[76],[15,73151],{"className":73152,"style":72},[71],[15,73154,73156,73159,73162,73165,73168,73171],{"className":73155},[57],[15,73157],{"className":73158,"style":154},[61],[15,73160,4554],{"className":73161},[66,67],[15,73163,34425],{"className":73164},[66],[15,73166],{"className":73167,"style":173},[71],[15,73169,34425],{"className":73170},[177],[15,73172],{"className":73173,"style":173},[71],[15,73175,73177,73180,73183,73186,73189],{"className":73176},[57],[15,73178],{"className":73179,"style":490},[61],[15,73181,446],{"className":73182,"style":16492},[66,67],[15,73184],{"className":73185,"style":173},[71],[15,73187,129],{"className":73188},[177],[15,73190],{"className":73191,"style":173},[71],[15,73193,73195,73198,73201],{"className":73194},[57],[15,73196],{"className":73197,"style":154},[61],[15,73199,123],{"className":73200},[66,67],[15,73202,1266],{"className":73203},[169],"。对 ",[15,73206,73208,73221],{"className":73207,"translate":19},[18],[15,73209,73211],{"className":73210},[23],[25,73212,73213],{"xmlns":27},[29,73214,73215,73219],{},[32,73216,73217],{},[35,73218,4554],{},[45,73220,4554],{"encoding":47},[15,73222,73224],{"className":73223,"ariaHidden":53},[52],[15,73225,73227,73230],{"className":73226},[57],[15,73228],{"className":73229,"style":86},[61],[15,73231,4554],{"className":73232},[66,67]," 求次梯度，在 ",[15,73235,73237,73250],{"className":73236,"translate":19},[18],[15,73238,73240],{"className":73239},[23],[25,73241,73242],{"xmlns":27},[29,73243,73244,73248],{},[32,73245,73246],{},[35,73247,61881],{},[45,73249,61881],{"encoding":47},[15,73251,73253],{"className":73252,"ariaHidden":53},[52],[15,73254,73256,73259],{"className":73255},[57],[15,73257],{"className":73258,"style":490},[61],[15,73260,61881],{"className":73261,"style":72},[66,67]," 的分布连续时它就是导数：",[15,73264,73266],{"className":73265,"translate":19},[294],[15,73267,73269,73338],{"className":73268,"translate":19},[18],[15,73270,73272],{"className":73271},[23],[25,73273,73274],{"xmlns":27,"display":303},[29,73275,73276,73335],{},[32,73277,73278,73285,73287,73289,73291,73293,73295,73297,73299,73301,73303,73305,73307,73309,73311,73313,73315,73317,73319,73321,73323,73325,73327,73329,73331,73333],{},[1057,73279,73280,73283],{},[35,73281,73282],{"mathvariant":138},"∂",[35,73284,4554],{},[35,73286,63103],{},[39,73288,120],{"stretchy":119},[35,73290,4554],{},[39,73292,335],{"separator":53},[35,73294,123],{},[39,73296,126],{"stretchy":119},[39,73298,129],{},[35,73300,28394],{},[39,73302,120],{"stretchy":119},[35,73304,61881],{},[39,73306,25767],{},[35,73308,4554],{},[39,73310,34425],{},[35,73312,123],{},[39,73314,126],{"stretchy":119},[39,73316,6600],{},[35,73318,28394],{},[39,73320,120],{"stretchy":119},[35,73322,61881],{},[39,73324,9278],{},[35,73326,4554],{},[39,73328,34425],{},[35,73330,123],{},[39,73332,126],{"stretchy":119},[39,73334,335],{"separator":53},[45,73336,73337],{"encoding":47},"\\partial_a r(a,x)=P(Y\u003Ca\\mid x)-P(Y>a\\mid x),",[15,73339,73341,73419,73443,73461,73482,73506,73524],{"className":73340,"ariaHidden":53},[52],[15,73342,73344,73347,73389,73392,73395,73398,73401,73404,73407,73410,73413,73416],{"className":73343},[57],[15,73345],{"className":73346,"style":154},[61],[15,73348,73350,73354],{"className":73349},[66],[15,73351,73282],{"className":73352,"style":73353},[66],"margin-right:0.0556em;",[15,73355,73357],{"className":73356},[198],[15,73358,73360,73381],{"className":73359},[202,1090],[15,73361,73363,73378],{"className":73362},[206],[15,73364,73366],{"className":73365,"style":2198},[210],[15,73367,73369,73372],{"style":73368},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[15,73370],{"className":73371,"style":218},[217],[15,73373,73375],{"className":73374},[222,223,224,225],[15,73376,4554],{"className":73377},[66,67,225],[15,73379,1114],{"className":73380},[1113],[15,73382,73384],{"className":73383},[206],[15,73385,73387],{"className":73386,"style":1121},[210],[15,73388],{},[15,73390,63103],{"className":73391,"style":23020},[66,67],[15,73393,120],{"className":73394},[162],[15,73396,4554],{"className":73397},[66,67],[15,73399,335],{"className":73400},[422],[15,73402],{"className":73403,"style":697},[71],[15,73405,123],{"className":73406},[66,67],[15,73408,126],{"className":73409},[169],[15,73411],{"className":73412,"style":173},[71],[15,73414,129],{"className":73415},[177],[15,73417],{"className":73418,"style":173},[71],[15,73420,73422,73425,73428,73431,73434,73437,73440],{"className":73421},[57],[15,73423],{"className":73424,"style":154},[61],[15,73426,28394],{"className":73427,"style":62509},[66,67],[15,73429,120],{"className":73430},[162],[15,73432,61881],{"className":73433,"style":72},[66,67],[15,73435],{"className":73436,"style":173},[71],[15,73438,25767],{"className":73439},[177],[15,73441],{"className":73442,"style":173},[71],[15,73444,73446,73449,73452,73455,73458],{"className":73445},[57],[15,73447],{"className":73448,"style":154},[61],[15,73450,4554],{"className":73451},[66,67],[15,73453],{"className":73454,"style":173},[71],[15,73456,34425],{"className":73457},[177],[15,73459],{"className":73460,"style":173},[71],[15,73462,73464,73467,73470,73473,73476,73479],{"className":73463},[57],[15,73465],{"className":73466,"style":154},[61],[15,73468,123],{"className":73469},[66,67],[15,73471,126],{"className":73472},[169],[15,73474],{"className":73475,"style":72},[71],[15,73477,6600],{"className":73478},[76],[15,73480],{"className":73481,"style":72},[71],[15,73483,73485,73488,73491,73494,73497,73500,73503],{"className":73484},[57],[15,73486],{"className":73487,"style":154},[61],[15,73489,28394],{"className":73490,"style":62509},[66,67],[15,73492,120],{"className":73493},[162],[15,73495,61881],{"className":73496,"style":72},[66,67],[15,73498],{"className":73499,"style":173},[71],[15,73501,9278],{"className":73502},[177],[15,73504],{"className":73505,"style":173},[71],[15,73507,73509,73512,73515,73518,73521],{"className":73508},[57],[15,73510],{"className":73511,"style":154},[61],[15,73513,4554],{"className":73514},[66,67],[15,73516],{"className":73517,"style":173},[71],[15,73519,34425],{"className":73520},[177],[15,73522],{"className":73523,"style":173},[71],[15,73525,73527,73530,73533,73536],{"className":73526},[57],[15,73528],{"className":73529,"style":154},[61],[15,73531,123],{"className":73532},[66,67],[15,73534,126],{"className":73535},[169],[15,73537,335],{"className":73538},[422],[11,73540,73541,73542,73570],{},"令 ",[15,73543,73545,73558],{"className":73544,"translate":19},[18],[15,73546,73548],{"className":73547},[23],[25,73549,73550],{"xmlns":27},[29,73551,73552,73556],{},[32,73553,73554],{},[1209,73555,1804],{},[45,73557,1804],{"encoding":47},[15,73559,73561],{"className":73560,"ariaHidden":53},[52],[15,73562,73564,73567],{"className":73563},[57],[15,73565],{"className":73566,"style":1836},[61],[15,73568,1804],{"className":73569},[66]," 属于次梯度，等价于",[15,73572,73574],{"className":73573,"translate":19},[294],[15,73575,73577,73643],{"className":73576,"translate":19},[18],[15,73578,73580],{"className":73579},[23],[25,73581,73582],{"xmlns":27,"display":303},[29,73583,73584,73640],{},[32,73585,73586,73588,73590,73592,73594,73596,73598,73600,73602,73604,73610,73612,73614,73616,73618,73620,73622,73624,73626,73628,73630,73632,73638],{},[35,73587,28394],{},[39,73589,120],{"stretchy":119},[35,73591,61881],{},[39,73593,24215],{},[35,73595,4554],{},[39,73597,34425],{},[35,73599,123],{},[39,73601,126],{"stretchy":119},[39,73603,1801],{},[6465,73605,73606,73608],{},[1209,73607,1211],{},[1209,73609,1263],{},[39,73611,335],{"separator":53},[71,73613],{"width":3902},[35,73615,28394],{},[39,73617,120],{"stretchy":119},[35,73619,61881],{},[39,73621,1801],{},[35,73623,4554],{},[39,73625,34425],{},[35,73627,123],{},[39,73629,126],{"stretchy":119},[39,73631,1801],{},[6465,73633,73634,73636],{},[1209,73635,1211],{},[1209,73637,1263],{},[35,73639,3571],{"mathvariant":138},[45,73641,73642],{"encoding":47},"P(Y\\le a\\mid x)\\ge \\frac12,\\qquad P(Y\\ge a\\mid x)\\ge \\frac12 .",[15,73644,73646,73670,73688,73709,73804,73822,73843],{"className":73645,"ariaHidden":53},[52],[15,73647,73649,73652,73655,73658,73661,73664,73667],{"className":73648},[57],[15,73650],{"className":73651,"style":154},[61],[15,73653,28394],{"className":73654,"style":62509},[66,67],[15,73656,120],{"className":73657},[162],[15,73659,61881],{"className":73660,"style":72},[66,67],[15,73662],{"className":73663,"style":173},[71],[15,73665,24215],{"className":73666},[177],[15,73668],{"className":73669,"style":173},[71],[15,73671,73673,73676,73679,73682,73685],{"className":73672},[57],[15,73674],{"className":73675,"style":154},[61],[15,73677,4554],{"className":73678},[66,67],[15,73680],{"className":73681,"style":173},[71],[15,73683,34425],{"className":73684},[177],[15,73686],{"className":73687,"style":173},[71],[15,73689,73691,73694,73697,73700,73703,73706],{"className":73690},[57],[15,73692],{"className":73693,"style":154},[61],[15,73695,123],{"className":73696},[66,67],[15,73698,126],{"className":73699},[169],[15,73701],{"className":73702,"style":173},[71],[15,73704,1801],{"className":73705},[177],[15,73707],{"className":73708,"style":173},[71],[15,73710,73712,73715,73777,73780,73783,73786,73789,73792,73795,73798,73801],{"className":73711},[57],[15,73713],{"className":73714,"style":49082},[61],[15,73716,73718,73721,73774],{"className":73717},[66],[15,73719],{"className":73720},[162,23412],[15,73722,73724],{"className":73723},[6465],[15,73725,73727,73766],{"className":73726},[202,1090],[15,73728,73730,73763],{"className":73729},[206],[15,73731,73733,73744,73752],{"className":73732,"style":23425},[210],[15,73734,73735,73738],{"style":23428},[15,73736],{"className":73737,"style":1368},[217],[15,73739,73741],{"className":73740},[66],[15,73742,1263],{"className":73743},[66],[15,73745,73746,73749],{"style":23440},[15,73747],{"className":73748,"style":1368},[217],[15,73750],{"className":73751,"style":4438},[23447],[15,73753,73754,73757],{"style":23450},[15,73755],{"className":73756,"style":1368},[217],[15,73758,73760],{"className":73759},[66],[15,73761,1211],{"className":73762},[66],[15,73764,1114],{"className":73765},[1113],[15,73767,73769],{"className":73768},[206],[15,73770,73772],{"className":73771,"style":23469},[210],[15,73773],{},[15,73775],{"className":73776},[169,23412],[15,73778,335],{"className":73779},[422],[15,73781],{"className":73782,"style":4059},[71],[15,73784],{"className":73785,"style":697},[71],[15,73787,28394],{"className":73788,"style":62509},[66,67],[15,73790,120],{"className":73791},[162],[15,73793,61881],{"className":73794,"style":72},[66,67],[15,73796],{"className":73797,"style":173},[71],[15,73799,1801],{"className":73800},[177],[15,73802],{"className":73803,"style":173},[71],[15,73805,73807,73810,73813,73816,73819],{"className":73806},[57],[15,73808],{"className":73809,"style":154},[61],[15,73811,4554],{"className":73812},[66,67],[15,73814],{"className":73815,"style":173},[71],[15,73817,34425],{"className":73818},[177],[15,73820],{"className":73821,"style":173},[71],[15,73823,73825,73828,73831,73834,73837,73840],{"className":73824},[57],[15,73826],{"className":73827,"style":154},[61],[15,73829,123],{"className":73830},[66,67],[15,73832,126],{"className":73833},[169],[15,73835],{"className":73836,"style":173},[71],[15,73838,1801],{"className":73839},[177],[15,73841],{"className":73842,"style":173},[71],[15,73844,73846,73849,73911],{"className":73845},[57],[15,73847],{"className":73848,"style":49082},[61],[15,73850,73852,73855,73908],{"className":73851},[66],[15,73853],{"className":73854},[162,23412],[15,73856,73858],{"className":73857},[6465],[15,73859,73861,73900],{"className":73860},[202,1090],[15,73862,73864,73897],{"className":73863},[206],[15,73865,73867,73878,73886],{"className":73866,"style":23425},[210],[15,73868,73869,73872],{"style":23428},[15,73870],{"className":73871,"style":1368},[217],[15,73873,73875],{"className":73874},[66],[15,73876,1263],{"className":73877},[66],[15,73879,73880,73883],{"style":23440},[15,73881],{"className":73882,"style":1368},[217],[15,73884],{"className":73885,"style":4438},[23447],[15,73887,73888,73891],{"style":23450},[15,73889],{"className":73890,"style":1368},[217],[15,73892,73894],{"className":73893},[66],[15,73895,1211],{"className":73896},[66],[15,73898,1114],{"className":73899},[1113],[15,73901,73903],{"className":73902},[206],[15,73904,73906],{"className":73905,"style":23469},[210],[15,73907],{},[15,73909],{"className":73910},[169,23412],[15,73912,3571],{"className":73913},[66],[11,73915,73916,73917,73945,73946,73974],{},"满足这两个不等式的 ",[15,73918,73920,73933],{"className":73919,"translate":19},[18],[15,73921,73923],{"className":73922},[23],[25,73924,73925],{"xmlns":27},[29,73926,73927,73931],{},[32,73928,73929],{},[35,73930,4554],{},[45,73932,4554],{"encoding":47},[15,73934,73936],{"className":73935,"ariaHidden":53},[52],[15,73937,73939,73942],{"className":73938},[57],[15,73940],{"className":73941,"style":86},[61],[15,73943,4554],{"className":73944},[66,67]," 就是后验中位数。直观上，中位数把后验概率一分为二，左右两侧的「质量」相等，因此向左或向右移动 ",[15,73947,73949,73962],{"className":73948,"translate":19},[18],[15,73950,73952],{"className":73951},[23],[25,73953,73954],{"xmlns":27},[29,73955,73956,73960],{},[32,73957,73958],{},[35,73959,4554],{},[45,73961,4554],{"encoding":47},[15,73963,73965],{"className":73964,"ariaHidden":53},[52],[15,73966,73968,73971],{"className":73967},[57],[15,73969],{"className":73970,"style":86},[61],[15,73972,4554],{"className":73973},[66,67]," 都不会降低期望绝对误差。",[11,73976,73977],{},"平方损失对异常值敏感，因为误差被平方；绝对损失只看见误差的大小。这个差异在决策层面的表现，就是均值与中位数的差异。",[1904,73979,73981,73982],{"id":73980},"_35-非对称损失阈值不再在-121212","3.5 非对称损失：阈值不再在 ",[15,73983,73985,74002],{"className":73984,"translate":19},[18],[15,73986,73988],{"className":73987},[23],[25,73989,73990],{"xmlns":27},[29,73991,73992,74000],{},[32,73993,73994,73996,73998],{},[1209,73995,1211],{},[35,73997,6623],{"mathvariant":138},[1209,73999,1263],{},[45,74001,71801],{"encoding":47},[15,74003,74005],{"className":74004,"ariaHidden":53},[52],[15,74006,74008,74011],{"className":74007},[57],[15,74009],{"className":74010,"style":154},[61],[15,74012,71801],{"className":74013},[66],[11,74015,74016,74017,74099,74100,74181,74182,72025],{},"现实中的两类错误很少同等昂贵。设预测为正类的代价为 ",[15,74018,74020,74043],{"className":74019,"translate":19},[18],[15,74021,74023],{"className":74022},[23],[25,74024,74025],{"xmlns":27},[29,74026,74027,74040],{},[32,74028,74029],{},[1057,74030,74031,74033],{},[35,74032,2256],{},[32,74034,74035,74038],{},[35,74036,74037],{"mathvariant":138},"F",[35,74039,28394],{"mathvariant":138},[45,74041,74042],{"encoding":47},"c_{\\mathrm{FP}}",[15,74044,74046],{"className":74045,"ariaHidden":53},[52],[15,74047,74049,74052],{"className":74048},[57],[15,74050],{"className":74051,"style":10192},[61],[15,74053,74055,74058],{"className":74054},[66],[15,74056,2256],{"className":74057},[66,67],[15,74059,74061],{"className":74060},[198],[15,74062,74064,74091],{"className":74063},[202,1090],[15,74065,74067,74088],{"className":74066},[206],[15,74068,74070],{"className":74069,"style":3636},[210],[15,74071,74072,74075],{"style":1584},[15,74073],{"className":74074,"style":218},[217],[15,74076,74078],{"className":74077},[222,223,224,225],[15,74079,74081],{"className":74080},[66,225],[15,74082,74084],{"className":74083},[66,225],[15,74085,74087],{"className":74086},[66,4427,225],"FP",[15,74089,1114],{"className":74090},[1113],[15,74092,74094],{"className":74093},[206],[15,74095,74097],{"className":74096,"style":1121},[210],[15,74098],{},"（假阳性），预测为负类的代价为 ",[15,74101,74103,74125],{"className":74102,"translate":19},[18],[15,74104,74106],{"className":74105},[23],[25,74107,74108],{"xmlns":27},[29,74109,74110,74122],{},[32,74111,74112],{},[1057,74113,74114,74116],{},[35,74115,2256],{},[32,74117,74118,74120],{},[35,74119,74037],{"mathvariant":138},[35,74121,28530],{"mathvariant":138},[45,74123,74124],{"encoding":47},"c_{\\mathrm{FN}}",[15,74126,74128],{"className":74127,"ariaHidden":53},[52],[15,74129,74131,74134],{"className":74130},[57],[15,74132],{"className":74133,"style":10192},[61],[15,74135,74137,74140],{"className":74136},[66],[15,74138,2256],{"className":74139},[66,67],[15,74141,74143],{"className":74142},[198],[15,74144,74146,74173],{"className":74145},[202,1090],[15,74147,74149,74170],{"className":74148},[206],[15,74150,74152],{"className":74151,"style":3636},[210],[15,74153,74154,74157],{"style":1584},[15,74155],{"className":74156,"style":218},[217],[15,74158,74160],{"className":74159},[222,223,224,225],[15,74161,74163],{"className":74162},[66,225],[15,74164,74166],{"className":74165},[66,225],[15,74167,74169],{"className":74168},[66,4427,225],"FN",[15,74171,1114],{"className":74172},[1113],[15,74174,74176],{"className":74175},[206],[15,74177,74179],{"className":74178,"style":1121},[210],[15,74180],{},"（假阴性），正确分类的代价为 ",[15,74183,74185,74198],{"className":74184,"translate":19},[18],[15,74186,74188],{"className":74187},[23],[25,74189,74190],{"xmlns":27},[29,74191,74192,74196],{},[32,74193,74194],{},[1209,74195,1804],{},[45,74197,1804],{"encoding":47},[15,74199,74201],{"className":74200,"ariaHidden":53},[52],[15,74202,74204,74207],{"className":74203},[57],[15,74205],{"className":74206,"style":1836},[61],[15,74208,1804],{"className":74209},[66],[15,74211,74213],{"className":74212,"translate":19},[294],[15,74214,74216,74310],{"className":74215,"translate":19},[18],[15,74217,74219],{"className":74218},[23],[25,74220,74221],{"xmlns":27,"display":303},[29,74222,74223,74307],{},[32,74224,74225,74227,74229,74231,74233,74235,74237,74239,74249,74251,74253,74255,74257,74259,74261,74263,74265,74267,74269,74271,74273,74275,74277,74279,74281,74291,74293,74295,74297,74299,74301,74303,74305],{},[35,74226,63103],{},[39,74228,120],{"stretchy":119},[1209,74230,1211],{},[39,74232,335],{"separator":53},[35,74234,123],{},[39,74236,126],{"stretchy":119},[39,74238,129],{},[1057,74240,74241,74243],{},[35,74242,2256],{},[32,74244,74245,74247],{},[35,74246,74037],{"mathvariant":138},[35,74248,28394],{"mathvariant":138},[1274,74250,2462],{},[35,74252,11],{},[39,74254,120],{"stretchy":119},[1209,74256,1804],{},[39,74258,34425],{},[35,74260,123],{},[39,74262,126],{"stretchy":119},[39,74264,335],{"separator":53},[71,74266],{"width":3902},[35,74268,63103],{},[39,74270,120],{"stretchy":119},[1209,74272,1804],{},[39,74274,335],{"separator":53},[35,74276,123],{},[39,74278,126],{"stretchy":119},[39,74280,129],{},[1057,74282,74283,74285],{},[35,74284,2256],{},[32,74286,74287,74289],{},[35,74288,74037],{"mathvariant":138},[35,74290,28530],{"mathvariant":138},[1274,74292,2462],{},[35,74294,11],{},[39,74296,120],{"stretchy":119},[1209,74298,1211],{},[39,74300,34425],{},[35,74302,123],{},[39,74304,126],{"stretchy":119},[35,74306,3571],{"mathvariant":138},[45,74308,74309],{"encoding":47},"r(1,x)=c_{\\mathrm{FP}}\\,p(0\\mid x),\\qquad\nr(0,x)=c_{\\mathrm{FN}}\\,p(1\\mid x).",[15,74311,74313,74349,74422,74473,74546],{"className":74312,"ariaHidden":53},[52],[15,74314,74316,74319,74322,74325,74328,74331,74334,74337,74340,74343,74346],{"className":74315},[57],[15,74317],{"className":74318,"style":154},[61],[15,74320,63103],{"className":74321,"style":23020},[66,67],[15,74323,120],{"className":74324},[162],[15,74326,1211],{"className":74327},[66],[15,74329,335],{"className":74330},[422],[15,74332],{"className":74333,"style":697},[71],[15,74335,123],{"className":74336},[66,67],[15,74338,126],{"className":74339},[169],[15,74341],{"className":74342,"style":173},[71],[15,74344,129],{"className":74345},[177],[15,74347],{"className":74348,"style":173},[71],[15,74350,74352,74355,74401,74404,74407,74410,74413,74416,74419],{"className":74351},[57],[15,74353],{"className":74354,"style":154},[61],[15,74356,74358,74361],{"className":74357},[66],[15,74359,2256],{"className":74360},[66,67],[15,74362,74364],{"className":74363},[198],[15,74365,74367,74393],{"className":74366},[202,1090],[15,74368,74370,74390],{"className":74369},[206],[15,74371,74373],{"className":74372,"style":3636},[210],[15,74374,74375,74378],{"style":1584},[15,74376],{"className":74377,"style":218},[217],[15,74379,74381],{"className":74380},[222,223,224,225],[15,74382,74384],{"className":74383},[66,225],[15,74385,74387],{"className":74386},[66,225],[15,74388,74087],{"className":74389},[66,4427,225],[15,74391,1114],{"className":74392},[1113],[15,74394,74396],{"className":74395},[206],[15,74397,74399],{"className":74398,"style":1121},[210],[15,74400],{},[15,74402],{"className":74403,"style":697},[71],[15,74405,11],{"className":74406},[66,67],[15,74408,120],{"className":74409},[162],[15,74411,1804],{"className":74412},[66],[15,74414],{"className":74415,"style":173},[71],[15,74417,34425],{"className":74418},[177],[15,74420],{"className":74421,"style":173},[71],[15,74423,74425,74428,74431,74434,74437,74440,74443,74446,74449,74452,74455,74458,74461,74464,74467,74470],{"className":74424},[57],[15,74426],{"className":74427,"style":154},[61],[15,74429,123],{"className":74430},[66,67],[15,74432,126],{"className":74433},[169],[15,74435,335],{"className":74436},[422],[15,74438],{"className":74439,"style":4059},[71],[15,74441],{"className":74442,"style":697},[71],[15,74444,63103],{"className":74445,"style":23020},[66,67],[15,74447,120],{"className":74448},[162],[15,74450,1804],{"className":74451},[66],[15,74453,335],{"className":74454},[422],[15,74456],{"className":74457,"style":697},[71],[15,74459,123],{"className":74460},[66,67],[15,74462,126],{"className":74463},[169],[15,74465],{"className":74466,"style":173},[71],[15,74468,129],{"className":74469},[177],[15,74471],{"className":74472,"style":173},[71],[15,74474,74476,74479,74525,74528,74531,74534,74537,74540,74543],{"className":74475},[57],[15,74477],{"className":74478,"style":154},[61],[15,74480,74482,74485],{"className":74481},[66],[15,74483,2256],{"className":74484},[66,67],[15,74486,74488],{"className":74487},[198],[15,74489,74491,74517],{"className":74490},[202,1090],[15,74492,74494,74514],{"className":74493},[206],[15,74495,74497],{"className":74496,"style":3636},[210],[15,74498,74499,74502],{"style":1584},[15,74500],{"className":74501,"style":218},[217],[15,74503,74505],{"className":74504},[222,223,224,225],[15,74506,74508],{"className":74507},[66,225],[15,74509,74511],{"className":74510},[66,225],[15,74512,74169],{"className":74513},[66,4427,225],[15,74515,1114],{"className":74516},[1113],[15,74518,74520],{"className":74519},[206],[15,74521,74523],{"className":74522,"style":1121},[210],[15,74524],{},[15,74526],{"className":74527,"style":697},[71],[15,74529,11],{"className":74530},[66,67],[15,74532,120],{"className":74533},[162],[15,74535,1211],{"className":74536},[66],[15,74538],{"className":74539,"style":173},[71],[15,74541,34425],{"className":74542},[177],[15,74544],{"className":74545,"style":173},[71],[15,74547,74549,74552,74555,74558],{"className":74548},[57],[15,74550],{"className":74551,"style":154},[61],[15,74553,123],{"className":74554},[66,67],[15,74556,126],{"className":74557},[169],[15,74559,3571],{"className":74560},[66],[11,74562,74563,74564,74671],{},"选择正类当且仅当 ",[15,74565,74567,74605],{"className":74566,"translate":19},[18],[15,74568,74570],{"className":74569},[23],[25,74571,74572],{"xmlns":27},[29,74573,74574,74602],{},[32,74575,74576,74578,74580,74582,74584,74586,74588,74590,74592,74594,74596,74598,74600],{},[35,74577,63103],{},[39,74579,120],{"stretchy":119},[1209,74581,1211],{},[39,74583,335],{"separator":53},[35,74585,123],{},[39,74587,126],{"stretchy":119},[39,74589,24215],{},[35,74591,63103],{},[39,74593,120],{"stretchy":119},[1209,74595,1804],{},[39,74597,335],{"separator":53},[35,74599,123],{},[39,74601,126],{"stretchy":119},[45,74603,74604],{"encoding":47},"r(1,x)\\le r(0,x)",[15,74606,74608,74644],{"className":74607,"ariaHidden":53},[52],[15,74609,74611,74614,74617,74620,74623,74626,74629,74632,74635,74638,74641],{"className":74610},[57],[15,74612],{"className":74613,"style":154},[61],[15,74615,63103],{"className":74616,"style":23020},[66,67],[15,74618,120],{"className":74619},[162],[15,74621,1211],{"className":74622},[66],[15,74624,335],{"className":74625},[422],[15,74627],{"className":74628,"style":697},[71],[15,74630,123],{"className":74631},[66,67],[15,74633,126],{"className":74634},[169],[15,74636],{"className":74637,"style":173},[71],[15,74639,24215],{"className":74640},[177],[15,74642],{"className":74643,"style":173},[71],[15,74645,74647,74650,74653,74656,74659,74662,74665,74668],{"className":74646},[57],[15,74648],{"className":74649,"style":154},[61],[15,74651,63103],{"className":74652,"style":23020},[66,67],[15,74654,120],{"className":74655},[162],[15,74657,1804],{"className":74658},[66],[15,74660,335],{"className":74661},[422],[15,74663],{"className":74664,"style":697},[71],[15,74666,123],{"className":74667},[66,67],[15,74669,126],{"className":74670},[169],"，即",[15,74673,74675],{"className":74674,"translate":19},[294],[15,74676,74678,74746],{"className":74677,"translate":19},[18],[15,74679,74681],{"className":74680},[23],[25,74682,74683],{"xmlns":27,"display":303},[29,74684,74685,74743],{},[32,74686,74687,74689,74691,74693,74695,74697,74699,74701,74703,74705,74741],{},[35,74688,11],{},[39,74690,120],{"stretchy":119},[1209,74692,1211],{},[39,74694,34425],{},[35,74696,123],{},[39,74698,126],{"stretchy":119},[1274,74700,24228],{},[39,74702,1801],{},[1274,74704,24228],{},[6465,74706,74707,74717],{},[1057,74708,74709,74711],{},[35,74710,2256],{},[32,74712,74713,74715],{},[35,74714,74037],{"mathvariant":138},[35,74716,28394],{"mathvariant":138},[32,74718,74719,74729,74731],{},[1057,74720,74721,74723],{},[35,74722,2256],{},[32,74724,74725,74727],{},[35,74726,74037],{"mathvariant":138},[35,74728,28394],{"mathvariant":138},[39,74730,1245],{},[1057,74732,74733,74735],{},[35,74734,2256],{},[32,74736,74737,74739],{},[35,74738,74037],{"mathvariant":138},[35,74740,28530],{"mathvariant":138},[35,74742,3571],{"mathvariant":138},[45,74744,74745],{"encoding":47},"p(1\\mid x)\\ \\ge\\ \\frac{c_{\\mathrm{FP}}}{c_{\\mathrm{FP}}+c_{\\mathrm{FN}}}.",[15,74747,74749,74773,74800],{"className":74748,"ariaHidden":53},[52],[15,74750,74752,74755,74758,74761,74764,74767,74770],{"className":74751},[57],[15,74753],{"className":74754,"style":154},[61],[15,74756,11],{"className":74757},[66,67],[15,74759,120],{"className":74760},[162],[15,74762,1211],{"className":74763},[66],[15,74765],{"className":74766,"style":173},[71],[15,74768,34425],{"className":74769},[177],[15,74771],{"className":74772,"style":173},[71],[15,74774,74776,74779,74782,74785,74788,74791,74794,74797],{"className":74775},[57],[15,74777],{"className":74778,"style":154},[61],[15,74780,123],{"className":74781},[66,67],[15,74783,126],{"className":74784},[169],[15,74786,24228],{"className":74787},[71],[15,74789],{"className":74790,"style":173},[71],[15,74792,1801],{"className":74793},[177],[15,74795,24228],{"className":74796},[71],[15,74798],{"className":74799,"style":173},[71],[15,74801,74803,74806,75009],{"className":74802},[57],[15,74804],{"className":74805,"style":52418},[61],[15,74807,74809,74812,75006],{"className":74808},[66],[15,74810],{"className":74811},[162,23412],[15,74813,74815],{"className":74814},[6465],[15,74816,74818,74998],{"className":74817},[202,1090],[15,74819,74821,74995],{"className":74820},[206],[15,74822,74824,74933,74941],{"className":74823,"style":39128},[210],[15,74825,74826,74829],{"style":23428},[15,74827],{"className":74828,"style":1368},[217],[15,74830,74832,74878,74881,74884,74887],{"className":74831},[66],[15,74833,74835,74838],{"className":74834},[66],[15,74836,2256],{"className":74837},[66,67],[15,74839,74841],{"className":74840},[198],[15,74842,74844,74870],{"className":74843},[202,1090],[15,74845,74847,74867],{"className":74846},[206],[15,74848,74850],{"className":74849,"style":3636},[210],[15,74851,74852,74855],{"style":1584},[15,74853],{"className":74854,"style":218},[217],[15,74856,74858],{"className":74857},[222,223,224,225],[15,74859,74861],{"className":74860},[66,225],[15,74862,74864],{"className":74863},[66,225],[15,74865,74087],{"className":74866},[66,4427,225],[15,74868,1114],{"className":74869},[1113],[15,74871,74873],{"className":74872},[206],[15,74874,74876],{"className":74875,"style":1121},[210],[15,74877],{},[15,74879],{"className":74880,"style":72},[71],[15,74882,1245],{"className":74883},[76],[15,74885],{"className":74886,"style":72},[71],[15,74888,74890,74893],{"className":74889},[66],[15,74891,2256],{"className":74892},[66,67],[15,74894,74896],{"className":74895},[198],[15,74897,74899,74925],{"className":74898},[202,1090],[15,74900,74902,74922],{"className":74901},[206],[15,74903,74905],{"className":74904,"style":3636},[210],[15,74906,74907,74910],{"style":1584},[15,74908],{"className":74909,"style":218},[217],[15,74911,74913],{"className":74912},[222,223,224,225],[15,74914,74916],{"className":74915},[66,225],[15,74917,74919],{"className":74918},[66,225],[15,74920,74169],{"className":74921},[66,4427,225],[15,74923,1114],{"className":74924},[1113],[15,74926,74928],{"className":74927},[206],[15,74929,74931],{"className":74930,"style":1121},[210],[15,74932],{},[15,74934,74935,74938],{"style":23440},[15,74936],{"className":74937,"style":1368},[217],[15,74939],{"className":74940,"style":4438},[23447],[15,74942,74943,74946],{"style":23450},[15,74944],{"className":74945,"style":1368},[217],[15,74947,74949],{"className":74948},[66],[15,74950,74952,74955],{"className":74951},[66],[15,74953,2256],{"className":74954},[66,67],[15,74956,74958],{"className":74957},[198],[15,74959,74961,74987],{"className":74960},[202,1090],[15,74962,74964,74984],{"className":74963},[206],[15,74965,74967],{"className":74966,"style":3636},[210],[15,74968,74969,74972],{"style":1584},[15,74970],{"className":74971,"style":218},[217],[15,74973,74975],{"className":74974},[222,223,224,225],[15,74976,74978],{"className":74977},[66,225],[15,74979,74981],{"className":74980},[66,225],[15,74982,74087],{"className":74983},[66,4427,225],[15,74985,1114],{"className":74986},[1113],[15,74988,74990],{"className":74989},[206],[15,74991,74993],{"className":74992,"style":1121},[210],[15,74994],{},[15,74996,1114],{"className":74997},[1113],[15,74999,75001],{"className":75000},[206],[15,75002,75004],{"className":75003,"style":52615},[210],[15,75005],{},[15,75007],{"className":75008},[169,23412],[15,75010,3571],{"className":75011},[66],[11,75013,75014,75015,75047,75048,75082],{},"阈值不再固定在 ",[15,75016,75018,75035],{"className":75017,"translate":19},[18],[15,75019,75021],{"className":75020},[23],[25,75022,75023],{"xmlns":27},[29,75024,75025,75033],{},[32,75026,75027,75029,75031],{},[1209,75028,1211],{},[35,75030,6623],{"mathvariant":138},[1209,75032,1263],{},[45,75034,71801],{"encoding":47},[15,75036,75038],{"className":75037,"ariaHidden":53},[52],[15,75039,75041,75044],{"className":75040},[57],[15,75042],{"className":75043,"style":154},[61],[15,75045,71801],{"className":75046},[66],"，而是由两类代价的比值决定。若漏诊的代价是误诊的十倍，阈值就降到 ",[15,75049,75051,75070],{"className":75050,"translate":19},[18],[15,75052,75054],{"className":75053},[23],[25,75055,75056],{"xmlns":27},[29,75057,75058,75067],{},[32,75059,75060,75062,75064],{},[1209,75061,1211],{},[35,75063,6623],{"mathvariant":138},[1209,75065,75066],{},"11",[45,75068,75069],{"encoding":47},"1\u002F11",[15,75071,75073],{"className":75072,"ariaHidden":53},[52],[15,75074,75076,75079],{"className":75075},[57],[15,75077],{"className":75078,"style":154},[61],[15,75080,75069],{"className":75081},[66]," 左右，模型会倾向于把更多样本判为正类。",[11,75084,75085,75086,75089],{},"这个式子也说明：类别不平衡本身不是问题，",[2040,75087,75088],{},"损失不对称","才是问题。重采样改变了训练分布，相当于改变了先验；如果最终要解决的是原分布上的决策问题，那么重采样之后还需要把阈值校正回来。",[11,75091,75092],{},"把常见损失与对应的贝叶斯行动放在一起：",[6150,75094,75095,75104],{},[6153,75096,75097],{},[6156,75098,75099,75101],{},[6159,75100,30588],{},[6159,75102,75103],{},"贝叶斯行动",[6169,75105,75106,75114,75121,75129,75361],{},[6156,75107,75108,75111],{},[6174,75109,75110],{},"0-1 损失",[6174,75112,75113],{},"后验众数",[6156,75115,75116,75119],{},[6174,75117,75118],{},"平方损失",[6174,75120,72658],{},[6156,75122,75123,75126],{},[6174,75124,75125],{},"绝对损失",[6174,75127,75128],{},"后验中位数",[6156,75130,75131,75134],{},[6174,75132,75133],{},"非对称 0-1 损失",[6174,75135,75136,75137],{},"阈值 ",[15,75138,75140,75190],{"className":75139,"translate":19},[18],[15,75141,75143],{"className":75142},[23],[25,75144,75145],{"xmlns":27},[29,75146,75147,75187],{},[32,75148,75149,75159,75161,75163,75173,75175,75185],{},[1057,75150,75151,75153],{},[35,75152,2256],{},[32,75154,75155,75157],{},[35,75156,74037],{"mathvariant":138},[35,75158,28394],{"mathvariant":138},[35,75160,6623],{"mathvariant":138},[39,75162,120],{"stretchy":119},[1057,75164,75165,75167],{},[35,75166,2256],{},[32,75168,75169,75171],{},[35,75170,74037],{"mathvariant":138},[35,75172,28394],{"mathvariant":138},[39,75174,1245],{},[1057,75176,75177,75179],{},[35,75178,2256],{},[32,75180,75181,75183],{},[35,75182,74037],{"mathvariant":138},[35,75184,28530],{"mathvariant":138},[39,75186,126],{"stretchy":119},[45,75188,75189],{"encoding":47},"c_{\\mathrm{FP}}\u002F(c_{\\mathrm{FP}}+c_{\\mathrm{FN}})",[15,75191,75193,75306],{"className":75192,"ariaHidden":53},[52],[15,75194,75196,75199,75245,75248,75251,75297,75300,75303],{"className":75195},[57],[15,75197],{"className":75198,"style":154},[61],[15,75200,75202,75205],{"className":75201},[66],[15,75203,2256],{"className":75204},[66,67],[15,75206,75208],{"className":75207},[198],[15,75209,75211,75237],{"className":75210},[202,1090],[15,75212,75214,75234],{"className":75213},[206],[15,75215,75217],{"className":75216,"style":3636},[210],[15,75218,75219,75222],{"style":1584},[15,75220],{"className":75221,"style":218},[217],[15,75223,75225],{"className":75224},[222,223,224,225],[15,75226,75228],{"className":75227},[66,225],[15,75229,75231],{"className":75230},[66,225],[15,75232,74087],{"className":75233},[66,4427,225],[15,75235,1114],{"className":75236},[1113],[15,75238,75240],{"className":75239},[206],[15,75241,75243],{"className":75242,"style":1121},[210],[15,75244],{},[15,75246,6623],{"className":75247},[66],[15,75249,120],{"className":75250},[162],[15,75252,75254,75257],{"className":75253},[66],[15,75255,2256],{"className":75256},[66,67],[15,75258,75260],{"className":75259},[198],[15,75261,75263,75289],{"className":75262},[202,1090],[15,75264,75266,75286],{"className":75265},[206],[15,75267,75269],{"className":75268,"style":3636},[210],[15,75270,75271,75274],{"style":1584},[15,75272],{"className":75273,"style":218},[217],[15,75275,75277],{"className":75276},[222,223,224,225],[15,75278,75280],{"className":75279},[66,225],[15,75281,75283],{"className":75282},[66,225],[15,75284,74087],{"className":75285},[66,4427,225],[15,75287,1114],{"className":75288},[1113],[15,75290,75292],{"className":75291},[206],[15,75293,75295],{"className":75294,"style":1121},[210],[15,75296],{},[15,75298],{"className":75299,"style":72},[71],[15,75301,1245],{"className":75302},[76],[15,75304],{"className":75305,"style":72},[71],[15,75307,75309,75312,75358],{"className":75308},[57],[15,75310],{"className":75311,"style":154},[61],[15,75313,75315,75318],{"className":75314},[66],[15,75316,2256],{"className":75317},[66,67],[15,75319,75321],{"className":75320},[198],[15,75322,75324,75350],{"className":75323},[202,1090],[15,75325,75327,75347],{"className":75326},[206],[15,75328,75330],{"className":75329,"style":3636},[210],[15,75331,75332,75335],{"style":1584},[15,75333],{"className":75334,"style":218},[217],[15,75336,75338],{"className":75337},[222,223,224,225],[15,75339,75341],{"className":75340},[66,225],[15,75342,75344],{"className":75343},[66,225],[15,75345,74169],{"className":75346},[66,4427,225],[15,75348,1114],{"className":75349},[1113],[15,75351,75353],{"className":75352},[206],[15,75354,75356],{"className":75355,"style":1121},[210],[15,75357],{},[15,75359,126],{"className":75360},[169],[6156,75362,75363,75366],{},[6174,75364,75365],{},"带拒绝的 0-1 损失",[6174,75367,75368,75369,75420],{},"最大后验低于 ",[15,75370,75372,75390],{"className":75371,"translate":19},[18],[15,75373,75375],{"className":75374},[23],[25,75376,75377],{"xmlns":27},[29,75378,75379,75387],{},[32,75380,75381,75383,75385],{},[1209,75382,1211],{},[39,75384,6600],{},[35,75386,2256],{},[45,75388,75389],{"encoding":47},"1-c",[15,75391,75393,75411],{"className":75392,"ariaHidden":53},[52],[15,75394,75396,75399,75402,75405,75408],{"className":75395},[57],[15,75397],{"className":75398,"style":30463},[61],[15,75400,1211],{"className":75401},[66],[15,75403],{"className":75404,"style":72},[71],[15,75406,6600],{"className":75407},[76],[15,75409],{"className":75410,"style":72},[71],[15,75412,75414,75417],{"className":75413},[57],[15,75415],{"className":75416,"style":86},[61],[15,75418,2256],{"className":75419},[66,67]," 时拒绝",[11,75422,75423,75424,75427,75428,75502,75503,75644,75645,75673],{},"最后一行是",[2040,75425,75426],{},"拒绝选项","（reject option）：除了预测类别，模型还可以说「我不确定」。若拒绝的代价为 ",[15,75429,75431,75457],{"className":75430,"translate":19},[18],[15,75432,75434],{"className":75433},[23],[25,75435,75436],{"xmlns":27},[29,75437,75438,75454],{},[32,75439,75440,75442,75444,75446,75448,75450,75452],{},[35,75441,2256],{},[39,75443,1187],{},[39,75445,120],{"stretchy":119},[1209,75447,1804],{},[39,75449,335],{"separator":53},[1209,75451,1211],{},[39,75453,126],{"stretchy":119},[45,75455,75456],{"encoding":47},"c\\in(0,1)",[15,75458,75460,75478],{"className":75459,"ariaHidden":53},[52],[15,75461,75463,75466,75469,75472,75475],{"className":75462},[57],[15,75464],{"className":75465,"style":3225},[61],[15,75467,2256],{"className":75468},[66,67],[15,75470],{"className":75471,"style":173},[71],[15,75473,1187],{"className":75474},[177],[15,75476],{"className":75477,"style":173},[71],[15,75479,75481,75484,75487,75490,75493,75496,75499],{"className":75480},[57],[15,75482],{"className":75483,"style":154},[61],[15,75485,120],{"className":75486},[162],[15,75488,1804],{"className":75489},[66],[15,75491,335],{"className":75492},[422],[15,75494],{"className":75495,"style":697},[71],[15,75497,1211],{"className":75498},[66],[15,75500,126],{"className":75501},[169],"，预测的期望代价为 ",[15,75504,75506,75544],{"className":75505,"translate":19},[18],[15,75507,75509],{"className":75508},[23],[25,75510,75511],{"xmlns":27},[29,75512,75513,75541],{},[32,75514,75515,75517,75519,75529,75531,75533,75535,75537,75539],{},[1209,75516,1211],{},[39,75518,6600],{},[1057,75520,75521,75527],{},[32,75522,75523,75525],{},[35,75524,23268],{},[39,75526,1170],{},[35,75528,1224],{},[35,75530,11],{},[39,75532,120],{"stretchy":119},[35,75534,1224],{},[39,75536,34425],{},[35,75538,123],{},[39,75540,126],{"stretchy":119},[45,75542,75543],{"encoding":47},"1-\\max_y p(y\\mid x)",[15,75545,75547,75565,75632],{"className":75546,"ariaHidden":53},[52],[15,75548,75550,75553,75556,75559,75562],{"className":75549},[57],[15,75551],{"className":75552,"style":30463},[61],[15,75554,1211],{"className":75555},[66],[15,75557],{"className":75558,"style":72},[71],[15,75560,6600],{"className":75561},[76],[15,75563],{"className":75564,"style":72},[71],[15,75566,75568,75571,75611,75614,75617,75620,75623,75626,75629],{"className":75567},[57],[15,75569],{"className":75570,"style":2956},[61],[15,75572,75574,75577],{"className":75573},[1339],[15,75575,23268],{"className":75576},[1339],[15,75578,75580],{"className":75579},[198],[15,75581,75583,75603],{"className":75582},[202,1090],[15,75584,75586,75600],{"className":75585},[206],[15,75587,75589],{"className":75588,"style":2198},[210],[15,75590,75591,75594],{"style":69553},[15,75592],{"className":75593,"style":218},[217],[15,75595,75597],{"className":75596},[222,223,224,225],[15,75598,1224],{"className":75599,"style":1515},[66,67,225],[15,75601,1114],{"className":75602},[1113],[15,75604,75606],{"className":75605},[206],[15,75607,75609],{"className":75608,"style":2715},[210],[15,75610],{},[15,75612],{"className":75613,"style":697},[71],[15,75615,11],{"className":75616},[66,67],[15,75618,120],{"className":75619},[162],[15,75621,1224],{"className":75622,"style":1515},[66,67],[15,75624],{"className":75625,"style":173},[71],[15,75627,34425],{"className":75628},[177],[15,75630],{"className":75631,"style":173},[71],[15,75633,75635,75638,75641],{"className":75634},[57],[15,75636],{"className":75637,"style":154},[61],[15,75639,123],{"className":75640},[66,67],[15,75642,126],{"className":75643},[169],"，拒绝的代价为 ",[15,75646,75648,75661],{"className":75647,"translate":19},[18],[15,75649,75651],{"className":75650},[23],[25,75652,75653],{"xmlns":27},[29,75654,75655,75659],{},[32,75656,75657],{},[35,75658,2256],{},[45,75660,2256],{"encoding":47},[15,75662,75664],{"className":75663,"ariaHidden":53},[52],[15,75665,75667,75670],{"className":75666},[57],[15,75668],{"className":75669,"style":86},[61],[15,75671,2256],{"className":75672},[66,67],"，因此最优规则是",[15,75675,75677],{"className":75676,"translate":19},[294],[15,75678,75680,75734],{"className":75679,"translate":19},[18],[15,75681,75683],{"className":75682},[23],[25,75684,75685],{"xmlns":27,"display":303},[29,75686,75687,75731],{},[32,75688,75689,75692,75694,75697,75699,75709,75711,75713,75715,75717,75719,75721,75723,75725,75727,75729],{},[1274,75690,75691],{},"拒绝",[71,75693],{"width":9786},[39,75695,75696],{},"⟺",[71,75698],{"width":9786},[1172,75700,75701,75707],{},[32,75702,75703,75705],{},[35,75704,23268],{},[39,75706,1170],{},[35,75708,1224],{},[35,75710,11],{},[39,75712,120],{"stretchy":119},[35,75714,1224],{},[39,75716,34425],{},[35,75718,123],{},[39,75720,126],{"stretchy":119},[39,75722,25767],{},[1209,75724,1211],{},[39,75726,6600],{},[35,75728,2256],{},[35,75730,3571],{"mathvariant":138},[45,75732,75733],{"encoding":47},"\\text{拒绝}\\quad\\Longleftrightarrow\\quad \\max_y p(y\\mid x)\u003C1-c.",[15,75735,75737,75765,75838,75859,75877],{"className":75736,"ariaHidden":53},[52],[15,75738,75740,75744,75750,75753,75756,75759,75762],{"className":75739},[57],[15,75741],{"className":75742,"style":75743},[61],"height:0.7073em;vertical-align:-0.024em;",[15,75745,75747],{"className":75746},[66,1711],[15,75748,75691],{"className":75749},[66,68610],[15,75751],{"className":75752,"style":10044},[71],[15,75754],{"className":75755,"style":173},[71],[15,75757,75696],{"className":75758},[177],[15,75760],{"className":75761,"style":10044},[71],[15,75763],{"className":75764,"style":173},[71],[15,75766,75768,75772,75817,75820,75823,75826,75829,75832,75835],{"className":75767},[57],[15,75769],{"className":75770,"style":75771},[61],"height:1.5861em;vertical-align:-0.8361em;",[15,75773,75775],{"className":75774},[1339,1351],[15,75776,75778,75808],{"className":75777},[202,1090],[15,75779,75781,75805],{"className":75780},[206],[15,75782,75784,75795],{"className":75783,"style":86},[210],[15,75785,75786,75789],{"style":48819},[15,75787],{"className":75788,"style":1368},[217],[15,75790,75792],{"className":75791},[222,223,224,225],[15,75793,1224],{"className":75794,"style":1515},[66,67,225],[15,75796,75797,75800],{"style":1386},[15,75798],{"className":75799,"style":1368},[217],[15,75801,75802],{},[15,75803,23268],{"className":75804},[1339],[15,75806,1114],{"className":75807},[1113],[15,75809,75811],{"className":75810},[206],[15,75812,75815],{"className":75813,"style":75814},[210],"height:0.8361em;",[15,75816],{},[15,75818],{"className":75819,"style":697},[71],[15,75821,11],{"className":75822},[66,67],[15,75824,120],{"className":75825},[162],[15,75827,1224],{"className":75828,"style":1515},[66,67],[15,75830],{"className":75831,"style":173},[71],[15,75833,34425],{"className":75834},[177],[15,75836],{"className":75837,"style":173},[71],[15,75839,75841,75844,75847,75850,75853,75856],{"className":75840},[57],[15,75842],{"className":75843,"style":154},[61],[15,75845,123],{"className":75846},[66,67],[15,75848,126],{"className":75849},[169],[15,75851],{"className":75852,"style":173},[71],[15,75854,25767],{"className":75855},[177],[15,75857],{"className":75858,"style":173},[71],[15,75860,75862,75865,75868,75871,75874],{"className":75861},[57],[15,75863],{"className":75864,"style":30463},[61],[15,75866,1211],{"className":75867},[66],[15,75869],{"className":75870,"style":72},[71],[15,75872,6600],{"className":75873},[76],[15,75875],{"className":75876,"style":72},[71],[15,75878,75880,75883,75886],{"className":75879},[57],[15,75881],{"className":75882,"style":86},[61],[15,75884,2256],{"className":75885},[66,67],[15,75887,3571],{"className":75888},[66],[11,75890,75891],{},"这就是 Chow 规则。它把「不确定性」直接翻译成了一个可执行的动作，而不是一个只用来画图的分数。",[1904,75893,75895],{"id":75894},"_36-贝叶斯风险","3.6 贝叶斯风险",[11,75897,75898],{},"把定理 3.1 中的最优值单独命名：",[15,75900,75902],{"className":75901,"translate":19},[294],[15,75903,75905,75985],{"className":75904,"translate":19},[18],[15,75906,75908],{"className":75907},[23],[25,75909,75910],{"xmlns":27,"display":303},[29,75911,75912,75982],{},[32,75913,75914,75920,75922,75932,75934,75936,75938,75940,75942,75948,75950,75966,75968,75970,75972,75974,75976,75978,75980],{},[131,75915,75916,75918],{},[35,75917,455],{},[39,75919,1162],{},[39,75921,129],{},[1172,75923,75924,75930],{},[32,75925,75926,75928],{},[35,75927,69676],{},[39,75929,1170],{},[35,75931,3883],{},[35,75933,455],{},[39,75935,120],{"stretchy":119},[35,75937,3883],{},[39,75939,126],{"stretchy":119},[39,75941,129],{},[1057,75943,75944,75946],{},[35,75945,62377],{"mathvariant":454},[35,75947,446],{},[39,75949,1194],{"fence":119,"stretchy":53,"minsize":4543,"maxsize":4543},[1172,75951,75952,75958],{},[32,75953,75954,75956],{},[35,75955,1178],{},[39,75957,1170],{},[32,75959,75960,75962,75964],{},[35,75961,4554],{},[39,75963,1187],{},[35,75965,62001],{"mathvariant":445},[35,75967,63103],{},[39,75969,120],{"stretchy":119},[35,75971,4554],{},[39,75973,335],{"separator":53},[35,75975,446],{},[39,75977,126],{"stretchy":119},[39,75979,1266],{"fence":119,"stretchy":53,"minsize":4543,"maxsize":4543},[35,75981,3571],{"mathvariant":138},[45,75983,75984],{"encoding":47},"R^*=\\inf_{\\delta}R(\\delta)\n=\\mathbb E_X\\Big[\\min_{a\\in\\mathcal A}r(a,X)\\Big].",[15,75986,75988,76032,76112],{"className":75987,"ariaHidden":53},[52],[15,75989,75991,75994,76023,76026,76029],{"className":75990},[57],[15,75992],{"className":75993,"style":1308},[61],[15,75995,75997,76000],{"className":75996},[66],[15,75998,455],{"className":75999,"style":62433},[66,67],[15,76001,76003],{"className":76002},[198],[15,76004,76006],{"className":76005},[202],[15,76007,76009],{"className":76008},[206],[15,76010,76012],{"className":76011,"style":1308},[210],[15,76013,76014,76017],{"style":397},[15,76015],{"className":76016,"style":218},[217],[15,76018,76020],{"className":76019},[222,223,224,225],[15,76021,1162],{"className":76022},[76,225],[15,76024],{"className":76025,"style":173},[71],[15,76027,129],{"className":76028},[177],[15,76030],{"className":76031,"style":173},[71],[15,76033,76035,76039,76088,76091,76094,76097,76100,76103,76106,76109],{"className":76034},[57],[15,76036],{"className":76037,"style":76038},[61],"height:1.5021em;vertical-align:-0.7521em;",[15,76040,76042],{"className":76041},[1339,1351],[15,76043,76045,76080],{"className":76044},[202,1090],[15,76046,76048,76077],{"className":76047},[206],[15,76049,76051,76065],{"className":76050,"style":840},[210],[15,76052,76053,76056],{"style":1364},[15,76054],{"className":76055,"style":1368},[217],[15,76057,76059],{"className":76058},[222,223,224,225],[15,76060,76062],{"className":76061},[66,225],[15,76063,3883],{"className":76064,"style":3944},[66,67,225],[15,76066,76067,76070],{"style":1386},[15,76068],{"className":76069,"style":1368},[217],[15,76071,76072],{},[15,76073,69694,76075],{"className":76074},[1339],[15,76076,116],{"style":69697},[15,76078,1114],{"className":76079},[1113],[15,76081,76083],{"className":76082},[206],[15,76084,76086],{"className":76085,"style":44020},[210],[15,76087],{},[15,76089],{"className":76090,"style":697},[71],[15,76092,455],{"className":76093,"style":62433},[66,67],[15,76095,120],{"className":76096},[162],[15,76098,3883],{"className":76099,"style":3944},[66,67],[15,76101,126],{"className":76102},[169],[15,76104],{"className":76105,"style":173},[71],[15,76107,129],{"className":76108},[177],[15,76110],{"className":76111,"style":173},[71],[15,76113,76115,76118,76158,76164,76167,76220,76223,76226,76229,76232,76235,76238,76241,76244,76250],{"className":76114},[57],[15,76116],{"className":76117,"style":68903},[61],[15,76119,76121,76124],{"className":76120},[66],[15,76122,62377],{"className":76123},[66,517],[15,76125,76127],{"className":76126},[198],[15,76128,76130,76150],{"className":76129},[202,1090],[15,76131,76133,76147],{"className":76132},[206],[15,76134,76136],{"className":76135,"style":3636},[210],[15,76137,76138,76141],{"style":1584},[15,76139],{"className":76140,"style":218},[217],[15,76142,76144],{"className":76143},[222,223,224,225],[15,76145,446],{"className":76146,"style":16492},[66,67,225],[15,76148,1114],{"className":76149},[1113],[15,76151,76153],{"className":76152},[206],[15,76154,76156],{"className":76155,"style":1121},[210],[15,76157],{},[15,76159,76161],{"className":76160},[66],[15,76162,1194],{"className":76163},[1422,4675],[15,76165],{"className":76166,"style":697},[71],[15,76168,76170],{"className":76169},[1339,1351],[15,76171,76173,76212],{"className":76172},[202,1090],[15,76174,76176,76209],{"className":76175},[206],[15,76177,76179,76199],{"className":76178,"style":1361},[210],[15,76180,76181,76184],{"style":24349},[15,76182],{"className":76183,"style":1368},[217],[15,76185,76187],{"className":76186},[222,223,224,225],[15,76188,76190,76193,76196],{"className":76189},[66,225],[15,76191,4554],{"className":76192},[66,67,225],[15,76194,1187],{"className":76195},[177,225],[15,76197,62001],{"className":76198},[66,494,225],[15,76200,76201,76204],{"style":1386},[15,76202],{"className":76203,"style":1368},[217],[15,76205,76206],{},[15,76207,1178],{"className":76208},[1339],[15,76210,1114],{"className":76211},[1113],[15,76213,76215],{"className":76214},[206],[15,76216,76218],{"className":76217,"style":68520},[210],[15,76219],{},[15,76221],{"className":76222,"style":697},[71],[15,76224,63103],{"className":76225,"style":23020},[66,67],[15,76227,120],{"className":76228},[162],[15,76230,4554],{"className":76231},[66,67],[15,76233,335],{"className":76234},[422],[15,76236],{"className":76237,"style":697},[71],[15,76239,446],{"className":76240,"style":16492},[66,67],[15,76242,126],{"className":76243},[169],[15,76245,76247],{"className":76246},[66],[15,76248,1266],{"className":76249},[1422,4675],[15,76251,3571],{"className":76252},[66],[11,76254,76255,2038,76314,76316,76317,76345],{},[15,76256,76258,76276],{"className":76257,"translate":19},[18],[15,76259,76261],{"className":76260},[23],[25,76262,76263],{"xmlns":27},[29,76264,76265,76273],{},[32,76266,76267],{},[131,76268,76269,76271],{},[35,76270,455],{},[39,76272,1162],{},[45,76274,76275],{"encoding":47},"R^*",[15,76277,76279],{"className":76278,"ariaHidden":53},[52],[15,76280,76282,76285],{"className":76281},[57],[15,76283],{"className":76284,"style":16244},[61],[15,76286,76288,76291],{"className":76287},[66],[15,76289,455],{"className":76290,"style":62433},[66,67],[15,76292,76294],{"className":76293},[198],[15,76295,76297],{"className":76296},[202],[15,76298,76300],{"className":76299},[206],[15,76301,76303],{"className":76302,"style":16244},[210],[15,76304,76305,76308],{"style":213},[15,76306],{"className":76307,"style":218},[217],[15,76309,76311],{"className":76310},[222,223,224,225],[15,76312,1162],{"className":76313},[76,225],[2040,76315,61850],{},"（Bayes risk），也叫贝叶斯错误率（分类情形）。它是这个问题在信息层面的下限：即使知道真实分布、知道每个 ",[15,76318,76320,76333],{"className":76319,"translate":19},[18],[15,76321,76323],{"className":76322},[23],[25,76324,76325],{"xmlns":27},[29,76326,76327,76331],{},[32,76328,76329],{},[35,76330,123],{},[45,76332,123],{"encoding":47},[15,76334,76336],{"className":76335,"ariaHidden":53},[52],[15,76337,76339,76342],{"className":76338},[57],[15,76340],{"className":76341,"style":86},[61],[15,76343,123],{"className":76344},[66,67]," 上的后验，也只能做到这里。它不是某个算法的性能指标，而是问题本身的属性。",[11,76347,76348],{},"对二分类 0-1 损失，把 3.2 节的规则代回去，得到",[15,76350,76352],{"className":76351,"translate":19},[294],[15,76353,76355,76413],{"className":76354,"translate":19},[18],[15,76356,76358],{"className":76357},[23],[25,76359,76360],{"xmlns":27,"display":303},[29,76361,76362,76410],{},[32,76363,76364,76370,76372,76374,76376,76378,76380,76382,76384,76386,76388,76390,76392,76394,76396,76398,76400,76402,76404,76406,76408],{},[131,76365,76366,76368],{},[35,76367,455],{},[39,76369,1162],{},[39,76371,129],{},[35,76373,62377],{"mathvariant":454},[39,76375,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,76377,1178],{},[39,76379,1170],{},[39,76381,4305],{"stretchy":119},[35,76383,71105],{},[39,76385,120],{"stretchy":119},[35,76387,446],{},[39,76389,126],{"stretchy":119},[39,76391,335],{"separator":53},[1209,76393,1211],{},[39,76395,6600],{},[35,76397,71105],{},[39,76399,120],{"stretchy":119},[35,76401,446],{},[39,76403,126],{"stretchy":119},[39,76405,4328],{"stretchy":119},[39,76407,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,76409,3571],{"mathvariant":138},[45,76411,76412],{"encoding":47},"R^*=\\mathbb E\\big[\\min\\{\\eta(X),1-\\eta(X)\\}\\big].",[15,76414,76416,76460,76514],{"className":76415,"ariaHidden":53},[52],[15,76417,76419,76422,76451,76454,76457],{"className":76418},[57],[15,76420],{"className":76421,"style":1308},[61],[15,76423,76425,76428],{"className":76424},[66],[15,76426,455],{"className":76427,"style":62433},[66,67],[15,76429,76431],{"className":76430},[198],[15,76432,76434],{"className":76433},[202],[15,76435,76437],{"className":76436},[206],[15,76438,76440],{"className":76439,"style":1308},[210],[15,76441,76442,76445],{"style":397},[15,76443],{"className":76444,"style":218},[217],[15,76446,76448],{"className":76447},[222,223,224,225],[15,76449,1162],{"className":76450},[76,225],[15,76452],{"className":76453,"style":173},[71],[15,76455,129],{"className":76456},[177],[15,76458],{"className":76459,"style":173},[71],[15,76461,76463,76466,76469,76475,76478,76481,76484,76487,76490,76493,76496,76499,76502,76505,76508,76511],{"className":76462},[57],[15,76464],{"className":76465,"style":4398},[61],[15,76467,62377],{"className":76468},[66,517],[15,76470,76472],{"className":76471},[66],[15,76473,1194],{"className":76474},[1422,1508],[15,76476],{"className":76477,"style":697},[71],[15,76479,1178],{"className":76480},[1339],[15,76482,4305],{"className":76483},[162],[15,76485,71105],{"className":76486,"style":1515},[66,67],[15,76488,120],{"className":76489},[162],[15,76491,446],{"className":76492,"style":16492},[66,67],[15,76494,126],{"className":76495},[169],[15,76497,335],{"className":76498},[422],[15,76500],{"className":76501,"style":697},[71],[15,76503,1211],{"className":76504},[66],[15,76506],{"className":76507,"style":72},[71],[15,76509,6600],{"className":76510},[76],[15,76512],{"className":76513,"style":72},[71],[15,76515,76517,76520,76523,76526,76529,76532,76538],{"className":76516},[57],[15,76518],{"className":76519,"style":4398},[61],[15,76521,71105],{"className":76522,"style":1515},[66,67],[15,76524,120],{"className":76525},[162],[15,76527,446],{"className":76528,"style":16492},[66,67],[15,76530,19275],{"className":76531},[169],[15,76533,76535],{"className":76534},[66],[15,76536,1266],{"className":76537},[1422,1508],[15,76539,3571],{"className":76540},[66],[11,76542,76543],{},"对多分类，则是",[15,76545,76547],{"className":76546,"translate":19},[294],[15,76548,76550,76610],{"className":76549,"translate":19},[18],[15,76551,76553],{"className":76552},[23],[25,76554,76555],{"xmlns":27,"display":303},[29,76556,76557,76607],{},[32,76558,76559,76565,76567,76569,76571,76573,76575,76591,76593,76595,76597,76599,76601,76603,76605],{},[131,76560,76561,76563],{},[35,76562,455],{},[39,76564,1162],{},[39,76566,129],{},[1209,76568,1211],{},[39,76570,6600],{},[35,76572,62377],{"mathvariant":454},[39,76574,1194],{"fence":119,"stretchy":53,"minsize":4543,"maxsize":4543},[1172,76576,76577,76583],{},[32,76578,76579,76581],{},[35,76580,23268],{},[39,76582,1170],{},[32,76584,76585,76587,76589],{},[35,76586,1224],{},[39,76588,1187],{},[35,76590,61881],{"mathvariant":445},[35,76592,11],{},[39,76594,120],{"stretchy":119},[35,76596,1224],{},[39,76598,34425],{},[35,76600,446],{},[39,76602,126],{"stretchy":119},[39,76604,1266],{"fence":119,"stretchy":53,"minsize":4543,"maxsize":4543},[35,76606,3571],{"mathvariant":138},[45,76608,76609],{"encoding":47},"R^*=1-\\mathbb E\\Big[\\max_{y\\in\\mathcal Y}p(y\\mid X)\\Big].",[15,76611,76613,76657,76675,76768],{"className":76612,"ariaHidden":53},[52],[15,76614,76616,76619,76648,76651,76654],{"className":76615},[57],[15,76617],{"className":76618,"style":1308},[61],[15,76620,76622,76625],{"className":76621},[66],[15,76623,455],{"className":76624,"style":62433},[66,67],[15,76626,76628],{"className":76627},[198],[15,76629,76631],{"className":76630},[202],[15,76632,76634],{"className":76633},[206],[15,76635,76637],{"className":76636,"style":1308},[210],[15,76638,76639,76642],{"style":397},[15,76640],{"className":76641,"style":218},[217],[15,76643,76645],{"className":76644},[222,223,224,225],[15,76646,1162],{"className":76647},[76,225],[15,76649],{"className":76650,"style":173},[71],[15,76652,129],{"className":76653},[177],[15,76655],{"className":76656,"style":173},[71],[15,76658,76660,76663,76666,76669,76672],{"className":76659},[57],[15,76661],{"className":76662,"style":30463},[61],[15,76664,1211],{"className":76665},[66],[15,76667],{"className":76668,"style":72},[71],[15,76670,6600],{"className":76671},[76],[15,76673],{"className":76674,"style":72},[71],[15,76676,76678,76682,76685,76691,76694,76747,76750,76753,76756,76759,76762,76765],{"className":76677},[57],[15,76679],{"className":76680,"style":76681},[61],"height:2.0304em;vertical-align:-0.8804em;",[15,76683,62377],{"className":76684},[66,517],[15,76686,76688],{"className":76687},[66],[15,76689,1194],{"className":76690},[1422,4675],[15,76692],{"className":76693,"style":697},[71],[15,76695,76697],{"className":76696},[1339,1351],[15,76698,76700,76739],{"className":76699},[202,1090],[15,76701,76703,76736],{"className":76702},[206],[15,76704,76706,76726],{"className":76705,"style":86},[210],[15,76707,76708,76711],{"style":24349},[15,76709],{"className":76710,"style":1368},[217],[15,76712,76714],{"className":76713},[222,223,224,225],[15,76715,76717,76720,76723],{"className":76716},[66,225],[15,76718,1224],{"className":76719,"style":1515},[66,67,225],[15,76721,1187],{"className":76722},[177,225],[15,76724,61881],{"className":76725,"style":28447},[66,494,225],[15,76727,76728,76731],{"style":1386},[15,76729],{"className":76730,"style":1368},[217],[15,76732,76733],{},[15,76734,23268],{"className":76735},[1339],[15,76737,1114],{"className":76738},[1113],[15,76740,76742],{"className":76741},[206],[15,76743,76745],{"className":76744,"style":67076},[210],[15,76746],{},[15,76748],{"className":76749,"style":697},[71],[15,76751,11],{"className":76752},[66,67],[15,76754,120],{"className":76755},[162],[15,76757,1224],{"className":76758,"style":1515},[66,67],[15,76760],{"className":76761,"style":173},[71],[15,76763,34425],{"className":76764},[177],[15,76766],{"className":76767,"style":173},[71],[15,76769,76771,76774,76777,76780,76786],{"className":76770},[57],[15,76772],{"className":76773,"style":69179},[61],[15,76775,446],{"className":76776,"style":16492},[66,67],[15,76778,126],{"className":76779},[169],[15,76781,76783],{"className":76782},[66],[15,76784,1266],{"className":76785},[1422,4675],[15,76787,3571],{"className":76788},[66],[11,76790,76791],{},"这两个式子在理论里出现得非常多，因为它们把「分类有多难」压缩成了一个标量。",[11,76793,76794,76795,76798],{},"进一步，二分类的",[2040,76796,76797],{},"超额风险","有一个很干净的形式：",[15,76800,76802],{"className":76801,"translate":19},[294],[15,76803,76805,76891],{"className":76804,"translate":19},[18],[15,76806,76808],{"className":76807},[23],[25,76809,76810],{"xmlns":27,"display":303},[29,76811,76812,76888],{},[32,76813,76814,76816,76818,76820,76822,76824,76830,76832,76834,76836,76838,76840,76842,76844,76846,76848,76850,76852,76854,76856,76858,76860,76862,76864,76866,76868,76870,76876,76878,76880,76882,76884,76886],{},[35,76815,455],{},[39,76817,120],{"stretchy":119},[35,76819,3883],{},[39,76821,126],{"stretchy":119},[39,76823,6600],{},[131,76825,76826,76828],{},[35,76827,455],{},[39,76829,1162],{},[39,76831,129],{},[35,76833,62377],{"mathvariant":454},[39,76835,1194],{"fence":119,"stretchy":53,"minsize":4543,"maxsize":4543},[35,76837,34425],{"mathvariant":138},[1209,76839,1263],{},[35,76841,71105],{},[39,76843,120],{"stretchy":119},[35,76845,446],{},[39,76847,126],{"stretchy":119},[39,76849,6600],{},[1209,76851,1211],{},[35,76853,34425],{"mathvariant":138},[1274,76855,2462],{},[1209,76857,1211],{"mathvariant":70360},[39,76859,4305],{"stretchy":119},[35,76861,3883],{},[39,76863,120],{"stretchy":119},[35,76865,446],{},[39,76867,126],{"stretchy":119},[39,76869,13014],{"mathvariant":138},[131,76871,76872,76874],{},[35,76873,3883],{},[39,76875,1162],{},[39,76877,120],{"stretchy":119},[35,76879,446],{},[39,76881,126],{"stretchy":119},[39,76883,4328],{"stretchy":119},[39,76885,1266],{"fence":119,"stretchy":53,"minsize":4543,"maxsize":4543},[35,76887,3571],{"mathvariant":138},[45,76889,76890],{"encoding":47},"R(\\delta)-R^*\n=\\mathbb E\\Big[|2\\eta(X)-1|\\,\\mathbf 1\\{\\delta(X)\\ne\\delta^*(X)\\}\\Big].",[15,76892,76894,76921,76965,77005,77078],{"className":76893,"ariaHidden":53},[52],[15,76895,76897,76900,76903,76906,76909,76912,76915,76918],{"className":76896},[57],[15,76898],{"className":76899,"style":154},[61],[15,76901,455],{"className":76902,"style":62433},[66,67],[15,76904,120],{"className":76905},[162],[15,76907,3883],{"className":76908,"style":3944},[66,67],[15,76910,126],{"className":76911},[169],[15,76913],{"className":76914,"style":72},[71],[15,76916,6600],{"className":76917},[76],[15,76919],{"className":76920,"style":72},[71],[15,76922,76924,76927,76956,76959,76962],{"className":76923},[57],[15,76925],{"className":76926,"style":1308},[61],[15,76928,76930,76933],{"className":76929},[66],[15,76931,455],{"className":76932,"style":62433},[66,67],[15,76934,76936],{"className":76935},[198],[15,76937,76939],{"className":76938},[202],[15,76940,76942],{"className":76941},[206],[15,76943,76945],{"className":76944,"style":1308},[210],[15,76946,76947,76950],{"style":397},[15,76948],{"className":76949,"style":218},[217],[15,76951,76953],{"className":76952},[222,223,224,225],[15,76954,1162],{"className":76955},[76,225],[15,76957],{"className":76958,"style":173},[71],[15,76960,129],{"className":76961},[177],[15,76963],{"className":76964,"style":173},[71],[15,76966,76968,76971,76974,76980,76984,76987,76990,76993,76996,76999,77002],{"className":76967},[57],[15,76969],{"className":76970,"style":69179},[61],[15,76972,62377],{"className":76973},[66,517],[15,76975,76977],{"className":76976},[66],[15,76978,1194],{"className":76979},[1422,4675],[15,76981,76983],{"className":76982},[66],"∣2",[15,76985,71105],{"className":76986,"style":1515},[66,67],[15,76988,120],{"className":76989},[162],[15,76991,446],{"className":76992,"style":16492},[66,67],[15,76994,126],{"className":76995},[169],[15,76997],{"className":76998,"style":72},[71],[15,77000,6600],{"className":77001},[76],[15,77003],{"className":77004,"style":72},[71],[15,77006,77008,77011,77015,77018,77021,77024,77027,77030,77033,77036,77039,77075],{"className":77007},[57],[15,77009],{"className":77010,"style":154},[61],[15,77012,77014],{"className":77013},[66],"1∣",[15,77016],{"className":77017,"style":697},[71],[15,77019,1211],{"className":77020},[66,70424],[15,77022,4305],{"className":77023},[162],[15,77025,3883],{"className":77026,"style":3944},[66,67],[15,77028,120],{"className":77029},[162],[15,77031,446],{"className":77032,"style":16492},[66,67],[15,77034,126],{"className":77035},[169],[15,77037],{"className":77038,"style":173},[71],[15,77040,77042,77069,77072],{"className":77041},[177],[15,77043,77045],{"className":77044},[177],[15,77046,77048],{"className":77047},[66,13081],[15,77049,77051],{"className":77050},[13085],[15,77052,77054,77057,77066],{"className":77053},[13089],[15,77055],{"className":77056,"style":3406},[61],[15,77058,77060],{"className":77059},[13096],[15,77061,77063],{"className":77062},[66],[15,77064,13103],{"className":77065},[177],[15,77067],{"className":77068},[13107],[15,77070],{"className":77071},[71,13111],[15,77073,129],{"className":77074},[177],[15,77076],{"className":77077,"style":173},[71],[15,77079,77081,77084,77113,77116,77119,77122,77128],{"className":77080},[57],[15,77082],{"className":77083,"style":69179},[61],[15,77085,77087,77090],{"className":77086},[66],[15,77088,3883],{"className":77089,"style":3944},[66,67],[15,77091,77093],{"className":77092},[198],[15,77094,77096],{"className":77095},[202],[15,77097,77099],{"className":77098},[206],[15,77100,77102],{"className":77101,"style":1308},[210],[15,77103,77104,77107],{"style":397},[15,77105],{"className":77106,"style":218},[217],[15,77108,77110],{"className":77109},[222,223,224,225],[15,77111,1162],{"className":77112},[76,225],[15,77114,120],{"className":77115},[162],[15,77117,446],{"className":77118,"style":16492},[66,67],[15,77120,19275],{"className":77121},[169],[15,77123,77125],{"className":77124},[66],[15,77126,1266],{"className":77127},[1422,4675],[15,77129,3571],{"className":77130},[66],[11,77132,77133,77135,77136,77164,77165,77276,77277,77305,77306,77387,77388,77469],{},[2040,77134,69043],{}," 对固定的 ",[15,77137,77139,77152],{"className":77138,"translate":19},[18],[15,77140,77142],{"className":77141},[23],[25,77143,77144],{"xmlns":27},[29,77145,77146,77150],{},[32,77147,77148],{},[35,77149,123],{},[45,77151,123],{"encoding":47},[15,77153,77155],{"className":77154,"ariaHidden":53},[52],[15,77156,77158,77161],{"className":77157},[57],[15,77159],{"className":77160,"style":86},[61],[15,77162,123],{"className":77163},[66,67],"，若 ",[15,77166,77168,77202],{"className":77167,"translate":19},[18],[15,77169,77171],{"className":77170},[23],[25,77172,77173],{"xmlns":27},[29,77174,77175,77199],{},[32,77176,77177,77179,77181,77183,77185,77187,77193,77195,77197],{},[35,77178,3883],{},[39,77180,120],{"stretchy":119},[35,77182,123],{},[39,77184,126],{"stretchy":119},[39,77186,129],{},[131,77188,77189,77191],{},[35,77190,3883],{},[39,77192,1162],{},[39,77194,120],{"stretchy":119},[35,77196,123],{},[39,77198,126],{"stretchy":119},[45,77200,77201],{"encoding":47},"\\delta(x)=\\delta^*(x)",[15,77203,77205,77232],{"className":77204,"ariaHidden":53},[52],[15,77206,77208,77211,77214,77217,77220,77223,77226,77229],{"className":77207},[57],[15,77209],{"className":77210,"style":154},[61],[15,77212,3883],{"className":77213,"style":3944},[66,67],[15,77215,120],{"className":77216},[162],[15,77218,123],{"className":77219},[66,67],[15,77221,126],{"className":77222},[169],[15,77224],{"className":77225,"style":173},[71],[15,77227,129],{"className":77228},[177],[15,77230],{"className":77231,"style":173},[71],[15,77233,77235,77238,77267,77270,77273],{"className":77234},[57],[15,77236],{"className":77237,"style":154},[61],[15,77239,77241,77244],{"className":77240},[66],[15,77242,3883],{"className":77243,"style":3944},[66,67],[15,77245,77247],{"className":77246},[198],[15,77248,77250],{"className":77249},[202],[15,77251,77253],{"className":77252},[206],[15,77254,77256],{"className":77255,"style":16244},[210],[15,77257,77258,77261],{"style":213},[15,77259],{"className":77260,"style":218},[217],[15,77262,77264],{"className":77263},[222,223,224,225],[15,77265,1162],{"className":77266},[76,225],[15,77268,120],{"className":77269},[162],[15,77271,123],{"className":77272},[66,67],[15,77274,126],{"className":77275},[169],"，逐点遗憾为 ",[15,77278,77280,77293],{"className":77279,"translate":19},[18],[15,77281,77283],{"className":77282},[23],[25,77284,77285],{"xmlns":27},[29,77286,77287,77291],{},[32,77288,77289],{},[1209,77290,1804],{},[45,77292,1804],{"encoding":47},[15,77294,77296],{"className":77295,"ariaHidden":53},[52],[15,77297,77299,77302],{"className":77298},[57],[15,77300],{"className":77301,"style":1836},[61],[15,77303,1804],{"className":77304},[66],"；若不等，则所选行动的后验概率是 ",[15,77307,77309,77339],{"className":77308,"translate":19},[18],[15,77310,77312],{"className":77311},[23],[25,77313,77314],{"xmlns":27},[29,77315,77316,77336],{},[32,77317,77318,77320,77322,77324,77326,77328,77330,77332,77334],{},[35,77319,23268],{},[39,77321,1170],{},[39,77323,4305],{"stretchy":119},[35,77325,71105],{},[39,77327,335],{"separator":53},[1209,77329,1211],{},[39,77331,6600],{},[35,77333,71105],{},[39,77335,4328],{"stretchy":119},[45,77337,77338],{"encoding":47},"\\max\\{\\eta,1-\\eta\\}",[15,77340,77342,77375],{"className":77341,"ariaHidden":53},[52],[15,77343,77345,77348,77351,77354,77357,77360,77363,77366,77369,77372],{"className":77344},[57],[15,77346],{"className":77347,"style":154},[61],[15,77349,23268],{"className":77350},[1339],[15,77352,4305],{"className":77353},[162],[15,77355,71105],{"className":77356,"style":1515},[66,67],[15,77358,335],{"className":77359},[422],[15,77361],{"className":77362,"style":697},[71],[15,77364,1211],{"className":77365},[66],[15,77367],{"className":77368,"style":72},[71],[15,77370,6600],{"className":77371},[76],[15,77373],{"className":77374,"style":72},[71],[15,77376,77378,77381,77384],{"className":77377},[57],[15,77379],{"className":77380,"style":154},[61],[15,77382,71105],{"className":77383,"style":1515},[66,67],[15,77385,4328],{"className":77386},[169],"，最优行动的后验概率是 ",[15,77389,77391,77421],{"className":77390,"translate":19},[18],[15,77392,77394],{"className":77393},[23],[25,77395,77396],{"xmlns":27},[29,77397,77398,77418],{},[32,77399,77400,77402,77404,77406,77408,77410,77412,77414,77416],{},[35,77401,1178],{},[39,77403,1170],{},[39,77405,4305],{"stretchy":119},[35,77407,71105],{},[39,77409,335],{"separator":53},[1209,77411,1211],{},[39,77413,6600],{},[35,77415,71105],{},[39,77417,4328],{"stretchy":119},[45,77419,77420],{"encoding":47},"\\min\\{\\eta,1-\\eta\\}",[15,77422,77424,77457],{"className":77423,"ariaHidden":53},[52],[15,77425,77427,77430,77433,77436,77439,77442,77445,77448,77451,77454],{"className":77426},[57],[15,77428],{"className":77429,"style":154},[61],[15,77431,1178],{"className":77432},[1339],[15,77434,4305],{"className":77435},[162],[15,77437,71105],{"className":77438,"style":1515},[66,67],[15,77440,335],{"className":77441},[422],[15,77443],{"className":77444,"style":697},[71],[15,77446,1211],{"className":77447},[66],[15,77449],{"className":77450,"style":72},[71],[15,77452,6600],{"className":77453},[76],[15,77455],{"className":77456,"style":72},[71],[15,77458,77460,77463,77466],{"className":77459},[57],[15,77461],{"className":77462,"style":154},[61],[15,77464,71105],{"className":77465,"style":1515},[66,67],[15,77467,4328],{"className":77468},[169],"，两者之差为",[15,77471,77473],{"className":77472,"translate":19},[294],[15,77474,77476,77542],{"className":77475,"translate":19},[18],[15,77477,77479],{"className":77478},[23],[25,77480,77481],{"xmlns":27,"display":303},[29,77482,77483,77539],{},[32,77484,77485,77487,77489,77491,77493,77495,77497,77499,77501,77503,77505,77507,77509,77511,77513,77515,77517,77519,77521,77523,77525,77527,77529,77531,77533,77535,77537],{},[35,77486,23268],{},[39,77488,1170],{},[39,77490,4305],{"stretchy":119},[35,77492,71105],{},[39,77494,335],{"separator":53},[1209,77496,1211],{},[39,77498,6600],{},[35,77500,71105],{},[39,77502,4328],{"stretchy":119},[39,77504,6600],{},[35,77506,1178],{},[39,77508,1170],{},[39,77510,4305],{"stretchy":119},[35,77512,71105],{},[39,77514,335],{"separator":53},[1209,77516,1211],{},[39,77518,6600],{},[35,77520,71105],{},[39,77522,4328],{"stretchy":119},[39,77524,129],{},[35,77526,34425],{"mathvariant":138},[1209,77528,1263],{},[35,77530,71105],{},[39,77532,6600],{},[1209,77534,1211],{},[35,77536,34425],{"mathvariant":138},[35,77538,3571],{"mathvariant":138},[45,77540,77541],{"encoding":47},"\\max\\{\\eta,1-\\eta\\}-\\min\\{\\eta,1-\\eta\\}=|2\\eta-1|.",[15,77543,77545,77578,77599,77632,77653,77674],{"className":77544,"ariaHidden":53},[52],[15,77546,77548,77551,77554,77557,77560,77563,77566,77569,77572,77575],{"className":77547},[57],[15,77549],{"className":77550,"style":154},[61],[15,77552,23268],{"className":77553},[1339],[15,77555,4305],{"className":77556},[162],[15,77558,71105],{"className":77559,"style":1515},[66,67],[15,77561,335],{"className":77562},[422],[15,77564],{"className":77565,"style":697},[71],[15,77567,1211],{"className":77568},[66],[15,77570],{"className":77571,"style":72},[71],[15,77573,6600],{"className":77574},[76],[15,77576],{"className":77577,"style":72},[71],[15,77579,77581,77584,77587,77590,77593,77596],{"className":77580},[57],[15,77582],{"className":77583,"style":154},[61],[15,77585,71105],{"className":77586,"style":1515},[66,67],[15,77588,4328],{"className":77589},[169],[15,77591],{"className":77592,"style":72},[71],[15,77594,6600],{"className":77595},[76],[15,77597],{"className":77598,"style":72},[71],[15,77600,77602,77605,77608,77611,77614,77617,77620,77623,77626,77629],{"className":77601},[57],[15,77603],{"className":77604,"style":154},[61],[15,77606,1178],{"className":77607},[1339],[15,77609,4305],{"className":77610},[162],[15,77612,71105],{"className":77613,"style":1515},[66,67],[15,77615,335],{"className":77616},[422],[15,77618],{"className":77619,"style":697},[71],[15,77621,1211],{"className":77622},[66],[15,77624],{"className":77625,"style":72},[71],[15,77627,6600],{"className":77628},[76],[15,77630],{"className":77631,"style":72},[71],[15,77633,77635,77638,77641,77644,77647,77650],{"className":77634},[57],[15,77636],{"className":77637,"style":154},[61],[15,77639,71105],{"className":77640,"style":1515},[66,67],[15,77642,4328],{"className":77643},[169],[15,77645],{"className":77646,"style":173},[71],[15,77648,129],{"className":77649},[177],[15,77651],{"className":77652,"style":173},[71],[15,77654,77656,77659,77662,77665,77668,77671],{"className":77655},[57],[15,77657],{"className":77658,"style":154},[61],[15,77660,76983],{"className":77661},[66],[15,77663,71105],{"className":77664,"style":1515},[66,67],[15,77666],{"className":77667,"style":72},[71],[15,77669,6600],{"className":77670},[76],[15,77672],{"className":77673,"style":72},[71],[15,77675,77677,77680],{"className":77676},[57],[15,77678],{"className":77679,"style":154},[61],[15,77681,77683],{"className":77682},[66],"1∣.",[11,77685,21789,77686,77714,77715],{},[15,77687,77689,77702],{"className":77688,"translate":19},[18],[15,77690,77692],{"className":77691},[23],[25,77693,77694],{"xmlns":27},[29,77695,77696,77700],{},[32,77697,77698],{},[35,77699,446],{},[45,77701,446],{"encoding":47},[15,77703,77705],{"className":77704,"ariaHidden":53},[52],[15,77706,77708,77711],{"className":77707},[57],[15,77709],{"className":77710,"style":490},[61],[15,77712,446],{"className":77713,"style":16492},[66,67]," 取期望即得。",[15,77716,77718,77731],{"className":77717,"translate":19},[18],[15,77719,77721],{"className":77720},[23],[25,77722,77723],{"xmlns":27},[29,77724,77725,77729],{},[32,77726,77727],{},[35,77728,13509],{"mathvariant":138},[45,77730,13512],{"encoding":47},[15,77732,77734],{"className":77733,"ariaHidden":53},[52],[15,77735,77737,77740],{"className":77736},[57],[15,77738],{"className":77739,"style":13522},[61],[15,77741,13509],{"className":77742},[66,13526],[11,77744,77745,77746,77821,77822,77854,77855,77883,77884,77912,77913,77941,77942,1124],{},"这个恒等式很有解释力：只有越过决策边界的样本才贡献错误，而且每个样本的权重是 ",[15,77747,77749,77779],{"className":77748,"translate":19},[18],[15,77750,77752],{"className":77751},[23],[25,77753,77754],{"xmlns":27},[29,77755,77756,77776],{},[32,77757,77758,77760,77762,77764,77766,77768,77770,77772,77774],{},[35,77759,34425],{"mathvariant":138},[1209,77761,1263],{},[35,77763,71105],{},[39,77765,120],{"stretchy":119},[35,77767,446],{},[39,77769,126],{"stretchy":119},[39,77771,6600],{},[1209,77773,1211],{},[35,77775,34425],{"mathvariant":138},[45,77777,77778],{"encoding":47},"|2\\eta(X)-1|",[15,77780,77782,77812],{"className":77781,"ariaHidden":53},[52],[15,77783,77785,77788,77791,77794,77797,77800,77803,77806,77809],{"className":77784},[57],[15,77786],{"className":77787,"style":154},[61],[15,77789,76983],{"className":77790},[66],[15,77792,71105],{"className":77793,"style":1515},[66,67],[15,77795,120],{"className":77796},[162],[15,77798,446],{"className":77799,"style":16492},[66,67],[15,77801,126],{"className":77802},[169],[15,77804],{"className":77805,"style":72},[71],[15,77807,6600],{"className":77808},[76],[15,77810],{"className":77811,"style":72},[71],[15,77813,77815,77818],{"className":77814},[57],[15,77816],{"className":77817,"style":154},[61],[15,77819,77014],{"className":77820},[66],"。后验接近 ",[15,77823,77825,77842],{"className":77824,"translate":19},[18],[15,77826,77828],{"className":77827},[23],[25,77829,77830],{"xmlns":27},[29,77831,77832,77840],{},[32,77833,77834,77836,77838],{},[1209,77835,1211],{},[35,77837,6623],{"mathvariant":138},[1209,77839,1263],{},[45,77841,71801],{"encoding":47},[15,77843,77845],{"className":77844,"ariaHidden":53},[52],[15,77846,77848,77851],{"className":77847},[57],[15,77849],{"className":77850,"style":154},[61],[15,77852,71801],{"className":77853},[66]," 的样本位于边界附近，即使分错，代价也小；后验接近 ",[15,77856,77858,77871],{"className":77857,"translate":19},[18],[15,77859,77861],{"className":77860},[23],[25,77862,77863],{"xmlns":27},[29,77864,77865,77869],{},[32,77866,77867],{},[1209,77868,1804],{},[45,77870,1804],{"encoding":47},[15,77872,77874],{"className":77873,"ariaHidden":53},[52],[15,77875,77877,77880],{"className":77876},[57],[15,77878],{"className":77879,"style":1836},[61],[15,77881,1804],{"className":77882},[66]," 或 ",[15,77885,77887,77900],{"className":77886,"translate":19},[18],[15,77888,77890],{"className":77889},[23],[25,77891,77892],{"xmlns":27},[29,77893,77894,77898],{},[32,77895,77896],{},[1209,77897,1211],{},[45,77899,1211],{"encoding":47},[15,77901,77903],{"className":77902,"ariaHidden":53},[52],[15,77904,77906,77909],{"className":77905},[57],[15,77907],{"className":77908,"style":1836},[61],[15,77910,1211],{"className":77911},[66]," 的样本一旦分错，代价就接近 ",[15,77914,77916,77929],{"className":77915,"translate":19},[18],[15,77917,77919],{"className":77918},[23],[25,77920,77921],{"xmlns":27},[29,77922,77923,77927],{},[32,77924,77925],{},[1209,77926,1211],{},[45,77928,1211],{"encoding":47},[15,77930,77932],{"className":77931,"ariaHidden":53},[52],[15,77933,77935,77938],{"className":77934},[57],[15,77936],{"className":77937,"style":1836},[61],[15,77939,1211],{"className":77940},[66],"。换句话说，",[2040,77943,77944],{},"贝叶斯错误率不是由边界的长度决定的，而是由边界附近的后验质量决定的",[11,77946,77947,77948,78103,78104,3247],{},"回归情形也有类似的恒等式。设 ",[15,77949,77951,77993],{"className":77950,"translate":19},[18],[15,77952,77954],{"className":77953},[23],[25,77955,77956],{"xmlns":27},[29,77957,77958,77990],{},[32,77959,77960,77966,77968,77970,77972,77974,77976,77978,77980,77982,77984,77986,77988],{},[131,77961,77962,77964],{},[35,77963,116],{},[39,77965,1162],{},[39,77967,120],{"stretchy":119},[35,77969,123],{},[39,77971,126],{"stretchy":119},[39,77973,129],{},[35,77975,62377],{"mathvariant":454},[39,77977,1194],{"stretchy":119},[35,77979,61881],{},[39,77981,34425],{},[35,77983,446],{},[39,77985,129],{},[35,77987,123],{},[39,77989,1266],{"stretchy":119},[45,77991,77992],{"encoding":47},"f^*(x)=\\mathbb E[Y\\mid X=x]",[15,77994,77996,78049,78073,78091],{"className":77995,"ariaHidden":53},[52],[15,77997,77999,78002,78031,78034,78037,78040,78043,78046],{"className":77998},[57],[15,78000],{"className":78001,"style":154},[61],[15,78003,78005,78008],{"className":78004},[66],[15,78006,116],{"className":78007,"style":158},[66,67],[15,78009,78011],{"className":78010},[198],[15,78012,78014],{"className":78013},[202],[15,78015,78017],{"className":78016},[206],[15,78018,78020],{"className":78019,"style":16244},[210],[15,78021,78022,78025],{"style":213},[15,78023],{"className":78024,"style":218},[217],[15,78026,78028],{"className":78027},[222,223,224,225],[15,78029,1162],{"className":78030},[76,225],[15,78032,120],{"className":78033},[162],[15,78035,123],{"className":78036},[66,67],[15,78038,126],{"className":78039},[169],[15,78041],{"className":78042,"style":173},[71],[15,78044,129],{"className":78045},[177],[15,78047],{"className":78048,"style":173},[71],[15,78050,78052,78055,78058,78061,78064,78067,78070],{"className":78051},[57],[15,78053],{"className":78054,"style":154},[61],[15,78056,62377],{"className":78057},[66,517],[15,78059,1194],{"className":78060},[162],[15,78062,61881],{"className":78063,"style":72},[66,67],[15,78065],{"className":78066,"style":173},[71],[15,78068,34425],{"className":78069},[177],[15,78071],{"className":78072,"style":173},[71],[15,78074,78076,78079,78082,78085,78088],{"className":78075},[57],[15,78077],{"className":78078,"style":490},[61],[15,78080,446],{"className":78081,"style":16492},[66,67],[15,78083],{"className":78084,"style":173},[71],[15,78086,129],{"className":78087},[177],[15,78089],{"className":78090,"style":173},[71],[15,78092,78094,78097,78100],{"className":78093},[57],[15,78095],{"className":78096,"style":154},[61],[15,78098,123],{"className":78099},[66,67],[15,78101,1266],{"className":78102},[169],"，则对任意 ",[15,78105,78107,78126],{"className":78106,"translate":19},[18],[15,78108,78110],{"className":78109},[23],[25,78111,78112],{"xmlns":27},[29,78113,78114,78123],{},[32,78115,78116],{},[4294,78117,78118,78120],{"accent":53},[35,78119,116],{},[39,78121,78122],{},"^",[45,78124,78125],{"encoding":47},"\\hat 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E\\big[(Y-\\hat f(X))^2\\big]-\\mathbb E\\big[(Y-f^*(X))^2\\big]\\\\\n&=\\mathbb E\\big[(\\hat f(X)-f^*(X))^2\\big]\n+2\\,\\mathbb E\\big[(Y-f^*(X))(f^*(X)-\\hat 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Name":79062},[1422,1508],[15,79064,79066,79069],{"style":79065},"top:-2.1421em;",[15,79067],{"className":79068,"style":1368},[217],[15,79070,79072,79075,79078,79081,79084,79087,79093,79096,79138,79141,79144,79147,79150,79153,79156,79185,79188,79191,79194,79223,79229,79232,79235,79238,79241,79244,79247,79253,79256,79259,79262,79265,79268,79297,79300,79303,79306,79309,79338,79341,79344,79347,79350,79353,79356,79398,79401,79404,79407,79413],{"className":79071},[66],[15,79073],{"className":79074},[66],[15,79076],{"className":79077,"style":173},[71],[15,79079,129],{"className":79080},[177],[15,79082],{"className":79083,"style":173},[71],[15,79085,62377],{"className":79086},[66,517],[15,79088,79090],{"className":79089},[66],[15,79091,1194],{"className":79092},[1422,1508],[15,79094,120],{"className":79095},[162],[15,79097,79099],{"className":79098},[66,78139],[15,79100,79102,79130],{"className":79101},[202,1090],[15,79103,79105,79127],{"className":79104},[206],[15,79106,79108,79116],{"className":79107,"style":78149},[210],[15,79109,79110,79113],{"style":1386},[15,79111],{"className":79112,"style":1368},[217],[15,79114,116],{"className":79115,"style":158},[66,67],[15,79117,79118,79121],{"style":78160},[15,79119],{"className":79120,"style":1368},[217],[15,79122,79124],{"className":79123,"style":78168},[78167],[15,79125,78122],{"className":79126},[66],[15,79128,1114],{"className":79129},[1113],[15,79131,79133],{"className":79132},[206],[15,79134,79136],{"className":79135,"style":4448},[210],[15,79137],{},[15,79139,120],{"className":79140},[162],[15,79142,446],{"className":79143,"style":16492},[66,67],[15,79145,126],{"className":79146},[169],[15,79148],{"className":79149,"style":72},[71],[15,79151,6600],{"className":79152},[76],[15,79154],{"className":79155,"style":72},[71],[15,79157,79159,79162],{"className":79158},[66],[15,79160,116],{"className":79161,"style":158},[66,67],[15,79163,79165],{"className":79164},[198],[15,79166,79168],{"className":79167},[202],[15,79169,79171],{"className":79170},[206],[15,79172,79174],{"className":79173,"style":1308},[210],[15,79175,79176,79179],{"style":397},[15,79177],{"className":79178,"style":218},[217],[15,79180,79182],{"className":79181},[222,223,224,225],[15,79183,1162],{"className":79184},[76,225],[15,79186,120],{"className":79187},[162],[15,79189,446],{"className":79190,"style":16492},[66,67],[15,79192,126],{"className":79193},[169],[15,79195,79197,79200],{"className":79196},[169],[15,79198,126],{"className":79199},[169],[15,79201,79203],{"className":79202},[198],[15,79204,79206],{"className":79205},[202],[15,79207,79209],{"className":79208},[206],[15,79210,79212],{"className":79211,"style":1650},[210],[15,79213,79214,79217],{"style":397},[15,79215],{"className":79216,"style":218},[217],[15,79218,79220],{"className":79219},[222,223,224,225],[15,79221,1263],{"className":79222},[66,225],[15,79224,79226],{"className":79225},[66],[15,79227,1266],{"className":79228},[1422,1508],[15,79230],{"className":79231,"style":72},[71],[15,79233,1245],{"className":79234},[76],[15,79236],{"className":79237,"style":72},[71],[15,79239,1263],{"className":79240},[66],[15,79242],{"className":79243,"style":697},[71],[15,79245,62377],{"className":79246},[66,517],[15,79248,79250],{"className":79249},[66],[15,79251,1194],{"className":79252},[1422,1508],[15,79254,120],{"className":79255},[162],[15,79257,61881],{"className":79258,"style":72},[66,67],[15,79260],{"className":79261,"style":72},[71],[15,79263,6600],{"className":79264},[76],[15,79266],{"className":79267,"style":72},[71],[15,79269,79271,79274],{"className":79270},[66],[15,79272,116],{"className":79273,"style":158},[66,67],[15,79275,79277],{"className":79276},[198],[15,79278,79280],{"className":79279},[202],[15,79281,79283],{"className":79282},[206],[15,79284,79286],{"className":79285,"style":1308},[210],[15,79287,79288,79291],{"style":397},[15,79289],{"className":79290,"style":218},[217],[15,79292,79294],{"className":79293},[222,223,224,225],[15,79295,1162],{"className":79296},[76,225],[15,79298,120],{"className":79299},[162],[15,79301,446],{"className":79302,"style":16492},[66,67],[15,79304,18766],{"className":79305},[169],[15,79307,120],{"className":79308},[162],[15,79310,79312,79315],{"className":79311},[66],[15,79313,116],{"className":79314,"style":158},[66,67],[15,79316,79318],{"className":79317},[198],[15,79319,79321],{"className":79320},[202],[15,79322,79324],{"className":79323},[206],[15,79325,79327],{"className":79326,"style":1308},[210],[15,79328,79329,79332],{"style":397},[15,79330],{"className":79331,"style":218},[217],[15,79333,79335],{"className":79334},[222,223,224,225],[15,79336,1162],{"className":79337},[76,225],[15,79339,120],{"className":79340},[162],[15,79342,446],{"className":79343,"style":16492},[66,67],[15,79345,126],{"className":79346},[169],[15,79348],{"className":79349,"style":72},[71],[15,79351,6600],{"className":79352},[76],[15,79354],{"className":79355,"style":72},[71],[15,79357,79359],{"className":79358},[66,78139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",[15,79431,79433,79446],{"className":79432,"translate":19},[18],[15,79434,79436],{"className":79435},[23],[25,79437,79438],{"xmlns":27},[29,79439,79440,79444],{},[32,79441,79442],{},[35,79443,446],{},[45,79445,446],{"encoding":47},[15,79447,79449],{"className":79448,"ariaHidden":53},[52],[15,79450,79452,79455],{"className":79451},[57],[15,79453],{"className":79454,"style":490},[61],[15,79456,446],{"className":79457,"style":16492},[66,67]," 取条件期望后，",[15,79460,79462,79504],{"className":79461,"translate":19},[18],[15,79463,79465],{"className":79464},[23],[25,79466,79467],{"xmlns":27},[29,79468,79469,79501],{},[32,79470,79471,79473,79475,79477,79479,79485,79487,79489,79491,79493,79495,79497,79499],{},[35,79472,62377],{"mathvariant":454},[39,79474,1194],{"stretchy":119},[35,79476,61881],{},[39,79478,6600],{},[131,79480,79481,79483],{},[35,79482,116],{},[39,79484,1162],{},[39,79486,120],{"stretchy":119},[35,79488,446],{},[39,79490,126],{"stretchy":119},[39,79492,34425],{},[35,79494,446],{},[39,79496,1266],{"stretchy":119},[39,79498,129],{},[1209,79500,1804],{},[45,79502,79503],{"encoding":47},"\\mathbb E[Y-f^*(X)\\mid X]=0",[15,79505,79507,79531,79584,79605],{"className":79506,"ariaHidden":53},[52],[15,79508,79510,79513,79516,79519,79522,79525,79528],{"className":79509},[57],[15,79511],{"className":79512,"style":154},[61],[15,79514,62377],{"className":79515},[66,517],[15,79517,1194],{"className":79518},[162],[15,79520,61881],{"className":79521,"style":72},[66,67],[15,79523],{"className":79524,"style":72},[71],[15,79526,6600],{"className":79527},[76],[15,79529],{"className":79530,"style":72},[71],[15,79532,79534,79537,79566,79569,79572,79575,79578,79581],{"className":79533},[57],[15,79535],{"className":79536,"style":154},[61],[15,79538,79540,79543],{"className":79539},[66],[15,79541,116],{"className":79542,"style":158},[66,67],[15,79544,79546],{"className":79545},[198],[15,79547,79549],{"className":79548},[202],[15,79550,79552],{"className":79551},[206],[15,79553,79555],{"className":79554,"style":16244},[210],[15,79556,79557,79560],{"style":213},[15,79558],{"className":79559,"style":218},[217],[15,79561,79563],{"className":79562},[222,223,224,225],[15,79564,1162],{"className":79565},[76,225],[15,79567,120],{"className":79568},[162],[15,79570,446],{"className":79571,"style":16492},[66,67],[15,79573,126],{"className":79574},[169],[15,79576],{"className":79577,"style":173},[71],[15,79579,34425],{"className":79580},[177],[15,79582],{"className":79583,"style":173},[71],[15,79585,79587,79590,79593,79596,79599,79602],{"className":79586},[57],[15,79588],{"className":79589,"style":154},[61],[15,79591,446],{"className":79592,"style":16492},[66,67],[15,79594,1266],{"className":79595},[169],[15,79597],{"className":79598,"style":173},[71],[15,79600,129],{"className":79601},[177],[15,79603],{"className":79604,"style":173},[71],[15,79606,79608,79611],{"className":79607},[57],[15,79609],{"className":79610,"style":1836},[61],[15,79612,1804],{"className":79613},[66],"，因此消失。",[15,79616,79618,79631],{"className":79617,"translate":19},[18],[15,79619,79621],{"className":79620},[23],[25,79622,79623],{"xmlns":27},[29,79624,79625,79629],{},[32,79626,79627],{},[35,79628,13509],{"mathvariant":138},[45,79630,13512],{"encoding":47},[15,79632,79634],{"className":79633,"ariaHidden":53},[52],[15,79635,79637,79640],{"className":79636},[57],[15,79638],{"className":79639,"style":13522},[61],[15,79641,13509],{"className":79642},[66,13526],[11,79644,79645],{},"这两个恒等式把「模型误差」翻译成了「决策遗憾」：分类器的错误只发生在越过边界的点上，回归器的误差则是函数逼近误差的平方。贝叶斯风险就是那个无法被任何模型消除的起点。",[1904,79647,79649],{"id":79648},"_37-map-不是贝叶斯行动","3.7 MAP 不是贝叶斯行动",[11,79651,79652],{},"这是最常见的混淆之一。MAP 是后验的众数，是一个「点」；贝叶斯行动是使后验期望损失最小的动作，依赖损失函数。两者只有在参数空间上取 0-1 损失时才重合。",[11,79654,79655,79656,79722,79723,79788,79789,79817,79818,79884],{},"更麻烦的是，MAP 对重参数化不协变。若 ",[15,79657,79659,79683],{"className":79658,"translate":19},[18],[15,79660,79662],{"className":79661},[23],[25,79663,79664],{"xmlns":27},[29,79665,79666,79680],{},[32,79667,79668,79670,79672,79674,79676,79678],{},[35,79669,17089],{},[39,79671,129],{},[35,79673,20177],{},[39,79675,120],{"stretchy":119},[35,79677,61614],{},[39,79679,126],{"stretchy":119},[45,79681,79682],{"encoding":47},"\\phi=h(\\theta)",[15,79684,79686,79704],{"className":79685,"ariaHidden":53},[52],[15,79687,79689,79692,79695,79698,79701],{"className":79688},[57],[15,79690],{"className":79691,"style":3406},[61],[15,79693,17089],{"className":79694},[66,67],[15,79696],{"className":79697,"style":173},[71],[15,79699,129],{"className":79700},[177],[15,79702],{"className":79703,"style":173},[71],[15,79705,79707,79710,79713,79716,79719],{"className":79706},[57],[15,79708],{"className":79709,"style":154},[61],[15,79711,20177],{"className":79712},[66,67],[15,79714,120],{"className":79715},[162],[15,79717,61614],{"className":79718,"style":23020},[66,67],[15,79720,126],{"className":79721},[169]," 是一个非线性变换，",[15,79724,79726,79749],{"className":79725,"translate":19},[18],[15,79727,79729],{"className":79728},[23],[25,79730,79731],{"xmlns":27},[29,79732,79733,79747],{},[32,79734,79735,79737,79739,79741,79743,79745],{},[35,79736,11],{},[39,79738,120],{"stretchy":119},[35,79740,61614],{},[39,79742,34425],{},[35,79744,32453],{},[39,79746,126],{"stretchy":119},[45,79748,65456],{"encoding":47},[15,79750,79752,79776],{"className":79751,"ariaHidden":53},[52],[15,79753,79755,79758,79761,79764,79767,79770,79773],{"className":79754},[57],[15,79756],{"className":79757,"style":154},[61],[15,79759,11],{"className":79760},[66,67],[15,79762,120],{"className":79763},[162],[15,79765,61614],{"className":79766,"style":23020},[66,67],[15,79768],{"className":79769,"style":173},[71],[15,79771,34425],{"className":79772},[177],[15,79774],{"className":79775,"style":173},[71],[15,79777,79779,79782,79785],{"className":79778},[57],[15,79780],{"className":79781,"style":154},[61],[15,79783,32453],{"className":79784,"style":23020},[66,67],[15,79786,126],{"className":79787},[169]," 的众数经过 ",[15,79790,79792,79805],{"className":79791,"translate":19},[18],[15,79793,79795],{"className":79794},[23],[25,79796,79797],{"xmlns":27},[29,79798,79799,79803],{},[32,79800,79801],{},[35,79802,20177],{},[45,79804,20177],{"encoding":47},[15,79806,79808],{"className":79807,"ariaHidden":53},[52],[15,79809,79811,79814],{"className":79810},[57],[15,79812],{"className":79813,"style":840},[61],[15,79815,20177],{"className":79816},[66,67]," 之后，一般不是 ",[15,79819,79821,79845],{"className":79820,"translate":19},[18],[15,79822,79824],{"className":79823},[23],[25,79825,79826],{"xmlns":27},[29,79827,79828,79842],{},[32,79829,79830,79832,79834,79836,79838,79840],{},[35,79831,11],{},[39,79833,120],{"stretchy":119},[35,79835,17089],{},[39,79837,34425],{},[35,79839,32453],{},[39,79841,126],{"stretchy":119},[45,79843,79844],{"encoding":47},"p(\\phi\\mid D)",[15,79846,79848,79872],{"className":79847,"ariaHidden":53},[52],[15,79849,79851,79854,79857,79860,79863,79866,79869],{"className":79850},[57],[15,79852],{"className":79853,"style":154},[61],[15,79855,11],{"className":79856},[66,67],[15,79858,120],{"className":79859},[162],[15,79861,17089],{"className":79862},[66,67],[15,79864],{"className":79865,"style":173},[71],[15,79867,34425],{"className":79868},[177],[15,79870],{"className":79871,"style":173},[71],[15,79873,79875,79878,79881],{"className":79874},[57],[15,79876],{"className":79877,"style":154},[61],[15,79879,32453],{"className":79880,"style":23020},[66,67],[15,79882,126],{"className":79883},[169]," 的众数，因为密度会多出一个 Jacobian 因子。后验中位数在单调变换下等变，后验均值在仿射变换下等变，而 MAP 连这个性质都没有。这也是为什么在贝叶斯推断里，除非有明确的 0-1 损失，否则「取众数」并不是一个默认正确的选择。",[95,79886,79888],{"id":79887},"_4-贝叶斯错误率","4. 贝叶斯错误率",[1904,79890,79892],{"id":79891},"_41-两类情形全变差与重叠面积","4.1 两类情形：全变差与重叠面积",[11,79894,79895,79896,80020,80021,80090],{},"设两类先验相等，类条件密度为 ",[15,79897,79899,79925],{"className":79898,"translate":19},[18],[15,79900,79902],{"className":79901},[23],[25,79903,79904],{"xmlns":27},[29,79905,79906,79922],{},[32,79907,79908,79914,79916],{},[1057,79909,79910,79912],{},[35,79911,11],{},[1209,79913,1804],{},[39,79915,335],{"separator":53},[1057,79917,79918,79920],{},[35,79919,11],{},[1209,79921,1211],{},[45,79923,79924],{"encoding":47},"p_0,p_1",[15,79926,79928],{"className":79927,"ariaHidden":53},[52],[15,79929,79931,79934,79974,79977,79980],{"className":79930},[57],[15,79932],{"className":79933,"style":471},[61],[15,79935,79937,79940],{"className":79936},[66],[15,79938,11],{"className":79939},[66,67],[15,79941,79943],{"className":79942},[198],[15,79944,79946,79966],{"className":79945},[202,1090],[15,79947,79949,79963],{"className":79948},[206],[15,79950,79952],{"className":79951,"style":2139},[210],[15,79953,79954,79957],{"style":1584},[15,79955],{"className":79956,"style":218},[217],[15,79958,79960],{"className":79959},[222,223,224,225],[15,79961,1804],{"className":79962},[66,225],[15,79964,1114],{"className":79965},[1113],[15,79967,79969],{"className":79968},[206],[15,79970,79972],{"className":79971,"style":1121},[210],[15,79973],{},[15,79975,335],{"className":79976},[422],[15,79978],{"className":79979,"style":697},[71],[15,79981,79983,79986],{"className":79982},[66],[15,79984,11],{"className":79985},[66,67],[15,79987,79989],{"className":79988},[198],[15,79990,79992,80012],{"className":79991},[202,1090],[15,79993,79995,80009],{"className":79994},[206],[15,79996,79998],{"className":79997,"style":2139},[210],[15,79999,80000,80003],{"style":1584},[15,80001],{"className":80002,"style":218},[217],[15,80004,80006],{"className":80005},[222,223,224,225],[15,80007,1211],{"className":80008},[66,225],[15,80010,1114],{"className":80011},[1113],[15,80013,80015],{"className":80014},[206],[15,80016,80018],{"className":80017,"style":1121},[210],[15,80019],{},"。由 3.2 节，贝叶斯规则在 ",[15,80022,80024,80051],{"className":80023,"translate":19},[18],[15,80025,80027],{"className":80026},[23],[25,80028,80029],{"xmlns":27},[29,80030,80031,80049],{},[32,80032,80033,80035,80037,80039,80041,80043,80045,80047],{},[35,80034,71105],{},[39,80036,120],{"stretchy":119},[35,80038,123],{},[39,80040,126],{"stretchy":119},[39,80042,129],{},[1209,80044,1211],{},[35,80046,6623],{"mathvariant":138},[1209,80048,1263],{},[45,80050,71836],{"encoding":47},[15,80052,80054,80081],{"className":80053,"ariaHidden":53},[52],[15,80055,80057,80060,80063,80066,80069,80072,80075,80078],{"className":80056},[57],[15,80058],{"className":80059,"style":154},[61],[15,80061,71105],{"className":80062,"style":1515},[66,67],[15,80064,120],{"className":80065},[162],[15,80067,123],{"className":80068},[66,67],[15,80070,126],{"className":80071},[169],[15,80073],{"className":80074,"style":173},[71],[15,80076,129],{"className":80077},[177],[15,80079],{"className":80080,"style":173},[71],[15,80082,80084,80087],{"className":80083},[57],[15,80085],{"className":80086,"style":154},[61],[15,80088,71801],{"className":80089},[66]," 处切换，其中",[15,80092,80094],{"className":80093,"translate":19},[294],[15,80095,80097,80165],{"className":80096,"translate":19},[18],[15,80098,80100],{"className":80099},[23],[25,80101,80102],{"xmlns":27,"display":303},[29,80103,80104,80162],{},[32,80105,80106,80108,80110,80112,80114,80116,80160],{},[35,80107,71105],{},[39,80109,120],{"stretchy":119},[35,80111,123],{},[39,80113,126],{"stretchy":119},[39,80115,129],{},[6465,80117,80118,80132],{},[32,80119,80120,80126,80128,80130],{},[1057,80121,80122,80124],{},[35,80123,11],{},[1209,80125,1211],{},[39,80127,120],{"stretchy":119},[35,80129,123],{},[39,80131,126],{"stretchy":119},[32,80133,80134,80140,80142,80144,80146,80148,80154,80156,80158],{},[1057,80135,80136,80138],{},[35,80137,11],{},[1209,80139,1804],{},[39,80141,120],{"stretchy":119},[35,80143,123],{},[39,80145,126],{"stretchy":119},[39,80147,1245],{},[1057,80149,80150,80152],{},[35,80151,11],{},[1209,80153,1211],{},[39,80155,120],{"stretchy":119},[35,80157,123],{},[39,80159,126],{"stretchy":119},[35,80161,3571],{"mathvariant":138},[45,80163,80164],{"encoding":47},"\\eta(x)=\\frac{p_1(x)}{p_0(x)+p_1(x)}.",[15,80166,80168,80195],{"className":80167,"ariaHidden":53},[52],[15,80169,80171,80174,80177,80180,80183,80186,80189,80192],{"className":80170},[57],[15,80172],{"className":80173,"style":154},[61],[15,80175,71105],{"className":80176,"style":1515},[66,67],[15,80178,120],{"className":80179},[162],[15,80181,123],{"className":80182},[66,67],[15,80184,126],{"className":80185},[169],[15,80187],{"className":80188,"style":173},[71],[15,80190,129],{"className":80191},[177],[15,80193],{"className":80194,"style":173},[71],[15,80196,80198,80201,80413],{"className":80197},[57],[15,80199],{"className":80200,"style":41187},[61],[15,80202,80204,80207,80410],{"className":80203},[66],[15,80205],{"className":80206},[162,23412],[15,80208,80210],{"className":80209},[6465],[15,80211,80213,80402],{"className":80212},[202,1090],[15,80214,80216,80399],{"className":80215},[206],[15,80217,80219,80334,80342],{"className":80218,"style":41206},[210],[15,80220,80221,80224],{"style":23428},[15,80222],{"className":80223,"style":1368},[217],[15,80225,80227,80267,80270,80273,80276,80279,80282,80285,80325,80328,80331],{"className":80226},[66],[15,80228,80230,80233],{"className":80229},[66],[15,80231,11],{"className":80232},[66,67],[15,80234,80236],{"className":80235},[198],[15,80237,80239,80259],{"className":80238},[202,1090],[15,80240,80242,80256],{"className":80241},[206],[15,80243,80245],{"className":80244,"style":2139},[210],[15,80246,80247,80250],{"style":1584},[15,80248],{"className":80249,"style":218},[217],[15,80251,80253],{"className":80252},[222,223,224,225],[15,80254,1804],{"className":80255},[66,225],[15,80257,1114],{"className":80258},[1113],[15,80260,80262],{"className":80261},[206],[15,80263,80265],{"className":80264,"style":1121},[210],[15,80266],{},[15,80268,120],{"className":80269},[162],[15,80271,123],{"className":80272},[66,67],[15,80274,126],{"className":80275},[169],[15,80277],{"className":80278,"style":72},[71],[15,80280,1245],{"className":80281},[76],[15,80283],{"className":80284,"style":72},[71],[15,80286,80288,80291],{"className":80287},[66],[15,80289,11],{"className":80290},[66,67],[15,80292,80294],{"className":80293},[198],[15,80295,80297,80317],{"className":80296},[202,1090],[15,80298,80300,80314],{"className":80299},[206],[15,80301,80303],{"className":80302,"style":2139},[210],[15,80304,80305,80308],{"style":1584},[15,80306],{"className":80307,"style":218},[217],[15,80309,80311],{"className":80310},[222,223,224,225],[15,80312,1211],{"className":80313},[66,225],[15,80315,1114],{"className":80316},[1113],[15,80318,80320],{"className":80319},[206],[15,80321,80323],{"className":80322,"style":1121},[210],[15,80324],{},[15,80326,120],{"className":80327},[162],[15,80329,123],{"className":80330},[66,67],[15,80332,126],{"className":80333},[169],[15,80335,80336,80339],{"style":23440},[15,80337],{"className":80338,"style":1368},[217],[15,80340],{"className":80341,"style":4438},[23447],[15,80343,80344,80347],{"style":23450},[15,80345],{"className":80346,"style":1368},[217],[15,80348,80350,80390,80393,80396],{"className":80349},[66],[15,80351,80353,80356],{"className":80352},[66],[15,80354,11],{"className":80355},[66,67],[15,80357,80359],{"className":80358},[198],[15,80360,80362,80382],{"className":80361},[202,1090],[15,80363,80365,80379],{"className":80364},[206],[15,80366,80368],{"className":80367,"style":2139},[210],[15,80369,80370,80373],{"style":1584},[15,80371],{"className":80372,"style":218},[217],[15,80374,80376],{"className":80375},[222,223,224,225],[15,80377,1211],{"className":80378},[66,225],[15,80380,1114],{"className":80381},[1113],[15,80383,80385],{"className":80384},[206],[15,80386,80388],{"className":80387,"style":1121},[210],[15,80389],{},[15,80391,120],{"className":80392},[162],[15,80394,123],{"className":80395},[66,67],[15,80397,126],{"className":80398},[169],[15,80400,1114],{"className":80401},[1113],[15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\\min\\left\\{\\tfrac12 p_0(x),\\ \\tfrac12 p_1(x)\\right\\}dx\\\\\n&=\\frac12\\int \\min\\{p_0(x),p_1(x)\\}\\,dx 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是全变差距离。公式的几何意义很直接：两个类条件分布的重叠面积越大，错误率越高；当两个分布完全重合时 ",[15,82729,82731,82759],{"className":82730,"translate":19},[18],[15,82732,82734],{"className":82733},[23],[25,82735,82736],{"xmlns":27},[29,82737,82738,82756],{},[32,82739,82740,82748,82750,82752,82754],{},[1255,82741,82742,82744,82746],{},[35,82743,11],{},[35,82745,48378],{},[39,82747,1162],{},[39,82749,129],{},[1209,82751,1211],{},[35,82753,6623],{"mathvariant":138},[1209,82755,1263],{},[45,82757,82758],{"encoding":47},"p_e^*=1\u002F2",[15,82760,82762,82829],{"className":82761,"ariaHidden":53},[52],[15,82763,82765,82769,82820,82823,82826],{"className":82764},[57],[15,82766],{"className":82767,"style":82768},[61],"height:0.9357em;vertical-align:-0.247em;",[15,82770,82772,82775],{"className":82771},[66],[15,82773,11],{"className":82774},[66,67],[15,82776,82778],{"className":82777},[198],[15,82779,82781,82812],{"className":82780},[202,1090],[15,82782,82784,82809],{"className":82783},[206],[15,82785,82787,82798],{"className":82786,"style":16244},[210],[15,82788,82789,82792],{"style":1653},[15,82790],{"className":82791,"style":218},[217],[15,82793,82795],{"className":82794},[222,223,224,225],[15,82796,48378],{"className":82797},[66,67,225],[15,82799,82800,82803],{"style":213},[15,82801],{"className":82802,"style":218},[217],[15,82804,82806],{"className":82805},[222,223,224,225],[15,82807,1162],{"className":82808},[76,225],[15,82810,1114],{"className":82811},[1113],[15,82813,82815],{"className":82814},[206],[15,82816,82818],{"className":82817,"style":1686},[210],[15,82819],{},[15,82821],{"className":82822,"style":173},[71],[15,82824,129],{"className":82825},[177],[15,82827],{"className":82828,"style":173},[71],[15,82830,82832,82835],{"className":82831},[57],[15,82833],{"className":82834,"style":154},[61],[15,82836,71801],{"className":82837},[66],"，当支撑集不相交时 ",[15,82840,82842,82866],{"className":82841,"translate":19},[18],[15,82843,82845],{"className":82844},[23],[25,82846,82847],{"xmlns":27},[29,82848,82849,82863],{},[32,82850,82851,82859,82861],{},[1255,82852,82853,82855,82857],{},[35,82854,11],{},[35,82856,48378],{},[39,82858,1162],{},[39,82860,129],{},[1209,82862,1804],{},[45,82864,82865],{"encoding":47},"p_e^*=0",[15,82867,82869,82935],{"className":82868,"ariaHidden":53},[52],[15,82870,82872,82875,82926,82929,82932],{"className":82871},[57],[15,82873],{"className":82874,"style":82768},[61],[15,82876,82878,82881],{"className":82877},[66],[15,82879,11],{"className":82880},[66,67],[15,82882,82884],{"className":82883},[198],[15,82885,82887,82918],{"className":82886},[202,1090],[15,82888,82890,82915],{"className":82889},[206],[15,82891,82893,82904],{"className":82892,"style":16244},[210],[15,82894,82895,82898],{"style":1653},[15,82896],{"className":82897,"style":218},[217],[15,82899,82901],{"className":82900},[222,223,224,225],[15,82902,48378],{"className":82903},[66,67,225],[15,82905,82906,82909],{"style":213},[15,82907],{"className":82908,"style":218},[217],[15,82910,82912],{"className":82911},[222,223,224,225],[15,82913,1162],{"className":82914},[76,225],[15,82916,1114],{"className":82917},[1113],[15,82919,82921],{"className":82920},[206],[15,82922,82924],{"className":82923,"style":1686},[210],[15,82925],{},[15,82927],{"className":82928,"style":173},[71],[15,82930,129],{"className":82931},[177],[15,82933],{"className":82934,"style":173},[71],[15,82936,82938,82941],{"className":82937},[57],[15,82939],{"className":82940,"style":1836},[61],[15,82942,1804],{"className":82943},[66],[11,82945,82946],{},"若先验不相等，公式改为",[15,82948,82950],{"className":82949,"translate":19},[294],[15,82951,82953,83033],{"className":82952,"translate":19},[18],[15,82954,82956],{"className":82955},[23],[25,82957,82958],{"xmlns":27,"display":303},[29,82959,82960,83030],{},[32,82961,82962,82970,82972,82974,82976,82978,82980,82986,82992,82994,82996,82998,83000,83002,83008,83014,83016,83018,83020,83022,83024,83026,83028],{},[1255,82963,82964,82966,82968],{},[35,82965,11],{},[35,82967,48378],{},[39,82969,1162],{},[39,82971,129],{},[39,82973,29727],{},[35,82975,1178],{},[39,82977,1170],{},[39,82979,4305],{"stretchy":119},[1057,82981,82982,82984],{},[35,82983,33489],{},[1209,82985,1804],{},[1057,82987,82988,82990],{},[35,82989,11],{},[1209,82991,1804],{},[39,82993,120],{"stretchy":119},[35,82995,123],{},[39,82997,126],{"stretchy":119},[39,82999,335],{"separator":53},[1274,83001,24228],{},[1057,83003,83004,83006],{},[35,83005,33489],{},[1209,83007,1211],{},[1057,83009,83010,83012],{},[35,83011,11],{},[1209,83013,1211],{},[39,83015,120],{"stretchy":119},[35,83017,123],{},[39,83019,126],{"stretchy":119},[39,83021,4328],{"stretchy":119},[1274,83023,2462],{},[35,83025,458],{},[35,83027,123],{},[39,83029,335],{"separator":53},[45,83031,83032],{"encoding":47},"p_e^*=\\int \\min\\{\\pi_0p_0(x),\\ \\pi_1p_1(x)\\}\\,dx,",[15,83034,83036,83102],{"className":83035,"ariaHidden":53},[52],[15,83037,83039,83042,83093,83096,83099],{"className":83038},[57],[15,83040],{"className":83041,"style":82015},[61],[15,83043,83045,83048],{"className":83044},[66],[15,83046,11],{"className":83047},[66,67],[15,83049,83051],{"className":83050},[198],[15,83052,83054,83085],{"className":83053},[202,1090],[15,83055,83057,83082],{"className":83056},[206],[15,83058,83060,83071],{"className":83059,"style":1308},[210],[15,83061,83062,83065],{"style":1653},[15,83063],{"className":83064,"style":218},[217],[15,83066,83068],{"className":83067},[222,223,224,225],[15,83069,48378],{"className":83070},[66,67,225],[15,83072,83073,83076],{"style":397},[15,83074],{"className":83075,"style":218},[217],[15,83077,83079],{"className":83078},[222,223,224,225],[15,83080,1162],{"className":83081},[76,225],[15,83083,1114],{"className":83084},[1113],[15,83086,83088],{"className":83087},[206],[15,83089,83091],{"className":83090,"style":1686},[210],[15,83092],{},[15,83094],{"className":83095,"style":173},[71],[15,83097,129],{"className":83098},[177],[15,83100],{"className":83101,"style":173},[71],[15,83103,83105,83108,83111,83114,83117,83120,83160,83200,83203,83206,83209,83212,83215,83218,83258,83298,83301,83304,83307,83310,83313,83316],{"className":83104},[57],[15,83106],{"className":83107,"style":65126},[61],[15,83109,29727],{"className":83110,"style":29791},[1339,1470,1471],[15,83112],{"className":83113,"style":697},[71],[15,83115,1178],{"className":83116},[1339],[15,83118,4305],{"className":83119},[162],[15,83121,83123,83126],{"className":83122},[66],[15,83124,33489],{"className":83125,"style":1515},[66,67],[15,83127,83129],{"className":83128},[198],[15,83130,83132,83152],{"className":83131},[202,1090],[15,83133,83135,83149],{"className":83134},[206],[15,83136,83138],{"className":83137,"style":2139},[210],[15,83139,83140,83143],{"style":1531},[15,83141],{"className":83142,"style":218},[217],[15,83144,83146],{"className":83145},[222,223,224,225],[15,83147,1804],{"className":83148},[66,225],[15,83150,1114],{"className":83151},[1113],[15,83153,83155],{"className":83154},[206],[15,83156,83158],{"className":83157,"style":1121},[210],[15,83159],{},[15,83161,83163,83166],{"className":83162},[66],[15,83164,11],{"className":83165},[66,67],[15,83167,83169],{"className":83168},[198],[15,83170,83172,83192],{"className":83171},[202,1090],[15,83173,83175,83189],{"className":83174},[206],[15,83176,83178],{"className":83177,"style":2139},[210],[15,83179,83180,83183],{"style":1584},[15,83181],{"className":83182,"style":218},[217],[15,83184,83186],{"className":83185},[222,223,224,225],[15,83187,1804],{"className":83188},[66,225],[15,83190,1114],{"className":83191},[1113],[15,83193,83195],{"className":83194},[206],[15,83196,83198],{"className":83197,"style":1121},[210],[15,83199],{},[15,83201,120],{"className":83202},[162],[15,83204,123],{"className":83205},[66,67],[15,83207,126],{"className":83208},[169],[15,83210,335],{"className":83211},[422],[15,83213,24228],{"className":83214},[71],[15,83216],{"className":83217,"style":697},[71],[15,83219,83221,83224],{"className":83220},[66],[15,83222,33489],{"className":83223,"style":1515},[66,67],[15,83225,83227],{"className":83226},[198],[15,83228,83230,83250],{"className":83229},[202,1090],[15,83231,83233,83247],{"className":83232},[206],[15,83234,83236],{"className":83235,"style":2139},[210],[15,83237,83238,83241],{"style":1531},[15,83239],{"className":83240,"style":218},[217],[15,83242,83244],{"className":83243},[222,223,224,225],[15,83245,1211],{"className":83246},[66,225],[15,83248,1114],{"className":83249},[1113],[15,83251,83253],{"className":83252},[206],[15,83254,83256],{"className":83255,"style":1121},[210],[15,83257],{},[15,83259,83261,83264],{"className":83260},[66],[15,83262,11],{"className":83263},[66,67],[15,83265,83267],{"className":83266},[198],[15,83268,83270,83290],{"className":83269},[202,1090],[15,83271,83273,83287],{"className":83272},[206],[15,83274,83276],{"className":83275,"style":2139},[210],[15,83277,83278,83281],{"style":1584},[15,83279],{"className":83280,"style":218},[217],[15,83282,83284],{"className":83283},[222,223,224,225],[15,83285,1211],{"className":83286},[66,225],[15,83288,1114],{"className":83289},[1113],[15,83291,83293],{"className":83292},[206],[15,83294,83296],{"className":83295,"style":1121},[210],[15,83297],{},[15,83299,120],{"className":83300},[162],[15,83302,123],{"className":83303},[66,67],[15,83305,19275],{"className":83306},[169],[15,83308],{"className":83309,"style":697},[71],[15,83311,458],{"className":83312},[66,67],[15,83314,123],{"className":83315},[66,67],[15,83317,335],{"className":83318},[422],[11,83320,83321],{},"它仍然是一个「重叠面积」，只是两侧的权重不同。先验越偏向某一类，另一类的判定区域就越小，错误率的构成也随之改变。",[1904,83323,83325,83326,83355],{"id":83324},"_42-高斯情形马氏距离与-qqq-函数","4.2 高斯情形：马氏距离与 ",[15,83327,83329,83343],{"className":83328,"translate":19},[18],[15,83330,83332],{"className":83331},[23],[25,83333,83334],{"xmlns":27},[29,83335,83336,83341],{},[32,83337,83338],{},[35,83339,83340],{},"Q",[45,83342,83340],{"encoding":47},[15,83344,83346],{"className":83345,"ariaHidden":53},[52],[15,83347,83349,83352],{"className":83348},[57],[15,83350],{"className":83351,"style":17374},[61],[15,83353,83340],{"className":83354},[66,67]," 函数",[11,83357,83358],{},"设",[15,83360,83362],{"className":83361,"translate":19},[294],[15,83363,83365,83441],{"className":83364,"translate":19},[18],[15,83366,83368],{"className":83367},[23],[25,83369,83370],{"xmlns":27,"display":303},[29,83371,83372,83438],{},[32,83373,83374,83376,83378,83380,83382,83384,83386,83388,83390,83397,83399,83402,83404,83406,83408,83410,83412,83414,83416,83418,83420,83422,83424,83430,83432,83434,83436],{},[35,83375,446],{},[39,83377,34425],{},[35,83379,61881],{},[39,83381,129],{},[1209,83383,1804],{},[39,83385,28386],{},[35,83387,28530],{"mathvariant":445},[39,83389,120],{"stretchy":119},[1057,83391,83392,83395],{},[35,83393,83394],{},"μ",[1209,83396,1804],{},[39,83398,335],{"separator":53},[35,83400,83401],{"mathvariant":138},"Σ",[39,83403,126],{"stretchy":119},[39,83405,335],{"separator":53},[71,83407],{"width":3902},[35,83409,446],{},[39,83411,34425],{},[35,83413,61881],{},[39,83415,129],{},[1209,83417,1211],{},[39,83419,28386],{},[35,83421,28530],{"mathvariant":445},[39,83423,120],{"stretchy":119},[1057,83425,83426,83428],{},[35,83427,83394],{},[1209,83429,1211],{},[39,83431,335],{"separator":53},[35,83433,83401],{"mathvariant":138},[39,83435,126],{"stretchy":119},[39,83437,335],{"separator":53},[45,83439,83440],{"encoding":47},"X\\mid 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\\Delta^2)",[15,85136,85138],{"className":85137,"ariaHidden":53},[52],[15,85139,85141,85144,85147,85150,85179,85182,85185,85188,85191,85220],{"className":85140},[57],[15,85142],{"className":85143,"style":9149},[61],[15,85145,28530],{"className":85146,"style":83507},[66,494],[15,85148,120],{"className":85149},[162],[15,85151,85153,85156],{"className":85152},[66],[15,85154,84660],{"className":85155},[66],[15,85157,85159],{"className":85158},[198],[15,85160,85162],{"className":85161},[202],[15,85163,85165],{"className":85164},[206],[15,85166,85168],{"className":85167,"style":6738},[210],[15,85169,85170,85173],{"style":213},[15,85171],{"className":85172,"style":218},[217],[15,85174,85176],{"className":85175},[222,223,224,225],[15,85177,1263],{"className":85178},[66,225],[15,85180,7108],{"className":85181},[66],[15,85183,335],{"className":85184},[422],[15,85186,24228],{"className":85187},[71],[15,85189],{"className":85190,"style":697},[71],[15,85192,85194,85197],{"className":85193},[66],[15,85195,84660],{"className":85196},[66],[15,85198,85200],{"className":85199},[198],[15,85201,85203],{"className":85202},[202],[15,85204,85206],{"className":85205},[206],[15,85207,85209],{"className":85208,"style":6738},[210],[15,85210,85211,85214],{"style":213},[15,85212],{"className":85213,"style":218},[217],[15,85215,85217],{"className":85216},[222,223,224,225],[15,85218,1263],{"className":85219},[66,225],[15,85221,126],{"className":85222},[169],"。贝叶斯规则在似然比为零处切换，因此",[15,85225,85227],{"className":85226,"translate":19},[294],[15,85228,85230,85323],{"className":85229,"translate":19},[18],[15,85231,85233],{"className":85232},[23],[25,85234,85235],{"xmlns":27,"display":303},[29,85236,85237,85320],{},[32,85238,85239,85247,85249,85251,85253,85255,85257,85263,85265,85267,85269,85275,85277,85279,85281,85283,85285,85288,85302,85304,85306,85318],{},[1255,85240,85241,85243,85245],{},[35,85242,11],{},[35,85244,48378],{},[39,85246,1162],{},[39,85248,129],{},[35,85250,28394],{},[39,85252,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,85254,28530],{"mathvariant":445},[39,85256,120],{"stretchy":119},[131,85258,85259,85261],{},[35,85260,84660],{"mathvariant":138},[1209,85262,1263],{},[35,85264,6623],{"mathvariant":138},[1209,85266,1263],{},[39,85268,335],{"separator":53},[131,85270,85271,85273],{},[35,85272,84660],{"mathvariant":138},[1209,85274,1263],{},[39,85276,126],{"stretchy":119},[39,85278,25767],{},[1209,85280,1804],{},[39,85282,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,85284,129],{},[35,85286,85287],{"mathvariant":138},"Φ",[32,85289,85290,85292,85294,85300],{},[39,85291,120],{"fence":53},[39,85293,6600],{},[6465,85295,85296,85298],{},[35,85297,84660],{"mathvariant":138},[1209,85299,1263],{},[39,85301,126],{"fence":53},[39,85303,129],{},[35,85305,83340],{},[32,85307,85308,85310,85316],{},[39,85309,120],{"fence":53},[6465,85311,85312,85314],{},[35,85313,84660],{"mathvariant":138},[1209,85315,1263],{},[39,85317,126],{"fence":53},[39,85319,335],{"separator":53},[45,85321,85322],{"encoding":47},"p_e^*=P\\big(\\mathcal N(\\Delta^2\u002F2,\\Delta^2)\u003C0\\big)\n=\\Phi\\left(-\\frac{\\Delta}{2}\\right)\n=Q\\left(\\frac{\\Delta}{2}\\right),",[15,85324,85326,85392,85492,85516,85618],{"className":85325,"ariaHidden":53},[52],[15,85327,85329,85332,85383,85386,85389],{"className":85328},[57],[15,85330],{"className":85331,"style":82015},[61],[15,85333,85335,85338],{"className":85334},[66],[15,85336,11],{"className":85337},[66,67],[15,85339,85341],{"className":85340},[198],[15,85342,85344,85375],{"className":85343},[202,1090],[15,85345,85347,85372],{"className":85346},[206],[15,85348,85350,85361],{"className":85349,"style":1308},[210],[15,85351,85352,85355],{"style":1653},[15,85353],{"className":85354,"style":218},[217],[15,85356,85358],{"className":85357},[222,223,224,225],[15,85359,48378],{"className":85360},[66,67,225],[15,85362,85363,85366],{"style":397},[15,85364],{"className":85365,"style":218},[217],[15,85367,85369],{"className":85368},[222,223,224,225],[15,85370,1162],{"className":85371},[76,225],[15,85373,1114],{"className":85374},[1113],[15,85376,85378],{"className":85377},[206],[15,85379,85381],{"className":85380,"style":1686},[210],[15,85382],{},[15,85384],{"className":85385,"style":173},[71],[15,85387,129],{"className":85388},[177],[15,85390],{"className":85391,"style":173},[71],[15,85393,85395,85398,85401,85407,85410,85413,85442,85445,85448,85451,85480,85483,85486,85489],{"className":85394},[57],[15,85396],{"className":85397,"style":11713},[61],[15,85399,28394],{"className":85400,"style":62509},[66,67],[15,85402,85404],{"className":85403},[66],[15,85405,120],{"className":85406},[1422,1508],[15,85408,28530],{"className":85409,"style":83507},[66,494],[15,85411,120],{"className":85412},[162],[15,85414,85416,85419],{"className":85415},[66],[15,85417,84660],{"className":85418},[66],[15,85420,85422],{"className":85421},[198],[15,85423,85425],{"className":85424},[202],[15,85426,85428],{"className":85427},[206],[15,85429,85431],{"className":85430,"style":1650},[210],[15,85432,85433,85436],{"style":397},[15,85434],{"className":85435,"style":218},[217],[15,85437,85439],{"className":85438},[222,223,224,225],[15,85440,1263],{"className":85441},[66,225],[15,85443,7108],{"className":85444},[66],[15,85446,335],{"className":85447},[422],[15,85449],{"className":85450,"style":697},[71],[15,85452,85454,85457],{"className":85453},[66],[15,85455,84660],{"className":85456},[66],[15,85458,85460],{"className":85459},[198],[15,85461,85463],{"className":85462},[202],[15,85464,85466],{"className":85465},[206],[15,85467,85469],{"className":85468,"style":1650},[210],[15,85470,85471,85474],{"style":397},[15,85472],{"className":85473,"style":218},[217],[15,85475,85477],{"className":85476},[222,223,224,225],[15,85478,1263],{"className":85479},[66,225],[15,85481,126],{"className":85482},[169],[15,85484],{"className":85485,"style":173},[71],[15,85487,25767],{"className":85488},[177],[15,85490],{"className":85491,"style":173},[71],[15,85493,85495,85498,85501,85507,85510,85513],{"className":85494},[57],[15,85496],{"className":85497,"style":4398},[61],[15,85499,1804],{"className":85500},[66],[15,85502,85504],{"className":85503},[66],[15,85505,126],{"className":85506},[1422,1508],[15,85508],{"className":85509,"style":173},[71],[15,85511,129],{"className":85512},[177],[15,85514],{"className":85515,"style":173},[71],[15,85517,85519,85522,85525,85528,85609,85612,85615],{"className":85518},[57],[15,85520],{"className":85521,"style":43968},[61],[15,85523,85287],{"className":85524},[66],[15,85526],{"className":85527,"style":697},[71],[15,85529,85531,85537,85540,85603],{"className":85530},[1413],[15,85532,85534],{"className":85533,"style":1418},[162,1417],[15,85535,120],{"className":85536},[1422,224],[15,85538,6600],{"className":85539},[66],[15,85541,85543,85546,85600],{"className":85542},[66],[15,85544],{"className":85545},[162,23412],[15,85547,85549],{"className":85548},[6465],[15,85550,85552,85592],{"className":85551},[202,1090],[15,85553,85555,85589],{"className":85554},[206],[15,85556,85559,85570,85578],{"className":85557,"style":85558},[210],"height:1.3603em;",[15,85560,85561,85564],{"style":23428},[15,85562],{"className":85563,"style":1368},[217],[15,85565,85567],{"className":85566},[66],[15,85568,1263],{"className":85569},[66],[15,85571,85572,85575],{"style":23440},[15,85573],{"className":85574,"style":1368},[217],[15,85576],{"className":85577,"style":4438},[23447],[15,85579,85580,85583],{"style":23450},[15,85581],{"className":85582,"style":1368},[217],[15,85584,85586],{"className":85585},[66],[15,85587,84660],{"className":85588},[66],[15,85590,1114],{"className":85591},[1113],[15,85593,85595],{"className":85594},[206],[15,85596,85598],{"className":85597,"style":23469},[210],[15,85599],{},[15,85601],{"className":85602},[169,23412],[15,85604,85606],{"className":85605,"style":1418},[169,1417],[15,85607,126],{"className":85608},[1422,224],[15,85610],{"className":85611,"style":173},[71],[15,85613,129],{"className":85614},[177],[15,85616],{"className":85617,"style":173},[71],[15,85619,85621,85624,85627,85630,85707,85710],{"className":85620},[57],[15,85622],{"className":85623,"style":43968},[61],[15,85625,83340],{"className":85626},[66,67],[15,85628],{"className":85629,"style":697},[71],[15,85631,85633,85639,85701],{"className":85632},[1413],[15,85634,85636],{"className":85635,"style":1418},[162,1417],[15,85637,120],{"className":85638},[1422,224],[15,85640,85642,85645,85698],{"className":85641},[66],[15,85643],{"className":85644},[162,23412],[15,85646,85648],{"className":85647},[6465],[15,85649,85651,85690],{"className":85650},[202,1090],[15,85652,85654,85687],{"className":85653},[206],[15,85655,85657,85668,85676],{"className":85656,"style":85558},[210],[15,85658,85659,85662],{"style":23428},[15,85660],{"className":85661,"style":1368},[217],[15,85663,85665],{"className":85664},[66],[15,85666,1263],{"className":85667},[66],[15,85669,85670,85673],{"style":23440},[15,85671],{"className":85672,"style":1368},[217],[15,85674],{"className":85675,"style":4438},[23447],[15,85677,85678,85681],{"style":23450},[15,85679],{"className":85680,"style":1368},[217],[15,85682,85684],{"className":85683},[66],[15,85685,84660],{"className":85686},[66],[15,85688,1114],{"className":85689},[1113],[15,85691,85693],{"className":85692},[206],[15,85694,85696],{"className":85695,"style":23469},[210],[15,85697],{},[15,85699],{"className":85700},[169,23412],[15,85702,85704],{"className":85703,"style":1418},[169,1417],[15,85705,126],{"className":85706},[1422,224],[15,85708],{"className":85709,"style":697},[71],[15,85711,335],{"className":85712},[422],[11,85714,425,85715,85744,85745,1124],{},[15,85716,85718,85732],{"className":85717,"translate":19},[18],[15,85719,85721],{"className":85720},[23],[25,85722,85723],{"xmlns":27},[29,85724,85725,85729],{},[32,85726,85727],{},[35,85728,85287],{"mathvariant":138},[45,85730,85731],{"encoding":47},"\\Phi",[15,85733,85735],{"className":85734,"ariaHidden":53},[52],[15,85736,85738,85741],{"className":85737},[57],[15,85739],{"className":85740,"style":490},[61],[15,85742,85287],{"className":85743},[66]," 是标准正态分布函数，",[15,85746,85748,85770],{"className":85747,"translate":19},[18],[15,85749,85751],{"className":85750},[23],[25,85752,85753],{"xmlns":27},[29,85754,85755,85767],{},[32,85756,85757,85759,85761,85763,85765],{},[35,85758,83340],{},[39,85760,129],{},[1209,85762,1211],{},[39,85764,6600],{},[35,85766,85287],{"mathvariant":138},[45,85768,85769],{"encoding":47},"Q=1-\\Phi",[15,85771,85773,85791,85809],{"className":85772,"ariaHidden":53},[52],[15,85774,85776,85779,85782,85785,85788],{"className":85775},[57],[15,85777],{"className":85778,"style":17374},[61],[15,85780,83340],{"className":85781},[66,67],[15,85783],{"className":85784,"style":173},[71],[15,85786,129],{"className":85787},[177],[15,85789],{"className":85790,"style":173},[71],[15,85792,85794,85797,85800,85803,85806],{"className":85793},[57],[15,85795],{"className":85796,"style":30463},[61],[15,85798,1211],{"className":85799},[66],[15,85801],{"className":85802,"style":72},[71],[15,85804,6600],{"className":85805},[76],[15,85807],{"className":85808,"style":72},[71],[15,85810,85812,85815],{"className":85811},[57],[15,85813],{"className":85814,"style":490},[61],[15,85816,85287],{"className":85817},[66],[11,85819,85820],{},"几个数值可以帮助建立直觉：",[6150,85822,85823,86176],{},[6153,85824,85825],{},[6156,85826,85827,85858,85985],{},[6159,85828,85829],{},[15,85830,85832,85846],{"className":85831,"translate":19},[18],[15,85833,85835],{"className":85834},[23],[25,85836,85837],{"xmlns":27},[29,85838,85839,85843],{},[32,85840,85841],{},[35,85842,84660],{"mathvariant":138},[45,85844,85845],{"encoding":47},"\\Delta",[15,85847,85849],{"className":85848,"ariaHidden":53},[52],[15,85850,85852,85855],{"className":85851},[57],[15,85853],{"className":85854,"style":490},[61],[15,85856,84660],{"className":85857},[66],[6159,85859,85860],{},[15,85861,85863,85897],{"className":85862,"translate":19},[18],[15,85864,85866],{"className":85865},[23],[25,85867,85868],{"xmlns":27},[29,85869,85870,85894],{},[32,85871,85872,85880,85882,85884,85886,85888,85890,85892],{},[1255,85873,85874,85876,85878],{},[35,85875,11],{},[35,85877,48378],{},[39,85879,1162],{},[39,85881,129],{},[35,85883,83340],{},[39,85885,120],{"stretchy":119},[35,85887,84660],{"mathvariant":138},[35,85889,6623],{"mathvariant":138},[1209,85891,1263],{},[39,85893,126],{"stretchy":119},[45,85895,85896],{"encoding":47},"p_e^*=Q(\\Delta\u002F2)",[15,85898,85900,85966],{"className":85899,"ariaHidden":53},[52],[15,85901,85903,85906,85957,85960,85963],{"className":85902},[57],[15,85904],{"className":85905,"style":82768},[61],[15,85907,85909,85912],{"className":85908},[66],[15,85910,11],{"className":85911},[66,67],[15,85913,85915],{"className":85914},[198],[15,85916,85918,85949],{"className":85917},[202,1090],[15,85919,85921,85946],{"className":85920},[206],[15,85922,85924,85935],{"className":85923,"style":16244},[210],[15,85925,85926,85929],{"style":1653},[15,85927],{"className":85928,"style":218},[217],[15,85930,85932],{"className":85931},[222,223,224,225],[15,85933,48378],{"className":85934},[66,67,225],[15,85936,85937,85940],{"style":213},[15,85938],{"className":85939,"style":218},[217],[15,85941,85943],{"className":85942},[222,223,224,225],[15,85944,1162],{"className":85945},[76,225],[15,85947,1114],{"className":85948},[1113],[15,85950,85952],{"className":85951},[206],[15,85953,85955],{"className":85954,"style":1686},[210],[15,85956],{},[15,85958],{"className":85959,"style":173},[71],[15,85961,129],{"className":85962},[177],[15,85964],{"className":85965,"style":173},[71],[15,85967,85969,85972,85975,85978,85982],{"className":85968},[57],[15,85970],{"className":85971,"style":154},[61],[15,85973,83340],{"className":85974},[66,67],[15,85976,120],{"className":85977},[162],[15,85979,85981],{"className":85980},[66],"Δ\u002F2",[15,85983,126],{"className":85984},[169],[6159,85986,85987,85988],{},"Bhattacharyya 上界 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很大时，",[15,87098,87100],{"className":87099,"translate":19},[294],[15,87101,87103,87167],{"className":87102,"translate":19},[18],[15,87104,87106],{"className":87105},[23],[25,87107,87108],{"xmlns":27,"display":303},[29,87109,87110,87164],{},[32,87111,87112,87114,87126,87128,87144,87162],{},[35,87113,83340],{},[32,87115,87116,87118,87124],{},[39,87117,120],{"fence":53},[6465,87119,87120,87122],{},[35,87121,84660],{"mathvariant":138},[1209,87123,1263],{},[39,87125,126],{"fence":53},[39,87127,28386],{},[6465,87129,87130,87132],{},[1209,87131,1263],{},[32,87133,87134,87136],{},[35,87135,84660],{"mathvariant":138},[48504,87137,87138],{},[32,87139,87140,87142],{},[1209,87141,1263],{},[35,87143,33489],{},[131,87145,87146,87148],{},[35,87147,48378],{},[32,87149,87150,87152,87158,87160],{},[39,87151,6600],{},[131,87153,87154,87156],{},[35,87155,84660],{"mathvariant":138},[1209,87157,1263],{},[35,87159,6623],{"mathvariant":138},[1209,87161,86026],{},[39,87163,335],{"separator":53},[45,87165,87166],{"encoding":47},"Q\\left(\\frac{\\Delta}{2}\\right)\\sim 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{},"与上界只差一个多项式因子。指数部分完全一致，这正是 Chernoff 界想要抓住的东西：错误率衰减的",[2040,87461,87462],{},"速率",[1904,87464,87466],{"id":87465},"_43-用散度给错误率定界","4.3 用散度给错误率定界",[11,87468,87469],{},"全变差是精确的，但它通常难以估计。实践中更常用的是 Hellinger 距离与 KL 散度。它们与全变差之间有一条标准的不等式链：",[15,87471,87473],{"className":87472,"translate":19},[294],[15,87474,87476,87596],{"className":87475,"translate":19},[18],[15,87477,87479],{"className":87478},[23],[25,87480,87481],{"xmlns":27,"display":303},[29,87482,87483,87593],{},[32,87484,87485,87487,87493,87495,87501,87511,87513,87515,87517,87523,87525,87531,87533,87535,87537,87543,87545,87551,87553,87559,87561,87563,87565,87567,87573,87575,87581,87583,87589,87591],{},[35,87486,1251],{"mathvariant":138},[1057,87488,87489,87491],{},[35,87490,28394],{},[1209,87492,1804],{},[39,87494,6600],{},[1057,87496,87497,87499],{},[35,87498,28394],{},[1209,87500,1211],{},[1057,87502,87503,87505],{},[35,87504,1251],{"mathvariant":138},[32,87506,87507,87509],{},[35,87508,32773],{"mathvariant":138},[35,87510,31727],{"mathvariant":138},[39,87512,24215],{},[35,87514,1061],{},[39,87516,120],{"stretchy":119},[1057,87518,87519,87521],{},[35,87520,28394],{},[1209,87522,1804],{},[39,87524,335],{"separator":53},[1057,87526,87527,87529],{},[35,87528,28394],{},[1209,87530,1211],{},[39,87532,126],{"stretchy":119},[39,87534,335],{"separator":53},[71,87536],{"width":3902},[131,87538,87539,87541],{},[35,87540,1061],{},[1209,87542,1263],{},[39,87544,120],{"stretchy":119},[1057,87546,87547,87549],{},[35,87548,28394],{},[1209,87550,1804],{},[39,87552,335],{"separator":53},[1057,87554,87555,87557],{},[35,87556,28394],{},[1209,87558,1211],{},[39,87560,126],{"stretchy":119},[39,87562,129],{},[1209,87564,1211],{},[39,87566,6600],{},[32,87568,87569,87571],{},[35,87570,63968],{"mathvariant":138},[35,87572,23251],{"mathvariant":138},[39,87574,120],{"stretchy":119},[1057,87576,87577,87579],{},[35,87578,28394],{},[1209,87580,1804],{},[39,87582,335],{"separator":53},[1057,87584,87585,87587],{},[35,87586,28394],{},[1209,87588,1211],{},[39,87590,126],{"stretchy":119},[39,87592,335],{"separator":53},[45,87594,87595],{"encoding":47},"\\|P_0-P_1\\|_{\\mathrm{TV}}\\le H(P_0,P_1),\n\\qquad\nH^2(P_0,P_1)=1-\\mathrm{BC}(P_0,P_1),",[15,87597,87599,87657,87758,87998,88016],{"className":87598,"ariaHidden":53},[52],[15,87600,87602,87605,87608,87648,87651,87654],{"className":87601},[57],[15,87603],{"className":87604,"style":154},[61],[15,87606,1251],{"className":87607},[66],[15,87609,87611,87614],{"className":87610},[66],[15,87612,28394],{"className":87613,"style":62509},[66,67],[15,87615,87617],{"className":87616},[198],[15,87618,87620,87640],{"className":87619},[202,1090],[15,87621,87623,87637],{"className":87622},[206],[15,87624,87626],{"className":87625,"style":2139},[210],[15,87627,87628,87631],{"style":82491},[15,87629],{"className":87630,"style":218},[217],[15,87632,87634],{"className":87633},[222,223,224,225],[15,87635,1804],{"className":87636},[66,225],[15,87638,1114],{"className":87639},[1113],[15,87641,87643],{"className":87642},[206],[15,87644,87646],{"className":87645,"style":1121},[210],[15,87647],{},[15,87649],{"className":87650,"style":72},[71],[15,87652,6600],{"className":87653},[76],[15,87655],{"className":87656,"style":72},[71],[15,87658,87660,87663,87703,87749,87752,87755],{"className":87659},[57],[15,87661],{"className":87662,"style":154},[61],[15,87664,87666,87669],{"className":87665},[66],[15,87667,28394],{"className":87668,"style":62509},[66,67],[15,87670,87672],{"className":87671},[198],[15,87673,87675,87695],{"className":87674},[202,1090],[15,87676,87678,87692],{"className":87677},[206],[15,87679,87681],{"className":87680,"style":2139},[210],[15,87682,87683,87686],{"style":82491},[15,87684],{"className":87685,"style":218},[217],[15,87687,87689],{"className":87688},[222,223,224,225],[15,87690,1211],{"className":87691},[66,225],[15,87693,1114],{"className":87694},[1113],[15,87696,87698],{"className":87697},[206],[15,87699,87701],{"className":87700,"style":1121},[210],[15,87702],{},[15,87704,87706,87709],{"className":87705},[66],[15,87707,1251],{"className":87708},[66],[15,87710,87712],{"className":87711},[198],[15,87713,87715,87741],{"className":87714},[202,1090],[15,87716,87718,87738],{"className":87717},[206],[15,87719,87721],{"className":87720,"style":3636},[210],[15,87722,87723,87726],{"style":1584},[15,87724],{"className":87725,"style":218},[217],[15,87727,87729],{"className":87728},[222,223,224,225],[15,87730,87732],{"className":87731},[66,225],[15,87733,87735],{"className":87734},[66,225],[15,87736,82602],{"className":87737,"style":1343},[66,4427,225],[15,87739,1114],{"className":87740},[1113],[15,87742,87744],{"className":87743},[206],[15,87745,87747],{"className":87746,"style":1121},[210],[15,87748],{},[15,87750],{"className":87751,"style":173},[71],[15,87753,24215],{"className":87754},[177],[15,87756],{"className":87757,"style":173},[71],[15,87759,87761,87764,87767,87770,87810,87813,87816,87856,87859,87862,87865,87868,87897,87900,87940,87943,87946,87986,87989,87992,87995],{"className":87760},[57],[15,87762],{"className":87763,"style":11209},[61],[15,87765,1061],{"className":87766,"style":7071},[66,67],[15,87768,120],{"className":87769},[162],[15,87771,87773,87776],{"className":87772},[66],[15,87774,28394],{"className":87775,"style":62509},[66,67],[15,87777,87779],{"className":87778},[198],[15,87780,87782,87802],{"className":87781},[202,1090],[15,87783,87785,87799],{"className":87784},[206],[15,87786,87788],{"className":87787,"style":2139},[210],[15,87789,87790,87793],{"style":82491},[15,87791],{"className":87792,"style":218},[217],[15,87794,87796],{"className":87795},[222,223,224,225],[15,87797,1804],{"className":87798},[66,225],[15,87800,1114],{"className":87801},[1113],[15,87803,87805],{"className":87804},[206],[15,87806,87808],{"className":87807,"style":1121},[210],[15,87809],{},[15,87811,335],{"className":87812},[422],[15,87814],{"className":87815,"style":697},[71],[15,87817,87819,87822],{"className":87818},[66],[15,87820,28394],{"className":87821,"style":62509},[66,67],[15,87823,87825],{"className":87824},[198],[15,87826,87828,87848],{"className":87827},[202,1090],[15,87829,87831,87845],{"className":87830},[206],[15,87832,87834],{"className":87833,"style":2139},[210],[15,87835,87836,87839],{"style":82491},[15,87837],{"className":87838,"style":218},[217],[15,87840,87842],{"className":87841},[222,223,224,225],[15,87843,1211],{"className":87844},[66,225],[15,87846,1114],{"className":87847},[1113],[15,87849,87851],{"className":87850},[206],[15,87852,87854],{"className":87853,"style":1121},[210],[15,87855],{},[15,87857,126],{"className":87858},[169],[15,87860,335],{"className":87861},[422],[15,87863],{"className":87864,"style":4059},[71],[15,87866],{"className":87867,"style":697},[71],[15,87869,87871,87874],{"className":87870},[66],[15,87872,1061],{"className":87873,"style":7071},[66,67],[15,87875,87877],{"className":87876},[198],[15,87878,87880],{"className":87879},[202],[15,87881,87883],{"className":87882},[206],[15,87884,87886],{"className":87885,"style":1650},[210],[15,87887,87888,87891],{"style":397},[15,87889],{"className":87890,"style":218},[217],[15,87892,87894],{"className":87893},[222,223,224,225],[15,87895,1263],{"className":87896},[66,225],[15,87898,120],{"className":87899},[162],[15,87901,87903,87906],{"className":87902},[66],[15,87904,28394],{"className":87905,"style":62509},[66,67],[15,87907,87909],{"className":87908},[198],[15,87910,87912,87932],{"className":87911},[202,1090],[15,87913,87915,87929],{"className":87914},[206],[15,87916,87918],{"className":87917,"style":2139},[210],[15,87919,87920,87923],{"style":82491},[15,87921],{"className":87922,"style":218},[217],[15,87924,87926],{"className":87925},[222,223,224,225],[15,87927,1804],{"className":87928},[66,225],[15,87930,1114],{"className":87931},[1113],[15,87933,87935],{"className":87934},[206],[15,87936,87938],{"className":87937,"style":1121},[210],[15,87939],{},[15,87941,335],{"className":87942},[422],[15,87944],{"className":87945,"style":697},[71],[15,87947,87949,87952],{"className":87948},[66],[15,87950,28394],{"className":87951,"style":62509},[66,67],[15,87953,87955],{"className":87954},[198],[15,87956,87958,87978],{"className":87957},[202,1090],[15,87959,87961,87975],{"className":87960},[206],[15,87962,87964],{"className":87963,"style":2139},[210],[15,87965,87966,87969],{"style":82491},[15,87967],{"className":87968,"style":218},[217],[15,87970,87972],{"className":87971},[222,223,224,225],[15,87973,1211],{"className":87974},[66,225],[15,87976,1114],{"className":87977},[1113],[15,87979,87981],{"className":87980},[206],[15,87982,87984],{"className":87983,"style":1121},[210],[15,87985],{},[15,87987,126],{"className":87988},[169],[15,87990],{"className":87991,"style":173},[71],[15,87993,129],{"className":87994},[177],[15,87996],{"className":87997,"style":173},[71],[15,87999,88001,88004,88007,88010,88013],{"className":88000},[57],[15,88002],{"className":88003,"style":30463},[61],[15,88005,1211],{"className":88006},[66],[15,88008],{"className":88009,"style":72},[71],[15,88011,6600],{"className":88012},[76],[15,88014],{"className":88015,"style":72},[71],[15,88017,88019,88022,88028,88031,88071,88074,88077,88117,88120],{"className":88018},[57],[15,88020],{"className":88021,"style":154},[61],[15,88023,88025],{"className":88024},[66],[15,88026,86459],{"className":88027},[66,4427],[15,88029,120],{"className":88030},[162],[15,88032,88034,88037],{"className":88033},[66],[15,88035,28394],{"className":88036,"style":62509},[66,67],[15,88038,88040],{"className":88039},[198],[15,88041,88043,88063],{"className":88042},[202,1090],[15,88044,88046,88060],{"className":88045},[206],[15,88047,88049],{"className":88048,"style":2139},[210],[15,88050,88051,88054],{"style":82491},[15,88052],{"className":88053,"style":218},[217],[15,88055,88057],{"className":88056},[222,223,224,225],[15,88058,1804],{"className":88059},[66,225],[15,88061,1114],{"className":88062},[1113],[15,88064,88066],{"className":88065},[206],[15,88067,88069],{"className":88068,"style":1121},[210],[15,88070],{},[15,88072,335],{"className":88073},[422],[15,88075],{"className":88076,"style":697},[71],[15,88078,88080,88083],{"className":88079},[66],[15,88081,28394],{"className":88082,"style":62509},[66,67],[15,88084,88086],{"className":88085},[198],[15,88087,88089,88109],{"className":88088},[202,1090],[15,88090,88092,88106],{"className":88091},[206],[15,88093,88095],{"className":88094,"style":2139},[210],[15,88096,88097,88100],{"style":82491},[15,88098],{"className":88099,"style":218},[217],[15,88101,88103],{"className":88102},[222,223,224,225],[15,88104,1211],{"className":88105},[66,225],[15,88107,1114],{"className":88108},[1113],[15,88110,88112],{"className":88111},[206],[15,88113,88115],{"className":88114,"style":1121},[210],[15,88116],{},[15,88118,126],{"className":88119},[169],[15,88121,335],{"className":88122},[422],[11,88124,88125],{},"以及",[15,88127,88129],{"className":88128,"translate":19},[294],[15,88130,88132,88206],{"className":88131,"translate":19},[18],[15,88133,88135],{"className":88134},[23],[25,88136,88137],{"xmlns":27,"display":303},[29,88138,88139,88203],{},[32,88140,88141,88147,88149,88155,88157,88163,88165,88167,88173,88183,88185,88191,88193,88199,88201],{},[131,88142,88143,88145],{},[35,88144,1061],{},[1209,88146,1263],{},[39,88148,120],{"stretchy":119},[1057,88150,88151,88153],{},[35,88152,28394],{},[1209,88154,1804],{},[39,88156,335],{"separator":53},[1057,88158,88159,88161],{},[35,88160,28394],{},[1209,88162,1211],{},[39,88164,126],{"stretchy":119},[39,88166,24215],{},[6465,88168,88169,88171],{},[1209,88170,1211],{},[1209,88172,1263],{},[1057,88174,88175,88177],{},[35,88176,32453],{},[32,88178,88179,88181],{},[35,88180,2853],{"mathvariant":138},[35,88182,1216],{"mathvariant":138},[39,88184,120],{"stretchy":119},[1057,88186,88187,88189],{},[35,88188,28394],{},[1209,88190,1804],{},[35,88192,1251],{"mathvariant":138},[1057,88194,88195,88197],{},[35,88196,28394],{},[1209,88198,1211],{},[39,88200,126],{"stretchy":119},[35,88202,3571],{"mathvariant":138},[45,88204,88205],{"encoding":47},"H^2(P_0,P_1)\\le \\frac12 D_{\\mathrm{KL}}(P_0\\|P_1).",[15,88207,88209,88345],{"className":88208,"ariaHidden":53},[52],[15,88210,88212,88215,88244,88247,88287,88290,88293,88333,88336,88339,88342],{"className":88211},[57],[15,88213],{"className":88214,"style":11209},[61],[15,88216,88218,88221],{"className":88217},[66],[15,88219,1061],{"className":88220,"style":7071},[66,67],[15,88222,88224],{"className":88223},[198],[15,88225,88227],{"className":88226},[202],[15,88228,88230],{"className":88229},[206],[15,88231,88233],{"className":88232,"style":1650},[210],[15,88234,88235,88238],{"style":397},[15,88236],{"className":88237,"style":218},[217],[15,88239,88241],{"className":88240},[222,223,224,225],[15,88242,1263],{"className":88243},[66,225],[15,88245,120],{"className":88246},[162],[15,88248,88250,88253],{"className":88249},[66],[15,88251,28394],{"className":88252,"style":62509},[66,67],[15,88254,88256],{"className":88255},[198],[15,88257,88259,88279],{"className":88258},[202,1090],[15,88260,88262,88276],{"className":88261},[206],[15,88263,88265],{"className":88264,"style":2139},[210],[15,88266,88267,88270],{"style":82491},[15,88268],{"className":88269,"style":218},[217],[15,88271,88273],{"className":88272},[222,223,224,225],[15,88274,1804],{"className":88275},[66,225],[15,88277,1114],{"className":88278},[1113],[15,88280,88282],{"className":88281},[206],[15,88283,88285],{"className":88284,"style":1121},[210],[15,88286],{},[15,88288,335],{"className":88289},[422],[15,88291],{"className":88292,"style":697},[71],[15,88294,88296,88299],{"className":88295},[66],[15,88297,28394],{"className":88298,"style":62509},[66,67],[15,88300,88302],{"className":88301},[198],[15,88303,88305,88325],{"className":88304},[202,1090],[15,88306,88308,88322],{"className":88307},[206],[15,88309,88311],{"className":88310,"style":2139},[210],[15,88312,88313,88316],{"style":82491},[15,88314],{"className":88315,"style":218},[217],[15,88317,88319],{"className":88318},[222,223,224,225],[15,88320,1211],{"className":88321},[66,225],[15,88323,1114],{"className":88324},[1113],[15,88326,88328],{"className":88327},[206],[15,88329,88331],{"className":88330,"style":1121},[210],[15,88332],{},[15,88334,126],{"className":88335},[169],[15,88337],{"className":88338,"style":173},[71],[15,88340,24215],{"className":88341},[177],[15,88343],{"className":88344,"style":173},[71],[15,88346,88348,88351,88413,88461,88464,88504,88507,88547,88550],{"className":88347},[57],[15,88349],{"className":88350,"style":49082},[61],[15,88352,88354,88357,88410],{"className":88353},[66],[15,88355],{"className":88356},[162,23412],[15,88358,88360],{"className":88359},[6465],[15,88361,88363,88402],{"className":88362},[202,1090],[15,88364,88366,88399],{"className":88365},[206],[15,88367,88369,88380,88388],{"className":88368,"style":23425},[210],[15,88370,88371,88374],{"style":23428},[15,88372],{"className":88373,"style":1368},[217],[15,88375,88377],{"className":88376},[66],[15,88378,1263],{"className":88379},[66],[15,88381,88382,88385],{"style":23440},[15,88383],{"className":88384,"style":1368},[217],[15,88386],{"className":88387,"style":4438},[23447],[15,88389,88390,88393],{"style":23450},[15,88391],{"className":88392,"style":1368},[217],[15,88394,88396],{"className":88395},[66],[15,88397,1211],{"className":88398},[66],[15,88400,1114],{"className":88401},[1113],[15,88403,88405],{"className":88404},[206],[15,88406,88408],{"className":88407,"style":23469},[210],[15,88409],{},[15,88411],{"className":88412},[169,23412],[15,88414,88416,88419],{"className":88415},[66],[15,88417,32453],{"className":88418,"style":23020},[66,67],[15,88420,88422],{"className":88421},[198],[15,88423,88425,88453],{"className":88424},[202,1090],[15,88426,88428,88450],{"className":88427},[206],[15,88429,88431],{"className":88430,"style":3636},[210],[15,88432,88434,88437],{"style":88433},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[15,88435],{"className":88436,"style":218},[217],[15,88438,88440],{"className":88439},[222,223,224,225],[15,88441,88443],{"className":88442},[66,225],[15,88444,88446],{"className":88445},[66,225],[15,88447,88449],{"className":88448},[66,4427,225],"KL",[15,88451,1114],{"className":88452},[1113],[15,88454,88456],{"className":88455},[206],[15,88457,88459],{"className":88458,"style":1121},[210],[15,88460],{},[15,88462,120],{"className":88463},[162],[15,88465,88467,88470],{"className":88466},[66],[15,88468,28394],{"className":88469,"style":62509},[66,67],[15,88471,88473],{"className":88472},[198],[15,88474,88476,88496],{"className":88475},[202,1090],[15,88477,88479,88493],{"className":88478},[206],[15,88480,88482],{"className":88481,"style":2139},[210],[15,88483,88484,88487],{"style":82491},[15,88485],{"className":88486,"style":218},[217],[15,88488,88490],{"className":88489},[222,223,224,225],[15,88491,1804],{"className":88492},[66,225],[15,88494,1114],{"className":88495},[1113],[15,88497,88499],{"className":88498},[206],[15,88500,88502],{"className":88501,"style":1121},[210],[15,88503],{},[15,88505,1251],{"className":88506},[66],[15,88508,88510,88513],{"className":88509},[66],[15,88511,28394],{"className":88512,"style":62509},[66,67],[15,88514,88516],{"className":88515},[198],[15,88517,88519,88539],{"className":88518},[202,1090],[15,88520,88522,88536],{"className":88521},[206],[15,88523,88525],{"className":88524,"style":2139},[210],[15,88526,88527,88530],{"style":82491},[15,88528],{"className":88529,"style":218},[217],[15,88531,88533],{"className":88532},[222,223,224,225],[15,88534,1211],{"className":88535},[66,225],[15,88537,1114],{"className":88538},[1113],[15,88540,88542],{"className":88541},[206],[15,88543,88545],{"className":88544,"style":1121},[210],[15,88546],{},[15,88548,126],{"className":88549},[169],[15,88551,3571],{"className":88552},[66],[11,88554,88555,88556,88776],{},"把第一条与 ",[15,88557,88559,88603],{"className":88558,"translate":19},[18],[15,88560,88562],{"className":88561},[23],[25,88563,88564],{"xmlns":27},[29,88565,88566,88600],{},[32,88567,88568,88576,88578,88586,88588,88590,88592,88598],{},[1255,88569,88570,88572,88574],{},[35,88571,11],{},[35,88573,48378],{},[39,88575,1162],{},[39,88577,129],{},[62983,88579,88580],{"scriptlevel":1804,"displaystyle":119},[6465,88581,88582,88584],{},[1209,88583,1211],{},[1209,88585,1263],{},[39,88587,120],{"stretchy":119},[1209,88589,1211],{},[39,88591,6600],{},[32,88593,88594,88596],{},[35,88595,32773],{"mathvariant":138},[35,88597,31727],{"mathvariant":138},[39,88599,126],{"stretchy":119},[45,88601,88602],{"encoding":47},"p_e^*=\\tfrac12(1-\\mathrm{TV})",[15,88604,88606,88672,88761],{"className":88605,"ariaHidden":53},[52],[15,88607,88609,88612,88663,88666,88669],{"className":88608},[57],[15,88610],{"className":88611,"style":82768},[61],[15,88613,88615,88618],{"className":88614},[66],[15,88616,11],{"className":88617},[66,67],[15,88619,88621],{"className":88620},[198],[15,88622,88624,88655],{"className":88623},[202,1090],[15,88625,88627,88652],{"className":88626},[206],[15,88628,88630,88641],{"className":88629,"style":16244},[210],[15,88631,88632,88635],{"style":1653},[15,88633],{"className":88634,"style":218},[217],[15,88636,88638],{"className":88637},[222,223,224,225],[15,88639,48378],{"className":88640},[66,67,225],[15,88642,88643,88646],{"style":213},[15,88644],{"className":88645,"style":218},[217],[15,88647,88649],{"className":88648},[222,223,224,225],[15,88650,1162],{"className":88651},[76,225],[15,88653,1114],{"className":88654},[1113],[15,88656,88658],{"className":88657},[206],[15,88659,88661],{"className":88660,"style":1686},[210],[15,88662],{},[15,88664],{"className":88665,"style":173},[71],[15,88667,129],{"className":88668},[177],[15,88670],{"className":88671,"style":173},[71],[15,88673,88675,88678,88746,88749,88752,88755,88758],{"className":88674},[57],[15,88676],{"className":88677,"style":30849},[61],[15,88679,88681,88684,88743],{"className":88680},[66],[15,88682],{"className":88683},[162,23412],[15,88685,88687],{"className":88686},[6465],[15,88688,88690,88735],{"className":88689},[202,1090],[15,88691,88693,88732],{"className":88692},[206],[15,88694,88696,88710,88718],{"className":88695,"style":30868},[210],[15,88697,88698,88701],{"style":30871},[15,88699],{"className":88700,"style":1368},[217],[15,88702,88704],{"className":88703},[222,223,224,225],[15,88705,88707],{"className":88706},[66,225],[15,88708,1263],{"className":88709},[66,225],[15,88711,88712,88715],{"style":23440},[15,88713],{"className":88714,"style":1368},[217],[15,88716],{"className":88717,"style":4438},[23447],[15,88719,88720,88723],{"style":30894},[15,88721],{"className":88722,"style":1368},[217],[15,88724,88726],{"className":88725},[222,223,224,225],[15,88727,88729],{"className":88728},[66,225],[15,88730,1211],{"className":88731},[66,225],[15,88733,1114],{"className":88734},[1113],[15,88736,88738],{"className":88737},[206],[15,88739,88741],{"className":88740,"style":30916},[210],[15,88742],{},[15,88744],{"className":88745},[169,23412],[15,88747,120],{"className":88748},[162],[15,88750,1211],{"className":88751},[66],[15,88753],{"className":88754,"style":72},[71],[15,88756,6600],{"className":88757},[76],[15,88759],{"className":88760,"style":72},[71],[15,88762,88764,88767,88773],{"className":88763},[57],[15,88765],{"className":88766,"style":154},[61],[15,88768,88770],{"className":88769},[66],[15,88771,82602],{"className":88772,"style":1343},[66,4427],[15,88774,126],{"className":88775},[169]," 结合，得到",[15,88778,88780],{"className":88779,"translate":19},[294],[15,88781,88783,88863],{"className":88782,"translate":19},[18],[15,88784,88786],{"className":88785},[23],[25,88787,88788],{"xmlns":27,"display":303},[29,88789,88790,88860],{},[32,88791,88792,88800,88802,88804,88806,88812,88858],{},[1255,88793,88794,88796,88798],{},[35,88795,11],{},[35,88797,48378],{},[39,88799,1162],{},[1274,88801,24228],{},[39,88803,1801],{},[1274,88805,24228],{},[6465,88807,88808,88810],{},[1209,88809,1211],{},[1209,88811,1263],{},[32,88813,88814,88816,88818,88820,88856],{},[39,88815,120],{"fence":53},[1209,88817,1211],{},[39,88819,6600],{},[48504,88821,88822],{},[6465,88823,88824,88854],{},[32,88825,88826,88836,88838,88844,88846,88852],{},[1057,88827,88828,88830],{},[35,88829,32453],{},[32,88831,88832,88834],{},[35,88833,2853],{"mathvariant":138},[35,88835,1216],{"mathvariant":138},[39,88837,120],{"stretchy":119},[1057,88839,88840,88842],{},[35,88841,28394],{},[1209,88843,1804],{},[35,88845,1251],{"mathvariant":138},[1057,88847,88848,88850],{},[35,88849,28394],{},[1209,88851,1211],{},[39,88853,126],{"stretchy":119},[1209,88855,1263],{},[39,88857,126],{"fence":53},[35,88859,3571],{"mathvariant":138},[45,88861,88862],{"encoding":47},"p_e^*\\ \\ge\\ \\frac12\\left(1-\\sqrt{\\frac{D_{\\mathrm{KL}}(P_0\\|P_1)}{2}}\\right).",[15,88864,88866,88938],{"className":88865,"ariaHidden":53},[52],[15,88867,88869,88872,88923,88926,88929,88932,88935],{"className":88868},[57],[15,88870],{"className":88871,"style":82015},[61],[15,88873,88875,88878],{"className":88874},[66],[15,88876,11],{"className":88877},[66,67],[15,88879,88881],{"className":88880},[198],[15,88882,88884,88915],{"className":88883},[202,1090],[15,88885,88887,88912],{"className":88886},[206],[15,88888,88890,88901],{"className":88889,"style":1308},[210],[15,88891,88892,88895],{"style":1653},[15,88893],{"className":88894,"style":218},[217],[15,88896,88898],{"className":88897},[222,223,224,225],[15,88899,48378],{"className":88900},[66,67,225],[15,88902,88903,88906],{"style":397},[15,88904],{"className":88905,"style":218},[217],[15,88907,88909],{"className":88908},[222,223,224,225],[15,88910,1162],{"className":88911},[76,225],[15,88913,1114],{"className":88914},[1113],[15,88916,88918],{"className":88917},[206],[15,88919,88921],{"className":88920,"style":1686},[210],[15,88922],{},[15,88924,24228],{"className":88925},[71],[15,88927],{"className":88928,"style":173},[71],[15,88930,1801],{"className":88931},[177],[15,88933,24228],{"className":88934},[71],[15,88936],{"className":88937,"style":173},[71],[15,88939,88941,88945,89007,89010,89285,89288],{"className":88940},[57],[15,88942],{"className":88943,"style":88944},[61],"height:3em;vertical-align:-1.25em;",[15,88946,88948,88951,89004],{"className":88947},[66],[15,88949],{"className":88950},[162,23412],[15,88952,88954],{"className":88953},[6465],[15,88955,88957,88996],{"className":88956},[202,1090],[15,88958,88960,88993],{"className":88959},[206],[15,88961,88963,88974,88982],{"className":88962,"style":23425},[210],[15,88964,88965,88968],{"style":23428},[15,88966],{"className":88967,"style":1368},[217],[15,88969,88971],{"className":88970},[66],[15,88972,1263],{"className":88973},[66],[15,88975,88976,88979],{"style":23440},[15,88977],{"className":88978,"style":1368},[217],[15,88980],{"className":88981,"style":4438},[23447],[15,88983,88984,88987],{"style":23450},[15,88985],{"className":88986,"style":1368},[217],[15,88988,88990],{"className":88989},[66],[15,88991,1211],{"className":88992},[66],[15,88994,1114],{"className":88995},[1113],[15,88997,88999],{"className":88998},[206],[15,89000,89002],{"className":89001,"style":23469},[210],[15,89003],{},[15,89005],{"className":89006},[169,23412],[15,89008],{"className":89009,"style":697},[71],[15,89011,89013,89019,89022,89025,89028,89031,89279],{"className":89012},[1413],[15,89014,89016],{"className":89015,"style":1418},[162,1417],[15,89017,120],{"className":89018},[1422,1423],[15,89020,1211],{"className":89021},[66],[15,89023],{"className":89024,"style":72},[71],[15,89026,6600],{"className":89027},[76],[15,89029],{"className":89030,"style":72},[71],[15,89032,89034],{"className":89033},[66,48546],[15,89035,89037,89270],{"className":89036},[202,1090],[15,89038,89040,89267],{"className":89039},[206],[15,89041,89044,89250],{"className":89042,"style":89043},[210],"height:1.7044em;",[15,89045,89048,89052],{"className":89046,"style":89047},[48560],"top:-4.4em;",[15,89049],{"className":89050,"style":89051},[217],"height:4.4em;",[15,89053,89056],{"className":89054,"style":89055},[66],"padding-left:1em;",[15,89057,89059,89062,89247],{"className":89058},[66],[15,89060],{"className":89061},[162,23412],[15,89063,89065],{"className":89064},[6465],[15,89066,89068,89239],{"className":89067},[202,1090],[15,89069,89071,89236],{"className":89070},[206],[15,89072,89074,89085,89093],{"className":89073,"style":41206},[210],[15,89075,89076,89079],{"style":23428},[15,89077],{"className":89078,"style":1368},[217],[15,89080,89082],{"className":89081},[66],[15,89083,1263],{"className":89084},[66],[15,89086,89087,89090],{"style":23440},[15,89088],{"className":89089,"style":1368},[217],[15,89091],{"className":89092,"style":4438},[23447],[15,89094,89095,89098],{"style":23450},[15,89096],{"className":89097,"style":1368},[217],[15,89099,89101,89147,89150,89190,89193,89233],{"className":89100},[66],[15,89102,89104,89107],{"className":89103},[66],[15,89105,32453],{"className":89106,"style":23020},[66,67],[15,89108,89110],{"className":89109},[198],[15,89111,89113,89139],{"className":89112},[202,1090],[15,89114,89116,89136],{"className":89115},[206],[15,89117,89119],{"className":89118,"style":3636},[210],[15,89120,89121,89124],{"style":88433},[15,89122],{"className":89123,"style":218},[217],[15,89125,89127],{"className":89126},[222,223,224,225],[15,89128,89130],{"className":89129},[66,225],[15,89131,89133],{"className":89132},[66,225],[15,89134,88449],{"className":89135},[66,4427,225],[15,89137,1114],{"className":89138},[1113],[15,89140,89142],{"className":89141},[206],[15,89143,89145],{"className":89144,"style":1121},[210],[15,89146],{},[15,89148,120],{"className":89149},[162],[15,89151,89153,89156],{"className":89152},[66],[15,89154,28394],{"className":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0 400000 2592",[34829,89265],{"d":89266},"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z",[15,89268,1114],{"className":89269},[1113],[15,89271,89273],{"className":89272},[206],[15,89274,89277],{"className":89275,"style":89276},[210],"height:0.7356em;",[15,89278],{},[15,89280,89282],{"className":89281,"style":1418},[169,1417],[15,89283,126],{"className":89284},[1422,1423],[15,89286],{"className":89287,"style":697},[71],[15,89289,3571],{"className":89290},[66],[11,89292,89293,89294,89297],{},"这是 Le Cam 两点法的核心不等式：如果两个分布很难区分（KL 很小），那么任何分类器的错误率都不可能低。注意方向——",[2040,89295,89296],{},"KL 给的是错误率的下界，不是上界","。KL 很大并不意味着问题容易，只意味着这个下界变得松甚至变得平凡。",[11,89299,89300],{},"上界则需要另一个量：",[15,89302,89304],{"className":89303,"translate":19},[294],[15,89305,89307,89363],{"className":89306,"translate":19},[18],[15,89308,89310],{"className":89309},[23],[25,89311,89312],{"xmlns":27,"display":303},[29,89313,89314,89360],{},[32,89315,89316,89324,89326,89332,89334,89340,89342,89348,89350,89356,89358],{},[1255,89317,89318,89320,89322],{},[35,89319,11],{},[35,89321,48378],{},[39,89323,1162],{},[39,89325,24215],{},[6465,89327,89328,89330],{},[1209,89329,1211],{},[1209,89331,1263],{},[1274,89333,2462],{},[32,89335,89336,89338],{},[35,89337,63968],{"mathvariant":138},[35,89339,23251],{"mathvariant":138},[39,89341,120],{"stretchy":119},[1057,89343,89344,89346],{},[35,89345,28394],{},[1209,89347,1804],{},[39,89349,335],{"separator":53},[1057,89351,89352,89354],{},[35,89353,28394],{},[1209,89355,1211],{},[39,89357,126],{"stretchy":119},[35,89359,3571],{"mathvariant":138},[45,89361,89362],{"encoding":47},"p_e^*\\le \\frac12\\,\\mathrm{BC}(P_0,P_1).",[15,89364,89366,89432],{"className":89365,"ariaHidden":53},[52],[15,89367,89369,89372,89423,89426,89429],{"className":89368},[57],[15,89370],{"className":89371,"style":82015},[61],[15,89373,89375,89378],{"className":89374},[66],[15,89376,11],{"className":89377},[66,67],[15,89379,89381],{"className":89380},[198],[15,89382,89384,89415],{"className":89383},[202,1090],[15,89385,89387,89412],{"className":89386},[206],[15,89388,89390,89401],{"className":89389,"style":1308},[210],[15,89391,89392,89395],{"style":1653},[15,89393],{"className":89394,"style":218},[217],[15,89396,89398],{"className":89397},[222,223,224,225],[15,89399,48378],{"className":89400},[66,67,225],[15,89402,89403,89406],{"style":397},[15,89404],{"className":89405,"style":218},[217],[15,89407,89409],{"className":89408},[222,223,224,225],[15,89410,1162],{"className":89411},[76,225],[15,89413,1114],{"className":89414},[1113],[15,89416,89418],{"className":89417},[206],[15,89419,89421],{"className":89420,"style":1686},[210],[15,89422],{},[15,89424],{"className":89425,"style":173},[71],[15,89427,24215],{"className":89428},[177],[15,89430],{"className":89431,"style":173},[71],[15,89433,89435,89438,89500,89503,89509,89512,89552,89555,89558,89598,89601],{"className":89434},[57],[15,89436],{"className":89437,"style":49082},[61],[15,89439,89441,89444,89497],{"className":89440},[66],[15,89442],{"className":89443},[162,23412],[15,89445,89447],{"className":89446},[6465],[15,89448,89450,89489],{"className":89449},[202,1090],[15,89451,89453,89486],{"className":89452},[206],[15,89454,89456,89467,89475],{"className":89455,"style":23425},[210],[15,89457,89458,89461],{"style":23428},[15,89459],{"className":89460,"style":1368},[217],[15,89462,89464],{"className":89463},[66],[15,89465,1263],{"className":89466},[66],[15,89468,89469,89472],{"style":23440},[15,89470],{"className":89471,"style":1368},[217],[15,89473],{"className":89474,"style":4438},[23447],[15,89476,89477,89480],{"style":23450},[15,89478],{"className":89479,"style":1368},[217],[15,89481,89483],{"className":89482},[66],[15,89484,1211],{"className":89485},[66],[15,89487,1114],{"className":89488},[1113],[15,89490,89492],{"className":89491},[206],[15,89493,89495],{"className":89494,"style":23469},[210],[15,89496],{},[15,89498],{"className":89499},[169,23412],[15,89501],{"className":89502,"style":697},[71],[15,89504,89506],{"className":89505},[66],[15,89507,86459],{"className":89508},[66,4427],[15,89510,120],{"className":89511},[162],[15,89513,89515,89518],{"className":89514},[66],[15,89516,28394],{"className":89517,"style":62509},[66,67],[15,89519,89521],{"className":89520},[198],[15,89522,89524,89544],{"className":89523},[202,1090],[15,89525,89527,89541],{"className":89526},[206],[15,89528,89530],{"className":89529,"style":2139},[210],[15,89531,89532,89535],{"style":82491},[15,89533],{"className":89534,"style":218},[217],[15,89536,89538],{"className":89537},[222,223,224,225],[15,89539,1804],{"className":89540},[66,225],[15,89542,1114],{"className":89543},[1113],[15,89545,89547],{"className":89546},[206],[15,89548,89550],{"className":89549,"style":1121},[210],[15,89551],{},[15,89553,335],{"className":89554},[422],[15,89556],{"className":89557,"style":697},[71],[15,89559,89561,89564],{"className":89560},[66],[15,89562,28394],{"className":89563,"style":62509},[66,67],[15,89565,89567],{"className":89566},[198],[15,89568,89570,89590],{"className":89569},[202,1090],[15,89571,89573,89587],{"className":89572},[206],[15,89574,89576],{"className":89575,"style":2139},[210],[15,89577,89578,89581],{"style":82491},[15,89579],{"className":89580,"style":218},[217],[15,89582,89584],{"className":89583},[222,223,224,225],[15,89585,1211],{"className":89586},[66,225],[15,89588,1114],{"className":89589},[1113],[15,89591,89593],{"className":89592},[206],[15,89594,89596],{"className":89595,"style":1121},[210],[15,89597],{},[15,89599,126],{"className":89600},[169],[15,89602,3571],{"className":89603},[66],[11,89605,89606,89607,89921],{},"这是因为 ",[15,89608,89610,89666],{"className":89609,"translate":19},[18],[15,89611,89613],{"className":89612},[23],[25,89614,89615],{"xmlns":27},[29,89616,89617,89663],{},[32,89618,89619,89621,89623,89625,89627,89633,89635,89641,89643,89645,89647],{},[39,89620,29727],{},[35,89622,1178],{},[39,89624,1170],{},[39,89626,4305],{"stretchy":119},[1057,89628,89629,89631],{},[35,89630,11],{},[1209,89632,1804],{},[39,89634,335],{"separator":53},[1057,89636,89637,89639],{},[35,89638,11],{},[1209,89640,1211],{},[39,89642,4328],{"stretchy":119},[39,89644,24215],{},[39,89646,29727],{},[48504,89648,89649],{},[32,89650,89651,89657],{},[1057,89652,89653,89655],{},[35,89654,11],{},[1209,89656,1804],{},[1057,89658,89659,89661],{},[35,89660,11],{},[1209,89662,1211],{},[45,89664,89665],{"encoding":47},"\\int\\min\\{p_0,p_1\\}\\le\\int\\sqrt{p_0p_1}",[15,89667,89669,89785],{"className":89668,"ariaHidden":53},[52],[15,89670,89672,89675,89678,89681,89684,89687,89727,89730,89733,89773,89776,89779,89782],{"className":89671},[57],[15,89673],{"className":89674,"style":34252},[61],[15,89676,29727],{"className":89677,"style":31338},[1339,1470,13696],[15,89679],{"className":89680,"style":697},[71],[15,89682,1178],{"className":89683},[1339],[15,89685,4305],{"className":89686},[162],[15,89688,89690,89693],{"className":89689},[66],[15,89691,11],{"className":89692},[66,67],[15,89694,89696],{"className":89695},[198],[15,89697,89699,89719],{"className":89698},[202,1090],[15,89700,89702,89716],{"className":89701},[206],[15,89703,89705],{"className":89704,"style":2139},[210],[15,89706,89707,89710],{"style":1584},[15,89708],{"className":89709,"style":218},[217],[15,89711,89713],{"className":89712},[222,223,224,225],[15,89714,1804],{"className":89715},[66,225],[15,89717,1114],{"className":89718},[1113],[15,89720,89722],{"className":89721},[206],[15,89723,89725],{"className":89724,"style":1121},[210],[15,89726],{},[15,89728,335],{"className":89729},[422],[15,89731],{"className":89732,"style":697},[71],[15,89734,89736,89739],{"className":89735},[66],[15,89737,11],{"className":89738},[66,67],[15,89740,89742],{"className":89741},[198],[15,89743,89745,89765],{"className":89744},[202,1090],[15,89746,89748,89762],{"className":89747},[206],[15,89749,89751],{"className":89750,"style":2139},[210],[15,89752,89753,89756],{"style":1584},[15,89754],{"className":89755,"style":218},[217],[15,89757,89759],{"className":89758},[222,223,224,225],[15,89760,1211],{"className":89761},[66,225],[15,89763,1114],{"className":89764},[1113],[15,89766,89768],{"className":89767},[206],[15,89769,89771],{"className":89770,"style":1121},[210],[15,89772],{},[15,89774,4328],{"className":89775},[169],[15,89777],{"className":89778,"style":173},[71],[15,89780,24215],{"className":89781},[177],[15,89783],{"className":89784,"style":173},[71],[15,89786,89788,89791,89794,89797],{"className":89787},[57],[15,89789],{"className":89790,"style":86933},[61],[15,89792,29727],{"className":89793,"style":31338},[1339,1470,13696],[15,89795],{"className":89796,"style":697},[71],[15,89798,89800],{"className":89799},[66,48546],[15,89801,89803,89913],{"className":89802},[202,1090],[15,89804,89806,89910],{"className":89805},[206],[15,89807,89809,89898],{"className":89808,"style":86952},[210],[15,89810,89812,89815],{"className":89811,"style":1386},[48560],[15,89813],{"className":89814,"style":1368},[217],[15,89816,89818,89858],{"className":89817,"style":48567},[66],[15,89819,89821,89824],{"className":89820},[66],[15,89822,11],{"className":89823},[66,67],[15,89825,89827],{"className":89826},[198],[15,89828,89830,89850],{"className":89829},[202,1090],[15,89831,89833,89847],{"className":89832},[206],[15,89834,89836],{"className":89835,"style":2139},[210],[15,89837,89838,89841],{"style":1584},[15,89839],{"className":89840,"style":218},[217],[15,89842,89844],{"className":89843},[222,223,224,225],[15,89845,1804],{"className":89846},[66,225],[15,89848,1114],{"className":89849},[1113],[15,89851,89853],{"className":89852},[206],[15,89854,89856],{"className":89855,"style":1121},[210],[15,89857],{},[15,89859,89861,89864],{"className":89860},[66],[15,89862,11],{"className":89863},[66,67],[15,89865,89867],{"className":89866},[198],[15,89868,89870,89890],{"className":89869},[202,1090],[15,89871,89873,89887],{"className":89872},[206],[15,89874,89876],{"className":89875,"style":2139},[210],[15,89877,89878,89881],{"style":1584},[15,89879],{"className":89880,"style":218},[217],[15,89882,89884],{"className":89883},[222,223,224,225],[15,89885,1211],{"className":89886},[66,225],[15,89888,1114],{"className":89889},[1113],[15,89891,89893],{"className":89892},[206],[15,89894,89896],{"className":89895,"style":1121},[210],[15,89897],{},[15,89899,89900,89903],{"style":87044},[15,89901],{"className":89902,"style":1368},[217],[15,89904,89906],{"className":89905,"style":48587},[48586],[34823,89907,89908],{"xmlns":34825,"width":48590,"height":48591,"viewBox":48592,"preserveAspectRatio":48593},[34829,89909],{"d":48596},[15,89911,1114],{"className":89912},[1113],[15,89914,89916],{"className":89915},[206],[15,89917,89919],{"className":89918,"style":87064},[210],[15,89920],{},"（算术-几何平均不等式）。Bhattacharyya 系数可以直接计算的情形（例如高斯），这个上界就能给出正确的指数衰减率。",[11,89923,89924,89925,89928,89929,89988,89989,89993],{},"把两个方向放在一起，可以看到散度在决策问题里的角色：它们不是「错误率」，而是错误率的",[2040,89926,89927],{},"证书","。全变差是唯一精确的那个，但它通常最难估计；Bhattacharyya 系数给出上界，KL 给出下界，Hellinger 则通过 ",[15,89930,89932,89954],{"className":89931,"translate":19},[18],[15,89933,89935],{"className":89934},[23],[25,89936,89937],{"xmlns":27},[29,89938,89939,89951],{},[32,89940,89941,89947,89949],{},[32,89942,89943,89945],{},[35,89944,32773],{"mathvariant":138},[35,89946,31727],{"mathvariant":138},[39,89948,24215],{},[35,89950,1061],{},[45,89952,89953],{"encoding":47},"\\mathrm{TV}\\le H",[15,89955,89957,89979],{"className":89956,"ariaHidden":53},[52],[15,89958,89960,89964,89970,89973,89976],{"className":89959},[57],[15,89961],{"className":89962,"style":89963},[61],"height:0.8193em;vertical-align:-0.136em;",[15,89965,89967],{"className":89966},[66],[15,89968,82602],{"className":89969,"style":1343},[66,4427],[15,89971],{"className":89972,"style":173},[71],[15,89974,24215],{"className":89975},[177],[15,89977],{"className":89978,"style":173},[71],[15,89980,89982,89985],{"className":89981},[57],[15,89983],{"className":89984,"style":490},[61],[15,89986,1061],{"className":89987,"style":7071},[66,67]," 把两边连起来。",[4554,89990,89992],{"href":89991},"\u002Fblog\u002F2026\u002F0011-about-the-f-divergence","关于 f-散度的那篇文章","，正是把这些证书放在同一个凸函数框架下；而贝叶斯错误率则告诉我们，这些证书最终约束的是什么。",[1904,89995,89997],{"id":89996},"_44-错误率住在哪里","4.4 错误率住在哪里",[11,89999,90000,90001,90145,90146,90220],{},"由 3.6 节的恒等式，只有满足 ",[15,90002,90004,90038],{"className":90003,"translate":19},[18],[15,90005,90007],{"className":90006},[23],[25,90008,90009],{"xmlns":27},[29,90010,90011,90035],{},[32,90012,90013,90015,90017,90019,90021,90023,90029,90031,90033],{},[35,90014,3883],{},[39,90016,120],{"stretchy":119},[35,90018,446],{},[39,90020,126],{"stretchy":119},[39,90022,13014],{"mathvariant":138},[131,90024,90025,90027],{},[35,90026,3883],{},[39,90028,1162],{},[39,90030,120],{"stretchy":119},[35,90032,446],{},[39,90034,126],{"stretchy":119},[45,90036,90037],{"encoding":47},"\\delta(X)\\ne\\delta^*(X)",[15,90039,90041,90101],{"className":90040,"ariaHidden":53},[52],[15,90042,90044,90047,90050,90053,90056,90059,90062,90098],{"className":90043},[57],[15,90045],{"className":90046,"style":154},[61],[15,90048,3883],{"className":90049,"style":3944},[66,67],[15,90051,120],{"className":90052},[162],[15,90054,446],{"className":90055,"style":16492},[66,67],[15,90057,126],{"className":90058},[169],[15,90060],{"className":90061,"style":173},[71],[15,90063,90065,90092,90095],{"className":90064},[177],[15,90066,90068],{"className":90067},[177],[15,90069,90071],{"className":90070},[66,13081],[15,90072,90074],{"className":90073},[13085],[15,90075,90077,90080,90089],{"className":90076},[13089],[15,90078],{"className":90079,"style":3406},[61],[15,90081,90083],{"className":90082},[13096],[15,90084,90086],{"className":90085},[66],[15,90087,13103],{"className":90088},[177],[15,90090],{"className":90091},[13107],[15,90093],{"className":90094},[71,13111],[15,90096,129],{"className":90097},[177],[15,90099],{"className":90100,"style":173},[71],[15,90102,90104,90107,90136,90139,90142],{"className":90103},[57],[15,90105],{"className":90106,"style":154},[61],[15,90108,90110,90113],{"className":90109},[66],[15,90111,3883],{"className":90112,"style":3944},[66,67],[15,90114,90116],{"className":90115},[198],[15,90117,90119],{"className":90118},[202],[15,90120,90122],{"className":90121},[206],[15,90123,90125],{"className":90124,"style":16244},[210],[15,90126,90127,90130],{"style":213},[15,90128],{"className":90129,"style":218},[217],[15,90131,90133],{"className":90132},[222,223,224,225],[15,90134,1162],{"className":90135},[76,225],[15,90137,120],{"className":90138},[162],[15,90140,446],{"className":90141,"style":16492},[66,67],[15,90143,126],{"className":90144},[169]," 的样本才贡献错误，且权重是 ",[15,90147,90149,90178],{"className":90148,"translate":19},[18],[15,90150,90152],{"className":90151},[23],[25,90153,90154],{"xmlns":27},[29,90155,90156,90176],{},[32,90157,90158,90160,90162,90164,90166,90168,90170,90172,90174],{},[35,90159,34425],{"mathvariant":138},[1209,90161,1263],{},[35,90163,71105],{},[39,90165,120],{"stretchy":119},[35,90167,446],{},[39,90169,126],{"stretchy":119},[39,90171,6600],{},[1209,90173,1211],{},[35,90175,34425],{"mathvariant":138},[45,90177,77778],{"encoding":47},[15,90179,90181,90211],{"className":90180,"ariaHidden":53},[52],[15,90182,90184,90187,90190,90193,90196,90199,90202,90205,90208],{"className":90183},[57],[15,90185],{"className":90186,"style":154},[61],[15,90188,76983],{"className":90189},[66],[15,90191,71105],{"className":90192,"style":1515},[66,67],[15,90194,120],{"className":90195},[162],[15,90197,446],{"className":90198,"style":16492},[66,67],[15,90200,126],{"className":90201},[169],[15,90203],{"className":90204,"style":72},[71],[15,90206,6600],{"className":90207},[76],[15,90209],{"className":90210,"style":72},[71],[15,90212,90214,90217],{"className":90213},[57],[15,90215],{"className":90216,"style":154},[61],[15,90218,77014],{"className":90219},[66],"。这意味着贝叶斯错误率不是由决策边界的「长度」或「复杂度」决定的，而是由边界附近的后验质量决定的。",[11,90222,90223,90224,90253,90254,77883,90282,90310,90311,90367],{},"高维空间里，决策边界可以非常复杂，但如果后验在边界两侧都很尖锐（",[15,90225,90227,90241],{"className":90226,"translate":19},[18],[15,90228,90230],{"className":90229},[23],[25,90231,90232],{"xmlns":27},[29,90233,90234,90238],{},[32,90235,90236],{},[35,90237,71105],{},[45,90239,90240],{"encoding":47},"\\eta",[15,90242,90244],{"className":90243,"ariaHidden":53},[52],[15,90245,90247,90250],{"className":90246},[57],[15,90248],{"className":90249,"style":471},[61],[15,90251,71105],{"className":90252,"style":1515},[66,67]," 接近 ",[15,90255,90257,90270],{"className":90256,"translate":19},[18],[15,90258,90260],{"className":90259},[23],[25,90261,90262],{"xmlns":27},[29,90263,90264,90268],{},[32,90265,90266],{},[1209,90267,1804],{},[45,90269,1804],{"encoding":47},[15,90271,90273],{"className":90272,"ariaHidden":53},[52],[15,90274,90276,90279],{"className":90275},[57],[15,90277],{"className":90278,"style":1836},[61],[15,90280,1804],{"className":90281},[66],[15,90283,90285,90298],{"className":90284,"translate":19},[18],[15,90286,90288],{"className":90287},[23],[25,90289,90290],{"xmlns":27},[29,90291,90292,90296],{},[32,90293,90294],{},[1209,90295,1211],{},[45,90297,1211],{"encoding":47},[15,90299,90301],{"className":90300,"ariaHidden":53},[52],[15,90302,90304,90307],{"className":90303},[57],[15,90305],{"className":90306,"style":1836},[61],[15,90308,1211],{"className":90309},[66],"），错误率仍然可以很低。反过来，一个简单的线性边界，如果穿过一大片 ",[15,90312,90314,90336],{"className":90313,"translate":19},[18],[15,90315,90317],{"className":90316},[23],[25,90318,90319],{"xmlns":27},[29,90320,90321,90333],{},[32,90322,90323,90325,90327,90329,90331],{},[35,90324,71105],{},[39,90326,31988],{},[1209,90328,1211],{},[35,90330,6623],{"mathvariant":138},[1209,90332,1263],{},[45,90334,90335],{"encoding":47},"\\eta\\approx 1\u002F2",[15,90337,90339,90358],{"className":90338,"ariaHidden":53},[52],[15,90340,90342,90346,90349,90352,90355],{"className":90341},[57],[15,90343],{"className":90344,"style":90345},[61],"height:0.6776em;vertical-align:-0.1944em;",[15,90347,71105],{"className":90348,"style":1515},[66,67],[15,90350],{"className":90351,"style":173},[71],[15,90353,31988],{"className":90354},[177],[15,90356],{"className":90357,"style":173},[71],[15,90359,90361,90364],{"className":90360},[57],[15,90362],{"className":90363,"style":154},[61],[15,90365,71801],{"className":90366},[66]," 的区域，错误率就会很高。这解释了为什么「边界复杂度」本身不是泛化误差的可靠指标：真正决定错误率的是后验在边界附近的分布，而不是边界的几何形状。",[1904,90369,90371],{"id":90370},"_45-回归的贝叶斯风险","4.5 回归的贝叶斯风险",[11,90373,90374],{},"平方损失下，贝叶斯风险为",[15,90376,90378],{"className":90377,"translate":19},[294],[15,90379,90381,90459],{"className":90380,"translate":19},[18],[15,90382,90384],{"className":90383},[23],[25,90385,90386],{"xmlns":27,"display":303},[29,90387,90388,90456],{},[32,90389,90390,90396,90398,90400,90402,90404,90406,90408,90414,90416,90418,90420,90426,90428,90430,90432,90434,90442,90444,90446,90448,90450,90452,90454],{},[131,90391,90392,90394],{},[35,90393,455],{},[39,90395,1162],{},[39,90397,129],{},[35,90399,62377],{"mathvariant":454},[39,90401,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,90403,120],{"stretchy":119},[35,90405,61881],{},[39,90407,6600],{},[131,90409,90410,90412],{},[35,90411,116],{},[39,90413,1162],{},[39,90415,120],{"stretchy":119},[35,90417,446],{},[39,90419,126],{"stretchy":119},[131,90421,90422,90424],{},[39,90423,126],{"stretchy":119},[1209,90425,1263],{},[39,90427,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,90429,129],{},[35,90431,62377],{"mathvariant":454},[39,90433,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[32,90435,90436,90438,90440],{},[35,90437,31727],{"mathvariant":138},[35,90439,4554],{"mathvariant":138},[35,90441,63103],{"mathvariant":138},[39,90443,120],{"stretchy":119},[35,90445,61881],{},[39,90447,34425],{},[35,90449,446],{},[39,90451,126],{"stretchy":119},[39,90453,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,90455,3571],{"mathvariant":138},[45,90457,90458],{"encoding":47},"R^*= \\mathbb E\\big[(Y-f^*(X))^2\\big]=\\mathbb E\\big[\\mathrm{Var}(Y\\mid X)\\big].",[15,90460,90462,90506,90536,90624,90660],{"className":90461,"ariaHidden":53},[52],[15,90463,90465,90468,90497,90500,90503],{"className":90464},[57],[15,90466],{"className":90467,"style":1308},[61],[15,90469,90471,90474],{"className":90470},[66],[15,90472,455],{"className":90473,"style":62433},[66,67],[15,90475,90477],{"className":90476},[198],[15,90478,90480],{"className":90479},[202],[15,90481,90483],{"className":90482},[206],[15,90484,90486],{"className":90485,"style":1308},[210],[15,90487,90488,90491],{"style":397},[15,90489],{"className":90490,"style":218},[217],[15,90492,90494],{"className":90493},[222,223,224,225],[15,90495,1162],{"className":90496},[76,225],[15,90498],{"className":90499,"style":173},[71],[15,90501,129],{"className":90502},[177],[15,90504],{"className":90505,"style":173},[71],[15,90507,90509,90512,90515,90521,90524,90527,90530,90533],{"className":90508},[57],[15,90510],{"className":90511,"style":4398},[61],[15,90513,62377],{"className":90514},[66,517],[15,90516,90518],{"className":90517},[66],[15,90519,1194],{"className":90520},[1422,1508],[15,90522,120],{"className":90523},[162],[15,90525,61881],{"className":90526,"style":72},[66,67],[15,90528],{"className":90529,"style":72},[71],[15,90531,6600],{"className":90532},[76],[15,90534],{"className":90535,"style":72},[71],[15,90537,90539,90542,90571,90574,90577,90580,90609,90615,90618,90621],{"className":90538},[57],[15,90540],{"className":90541,"style":11713},[61],[15,90543,90545,90548],{"className":90544},[66],[15,90546,116],{"className":90547,"style":158},[66,67],[15,90549,90551],{"className":90550},[198],[15,90552,90554],{"className":90553},[202],[15,90555,90557],{"className":90556},[206],[15,90558,90560],{"className":90559,"style":1308},[210],[15,90561,90562,90565],{"style":397},[15,90563],{"className":90564,"style":218},[217],[15,90566,90568],{"className":90567},[222,223,224,225],[15,90569,1162],{"className":90570},[76,225],[15,90572,120],{"className":90573},[162],[15,90575,446],{"className":90576,"style":16492},[66,67],[15,90578,126],{"className":90579},[169],[15,90581,90583,90586],{"className":90582},[169],[15,90584,126],{"className":90585},[169],[15,90587,90589],{"className":90588},[198],[15,90590,90592],{"className":90591},[202],[15,90593,90595],{"className":90594},[206],[15,90596,90598],{"className":90597,"style":1650},[210],[15,90599,90600,90603],{"style":397},[15,90601],{"className":90602,"style":218},[217],[15,90604,90606],{"className":90605},[222,223,224,225],[15,90607,1263],{"className":90608},[66,225],[15,90610,90612],{"className":90611},[66],[15,90613,1266],{"className":90614},[1422,1508],[15,90616],{"className":90617,"style":173},[71],[15,90619,129],{"className":90620},[177],[15,90622],{"className":90623,"style":173},[71],[15,90625,90627,90630,90633,90639,90645,90648,90651,90654,90657],{"className":90626},[57],[15,90628],{"className":90629,"style":4398},[61],[15,90631,62377],{"className":90632},[66,517],[15,90634,90636],{"className":90635},[66],[15,90637,1194],{"className":90638},[1422,1508],[15,90640,90642],{"className":90641},[66],[15,90643,72399],{"className":90644},[66,4427],[15,90646,120],{"className":90647},[162],[15,90649,61881],{"className":90650,"style":72},[66,67],[15,90652],{"className":90653,"style":173},[71],[15,90655,34425],{"className":90656},[177],[15,90658],{"className":90659,"style":173},[71],[15,90661,90663,90666,90669,90672,90678],{"className":90662},[57],[15,90664],{"className":90665,"style":4398},[61],[15,90667,446],{"className":90668,"style":16492},[66,67],[15,90670,126],{"className":90671},[169],[15,90673,90675],{"className":90674},[66],[15,90676,1266],{"className":90677},[1422,1508],[15,90679,3571],{"className":90680},[66],[11,90682,90683,90684,90712,90713,90826],{},"这是数据生成过程本身的噪声，任何模型都消不掉。对一个在训练集 ",[15,90685,90687,90700],{"className":90686,"translate":19},[18],[15,90688,90690],{"className":90689},[23],[25,90691,90692],{"xmlns":27},[29,90693,90694,90698],{},[32,90695,90696],{},[35,90697,32453],{},[45,90699,32453],{"encoding":47},[15,90701,90703],{"className":90702,"ariaHidden":53},[52],[15,90704,90706,90709],{"className":90705},[57],[15,90707],{"className":90708,"style":490},[61],[15,90710,32453],{"className":90711,"style":23020},[66,67]," 上学到的回归器 ",[15,90714,90716,90738],{"className":90715,"translate":19},[18],[15,90717,90719],{"className":90718},[23],[25,90720,90721],{"xmlns":27},[29,90722,90723,90735],{},[32,90724,90725],{},[1057,90726,90727,90733],{},[4294,90728,90729,90731],{"accent":53},[35,90730,116],{},[39,90732,78122],{},[35,90734,32453],{},[45,90736,90737],{"encoding":47},"\\hat f_D",[15,90739,90741],{"className":90740,"ariaHidden":53},[52],[15,90742,90744,90747],{"className":90743},[57],[15,90745],{"className":90746,"style":78135},[61],[15,90748,90750,90792],{"className":90749},[66],[15,90751,90753],{"className":90752},[66,78139],[15,90754,90756,90784],{"className":90755},[202,1090],[15,90757,90759,90781],{"className":90758},[206],[15,90760,90762,90770],{"className":90761,"style":78149},[210],[15,90763,90764,90767],{"style":1386},[15,90765],{"className":90766,"style":1368},[217],[15,90768,116],{"className":90769,"style":158},[66,67],[15,90771,90772,90775],{"style":78160},[15,90773],{"className":90774,"style":1368},[217],[15,90776,90778],{"className":90777,"style":78168},[78167],[15,90779,78122],{"className":90780},[66],[15,90782,1114],{"className":90783},[1113],[15,90785,90787],{"className":90786},[206],[15,90788,90790],{"className":90789,"style":4448},[210],[15,90791],{},[15,90793,90795],{"className":90794},[198],[15,90796,90798,90818],{"className":90797},[202,1090],[15,90799,90801,90815],{"className":90800},[206],[15,90802,90804],{"className":90803,"style":3636},[210],[15,90805,90806,90809],{"style":9913},[15,90807],{"className":90808,"style":218},[217],[15,90810,90812],{"className":90811},[222,223,224,225],[15,90813,32453],{"className":90814,"style":23020},[66,67,225],[15,90816,1114],{"className":90817},[1113],[15,90819,90821],{"className":90820},[206],[15,90822,90824],{"className":90823,"style":1121},[210],[15,90825],{},"，有标准的偏差-方差分解：",[15,90828,90830],{"className":90829,"translate":19},[294],[15,90831,90833,91011],{"className":90832,"translate":19},[18],[15,90834,90836],{"className":90835},[23],[25,90837,90838],{"xmlns":27,"display":303},[29,90839,90840,91008],{},[32,90841,90842,90848,90850,90852,90854,90856,90866,90868,90870,90872,90878,90880,90882,90884,90900,90902,90961,90963,91006],{},[1057,90843,90844,90846],{},[35,90845,62377],{"mathvariant":454},[35,90847,32453],{},[39,90849,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,90851,120],{"stretchy":119},[35,90853,61881],{},[39,90855,6600],{},[1057,90857,90858,90864],{},[4294,90859,90860,90862],{"accent":53},[35,90861,116],{},[39,90863,78122],{},[35,90865,32453],{},[39,90867,120],{"stretchy":119},[35,90869,446],{},[39,90871,126],{"stretchy":119},[131,90873,90874,90876],{},[39,90875,126],{"stretchy":119},[1209,90877,1263],{},[39,90879,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,90881,129],{},[71,90883],{"linebreak":10451},[1172,90885,90886,90898],{},[1172,90887,90888,90896],{},[1255,90889,90890,90892,90894],{},[35,90891,6632],{},[35,90893,28513],{},[1209,90895,1263],{},[39,90897,69838],{"stretchy":53},[1274,90899,61850],{},[39,90901,1245],{},[1172,90903,90904,90954],{},[1172,90905,90906,90952],{},[32,90907,90908,90910,90916,90918,90920,90922,90924,90930,90940,90942,90944,90946],{},[39,90909,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[131,90911,90912,90914],{},[35,90913,116],{},[39,90915,1162],{},[39,90917,120],{"stretchy":119},[35,90919,446],{},[39,90921,126],{"stretchy":119},[39,90923,6600],{},[1057,90925,90926,90928],{},[35,90927,62377],{"mathvariant":454},[35,90929,32453],{},[1057,90931,90932,90938],{},[4294,90933,90934,90936],{"accent":53},[35,90935,116],{},[39,90937,78122],{},[35,90939,32453],{},[39,90941,120],{"stretchy":119},[35,90943,446],{},[39,90945,126],{"stretchy":119},[131,90947,90948,90950],{},[39,90949,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,90951,1263],{},[39,90953,69838],{"stretchy":53},[131,90955,90956,90959],{},[1274,90957,90958],{},"偏差",[1209,90960,1263],{},[39,90962,1245],{},[1172,90964,90965,91003],{},[1172,90966,90967,91001],{},[32,90968,90969,90981,90983,90993,90995,90997,90999],{},[1057,90970,90971,90979],{},[32,90972,90973,90975,90977],{},[35,90974,31727],{"mathvariant":138},[35,90976,4554],{"mathvariant":138},[35,90978,63103],{"mathvariant":138},[35,90980,32453],{},[39,90982,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,90984,90985,90991],{},[4294,90986,90987,90989],{"accent":53},[35,90988,116],{},[39,90990,78122],{},[35,90992,32453],{},[39,90994,120],{"stretchy":119},[35,90996,446],{},[39,90998,126],{"stretchy":119},[39,91000,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,91002,69838],{"stretchy":53},[1274,91004,91005],{},"方差",[35,91007,3571],{"mathvariant":138},[45,91009,91010],{"encoding":47},"\\mathbb E_D\\big[(Y-\\hat f_D(X))^2\\big]\n= \\\\ \\underbrace{\\sigma_\\varepsilon^2}_{\\text{贝叶斯风险}}\n+\\underbrace{\\big(f^*(X)-\\mathbb E_D\\hat f_D(X)\\big)^2}_{\\text{偏差}^2}\n+\\underbrace{\\mathrm{Var}_D\\big(\\hat 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贝叶斯决策如何长成算法",[1904,92038,92040],{"id":92039},"_51-生成式与判别式","5.1 生成式与判别式",[11,92042,92043,92044,92168,92169,92197,92198,92264,92265,92309,92310,92313,92314,92317],{},"贝叶斯规则只依赖后验概率 ",[15,92045,92047,92084],{"className":92046,"translate":19},[18],[15,92048,92050],{"className":92049},[23],[25,92051,92052],{"xmlns":27},[29,92053,92054,92082],{},[32,92055,92056,92058,92060,92062,92064,92066,92068,92070,92072,92074,92076,92078,92080],{},[35,92057,71105],{},[39,92059,120],{"stretchy":119},[35,92061,123],{},[39,92063,126],{"stretchy":119},[39,92065,129],{},[35,92067,11],{},[39,92069,120],{"stretchy":119},[35,92071,61881],{},[39,92073,129],{},[1209,92075,1211],{},[39,92077,34425],{},[35,92079,123],{},[39,92081,126],{"stretchy":119},[45,92083,71132],{"encoding":47},[15,92085,92087,92114,92138,92156],{"className":92086,"ariaHidden":53},[52],[15,92088,92090,92093,92096,92099,92102,92105,92108,92111],{"className":92089},[57],[15,92091],{"className":92092,"style":154},[61],[15,92094,71105],{"className":92095,"style":1515},[66,67],[15,92097,120],{"className":92098},[162],[15,92100,123],{"className":92101},[66,67],[15,92103,126],{"className":92104},[169],[15,92106],{"className":92107,"style":173},[71],[15,92109,129],{"className":92110},[177],[15,92112],{"className":92113,"style":173},[71],[15,92115,92117,92120,92123,92126,92129,92132,92135],{"className":92116},[57],[15,92118],{"className":92119,"style":154},[61],[15,92121,11],{"className":92122},[66,67],[15,92124,120],{"className":92125},[162],[15,92127,61881],{"className":92128,"style":72},[66,67],[15,92130],{"className":92131,"style":173},[71],[15,92133,129],{"className":92134},[177],[15,92136],{"className":92137,"style":173},[71],[15,92139,92141,92144,92147,92150,92153],{"className":92140},[57],[15,92142],{"className":92143,"style":154},[61],[15,92145,1211],{"className":92146},[66],[15,92148],{"className":92149,"style":173},[71],[15,92151,34425],{"className":92152},[177],[15,92154],{"className":92155,"style":173},[71],[15,92157,92159,92162,92165],{"className":92158},[57],[15,92160],{"className":92161,"style":154},[61],[15,92163,123],{"className":92164},[66,67],[15,92166,126],{"className":92167},[169],"。因此有两条路可以走：直接建模 ",[15,92170,92172,92185],{"className":92171,"translate":19},[18],[15,92173,92175],{"className":92174},[23],[25,92176,92177],{"xmlns":27},[29,92178,92179,92183],{},[32,92180,92181],{},[35,92182,71105],{},[45,92184,90240],{"encoding":47},[15,92186,92188],{"className":92187,"ariaHidden":53},[52],[15,92189,92191,92194],{"className":92190},[57],[15,92192],{"className":92193,"style":471},[61],[15,92195,71105],{"className":92196,"style":1515},[66,67],"，或者分别建模类条件密度 ",[15,92199,92201,92225],{"className":92200,"translate":19},[18],[15,92202,92204],{"className":92203},[23],[25,92205,92206],{"xmlns":27},[29,92207,92208,92222],{},[32,92209,92210,92212,92214,92216,92218,92220],{},[35,92211,11],{},[39,92213,120],{"stretchy":119},[35,92215,123],{},[39,92217,34425],{},[35,92219,1224],{},[39,92221,126],{"stretchy":119},[45,92223,92224],{"encoding":47},"p(x\\mid y)",[15,92226,92228,92252],{"className":92227,"ariaHidden":53},[52],[15,92229,92231,92234,92237,92240,92243,92246,92249],{"className":92230},[57],[15,92232],{"className":92233,"style":154},[61],[15,92235,11],{"className":92236},[66,67],[15,92238,120],{"className":92239},[162],[15,92241,123],{"className":92242},[66,67],[15,92244],{"className":92245,"style":173},[71],[15,92247,34425],{"className":92248},[177],[15,92250],{"className":92251,"style":173},[71],[15,92253,92255,92258,92261],{"className":92254},[57],[15,92256],{"className":92257,"style":154},[61],[15,92259,1224],{"className":92260,"style":1515},[66,67],[15,92262,126],{"className":92263},[169]," 与先验 ",[15,92266,92268,92288],{"className":92267,"translate":19},[18],[15,92269,92271],{"className":92270},[23],[25,92272,92273],{"xmlns":27},[29,92274,92275,92285],{},[32,92276,92277,92279,92281,92283],{},[35,92278,11],{},[39,92280,120],{"stretchy":119},[35,92282,1224],{},[39,92284,126],{"stretchy":119},[45,92286,92287],{"encoding":47},"p(y)",[15,92289,92291],{"className":92290,"ariaHidden":53},[52],[15,92292,92294,92297,92300,92303,92306],{"className":92293},[57],[15,92295],{"className":92296,"style":154},[61],[15,92298,11],{"className":92299},[66,67],[15,92301,120],{"className":92302},[162],[15,92304,1224],{"className":92305,"style":1515},[66,67],[15,92307,126],{"className":92308},[169],"，再用贝叶斯公式合成。前者叫",[2040,92311,92312],{},"判别式","（discriminative），后者叫",[2040,92315,92316],{},"生成式","（generative）。",[11,92319,92320],{},"生成式的合成公式是",[15,92322,92324],{"className":92323,"translate":19},[294],[15,92325,92327,92413],{"className":92326,"translate":19},[18],[15,92328,92330],{"className":92329},[23],[25,92331,92332],{"xmlns":27,"display":303},[29,92333,92334,92410],{},[32,92335,92336,92338,92340,92342,92344,92346,92408],{},[35,92337,71105],{},[39,92339,120],{"stretchy":119},[35,92341,123],{},[39,92343,126],{"stretchy":119},[39,92345,129],{},[6465,92347,92348,92368],{},[32,92349,92350,92356,92362,92364,92366],{},[1057,92351,92352,92354],{},[35,92353,33489],{},[1209,92355,1211],{},[1057,92357,92358,92360],{},[35,92359,11],{},[1209,92361,1211],{},[39,92363,120],{"stretchy":119},[35,92365,123],{},[39,92367,126],{"stretchy":119},[32,92369,92370,92376,92382,92384,92386,92388,92390,92396,92402,92404,92406],{},[1057,92371,92372,92374],{},[35,92373,33489],{},[1209,92375,1804],{},[1057,92377,92378,92380],{},[35,92379,11],{},[1209,92381,1804],{},[39,92383,120],{"stretchy":119},[35,92385,123],{},[39,92387,126],{"stretchy":119},[39,92389,1245],{},[1057,92391,92392,92394],{},[35,92393,33489],{},[1209,92395,1211],{},[1057,92397,92398,92400],{},[35,92399,11],{},[1209,92401,1211],{},[39,92403,120],{"stretchy":119},[35,92405,123],{},[39,92407,126],{"stretchy":119},[35,92409,3571],{"mathvariant":138},[45,92411,92412],{"encoding":47},"\\eta(x)=\\frac{\\pi_1p_1(x)}{\\pi_0p_0(x)+\\pi_1p_1(x)}.",[15,92414,92416,92443],{"className":92415,"ariaHidden":53},[52],[15,92417,92419,92422,92425,92428,92431,92434,92437,92440],{"className":92418},[57],[15,92420],{"className":92421,"style":154},[61],[15,92423,71105],{"className":92424,"style":1515},[66,67],[15,92426,120],{"className":92427},[162],[15,92429,123],{"className":92430},[66,67],[15,92432,126],{"className":92433},[169],[15,92435],{"className":92436,"style":173},[71],[15,92438,129],{"className":92439},[177],[15,92441],{"className":92442,"style":173},[71],[15,92444,92446,92449,92781],{"className":92445},[57],[15,92447],{"className":92448,"style":41187},[61],[15,92450,92452,92455,92778],{"className":92451},[66],[15,92453],{"className":92454},[162,23412],[15,92456,92458],{"className":92457},[6465],[15,92459,92461,92770],{"className":92460},[202,1090],[15,92462,92464,92767],{"className":92463},[206],[15,92465,92467,92662,92670],{"className":92466,"style":41206},[210],[15,92468,92469,92472],{"style":23428},[15,92470],{"className":92471,"style":1368},[217],[15,92473,92475,92515,92555,92558,92561,92564,92567,92570,92573,92613,92653,92656,92659],{"className":92474},[66],[15,92476,92478,92481],{"className":92477},[66],[15,92479,33489],{"className":92480,"style":1515},[66,67],[15,92482,92484],{"className":92483},[198],[15,92485,92487,92507],{"className":92486},[202,1090],[15,92488,92490,92504],{"className":92489},[206],[15,92491,92493],{"className":92492,"style":2139},[210],[15,92494,92495,92498],{"style":1531},[15,92496],{"className":92497,"style":218},[217],[15,92499,92501],{"className":92500},[222,223,224,225],[15,92502,1804],{"className":92503},[66,225],[15,92505,1114],{"className":92506},[1113],[15,92508,92510],{"className":92509},[206],[15,92511,92513],{"className":92512,"style":1121},[210],[15,92514],{},[15,92516,92518,92521],{"className":92517},[66],[15,92519,11],{"className":92520},[66,67],[15,92522,92524],{"className":92523},[198],[15,92525,92527,92547],{"className":92526},[202,1090],[15,92528,92530,92544],{"className":92529},[206],[15,92531,92533],{"className":92532,"style":2139},[210],[15,92534,92535,92538],{"style":1584},[15,92536],{"className":92537,"style":218},[217],[15,92539,92541],{"className":92540},[222,223,224,225],[15,92542,1804],{"className":92543},[66,225],[15,92545,1114],{"className":92546},[1113],[15,92548,92550],{"className":92549},[206],[15,92551,92553],{"className":92552,"style":1121},[210],[15,92554],{},[15,92556,120],{"className":92557},[162],[15,92559,123],{"className":92560},[66,67],[15,92562,126],{"className":92563},[169],[15,92565],{"className":92566,"style":72},[71],[15,92568,1245],{"className":92569},[76],[15,92571],{"className":92572,"style":72},[71],[15,92574,92576,92579],{"className":92575},[66],[15,92577,33489],{"className":92578,"style":1515},[66,67],[15,92580,92582],{"className":92581},[198],[15,92583,92585,92605],{"className":92584},[202,1090],[15,92586,92588,92602],{"className":92587},[206],[15,92589,92591],{"className":92590,"style":2139},[210],[15,92592,92593,92596],{"style":1531},[15,92594],{"className":92595,"style":218},[217],[15,92597,92599],{"className":92598},[222,223,224,225],[15,92600,1211],{"className":92601},[66,225],[15,92603,1114],{"className":92604},[1113],[15,92606,92608],{"className":92607},[206],[15,92609,92611],{"className":92610,"style":1121},[210],[15,92612],{},[15,92614,92616,92619],{"className":92615},[66],[15,92617,11],{"className":92618},[66,67],[15,92620,92622],{"className":92621},[198],[15,92623,92625,92645],{"className":92624},[202,1090],[15,92626,92628,92642],{"className":92627},[206],[15,92629,92631],{"className":92630,"style":2139},[210],[15,92632,92633,92636],{"style":1584},[15,92634],{"className":92635,"style":218},[217],[15,92637,92639],{"className":92638},[222,223,224,225],[15,92640,1211],{"className":92641},[66,225],[15,92643,1114],{"className":92644},[1113],[15,92646,92648],{"className":92647},[206],[15,92649,92651],{"className":92650,"style":1121},[210],[15,92652],{},[15,92654,120],{"className":92655},[162],[15,92657,123],{"className":92658},[66,67],[15,92660,126],{"className":92661},[169],[15,92663,92664,92667],{"style":23440},[15,92665],{"className":92666,"style":1368},[217],[15,92668],{"className":92669,"style":4438},[23447],[15,92671,92672,92675],{"style":23450},[15,92673],{"className":92674,"style":1368},[217],[15,92676,92678,92718,92758,92761,92764],{"className":92677},[66],[15,92679,92681,92684],{"className":92680},[66],[15,92682,33489],{"className":92683,"style":1515},[66,67],[15,92685,92687],{"className":92686},[198],[15,92688,92690,92710],{"className":92689},[202,1090],[15,92691,92693,92707],{"className":92692},[206],[15,92694,92696],{"className":92695,"style":2139},[210],[15,92697,92698,92701],{"style":1531},[15,92699],{"className":92700,"style":218},[217],[15,92702,92704],{"className":92703},[222,223,224,225],[15,92705,1211],{"className":92706},[66,225],[15,92708,1114],{"className":92709},[1113],[15,92711,92713],{"className":92712},[206],[15,92714,92716],{"className":92715,"style":1121},[210],[15,92717],{},[15,92719,92721,92724],{"className":92720},[66],[15,92722,11],{"className":92723},[66,67],[15,92725,92727],{"className":92726},[198],[15,92728,92730,92750],{"className":92729},[202,1090],[15,92731,92733,92747],{"className":92732},[206],[15,92734,92736],{"className":92735,"style":2139},[210],[15,92737,92738,92741],{"style":1584},[15,92739],{"className":92740,"style":218},[217],[15,92742,92744],{"className":92743},[222,223,224,225],[15,92745,1211],{"className":92746},[66,225],[15,92748,1114],{"className":92749},[1113],[15,92751,92753],{"className":92752},[206],[15,92754,92756],{"className":92755,"style":1121},[210],[15,92757],{},[15,92759,120],{"className":92760},[162],[15,92762,123],{"className":92763},[66,67],[15,92765,126],{"className":92766},[169],[15,92768,1114],{"className":92769},[1113],[15,92771,92773],{"className":92772},[206],[15,92774,92776],{"className":92775,"style":41277},[210],[15,92777],{},[15,92779],{"className":92780},[169,23412],[15,92782,3571],{"className":92783},[66],[11,92785,92786,92787,92816,92817,92845],{},"只要两边的密度估计足够准，这个式子就给出贝叶斯后验。线性判别分析（LDA）假设两类共享协方差 ",[15,92788,92790,92804],{"className":92789,"translate":19},[18],[15,92791,92793],{"className":92792},[23],[25,92794,92795],{"xmlns":27},[29,92796,92797,92801],{},[32,92798,92799],{},[35,92800,83401],{"mathvariant":138},[45,92802,92803],{"encoding":47},"\\Sigma",[15,92805,92807],{"className":92806,"ariaHidden":53},[52],[15,92808,92810,92813],{"className":92809},[57],[15,92811],{"className":92812,"style":490},[61],[15,92814,83401],{"className":92815},[66],"，此时对数似然比是 ",[15,92818,92820,92833],{"className":92819,"translate":19},[18],[15,92821,92823],{"className":92822},[23],[25,92824,92825],{"xmlns":27},[29,92826,92827,92831],{},[32,92828,92829],{},[35,92830,123],{},[45,92832,123],{"encoding":47},[15,92834,92836],{"className":92835,"ariaHidden":53},[52],[15,92837,92839,92842],{"className":92838},[57],[15,92840],{"className":92841,"style":86},[61],[15,92843,123],{"className":92844},[66,67]," 的线性函数：",[15,92847,92849],{"className":92848,"translate":19},[294],[15,92850,92852,92941],{"className":92851,"translate":19},[18],[15,92853,92855],{"className":92854},[23],[25,92856,92857],{"xmlns":27,"display":303},[29,92858,92859,92938],{},[32,92860,92861,92863,92865,92895,92897,92899,92905,92907,92913,92919,92929,92931,92933,92936],{},[35,92862,27068],{},[39,92864,1170],{},[6465,92866,92867,92881],{},[32,92868,92869,92875,92877,92879],{},[1057,92870,92871,92873],{},[35,92872,11],{},[1209,92874,1211],{},[39,92876,120],{"stretchy":119},[35,92878,123],{},[39,92880,126],{"stretchy":119},[32,92882,92883,92889,92891,92893],{},[1057,92884,92885,92887],{},[35,92886,11],{},[1209,92888,1804],{},[39,92890,120],{"stretchy":119},[35,92892,123],{},[39,92894,126],{"stretchy":119},[39,92896,129],{},[39,92898,120],{"stretchy":119},[1057,92900,92901,92903],{},[35,92902,83394],{},[1209,92904,1211],{},[39,92906,6600],{},[1057,92908,92909,92911],{},[35,92910,83394],{},[1209,92912,1804],{},[131,92914,92915,92917],{},[39,92916,126],{"stretchy":119},[35,92918,139],{"mathvariant":138},[131,92920,92921,92923],{},[35,92922,83401],{"mathvariant":138},[32,92924,92925,92927],{},[39,92926,6600],{},[1209,92928,1211],{},[35,92930,123],{},[39,92932,1245],{},[1274,92934,92935],{},"常数",[39,92937,335],{"separator":53},[45,92939,92940],{"encoding":47},"\\log\\frac{p_1(x)}{p_0(x)}\n=(\\mu_1-\\mu_0)^\\top\\Sigma^{-1}x+\\text{常数},",[15,92942,92944,93121,93179,93301],{"className":92943,"ariaHidden":53},[52],[15,92945,92947,92950,92955,92958,93112,93115,93118],{"className":92946},[57],[15,92948],{"className":92949,"style":41187},[61],[15,92951,27278,92953],{"className":92952},[1339],[15,92954,1344],{"style":1343},[15,92956],{"className":92957,"style":697},[71],[15,92959,92961,92964,93109],{"className":92960},[66],[15,92962],{"className":92963},[162,23412],[15,92965,92967],{"className":92966},[6465],[15,92968,92970,93101],{"className":92969},[202,1090],[15,92971,92973,93098],{"className":92972},[206],[15,92974,92976,93033,93041],{"className":92975,"style":41206},[210],[15,92977,92978,92981],{"style":23428},[15,92979],{"className":92980,"style":1368},[217],[15,92982,92984,93024,93027,93030],{"className":92983},[66],[15,92985,92987,92990],{"className":92986},[66],[15,92988,11],{"className":92989},[66,67],[15,92991,92993],{"className":92992},[198],[15,92994,92996,93016],{"className":92995},[202,1090],[15,92997,92999,93013],{"className":92998},[206],[15,93000,93002],{"className":93001,"style":2139},[210],[15,93003,93004,93007],{"style":1584},[15,93005],{"className":93006,"style":218},[217],[15,93008,93010],{"className":93009},[222,223,224,225],[15,93011,1804],{"className":93012},[66,225],[15,93014,1114],{"className":93015},[1113],[15,93017,93019],{"className":93018},[206],[15,93020,93022],{"className":93021,"style":1121},[210],[15,93023],{},[15,93025,120],{"className":93026},[162],[15,93028,123],{"className":93029},[66,67],[15,93031,126],{"className":93032},[169],[15,93034,93035,93038],{"style":23440},[15,93036],{"className":93037,"style":1368},[217],[15,93039],{"className":93040,"style":4438},[23447],[15,93042,93043,93046],{"style":23450},[15,93044],{"className":93045,"style":1368},[217],[15,93047,93049,93089,93092,93095],{"className":93048},[66],[15,93050,93052,93055],{"className":93051},[66],[15,93053,11],{"className":93054},[66,67],[15,93056,93058],{"className":93057},[198],[15,93059,93061,93081],{"className":93060},[202,1090],[15,93062,93064,93078],{"className":93063},[206],[15,93065,93067],{"className":93066,"style":2139},[210],[15,93068,93069,93072],{"style":1584},[15,93070],{"className":93071,"style":218},[217],[15,93073,93075],{"className":93074},[222,223,224,225],[15,93076,1211],{"className":93077},[66,225],[15,93079,1114],{"className":93080},[1113],[15,93082,93084],{"className":93083},[206],[15,93085,93087],{"className":93086,"style":1121},[210],[15,93088],{},[15,93090,120],{"className":93091},[162],[15,93093,123],{"className":93094},[66,67],[15,93096,126],{"className":93097},[169],[15,93099,1114],{"className":93100},[1113],[15,93102,93104],{"className":93103},[206],[15,93105,93107],{"className":93106,"style":41277},[210],[15,93108],{},[15,93110],{"className":93111},[169,23412],[15,93113],{"className":93114,"style":173},[71],[15,93116,129],{"className":93117},[177],[15,93119],{"className":93120,"style":173},[71],[15,93122,93124,93127,93130,93170,93173,93176],{"className":93123},[57],[15,93125],{"className":93126,"style":154},[61],[15,93128,120],{"className":93129},[162],[15,93131,93133,93136],{"className":93132},[66],[15,93134,83394],{"className":93135},[66,67],[15,93137,93139],{"className":93138},[198],[15,93140,93142,93162],{"className":93141},[202,1090],[15,93143,93145,93159],{"className":93144},[206],[15,93146,93148],{"className":93147,"style":2139},[210],[15,93149,93150,93153],{"style":1584},[15,93151],{"className":93152,"style":218},[217],[15,93154,93156],{"className":93155},[222,223,224,225],[15,93157,1211],{"className":93158},[66,225],[15,93160,1114],{"className":93161},[1113],[15,93163,93165],{"className":93164},[206],[15,93166,93168],{"className":93167,"style":1121},[210],[15,93169],{},[15,93171],{"className":93172,"style":72},[71],[15,93174,6600],{"className":93175},[76],[15,93177],{"className":93178,"style":72},[71],[15,93180,93182,93185,93225,93254,93289,93292,93295,93298],{"className":93181},[57],[15,93183],{"className":93184,"style":375},[61],[15,93186,93188,93191],{"className":93187},[66],[15,93189,83394],{"className":93190},[66,67],[15,93192,93194],{"className":93193},[198],[15,93195,93197,93217],{"className":93196},[202,1090],[15,93198,93200,93214],{"className":93199},[206],[15,93201,93203],{"className":93202,"style":2139},[210],[15,93204,93205,93208],{"style":1584},[15,93206],{"className":93207,"style":218},[217],[15,93209,93211],{"className":93210},[222,223,224,225],[15,93212,1804],{"className":93213},[66,225],[15,93215,1114],{"className":93216},[1113],[15,93218,93220],{"className":93219},[206],[15,93221,93223],{"className":93222,"style":1121},[210],[15,93224],{},[15,93226,93228,93231],{"className":93227},[169],[15,93229,126],{"className":93230},[169],[15,93232,93234],{"className":93233},[198],[15,93235,93237],{"className":93236},[202],[15,93238,93240],{"className":93239},[206],[15,93241,93243],{"className":93242,"style":394},[210],[15,93244,93245,93248],{"style":397},[15,93246],{"className":93247,"style":218},[217],[15,93249,93251],{"className":93250},[222,223,224,225],[15,93252,139],{"className":93253},[66,225],[15,93255,93257,93260],{"className":93256},[66],[15,93258,83401],{"className":93259},[66],[15,93261,93263],{"className":93262},[198],[15,93264,93266],{"className":93265},[202],[15,93267,93269],{"className":93268},[206],[15,93270,93272],{"className":93271,"style":1650},[210],[15,93273,93274,93277],{"style":397},[15,93275],{"className":93276,"style":218},[217],[15,93278,93280],{"className":93279},[222,223,224,225],[15,93281,93283,93286],{"className":93282},[66,225],[15,93284,6600],{"className":93285},[66,225],[15,93287,1211],{"className":93288},[66,225],[15,93290,123],{"className":93291},[66,67],[15,93293],{"className":93294,"style":72},[71],[15,93296,1245],{"className":93297},[76],[15,93299],{"className":93300,"style":72},[71],[15,93302,93304,93307,93313],{"className":93303},[57],[15,93305],{"className":93306,"style":17374},[61],[15,93308,93310],{"className":93309},[66,1711],[15,93311,92935],{"className":93312},[66,68610],[15,93314,335],{"className":93315},[422],[11,93317,93318,93319,93443,93444,93678,93679,93744],{},"决策边界是一个超平面。二次判别分析（QDA）允许两类有不同的协方差 ",[15,93320,93322,93348],{"className":93321,"translate":19},[18],[15,93323,93325],{"className":93324},[23],[25,93326,93327],{"xmlns":27},[29,93328,93329,93345],{},[32,93330,93331,93337,93339],{},[1057,93332,93333,93335],{},[35,93334,83401],{"mathvariant":138},[1209,93336,1804],{},[39,93338,335],{"separator":53},[1057,93340,93341,93343],{},[35,93342,83401],{"mathvariant":138},[1209,93344,1211],{},[45,93346,93347],{"encoding":47},"\\Sigma_0,\\Sigma_1",[15,93349,93351],{"className":93350,"ariaHidden":53},[52],[15,93352,93354,93357,93397,93400,93403],{"className":93353},[57],[15,93355],{"className":93356,"style":17374},[61],[15,93358,93360,93363],{"className":93359},[66],[15,93361,83401],{"className":93362},[66],[15,93364,93366],{"className":93365},[198],[15,93367,93369,93389],{"className":93368},[202,1090],[15,93370,93372,93386],{"className":93371},[206],[15,93373,93375],{"className":93374,"style":2139},[210],[15,93376,93377,93380],{"style":1584},[15,93378],{"className":93379,"style":218},[217],[15,93381,93383],{"className":93382},[222,223,224,225],[15,93384,1804],{"className":93385},[66,225],[15,93387,1114],{"className":93388},[1113],[15,93390,93392],{"className":93391},[206],[15,93393,93395],{"className":93394,"style":1121},[210],[15,93396],{},[15,93398,335],{"className":93399},[422],[15,93401],{"className":93402,"style":697},[71],[15,93404,93406,93409],{"className":93405},[66],[15,93407,83401],{"className":93408},[66],[15,93410,93412],{"className":93411},[198],[15,93413,93415,93435],{"className":93414},[202,1090],[15,93416,93418,93432],{"className":93417},[206],[15,93419,93421],{"className":93420,"style":2139},[210],[15,93422,93423,93426],{"style":1584},[15,93424],{"className":93425,"style":218},[217],[15,93427,93429],{"className":93428},[222,223,224,225],[15,93430,1211],{"className":93431},[66,225],[15,93433,1114],{"className":93434},[1113],[15,93436,93438],{"className":93437},[206],[15,93439,93441],{"className":93440,"style":1121},[210],[15,93442],{},"，对数似然比出现 ",[15,93445,93447,93497],{"className":93446,"translate":19},[18],[15,93448,93450],{"className":93449},[23],[25,93451,93452],{"xmlns":27},[29,93453,93454,93494],{},[32,93455,93456,93462,93464,93476,93478,93490,93492],{},[131,93457,93458,93460],{},[35,93459,123],{},[35,93461,139],{"mathvariant":138},[39,93463,120],{"stretchy":119},[1255,93465,93466,93468,93470],{},[35,93467,83401],{"mathvariant":138},[1209,93469,1211],{},[32,93471,93472,93474],{},[39,93473,6600],{},[1209,93475,1211],{},[39,93477,6600],{},[1255,93479,93480,93482,93484],{},[35,93481,83401],{"mathvariant":138},[1209,93483,1804],{},[32,93485,93486,93488],{},[39,93487,6600],{},[1209,93489,1211],{},[39,93491,126],{"stretchy":119},[35,93493,123],{},[45,93495,93496],{"encoding":47},"x^\\top(\\Sigma_1^{-1}-\\Sigma_0^{-1})x",[15,93498,93500,93609],{"className":93499,"ariaHidden":53},[52],[15,93501,93503,93507,93536,93539,93600,93603,93606],{"className":93502},[57],[15,93504],{"className":93505,"style":93506},[61],"height:1.1205em;vertical-align:-0.2663em;",[15,93508,93510,93513],{"className":93509},[66],[15,93511,123],{"className":93512},[66,67],[15,93514,93516],{"className":93515},[198],[15,93517,93519],{"className":93518},[202],[15,93520,93522],{"className":93521},[206],[15,93523,93525],{"className":93524,"style":187},[210],[15,93526,93527,93530],{"style":213},[15,93528],{"className":93529,"style":218},[217],[15,93531,93533],{"className":93532},[222,223,224,225],[15,93534,139],{"className":93535},[66,225],[15,93537,120],{"className":93538},[162],[15,93540,93542,93545],{"className":93541},[66],[15,93543,83401],{"className":93544},[66],[15,93546,93548],{"className":93547},[198],[15,93549,93551,93591],{"className":93550},[202,1090],[15,93552,93554,93588],{"className":93553},[206],[15,93555,93558,93570],{"className":93556,"style":93557},[210],"height:0.8542em;",[15,93559,93561,93564],{"style":93560},"top:-2.4337em;margin-left:0em;margin-right:0.05em;",[15,93562],{"className":93563,"style":218},[217],[15,93565,93567],{"className":93566},[222,223,224,225],[15,93568,1211],{"className":93569},[66,225],[15,93571,93573,93576],{"style":93572},"top:-3.1031em;margin-right:0.05em;",[15,93574],{"className":93575,"style":218},[217],[15,93577,93579],{"className":93578},[222,223,224,225],[15,93580,93582,93585],{"className":93581},[66,225],[15,93583,6600],{"className":93584},[66,225],[15,93586,1211],{"className":93587},[66,225],[15,93589,1114],{"className":93590},[1113],[15,93592,93594],{"className":93593},[206],[15,93595,93598],{"className":93596,"style":93597},[210],"height:0.2663em;",[15,93599],{},[15,93601],{"className":93602,"style":72},[71],[15,93604,6600],{"className":93605},[76],[15,93607],{"className":93608,"style":72},[71],[15,93610,93612,93615,93672,93675],{"className":93611},[57],[15,93613],{"className":93614,"style":93506},[61],[15,93616,93618,93621],{"className":93617},[66],[15,93619,83401],{"className":93620},[66],[15,93622,93624],{"className":93623},[198],[15,93625,93627,93664],{"className":93626},[202,1090],[15,93628,93630,93661],{"className":93629},[206],[15,93631,93633,93644],{"className":93632,"style":93557},[210],[15,93634,93635,93638],{"style":93560},[15,93636],{"className":93637,"style":218},[217],[15,93639,93641],{"className":93640},[222,223,224,225],[15,93642,1804],{"className":93643},[66,225],[15,93645,93646,93649],{"style":93572},[15,93647],{"className":93648,"style":218},[217],[15,93650,93652],{"className":93651},[222,223,224,225],[15,93653,93655,93658],{"className":93654},[66,225],[15,93656,6600],{"className":93657},[66,225],[15,93659,1211],{"className":93660},[66,225],[15,93662,1114],{"className":93663},[1113],[15,93665,93667],{"className":93666},[206],[15,93668,93670],{"className":93669,"style":93597},[210],[15,93671],{},[15,93673,126],{"className":93674},[169],[15,93676,123],{"className":93677},[66,67]," 这样的二次项，边界变成二次曲面。朴素贝叶斯再加一层条件独立假设，把 ",[15,93680,93682,93705],{"className":93681,"translate":19},[18],[15,93683,93685],{"className":93684},[23],[25,93686,93687],{"xmlns":27},[29,93688,93689,93703],{},[32,93690,93691,93693,93695,93697,93699,93701],{},[35,93692,11],{},[39,93694,120],{"stretchy":119},[35,93696,123],{},[39,93698,34425],{},[35,93700,1224],{},[39,93702,126],{"stretchy":119},[45,93704,92224],{"encoding":47},[15,93706,93708,93732],{"className":93707,"ariaHidden":53},[52],[15,93709,93711,93714,93717,93720,93723,93726,93729],{"className":93710},[57],[15,93712],{"className":93713,"style":154},[61],[15,93715,11],{"className":93716},[66,67],[15,93718,120],{"className":93719},[162],[15,93721,123],{"className":93722},[66,67],[15,93724],{"className":93725,"style":173},[71],[15,93727,34425],{"className":93728},[177],[15,93730],{"className":93731,"style":173},[71],[15,93733,93735,93738,93741],{"className":93734},[57],[15,93736],{"className":93737,"style":154},[61],[15,93739,1224],{"className":93740,"style":1515},[66,67],[15,93742,126],{"className":93743},[169]," 拆成各个特征的乘积，用很大的偏差换取很小的方差。",[11,93746,93747,93748,93776,93777,93842],{},"两条路的目标是同一个 ",[15,93749,93751,93764],{"className":93750,"translate":19},[18],[15,93752,93754],{"className":93753},[23],[25,93755,93756],{"xmlns":27},[29,93757,93758,93762],{},[32,93759,93760],{},[35,93761,71105],{},[45,93763,90240],{"encoding":47},[15,93765,93767],{"className":93766,"ariaHidden":53},[52],[15,93768,93770,93773],{"className":93769},[57],[15,93771],{"className":93772,"style":471},[61],[15,93774,71105],{"className":93775,"style":1515},[66,67],"，但统计性质不同。生成式利用了 ",[15,93778,93780,93803],{"className":93779,"translate":19},[18],[15,93781,93783],{"className":93782},[23],[25,93784,93785],{"xmlns":27},[29,93786,93787,93801],{},[32,93788,93789,93791,93793,93795,93797,93799],{},[35,93790,11],{},[39,93792,120],{"stretchy":119},[35,93794,123],{},[39,93796,34425],{},[35,93798,1224],{},[39,93800,126],{"stretchy":119},[45,93802,92224],{"encoding":47},[15,93804,93806,93830],{"className":93805,"ariaHidden":53},[52],[15,93807,93809,93812,93815,93818,93821,93824,93827],{"className":93808},[57],[15,93810],{"className":93811,"style":154},[61],[15,93813,11],{"className":93814},[66,67],[15,93816,120],{"className":93817},[162],[15,93819,123],{"className":93820},[66,67],[15,93822],{"className":93823,"style":173},[71],[15,93825,34425],{"className":93826},[177],[15,93828],{"className":93829,"style":173},[71],[15,93831,93833,93836,93839],{"className":93832},[57],[15,93834],{"className":93835,"style":154},[61],[15,93837,1224],{"className":93838,"style":1515},[66,67],[15,93840,126],{"className":93841},[169]," 的结构，在模型正确时样本效率更高；判别式直接优化决策边界，在模型误设时通常更稳健。这不是「谁更好」的问题，而是「你愿意把假设放在哪一侧」的问题。",[11,93844,93845,93846,93849,93850,93878,93879,93952,93953,94011],{},"这里说的是 ",[2040,93847,93848],{},"oracle 贝叶斯规则","：它假设 ",[15,93851,93853,93866],{"className":93852,"translate":19},[18],[15,93854,93856],{"className":93855},[23],[25,93857,93858],{"xmlns":27},[29,93859,93860,93864],{},[32,93861,93862],{},[35,93863,71105],{},[45,93865,90240],{"encoding":47},[15,93867,93869],{"className":93868,"ariaHidden":53},[52],[15,93870,93872,93875],{"className":93871},[57],[15,93873],{"className":93874,"style":471},[61],[15,93876,71105],{"className":93877,"style":1515},[66,67]," 已知。实践中我们把估计的 ",[15,93880,93882,93900],{"className":93881,"translate":19},[18],[15,93883,93885],{"className":93884},[23],[25,93886,93887],{"xmlns":27},[29,93888,93889,93897],{},[32,93890,93891],{},[4294,93892,93893,93895],{"accent":53},[35,93894,71105],{},[39,93896,78122],{},[45,93898,93899],{"encoding":47},"\\hat\\eta",[15,93901,93903],{"className":93902,"ariaHidden":53},[52],[15,93904,93906,93909],{"className":93905},[57],[15,93907],{"className":93908,"style":3406},[61],[15,93910,93912],{"className":93911},[66,78139],[15,93913,93915,93944],{"className":93914},[202,1090],[15,93916,93918,93941],{"className":93917},[206],[15,93919,93921,93929],{"className":93920,"style":840},[210],[15,93922,93923,93926],{"style":1386},[15,93924],{"className":93925,"style":1368},[217],[15,93927,71105],{"className":93928,"style":1515},[66,67],[15,93930,93931,93934],{"style":1386},[15,93932],{"className":93933,"style":1368},[217],[15,93935,93938],{"className":93936,"style":93937},[78167],"left:-0.1944em;",[15,93939,78122],{"className":93940},[66],[15,93942,1114],{"className":93943},[1113],[15,93945,93947],{"className":93946},[206],[15,93948,93950],{"className":93949,"style":4448},[210],[15,93951],{}," 代入，得到的只是 plug-in 分类器；文献里的「贝叶斯分类器」有时指这个 oracle 规则，有时指朴素贝叶斯之类的具体算法，读的时候要分清。plug-in 分类器的风险与 ",[15,93954,93956,93973],{"className":93955,"translate":19},[18],[15,93957,93959],{"className":93958},[23],[25,93960,93961],{"xmlns":27},[29,93962,93963,93971],{},[32,93964,93965],{},[131,93966,93967,93969],{},[35,93968,455],{},[39,93970,1162],{},[45,93972,76275],{"encoding":47},[15,93974,93976],{"className":93975,"ariaHidden":53},[52],[15,93977,93979,93982],{"className":93978},[57],[15,93980],{"className":93981,"style":16244},[61],[15,93983,93985,93988],{"className":93984},[66],[15,93986,455],{"className":93987,"style":62433},[66,67],[15,93989,93991],{"className":93990},[198],[15,93992,93994],{"className":93993},[202],[15,93995,93997],{"className":93996},[206],[15,93998,94000],{"className":93999,"style":16244},[210],[15,94001,94002,94005],{"style":213},[15,94003],{"className":94004,"style":218},[217],[15,94006,94008],{"className":94007},[222,223,224,225],[15,94009,1162],{"className":94010},[76,225]," 之间的差距，正是第 7 节要分解的估计误差。",[11,94013,94014,94015,94059,94060,94092,94093,94121,94122,1124],{},"值得一提的是，朴素贝叶斯的条件独立假设几乎总是错的，但它的分类性能常常不差。原因在于 0-1 损失只关心 ",[15,94016,94018,94038],{"className":94017,"translate":19},[18],[15,94019,94021],{"className":94020},[23],[25,94022,94023],{"xmlns":27},[29,94024,94025,94035],{},[32,94026,94027,94029,94031,94033],{},[35,94028,71105],{},[39,94030,120],{"stretchy":119},[35,94032,123],{},[39,94034,126],{"stretchy":119},[45,94036,94037],{"encoding":47},"\\eta(x)",[15,94039,94041],{"className":94040,"ariaHidden":53},[52],[15,94042,94044,94047,94050,94053,94056],{"className":94043},[57],[15,94045],{"className":94046,"style":154},[61],[15,94048,71105],{"className":94049,"style":1515},[66,67],[15,94051,120],{"className":94052},[162],[15,94054,123],{"className":94055},[66,67],[15,94057,126],{"className":94058},[169]," 是否越过 ",[15,94061,94063,94080],{"className":94062,"translate":19},[18],[15,94064,94066],{"className":94065},[23],[25,94067,94068],{"xmlns":27},[29,94069,94070,94078],{},[32,94071,94072,94074,94076],{},[1209,94073,1211],{},[35,94075,6623],{"mathvariant":138},[1209,94077,1263],{},[45,94079,71801],{"encoding":47},[15,94081,94083],{"className":94082,"ariaHidden":53},[52],[15,94084,94086,94089],{"className":94085},[57],[15,94087],{"className":94088,"style":154},[61],[15,94090,71801],{"className":94091},[66],"，而不要求 ",[15,94094,94096,94109],{"className":94095,"translate":19},[18],[15,94097,94099],{"className":94098},[23],[25,94100,94101],{"xmlns":27},[29,94102,94103,94107],{},[32,94104,94105],{},[35,94106,71105],{},[45,94108,90240],{"encoding":47},[15,94110,94112],{"className":94111,"ariaHidden":53},[52],[15,94113,94115,94118],{"className":94114},[57],[15,94116],{"className":94117,"style":471},[61],[15,94119,71105],{"className":94120,"style":1515},[66,67]," 的数值完全准确；只要假设带来的偏差没有把决策边界推得太远，分类结果就可以是对的。这个现象也提醒我们：",[2040,94123,94124],{},"概率估计的质量与决策的质量不是同一件事",[11,94126,94127,94128,94171,94172,94175,94176,37639],{},"这个区别在损失函数上有一个精确的对应。0-1 损失只关心 ",[15,94129,94131,94150],{"className":94130,"translate":19},[18],[15,94132,94134],{"className":94133},[23],[25,94135,94136],{"xmlns":27},[29,94137,94138,94148],{},[32,94139,94140,94142,94144,94146],{},[35,94141,71105],{},[39,94143,120],{"stretchy":119},[35,94145,123],{},[39,94147,126],{"stretchy":119},[45,94149,94037],{"encoding":47},[15,94151,94153],{"className":94152,"ariaHidden":53},[52],[15,94154,94156,94159,94162,94165,94168],{"className":94155},[57],[15,94157],{"className":94158,"style":154},[61],[15,94160,71105],{"className":94161,"style":1515},[66,67],[15,94163,120],{"className":94164},[162],[15,94166,123],{"className":94167},[66,67],[15,94169,126],{"className":94170},[169]," 是否越过阈值，因此它对概率的微小误差不敏感；交叉熵（对数损失）则是一个",[2040,94173,94174],{},"严格 proper scoring rule","，它的总体最优解恰好是真实的条件分布 ",[15,94177,94179,94202],{"className":94178,"translate":19},[18],[15,94180,94182],{"className":94181},[23],[25,94183,94184],{"xmlns":27},[29,94185,94186,94200],{},[32,94187,94188,94190,94192,94194,94196,94198],{},[35,94189,11],{},[39,94191,120],{"stretchy":119},[35,94193,1224],{},[39,94195,34425],{},[35,94197,123],{},[39,94199,126],{"stretchy":119},[45,94201,68202],{"encoding":47},[15,94203,94205,94229],{"className":94204,"ariaHidden":53},[52],[15,94206,94208,94211,94214,94217,94220,94223,94226],{"className":94207},[57],[15,94209],{"className":94210,"style":154},[61],[15,94212,11],{"className":94213},[66,67],[15,94215,120],{"className":94216},[162],[15,94218,1224],{"className":94219,"style":1515},[66,67],[15,94221],{"className":94222,"style":173},[71],[15,94224,34425],{"className":94225},[177],[15,94227],{"className":94228,"style":173},[71],[15,94230,94232,94235,94238],{"className":94231},[57],[15,94233],{"className":94234,"style":154},[61],[15,94236,123],{"className":94237},[66,67],[15,94239,126],{"className":94240},[169],[15,94242,94244],{"className":94243,"translate":19},[294],[15,94245,94247,94356],{"className":94246,"translate":19},[18],[15,94248,94250],{"className":94249},[23],[25,94251,94252],{"xmlns":27,"display":303},[29,94253,94254,94353],{},[32,94255,94256,94258,94260,94262,94264,94266,94269,94271,94273,94275,94277,94279,94281,94283,94285,94287,94289,94291,94293,94295,94297,94299,94301,94303,94305,94307,94309,94311,94313,94319,94321,94323,94325,94327,94329,94331,94333,94335,94337,94339,94341,94343,94345,94347,94349,94351],{},[35,94257,62377],{"mathvariant":454},[39,94259,1194],{"stretchy":119},[39,94261,6600],{},[35,94263,27068],{},[39,94265,1170],{},[35,94267,94268],{},"q",[39,94270,120],{"stretchy":119},[35,94272,61881],{},[39,94274,34425],{},[35,94276,446],{},[39,94278,126],{"stretchy":119},[39,94280,1266],{"stretchy":119},[39,94282,129],{},[35,94284,62377],{"mathvariant":454},[39,94286,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,94288,1061],{"mathvariant":454},[39,94290,120],{"stretchy":119},[35,94292,11],{},[39,94294,120],{"stretchy":119},[39,94296,3266],{},[39,94298,34425],{},[35,94300,446],{},[39,94302,126],{"stretchy":119},[39,94304,126],{"stretchy":119},[39,94306,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,94308,1245],{},[35,94310,62377],{"mathvariant":454},[39,94312,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[32,94314,94315,94317],{},[35,94316,2853],{"mathvariant":138},[35,94318,1216],{"mathvariant":138},[39,94320,120],{"stretchy":119},[35,94322,11],{},[39,94324,120],{"stretchy":119},[39,94326,3266],{},[39,94328,34425],{},[35,94330,446],{},[39,94332,126],{"stretchy":119},[35,94334,1251],{"mathvariant":138},[35,94336,94268],{},[39,94338,120],{"stretchy":119},[39,94340,3266],{},[39,94342,34425],{},[35,94344,446],{},[39,94346,126],{"stretchy":119},[39,94348,126],{"stretchy":119},[39,94350,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,94352,335],{"separator":53},[45,94354,94355],{"encoding":47},"\\mathbb E[-\\log q(Y\\mid X)]\n=\\mathbb E\\big[\\mathbb H(p(\\cdot\\mid X))\\big]\n+\\mathbb E\\big[\\mathrm{KL}(p(\\cdot\\mid X)\\|q(\\cdot\\mid X))\\big],",[15,94357,94359,94403,94425,94464,94491,94533,94566],{"className":94358,"ariaHidden":53},[52],[15,94360,94362,94365,94368,94371,94374,94377,94382,94385,94388,94391,94394,94397,94400],{"className":94361},[57],[15,94363],{"className":94364,"style":154},[61],[15,94366,62377],{"className":94367},[66,517],[15,94369,1194],{"className":94370},[162],[15,94372,6600],{"className":94373},[66],[15,94375],{"className":94376,"style":697},[71],[15,94378,27278,94380],{"className":94379},[1339],[15,94381,1344],{"style":1343},[15,94383],{"className":94384,"style":697},[71],[15,94386,94268],{"className":94387,"style":1515},[66,67],[15,94389,120],{"className":94390},[162],[15,94392,61881],{"className":94393,"style":72},[66,67],[15,94395],{"className":94396,"style":173},[71],[15,94398,34425],{"className":94399},[177],[15,94401],{"className":94402,"style":173},[71],[15,94404,94406,94409,94412,94416,94419,94422],{"className":94405},[57],[15,94407],{"className":94408,"style":154},[61],[15,94410,446],{"className":94411,"style":16492},[66,67],[15,94413,94415],{"className":94414},[169],")]",[15,94417],{"className":94418,"style":173},[71],[15,94420,129],{"className":94421},[177],[15,94423],{"className":94424,"style":173},[71],[15,94426,94428,94431,94434,94440,94443,94446,94449,94452,94455,94458,94461],{"className":94427},[57],[15,94429],{"className":94430,"style":4398},[61],[15,94432,62377],{"className":94433},[66,517],[15,94435,94437],{"className":94436},[66],[15,94438,1194],{"className":94439},[1422,1508],[15,94441,1061],{"className":94442},[66,517],[15,94444,120],{"className":94445},[162],[15,94447,11],{"className":94448},[66,67],[15,94450,120],{"className":94451},[162],[15,94453,3266],{"className":94454},[66],[15,94456],{"className":94457,"style":173},[71],[15,94459,34425],{"className":94460},[177],[15,94462],{"className":94463,"style":173},[71],[15,94465,94467,94470,94473,94476,94482,94485,94488],{"className":94466},[57],[15,94468],{"className":94469,"style":4398},[61],[15,94471,446],{"className":94472,"style":16492},[66,67],[15,94474,18766],{"className":94475},[169],[15,94477,94479],{"className":94478},[66],[15,94480,1266],{"className":94481},[1422,1508],[15,94483],{"className":94484,"style":72},[71],[15,94486,1245],{"className":94487},[76],[15,94489],{"className":94490,"style":72},[71],[15,94492,94494,94497,94500,94506,94512,94515,94518,94521,94524,94527,94530],{"className":94493},[57],[15,94495],{"className":94496,"style":4398},[61],[15,94498,62377],{"className":94499},[66,517],[15,94501,94503],{"className":94502},[66],[15,94504,1194],{"className":94505},[1422,1508],[15,94507,94509],{"className":94508},[66],[15,94510,88449],{"className":94511},[66,4427],[15,94513,120],{"className":94514},[162],[15,94516,11],{"className":94517},[66,67],[15,94519,120],{"className":94520},[162],[15,94522,3266],{"className":94523},[66],[15,94525],{"className":94526,"style":173},[71],[15,94528,34425],{"className":94529},[177],[15,94531],{"className":94532,"style":173},[71],[15,94534,94536,94539,94542,94545,94548,94551,94554,94557,94560,94563],{"className":94535},[57],[15,94537],{"className":94538,"style":154},[61],[15,94540,446],{"className":94541,"style":16492},[66,67],[15,94543,126],{"className":94544},[169],[15,94546,1251],{"className":94547},[66],[15,94549,94268],{"className":94550,"style":1515},[66,67],[15,94552,120],{"className":94553},[162],[15,94555,3266],{"className":94556},[66],[15,94558],{"className":94559,"style":173},[71],[15,94561,34425],{"className":94562},[177],[15,94564],{"className":94565,"style":173},[71],[15,94567,94569,94572,94575,94578,94584],{"className":94568},[57],[15,94570],{"className":94571,"style":4398},[61],[15,94573,446],{"className":94574,"style":16492},[66,67],[15,94576,18766],{"className":94577},[169],[15,94579,94581],{"className":94580},[66],[15,94582,1266],{"className":94583},[1422,1508],[15,94585,335],{"className":94586},[422],[11,94588,94589,94590,94618,94619,94707],{},"右边第一项与 ",[15,94591,94593,94606],{"className":94592,"translate":19},[18],[15,94594,94596],{"className":94595},[23],[25,94597,94598],{"xmlns":27},[29,94599,94600,94604],{},[32,94601,94602],{},[35,94603,94268],{},[45,94605,94268],{"encoding":47},[15,94607,94609],{"className":94608,"ariaHidden":53},[52],[15,94610,94612,94615],{"className":94611},[57],[15,94613],{"className":94614,"style":471},[61],[15,94616,94268],{"className":94617,"style":1515},[66,67]," 无关，第二项在 ",[15,94620,94622,94650],{"className":94621,"translate":19},[18],[15,94623,94625],{"className":94624},[23],[25,94626,94627],{"xmlns":27},[29,94628,94629,94647],{},[32,94630,94631,94633,94635,94637,94639,94641,94643,94645],{},[35,94632,94268],{},[39,94634,129],{},[35,94636,11],{},[39,94638,120],{"stretchy":119},[39,94640,3266],{},[39,94642,34425],{},[35,94644,446],{},[39,94646,126],{"stretchy":119},[45,94648,94649],{"encoding":47},"q=p(\\cdot\\mid X)",[15,94651,94653,94671,94695],{"className":94652,"ariaHidden":53},[52],[15,94654,94656,94659,94662,94665,94668],{"className":94655},[57],[15,94657],{"className":94658,"style":471},[61],[15,94660,94268],{"className":94661,"style":1515},[66,67],[15,94663],{"className":94664,"style":173},[71],[15,94666,129],{"className":94667},[177],[15,94669],{"className":94670,"style":173},[71],[15,94672,94674,94677,94680,94683,94686,94689,94692],{"className":94673},[57],[15,94675],{"className":94676,"style":154},[61],[15,94678,11],{"className":94679},[66,67],[15,94681,120],{"className":94682},[162],[15,94684,3266],{"className":94685},[66],[15,94687],{"className":94688,"style":173},[71],[15,94690,34425],{"className":94691},[177],[15,94693],{"className":94694,"style":173},[71],[15,94696,94698,94701,94704],{"className":94697},[57],[15,94699],{"className":94700,"style":154},[61],[15,94702,446],{"className":94703,"style":16492},[66,67],[15,94705,126],{"className":94706},[169]," 时为零。所以，在总体意义上，用交叉熵训练模型是在要求它给出校准的概率；用 0-1 损失训练，只要求它给出正确的分类。两者目标不同，不能互相替代。",[1904,94709,94711],{"id":94710},"_52-正则化就是先验","5.2 正则化就是先验",[11,94713,94714,94715,94985,94986,95045],{},"设参数化模型 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x_i,\\theta)",[15,94776,94778,94802,94823,94927],{"className":94777,"ariaHidden":53},[52],[15,94779,94781,94784,94787,94790,94793,94796,94799],{"className":94780},[57],[15,94782],{"className":94783,"style":154},[61],[15,94785,11],{"className":94786},[66,67],[15,94788,120],{"className":94789},[162],[15,94791,32453],{"className":94792,"style":23020},[66,67],[15,94794],{"className":94795,"style":173},[71],[15,94797,34425],{"className":94798},[177],[15,94800],{"className":94801,"style":173},[71],[15,94803,94805,94808,94811,94814,94817,94820],{"className":94804},[57],[15,94806],{"className":94807,"style":154},[61],[15,94809,61614],{"className":94810,"style":23020},[66,67],[15,94812,126],{"className":94813},[169],[15,94815],{"className":94816,"style":173},[71],[15,94818,129],{"className":94819},[177],[15,94821],{"className":94822,"style":173},[71],[15,94824,94826,94829,94869,94872,94875,94878,94918,94921,94924],{"className":94825},[57],[15,94827],{"className":94828,"style":16861},[61],[15,94830,94832,94835],{"className":94831},[1339],[15,94833,94745],{"className":94834,"style":13697},[1339,1470,13696],[15,94836,94838],{"className":94837},[198],[15,94839,94841,94861],{"className":94840},[202,1090],[15,94842,94844,94858],{"className":94843},[206],[15,94845,94847],{"className":94846,"style":16880},[210],[15,94848,94849,94852],{"style":13713},[15,94850],{"className":94851,"style":218},[217],[15,94853,94855],{"className":94854},[222,223,224,225],[15,94856,1205],{"className":94857},[66,67,225],[15,94859,1114],{"className":94860},[1113],[15,94862,94864],{"className":94863},[206],[15,94865,94867],{"className":94866,"style":16901},[210],[15,94868],{},[15,94870],{"className":94871,"style":697},[71],[15,94873,11],{"className":94874},[66,67],[15,94876,120],{"className":94877},[162],[15,94879,94881,94884],{"className":94880},[66],[15,94882,1224],{"className":94883,"style":1515},[66,67],[15,94885,94887],{"className":94886},[198],[15,94888,94890,94910],{"className":94889},[202,1090],[15,94891,94893,94907],{"className":94892},[206],[15,94894,94896],{"className":94895,"style":1528},[210],[15,94897,94898,94901],{"style":1531},[15,94899],{"className":94900,"style":218},[217],[15,94902,94904],{"className":94903},[222,223,224,225],[15,94905,1205],{"className":94906},[66,67,225],[15,94908,1114],{"className":94909},[1113],[15,94911,94913],{"className":94912},[206],[15,94914,94916],{"className":94915,"style":1121},[210],[15,94917],{},[15,94919],{"className":94920,"style":173},[71],[15,94922,34425],{"className":94923},[177],[15,94925],{"className":94926,"style":173},[71],[15,94928,94930,94933,94973,94976,94979,94982],{"className":94929},[57],[15,94931],{"className":94932,"style":154},[61],[15,94934,94936,94939],{"className":94935},[66],[15,94937,123],{"className":94938},[66,67],[15,94940,94942],{"className":94941},[198],[15,94943,94945,94965],{"className":94944},[202,1090],[15,94946,94948,94962],{"className":94947},[206],[15,94949,94951],{"className":94950,"style":1528},[210],[15,94952,94953,94956],{"style":1584},[15,94954],{"className":94955,"style":218},[217],[15,94957,94959],{"className":94958},[222,223,224,225],[15,94960,1205],{"className":94961},[66,67,225],[15,94963,1114],{"className":94964},[1113],[15,94966,94968],{"className":94967},[206],[15,94969,94971],{"className":94970,"style":1121},[210],[15,94972],{},[15,94974,335],{"className":94975},[422],[15,94977],{"className":94978,"style":697},[71],[15,94980,61614],{"className":94981,"style":23020},[66,67],[15,94983,126],{"className":94984},[169],"，噪声为高斯，方差 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p(D\\mid\\theta)\n=\\frac{1}{2\\sigma^2}\\sum_i\\big(y_i-f_\\theta(x_i)\\big)^2+\\text{常数}.",[15,95143,95145,95183,95204,95407,95540],{"className":95144,"ariaHidden":53},[52],[15,95146,95148,95151,95154,95157,95162,95165,95168,95171,95174,95177,95180],{"className":95147},[57],[15,95149],{"className":95150,"style":154},[61],[15,95152,6600],{"className":95153},[66],[15,95155],{"className":95156,"style":697},[71],[15,95158,27278,95160],{"className":95159},[1339],[15,95161,1344],{"style":1343},[15,95163],{"className":95164,"style":697},[71],[15,95166,11],{"className":95167},[66,67],[15,95169,120],{"className":95170},[162],[15,95172,32453],{"className":95173,"style":23020},[66,67],[15,95175],{"className":95176,"style":173},[71],[15,95178,34425],{"className":95179},[177],[15,95181],{"className":95182,"style":173},[71],[15,95184,95186,95189,95192,95195,95198,95201],{"className":95185},[57],[15,95187],{"className":95188,"style":154},[61],[15,95190,61614],{"className":95191,"style":23020},[66,67],[15,95193,126],{"className":95194},[169],[15,95196],{"className":95197,"style":173},[71],[15,95199,129],{"className":95200},[177],[15,95202],{"className":95203,"style":173},[71],[15,95205,95207,95211,95302,95305,95349,95352,95358,95398,95401,95404],{"className":95206},[57],[15,95208],{"className":95209,"style":95210},[61],"height:2.5991em;vertical-align:-1.2777em;",[15,95212,95214,95217,95299],{"className":95213},[66],[15,95215],{"className":95216},[162,23412],[15,95218,95220],{"className":95219},[6465],[15,95221,95223,95291],{"className":95222},[202,1090],[15,95224,95226,95288],{"className":95225},[206],[15,95227,95229,95269,95277],{"className":95228,"style":23425},[210],[15,95230,95231,95234],{"style":23428},[15,95232],{"className":95233,"style":1368},[217],[15,95235,95237,95240],{"className":95236},[66],[15,95238,1263],{"className":95239},[66],[15,95241,95243,95246],{"className":95242},[66],[15,95244,6632],{"className":95245,"style":1515},[66,67],[15,95247,95249],{"className":95248},[198],[15,95250,95252],{"className":95251},[202],[15,95253,95255],{"className":95254},[206],[15,95256,95258],{"className":95257,"style":46247},[210],[15,95259,95260,95263],{"style":46250},[15,95261],{"className":95262,"style":218},[217],[15,95264,95266],{"className":95265},[222,223,224,225],[15,95267,1263],{"className":95268},[66,225],[15,95270,95271,95274],{"style":23440},[15,95272],{"className":95273,"style":1368},[217],[15,95275],{"className":95276,"style":4438},[23447],[15,95278,95279,95282],{"style":23450},[15,95280],{"className":95281,"style":1368},[217],[15,95283,95285],{"className":95284},[66],[15,95286,1211],{"className":95287},[66],[15,95289,1114],{"className":95290},[1113],[15,95292,95294],{"className":95293},[206],[15,95295,95297],{"className":95296,"style":23469},[210],[15,95298],{},[15,95300],{"className":95301},[169,23412],[15,95303],{"className":95304,"style":697},[71],[15,95306,95308],{"className":95307},[1339,1351],[15,95309,95311,95341],{"className":95310},[202,1090],[15,95312,95314,95338],{"className":95313},[206],[15,95315,95317,95328],{"className":95316,"style":4688},[210],[15,95318,95319,95322],{"style":1439},[15,95320],{"className":95321,"style":1443},[217],[15,95323,95325],{"className":95324},[222,223,224,225],[15,95326,1205],{"className":95327},[66,67,225],[15,95329,95330,95333],{"style":1461},[15,95331],{"className":95332,"style":1443},[217],[15,95334,95335],{},[15,95336,1200],{"className":95337},[1339,1470,1471],[15,95339,1114],{"className":95340},[1113],[15,95342,95344],{"className":95343},[206],[15,95345,95347],{"className":95346,"style":1493},[210],[15,95348],{},[15,95350],{"className":95351,"style":697},[71],[15,95353,95355],{"className":95354},[66],[15,95356,120],{"className":95357},[1422,1508],[15,95359,95361,95364],{"className":95360},[66],[15,95362,1224],{"className":95363,"style":1515},[66,67],[15,95365,95367],{"className":95366},[198],[15,95368,95370,95390],{"className":95369},[202,1090],[15,95371,95373,95387],{"className":95372},[206],[15,95374,95376],{"className":95375,"style":1528},[210],[15,95377,95378,95381],{"style":1531},[15,95379],{"className":95380,"style":218},[217],[15,95382,95384],{"className":95383},[222,223,224,225],[15,95385,1205],{"className":95386},[66,67,225],[15,95388,1114],{"className":95389},[1113],[15,95391,95393],{"className":95392},[206],[15,95394,95396],{"className":95395,"style":1121},[210],[15,95397],{},[15,95399],{"className":95400,"style":72},[71],[15,95402,6600],{"className":95403},[76],[15,95405],{"className":95406,"style":72},[71],[15,95408,95410,95413,95453,95456,95496,95499,95531,95534,95537],{"className":95409},[57],[15,95411],{"className":95412,"style":30180},[61],[15,95414,95416,95419],{"className":95415},[66],[15,95417,116],{"className":95418,"style":158},[66,67],[15,95420,95422],{"className":95421},[198],[15,95423,95425,95445],{"className":95424},[202,1090],[15,95426,95428,95442],{"className":95427},[206],[15,95429,95431],{"className":95430,"style":1097},[210],[15,95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",[15,96619,96621,96653],{"className":96620,"translate":19},[18],[15,96622,96624],{"className":96623},[23],[25,96625,96626],{"xmlns":27},[29,96627,96628,96650],{},[32,96629,96630,96636,96638,96640,96642,96648],{},[131,96631,96632,96634],{},[35,96633,6632],{},[1209,96635,1263],{},[35,96637,6623],{"mathvariant":138},[39,96639,120],{"stretchy":119},[35,96641,37],{},[131,96643,96644,96646],{},[35,96645,95586],{},[1209,96647,1263],{},[39,96649,126],{"stretchy":119},[45,96651,96652],{"encoding":47},"\\sigma^2\u002F(n\\tau^2)",[15,96654,96656],{"className":96655,"ariaHidden":53},[52],[15,96657,96659,96662,96691,96694,96697,96700,96729],{"className":96658},[57],[15,96660],{"className":96661,"style":9149},[61],[15,96663,96665,96668],{"className":96664},[66],[15,96666,6632],{"className":96667,"style":1515},[66,67],[15,96669,96671],{"className":96670},[198],[15,96672,96674],{"className":96673},[202],[15,96675,96677],{"className":96676},[206],[15,96678,96680],{"className":96679,"style":6738},[210],[15,96681,96682,96685],{"style":213},[15,96683],{"className":96684,"style":218},[217],[15,96686,96688],{"className":96687},[222,223,224,225],[15,96689,1263],{"className":96690},[66,225],[15,96692,6623],{"className":96693},[66],[15,96695,120],{"className":96696},[162],[15,96698,37],{"className":96699},[66,67],[15,96701,96703,96706],{"className":96702},[66],[15,96704,95586],{"className":96705,"style":95644},[66,67],[15,96707,96709],{"className":96708},[198],[15,96710,96712],{"className":96711},[202],[15,96713,96715],{"className":96714},[206],[15,96716,96718],{"className":96717,"style":6738},[210],[15,96719,96720,96723],{"style":213},[15,96721],{"className":96722,"style":218},[217],[15,96724,96726],{"className":96725},[222,223,224,225],[15,96727,1263],{"className":96728},[66,225],[15,96730,126],{"className":96731},[169],"。若先验换成 Laplace 分布 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",[15,97025,97027,97049],{"className":97026,"translate":19},[18],[15,97028,97030],{"className":97029},[23],[25,97031,97032],{"xmlns":27},[29,97033,97034,97046],{},[32,97035,97036,97038,97040],{},[35,97037,1251],{"mathvariant":138},[35,97039,61614],{},[1057,97041,97042,97044],{},[35,97043,1251],{"mathvariant":138},[1209,97045,1211],{},[45,97047,97048],{"encoding":47},"\\|\\theta\\|_1",[15,97050,97052],{"className":97051,"ariaHidden":53},[52],[15,97053,97055,97058,97061,97064],{"className":97054},[57],[15,97056],{"className":97057,"style":154},[61],[15,97059,1251],{"className":97060},[66],[15,97062,61614],{"className":97063,"style":23020},[66,67],[15,97065,97067,97070],{"className":97066},[66],[15,97068,1251],{"className":97069},[66],[15,97071,97073],{"className":97072},[198],[15,97074,97076,97096],{"className":97075},[202,1090],[15,97077,97079,97093],{"className":97078},[206],[15,97080,97082],{"className":97081,"style":2139},[210],[15,97083,97084,97087],{"style":1584},[15,97085],{"className":97086,"style":218},[217],[15,97088,97090],{"className":97089},[222,223,224,225],[15,97091,1211],{"className":97092},[66,225],[15,97094,1114],{"className":97095},[1113],[15,97097,97099],{"className":97098},[206],[15,97100,97102],{"className":97101,"style":1121},[210],[15,97103],{},"，MAP 就变成 Lasso。",[6150,97106,97107,97119],{},[6153,97108,97109],{},[6156,97110,97111,97113,97116],{},[6159,97112,65214],{},[6159,97114,97115],{},"负对数先验",[6159,97117,97118],{},"等价惩罚",[6169,97120,97121,97131,97352,97504],{},[6156,97122,97123,97126,97128],{},[6174,97124,97125],{},"均匀先验",[6174,97127,92935],{},[6174,97129,97130],{},"无（MLE）",[6156,97132,97133,97223,97349],{},[6174,97134,97135,97136],{},"高斯 ",[15,97137,97139,97167],{"className":97138,"translate":19},[18],[15,97140,97142],{"className":97141},[23],[25,97143,97144],{"xmlns":27},[29,97145,97146,97164],{},[32,97147,97148,97150,97152,97154,97156,97162],{},[35,97149,28530],{"mathvariant":445},[39,97151,120],{"stretchy":119},[1209,97153,1804],{},[39,97155,335],{"separator":53},[131,97157,97158,97160],{},[35,97159,95586],{},[1209,97161,1263],{},[39,97163,126],{"stretchy":119},[45,97165,97166],{"encoding":47},"\\mathcal N(0,\\tau^2)",[15,97168,97170],{"className":97169,"ariaHidden":53},[52],[15,97171,97173,97176,97179,97182,97185,97188,97191,97220],{"className":97172},[57],[15,97174],{"className":97175,"style":9149},[61],[15,97177,28530],{"className":97178,"style":83507},[66,494],[15,97180,120],{"className":97181},[162],[15,97183,1804],{"className":97184},[66],[15,97186,335],{"className":97187},[422],[15,97189],{"className":97190,"style":697},[71],[15,97192,97194,97197],{"className":97193},[66],[15,97195,95586],{"className":97196,"style":95644},[66,67],[15,97198,97200],{"className":97199},[198],[15,97201,97203],{"className":97202},[202],[15,97204,97206],{"className":97205},[206],[15,97207,97209],{"className":97208,"style":6738},[210],[15,97210,97211,97214],{"style":213},[15,97212],{"className":97213,"style":218},[217],[15,97215,97217],{"className":97216},[222,223,224,225],[15,97218,1263],{"className":97219},[66,225],[15,97221,126],{"className":97222},[169],[6174,97224,97225],{},[15,97226,97228,97264],{"className":97227,"translate":19},[18],[15,97229,97231],{"className":97230},[23],[25,97232,97233],{"xmlns":27},[29,97234,97235,97261],{},[32,97236,97237,97239,97241,97247,97249,97251,97253,97259],{},[35,97238,1251],{"mathvariant":138},[35,97240,61614],{},[131,97242,97243,97245],{},[35,97244,1251],{"mathvariant":138},[1209,97246,1263],{},[35,97248,6623],{"mathvariant":138},[39,97250,120],{"stretchy":119},[1209,97252,1263],{},[131,97254,97255,97257],{},[35,97256,95586],{},[1209,97258,1263],{},[39,97260,126],{"stretchy":119},[45,97262,97263],{"encoding":47},"\\|\\theta\\|^2\u002F(2\\tau^2)",[15,97265,97267],{"className":97266,"ariaHidden":53},[52],[15,97268,97270,97273,97276,97279,97308,97311,97314,97317,97346],{"className":97269},[57],[15,97271],{"className":97272,"style":9149},[61],[15,97274,1251],{"className":97275},[66],[15,97277,61614],{"className":97278,"style":23020},[66,67],[15,97280,97282,97285],{"className":97281},[66],[15,97283,1251],{"className":97284},[66],[15,97286,97288],{"className":97287},[198],[15,97289,97291],{"className":97290},[202],[15,97292,97294],{"className":97293},[206],[15,97295,97297],{"className":97296,"style":6738},[210],[15,97298,97299,97302],{"style":213},[15,97300],{"className":97301,"style":218},[217],[15,97303,97305],{"className":97304},[222,223,224,225],[15,97306,1263],{"className":97307},[66,225],[15,97309,6623],{"className":97310},[66],[15,97312,120],{"className":97313},[162],[15,97315,1263],{"className":97316},[66],[15,97318,97320,97323],{"className":97319},[66],[15,97321,95586],{"className":97322,"style":95644},[66,67],[15,97324,97326],{"className":97325},[198],[15,97327,97329],{"className":97328},[202],[15,97330,97332],{"className":97331},[206],[15,97333,97335],{"className":97334,"style":6738},[210],[15,97336,97337,97340],{"style":213},[15,97338],{"className":97339,"style":218},[217],[15,97341,97343],{"className":97342},[222,223,224,225],[15,97344,1263],{"className":97345},[66,225],[15,97347,126],{"className":97348},[169],[6174,97350,97351],{},"岭回归、权重衰减",[6156,97353,97354,97409,97501],{},[6174,97355,97356,97357],{},"Laplace ",[15,97358,97360,97382],{"className":97359,"translate":19},[18],[15,97361,97363],{"className":97362},[23],[25,97364,97365],{"xmlns":27},[29,97366,97367,97379],{},[32,97368,97369,97371,97373,97375,97377],{},[39,97370,120],{"stretchy":119},[1209,97372,1804],{},[39,97374,335],{"separator":53},[35,97376,4585],{},[39,97378,126],{"stretchy":119},[45,97380,97381],{"encoding":47},"(0,b)",[15,97383,97385],{"className":97384,"ariaHidden":53},[52],[15,97386,97388,97391,97394,97397,97400,97403,97406],{"className":97387},[57],[15,97389],{"className":97390,"style":154},[61],[15,97392,120],{"className":97393},[162],[15,97395,1804],{"className":97396},[66],[15,97398,335],{"className":97399},[422],[15,97401],{"className":97402,"style":697},[71],[15,97404,4585],{"className":97405},[66,67],[15,97407,126],{"className":97408},[169],[6174,97410,97411],{},[15,97412,97414,97440],{"className":97413,"translate":19},[18],[15,97415,97417],{"className":97416},[23],[25,97418,97419],{"xmlns":27},[29,97420,97421,97437],{},[32,97422,97423,97425,97427,97433,97435],{},[35,97424,1251],{"mathvariant":138},[35,97426,61614],{},[1057,97428,97429,97431],{},[35,97430,1251],{"mathvariant":138},[1209,97432,1211],{},[35,97434,6623],{"mathvariant":138},[35,97436,4585],{},[45,97438,97439],{"encoding":47},"\\|\\theta\\|_1\u002Fb",[15,97441,97443],{"className":97442,"ariaHidden":53},[52],[15,97444,97446,97449,97452,97455,97495,97498],{"className":97445},[57],[15,97447],{"className":97448,"style":154},[61],[15,97450,1251],{"className":97451},[66],[15,97453,61614],{"className":97454,"style":23020},[66,67],[15,97456,97458,97461],{"className":97457},[66],[15,97459,1251],{"className":97460},[66],[15,97462,97464],{"className":97463},[198],[15,97465,97467,97487],{"className":97466},[202,1090],[15,97468,97470,97484],{"className":97469},[206],[15,97471,97473],{"className":97472,"style":2139},[210],[15,97474,97475,97478],{"style":1584},[15,97476],{"className":97477,"style":218},[217],[15,97479,97481],{"className":97480},[222,223,224,225],[15,97482,1211],{"className":97483},[66,225],[15,97485,1114],{"className":97486},[1113],[15,97488,97490],{"className":97489},[206],[15,97491,97493],{"className":97492,"style":1121},[210],[15,97494],{},[15,97496,6623],{"className":97497},[66],[15,97499,4585],{"className":97500},[66,67],[6174,97502,97503],{},"Lasso",[6156,97505,97506,97509,97512],{},[6174,97507,97508],{},"稀疏诱导先验",[6174,97510,97511],{},"非凸惩罚",[6174,97513,97514],{},"特征选择",[11,97516,97517],{},"这张表解释了一个常见的困惑：为什么正则化看起来像是「人为加上去的技巧」，却总是有效。它其实是在补充一个先验：参数不应该太大，或者不应该同时有很多非零分量。正则项不是外来的约束，而是模型假设的一部分，只是以惩罚的形式出现。",[11,97519,97520],{},"需要谨慎的是，深度网络里的权重衰减与高斯先验的对应只在全批量 MAP 的意义上成立。SGD 的噪声、批量大小、参数化方式（例如 BatchNorm 的缩放不变性）都会改变有效的先验。把它当成一个有用的类比，而不是一个严格的等价。",[1904,97522,97524],{"id":97523},"_53-似然比roc-与代价","5.3 似然比、ROC 与代价",[11,97526,97527],{},"二分类的贝叶斯规则可以写成似然比检验：",[15,97529,97531],{"className":97530,"translate":19},[294],[15,97532,97534,97640],{"className":97533,"translate":19},[18],[15,97535,97537],{"className":97536},[23],[25,97538,97539],{"xmlns":27,"display":303},[29,97540,97541,97637],{},[32,97542,97543,97545,97547,97549,97551,97553,97555,97557,97559,97561,97591,97593,97635],{},[35,97544,3883],{},[39,97546,120],{"stretchy":119},[35,97548,123],{},[39,97550,126],{"stretchy":119},[39,97552,129],{},[1209,97554,1211],{},[71,97556],{"width":9786},[39,97558,75696],{},[71,97560],{"width":9786},[6465,97562,97563,97577],{},[32,97564,97565,97571,97573,97575],{},[1057,97566,97567,97569],{},[35,97568,11],{},[1209,97570,1211],{},[39,97572,120],{"stretchy":119},[35,97574,123],{},[39,97576,126],{"stretchy":119},[32,97578,97579,97585,97587,97589],{},[1057,97580,97581,97583],{},[35,97582,11],{},[1209,97584,1804],{},[39,97586,120],{"stretchy":119},[35,97588,123],{},[39,97590,126],{"stretchy":119},[39,97592,1801],{},[6465,97594,97595,97615],{},[32,97596,97597,97603,97605],{},[1057,97598,97599,97601],{},[35,97600,33489],{},[1209,97602,1804],{},[1274,97604,2462],{},[1057,97606,97607,97609],{},[35,97608,2256],{},[32,97610,97611,97613],{},[35,97612,74037],{"mathvariant":138},[35,97614,28394],{"mathvariant":138},[32,97616,97617,97623,97625],{},[1057,97618,97619,97621],{},[35,97620,33489],{},[1209,97622,1211],{},[1274,97624,2462],{},[1057,97626,97627,97629],{},[35,97628,2256],{},[32,97630,97631,97633],{},[35,97632,74037],{"mathvariant":138},[35,97634,28530],{"mathvariant":138},[35,97636,3571],{"mathvariant":138},[45,97638,97639],{"encoding":47},"\\delta(x)=1\n\\quad\\Longleftrightarrow\\quad\n\\frac{p_1(x)}{p_0(x)}\n\\ge\n\\frac{\\pi_0\\,c_{\\mathrm{FP}}}{\\pi_1\\,c_{\\mathrm{FN}}}.",[15,97641,97643,97670,97695,97864],{"className":97642,"ariaHidden":53},[52],[15,97644,97646,97649,97652,97655,97658,97661,97664,97667],{"className":97645},[57],[15,97647],{"className":97648,"style":154},[61],[15,97650,3883],{"className":97651,"style":3944},[66,67],[15,97653,120],{"className":97654},[162],[15,97656,123],{"className":97657},[66,67],[15,97659,126],{"className":97660},[169],[15,97662],{"className":97663,"style":173},[71],[15,97665,129],{"className":97666},[177],[15,97668],{"className":97669,"style":173},[71],[15,97671,97673,97677,97680,97683,97686,97689,97692],{"className":97672},[57],[15,97674],{"className":97675,"style":97676},[61],"height:0.6684em;vertical-align:-0.024em;",[15,97678,1211],{"className":97679},[66],[15,97681],{"className":97682,"style":10044},[71],[15,97684],{"className":97685,"style":173},[71],[15,97687,75696],{"className":97688},[177],[15,97690],{"className":97691,"style":10044},[71],[15,97693],{"className":97694,"style":173},[71],[15,97696,97698,97701,97855,97858,97861],{"className":97697},[57],[15,97699],{"className":97700,"style":41187},[61],[15,97702,97704,97707,97852],{"className":97703},[66],[15,97705],{"className":97706},[162,23412],[15,97708,97710],{"className":97709},[6465],[15,97711,97713,97844],{"className":97712},[202,1090],[15,97714,97716,97841],{"className":97715},[206],[15,97717,97719,97776,97784],{"className":97718,"style":41206},[210],[15,97720,97721,97724],{"style":23428},[15,97722],{"className":97723,"style":1368},[217],[15,97725,97727,97767,97770,97773],{"className":97726},[66],[15,97728,97730,97733],{"className":97729},[66],[15,97731,11],{"className":97732},[66,67],[15,97734,97736],{"className":97735},[198],[15,97737,97739,97759],{"className":97738},[202,1090],[15,97740,97742,97756],{"className":97741},[206],[15,97743,97745],{"className":97744,"style":2139},[210],[15,97746,97747,97750],{"style":1584},[15,97748],{"className":97749,"style":218},[217],[15,97751,97753],{"className":97752},[222,223,224,225],[15,97754,1804],{"className":97755},[66,225],[15,97757,1114],{"className":97758},[1113],[15,97760,97762],{"className":97761},[206],[15,97763,97765],{"className":97764,"style":1121},[210],[15,97766],{},[15,97768,120],{"className":97769},[162],[15,97771,123],{"className":97772},[66,67],[15,97774,126],{"className":97775},[169],[15,97777,97778,97781],{"style":23440},[15,97779],{"className":97780,"style":1368},[217],[15,97782],{"className":97783,"style":4438},[23447],[15,97785,97786,97789],{"style":23450},[15,97787],{"className":97788,"style":1368},[217],[15,97790,97792,97832,97835,97838],{"className":97791},[66],[15,97793,97795,97798],{"className":97794},[66],[15,97796,11],{"className":97797},[66,67],[15,97799,97801],{"className":97800},[198],[15,97802,97804,97824],{"className":97803},[202,1090],[15,97805,97807,97821],{"className":97806},[206],[15,97808,97810],{"className":97809,"style":2139},[210],[15,97811,97812,97815],{"style":1584},[15,97813],{"className":97814,"style":218},[217],[15,97816,97818],{"className":97817},[222,223,224,225],[15,97819,1211],{"className":97820},[66,225],[15,97822,1114],{"className":97823},[1113],[15,97825,97827],{"className":97826},[206],[15,97828,97830],{"className":97829,"style":1121},[210],[15,97831],{},[15,97833,120],{"className":97834},[162],[15,97836,123],{"className":97837},[66,67],[15,97839,126],{"className":97840},[169],[15,97842,1114],{"className":97843},[1113],[15,97845,97847],{"className":97846},[206],[15,97848,97850],{"className":97849,"style":41277},[210],[15,97851],{},[15,97853],{"className":97854},[169,23412],[15,97856],{"className":97857,"style":173},[71],[15,97859,1801],{"className":97860},[177],[15,97862],{"className":97863,"style":173},[71],[15,97865,97867,97870,98104],{"className":97866},[57],[15,97868],{"className":97869,"style":52418},[61],[15,97871,97873,97876,98101],{"className":97872},[66],[15,97874],{"className":97875},[162,23412],[15,97877,97879],{"className":97878},[6465],[15,97880,97882,98093],{"className":97881},[202,1090],[15,97883,97885,98090],{"className":97884},[206],[15,97886,97888,97985,97993],{"className":97887,"style":39128},[210],[15,97889,97890,97893],{"style":23428},[15,97891],{"className":97892,"style":1368},[217],[15,97894,97896,97936,97939],{"className":97895},[66],[15,97897,97899,97902],{"className":97898},[66],[15,97900,33489],{"className":97901,"style":1515},[66,67],[15,97903,97905],{"className":97904},[198],[15,97906,97908,97928],{"className":97907},[202,1090],[15,97909,97911,97925],{"className":97910},[206],[15,97912,97914],{"className":97913,"style":2139},[210],[15,97915,97916,97919],{"style":1531},[15,97917],{"className":97918,"style":218},[217],[15,97920,97922],{"className":97921},[222,223,224,225],[15,97923,1211],{"className":97924},[66,225],[15,97926,1114],{"className":97927},[1113],[15,97929,97931],{"className":97930},[206],[15,97932,97934],{"className":97933,"style":1121},[210],[15,97935],{},[15,97937],{"className":97938,"style":697},[71],[15,97940,97942,97945],{"className":97941},[66],[15,97943,2256],{"className":97944},[66,67],[15,97946,97948],{"className":97947},[198],[15,97949,97951,97977],{"className":97950},[202,1090],[15,97952,97954,97974],{"className":97953},[206],[15,97955,97957],{"className":97956,"style":3636},[210],[15,97958,97959,97962],{"style":1584},[15,97960],{"className":97961,"style":218},[217],[15,97963,97965],{"className":97964},[222,223,224,225],[15,97966,97968],{"className":97967},[66,225],[15,97969,97971],{"className":97970},[66,225],[15,97972,74169],{"className":97973},[66,4427,225],[15,97975,1114],{"className":97976},[1113],[15,97978,97980],{"className":97979},[206],[15,97981,97983],{"className":97982,"style":1121},[210],[15,97984],{},[15,97986,97987,97990],{"style":23440},[15,97988],{"className":97989,"style":1368},[217],[15,97991],{"className":97992,"style":4438},[23447],[15,97994,97995,97998],{"style":23450},[15,97996],{"className":97997,"style":1368},[217],[15,97999,98001,98041,98044],{"className":98000},[66],[15,98002,98004,98007],{"className":98003},[66],[15,98005,33489],{"className":98006,"style":1515},[66,67],[15,98008,98010],{"className":98009},[198],[15,98011,98013,98033],{"className":98012},[202,1090],[15,98014,98016,98030],{"className":98015},[206],[15,98017,98019],{"className":98018,"style":2139},[210],[15,98020,98021,98024],{"style":1531},[15,98022],{"className":98023,"style":218},[217],[15,98025,98027],{"className":98026},[222,223,224,225],[15,98028,1804],{"className":98029},[66,225],[15,98031,1114],{"className":98032},[1113],[15,98034,98036],{"className":98035},[206],[15,98037,98039],{"className":98038,"style":1121},[210],[15,98040],{},[15,98042],{"className":98043,"style":697},[71],[15,98045,98047,98050],{"className":98046},[66],[15,98048,2256],{"className":98049},[66,67],[15,98051,98053],{"className":98052},[198],[15,98054,98056,98082],{"className":98055},[202,1090],[15,98057,98059,98079],{"className":98058},[206],[15,98060,98062],{"className":98061,"style":3636},[210],[15,98063,98064,98067],{"style":1584},[15,98065],{"className":98066,"style":218},[217],[15,98068,98070],{"className":98069},[222,223,224,225],[15,98071,98073],{"className":98072},[66,225],[15,98074,98076],{"className":98075},[66,225],[15,98077,74087],{"className":98078},[66,4427,225],[15,98080,1114],{"className":98081},[1113],[15,98083,98085],{"className":98084},[206],[15,98086,98088],{"className":98087,"style":1121},[210],[15,98089],{},[15,98091,1114],{"className":98092},[1113],[15,98094,98096],{"className":98095},[206],[15,98097,98099],{"className":98098,"style":52615},[210],[15,98100],{},[15,98102],{"className":98103},[169,23412],[15,98105,3571],{"className":98106},[66],[11,98108,98109],{},"左边是数据提供的证据，右边是先验与代价决定的阈值。先验偏向负类，阈值升高；漏诊更贵，阈值降低。这个分解把「数据说什么」与「我们更怕哪种错误」分开了。",[11,98111,98112,98113,98142,98143,98197,98198,98252],{},"把阈值从低到高扫一遍，就得到 ROC 曲线。设阈值为 ",[15,98114,98116,98129],{"className":98115,"translate":19},[18],[15,98117,98119],{"className":98118},[23],[25,98120,98121],{"xmlns":27},[29,98122,98123,98127],{},[32,98124,98125],{},[35,98126,9207],{},[45,98128,9207],{"encoding":47},[15,98130,98132],{"className":98131,"ariaHidden":53},[52],[15,98133,98135,98139],{"className":98134},[57],[15,98136],{"className":98137,"style":98138},[61],"height:0.6151em;",[15,98140,9207],{"className":98141},[66,67],"，记真正率为 ",[15,98144,98146,98172],{"className":98145,"translate":19},[18],[15,98147,98149],{"className":98148},[23],[25,98150,98151],{"xmlns":27},[29,98152,98153,98169],{},[32,98154,98155,98163,98165,98167],{},[32,98156,98157,98159,98161],{},[35,98158,32773],{"mathvariant":138},[35,98160,28394],{"mathvariant":138},[35,98162,455],{"mathvariant":138},[39,98164,120],{"stretchy":119},[35,98166,9207],{},[39,98168,126],{"stretchy":119},[45,98170,98171],{"encoding":47},"\\mathrm{TPR}(t)",[15,98173,98175],{"className":98174,"ariaHidden":53},[52],[15,98176,98178,98181,98188,98191,98194],{"className":98177},[57],[15,98179],{"className":98180,"style":154},[61],[15,98182,98184],{"className":98183},[66],[15,98185,98187],{"className":98186},[66,4427],"TPR",[15,98189,120],{"className":98190},[162],[15,98192,9207],{"className":98193},[66,67],[15,98195,126],{"className":98196},[169],"、假正率为 ",[15,98199,98201,98227],{"className":98200,"translate":19},[18],[15,98202,98204],{"className":98203},[23],[25,98205,98206],{"xmlns":27},[29,98207,98208,98224],{},[32,98209,98210,98218,98220,98222],{},[32,98211,98212,98214,98216],{},[35,98213,74037],{"mathvariant":138},[35,98215,28394],{"mathvariant":138},[35,98217,455],{"mathvariant":138},[39,98219,120],{"stretchy":119},[35,98221,9207],{},[39,98223,126],{"stretchy":119},[45,98225,98226],{"encoding":47},"\\mathrm{FPR}(t)",[15,98228,98230],{"className":98229,"ariaHidden":53},[52],[15,98231,98233,98236,98243,98246,98249],{"className":98232},[57],[15,98234],{"className":98235,"style":154},[61],[15,98237,98239],{"className":98238},[66],[15,98240,98242],{"className":98241},[66,4427],"FPR",[15,98244,120],{"className":98245},[162],[15,98247,9207],{"className":98248},[66,67],[15,98250,126],{"className":98251},[169],"，则期望代价为",[15,98254,98256],{"className":98255,"translate":19},[294],[15,98257,98259,98361],{"className":98258,"translate":19},[18],[15,98260,98262],{"className":98261},[23],[25,98263,98264],{"xmlns":27,"display":303},[29,98265,98266,98358],{},[32,98267,98268,98270,98272,98274,98276,98278,98284,98286,98296,98298,98306,98308,98310,98312,98314,98320,98322,98332,98334,98336,98338,98340,98348,98350,98352,98354,98356],{},[35,98269,23251],{},[39,98271,120],{"stretchy":119},[35,98273,9207],{},[39,98275,126],{"stretchy":119},[39,98277,129],{},[1057,98279,98280,98282],{},[35,98281,33489],{},[1209,98283,1804],{},[1274,98285,2462],{},[1057,98287,98288,98290],{},[35,98289,2256],{},[32,98291,98292,98294],{},[35,98293,74037],{"mathvariant":138},[35,98295,28394],{"mathvariant":138},[1274,98297,2462],{},[32,98299,98300,98302,98304],{},[35,98301,74037],{"mathvariant":138},[35,98303,28394],{"mathvariant":138},[35,98305,455],{"mathvariant":138},[39,98307,120],{"stretchy":119},[35,98309,9207],{},[39,98311,126],{"stretchy":119},[39,98313,1245],{},[1057,98315,98316,98318],{},[35,98317,33489],{},[1209,98319,1211],{},[1274,98321,2462],{},[1057,98323,98324,98326],{},[35,98325,2256],{},[32,98327,98328,98330],{},[35,98329,74037],{"mathvariant":138},[35,98331,28530],{"mathvariant":138},[1274,98333,2462],{},[39,98335,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,98337,1211],{},[39,98339,6600],{},[32,98341,98342,98344,98346],{},[35,98343,32773],{"mathvariant":138},[35,98345,28394],{"mathvariant":138},[35,98347,455],{"mathvariant":138},[39,98349,120],{"stretchy":119},[35,98351,9207],{},[39,98353,126],{"stretchy":119},[39,98355,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,98357,3571],{"mathvariant":138},[45,98359,98360],{"encoding":47},"C(t)=\\pi_0\\,c_{\\mathrm{FP}}\\,\\mathrm{FPR}(t)\n+\\pi_1\\,c_{\\mathrm{FN}}\\,\\big(1-\\mathrm{TPR}(t)\\big).",[15,98362,98364,98391,98513,98629],{"className":98363,"ariaHidden":53},[52],[15,98365,98367,98370,98373,98376,98379,98382,98385,98388],{"className":98366},[57],[15,98368],{"className":98369,"style":154},[61],[15,98371,23251],{"className":98372,"style":2901},[66,67],[15,98374,120],{"className":98375},[162],[15,98377,9207],{"className":98378},[66,67],[15,98380,126],{"className":98381},[169],[15,98383],{"className":98384,"style":173},[71],[15,98386,129],{"className":98387},[177],[15,98389],{"className":98390,"style":173},[71],[15,98392,98394,98397,98437,98440,98486,98489,98495,98498,98501,98504,98507,98510],{"className":98393},[57],[15,98395],{"className":98396,"style":154},[61],[15,98398,98400,98403],{"className":98399},[66],[15,98401,33489],{"className":98402,"style":1515},[66,67],[15,98404,98406],{"className":98405},[198],[15,98407,98409,98429],{"className":98408},[202,1090],[15,98410,98412,98426],{"className":98411},[206],[15,98413,98415],{"className":98414,"style":2139},[210],[15,98416,98417,98420],{"style":1531},[15,98418],{"className":98419,"style":218},[217],[15,98421,98423],{"className":98422},[222,223,224,225],[15,98424,1804],{"className":98425},[66,225],[15,98427,1114],{"className":98428},[1113],[15,98430,98432],{"className":98431},[206],[15,98433,98435],{"className":98434,"style":1121},[210],[15,98436],{},[15,98438],{"className":98439,"style":697},[71],[15,98441,98443,98446],{"className":98442},[66],[15,98444,2256],{"className":98445},[66,67],[15,98447,98449],{"className":98448},[198],[15,98450,98452,98478],{"className":98451},[202,1090],[15,98453,98455,98475],{"className":98454},[206],[15,98456,98458],{"className":98457,"style":3636},[210],[15,98459,98460,98463],{"style":1584},[15,98461],{"className":98462,"style":218},[217],[15,98464,98466],{"className":98465},[222,223,224,225],[15,98467,98469],{"className":98468},[66,225],[15,98470,98472],{"className":98471},[66,225],[15,98473,74087],{"className":98474},[66,4427,225],[15,98476,1114],{"className":98477},[1113],[15,98479,98481],{"className":98480},[206],[15,98482,98484],{"className":98483,"style":1121},[210],[15,98485],{},[15,98487],{"className":98488,"style":697},[71],[15,98490,98492],{"className":98491},[66],[15,98493,98242],{"className":98494},[66,4427],[15,98496,120],{"className":98497},[162],[15,98499,9207],{"className":98500},[66,67],[15,98502,126],{"className":98503},[169],[15,98505],{"className":98506,"style":72},[71],[15,98508,1245],{"className":98509},[76],[15,98511],{"className":98512,"style":72},[71],[15,98514,98516,98519,98559,98562,98608,98611,98617,98620,98623,98626],{"className":98515},[57],[15,98517],{"className":98518,"style":4398},[61],[15,98520,98522,98525],{"className":98521},[66],[15,98523,33489],{"className":98524,"style":1515},[66,67],[15,98526,98528],{"className":98527},[198],[15,98529,98531,98551],{"className":98530},[202,1090],[15,98532,98534,98548],{"className":98533},[206],[15,98535,98537],{"className":98536,"style":2139},[210],[15,98538,98539,98542],{"style":1531},[15,98540],{"className":98541,"style":218},[217],[15,98543,98545],{"className":98544},[222,223,224,225],[15,98546,1211],{"className":98547},[66,225],[15,98549,1114],{"className":98550},[1113],[15,98552,98554],{"className":98553},[206],[15,98555,98557],{"className":98556,"style":1121},[210],[15,98558],{},[15,98560],{"className":98561,"style":697},[71],[15,98563,98565,98568],{"className":98564},[66],[15,98566,2256],{"className":98567},[66,67],[15,98569,98571],{"className":98570},[198],[15,98572,98574,98600],{"className":98573},[202,1090],[15,98575,98577,98597],{"className":98576},[206],[15,98578,98580],{"className":98579,"style":3636},[210],[15,98581,98582,98585],{"style":1584},[15,98583],{"className":98584,"style":218},[217],[15,98586,98588],{"className":98587},[222,223,224,225],[15,98589,98591],{"className":98590},[66,225],[15,98592,98594],{"className":98593},[66,225],[15,98595,74169],{"className":98596},[66,4427,225],[15,98598,1114],{"className":98599},[1113],[15,98601,98603],{"className":98602},[206],[15,98604,98606],{"className":98605,"style":1121},[210],[15,98607],{},[15,98609],{"className":98610,"style":697},[71],[15,98612,98614],{"className":98613},[66],[15,98615,120],{"className":98616},[1422,1508],[15,98618,1211],{"className":98619},[66],[15,98621],{"className":98622,"style":72},[71],[15,98624,6600],{"className":98625},[76],[15,98627],{"className":98628,"style":72},[71],[15,98630,98632,98635,98641,98644,98647,98650,98656],{"className":98631},[57],[15,98633],{"className":98634,"style":4398},[61],[15,98636,98638],{"className":98637},[66],[15,98639,98187],{"className":98640},[66,4427],[15,98642,120],{"className":98643},[162],[15,98645,9207],{"className":98646},[66,67],[15,98648,126],{"className":98649},[169],[15,98651,98653],{"className":98652},[66],[15,98654,126],{"className":98655},[1422,1508],[15,98657,3571],{"className":98658},[66],[11,98660,98661],{},"贝叶斯决策做的不是「选一个看起来最好的点」，而是根据先验与代价，在 ROC 上选一个使这个线性函数最小的点。等代价线的斜率是",[15,98663,98665],{"className":98664,"translate":19},[294],[15,98666,98668,98756],{"className":98667,"translate":19},[18],[15,98669,98671],{"className":98670},[23],[25,98672,98673],{"xmlns":27,"display":303},[29,98674,98675,98753],{},[32,98676,98677,98707,98709,98751],{},[6465,98678,98679,98693],{},[32,98680,98681,98683,98685],{},[35,98682,458],{},[1274,98684,2462],{},[32,98686,98687,98689,98691],{},[35,98688,32773],{"mathvariant":138},[35,98690,28394],{"mathvariant":138},[35,98692,455],{"mathvariant":138},[32,98694,98695,98697,98699],{},[35,98696,458],{},[1274,98698,2462],{},[32,98700,98701,98703,98705],{},[35,98702,74037],{"mathvariant":138},[35,98704,28394],{"mathvariant":138},[35,98706,455],{"mathvariant":138},[39,98708,129],{},[6465,98710,98711,98731],{},[32,98712,98713,98719,98721],{},[1057,98714,98715,98717],{},[35,98716,33489],{},[1209,98718,1804],{},[1274,98720,2462],{},[1057,98722,98723,98725],{},[35,98724,2256],{},[32,98726,98727,98729],{},[35,98728,74037],{"mathvariant":138},[35,98730,28394],{"mathvariant":138},[32,98732,98733,98739,98741],{},[1057,98734,98735,98737],{},[35,98736,33489],{},[1209,98738,1211],{},[1274,98740,2462],{},[1057,98742,98743,98745],{},[35,98744,2256],{},[32,98746,98747,98749],{},[35,98748,74037],{"mathvariant":138},[35,98750,28530],{"mathvariant":138},[35,98752,3571],{"mathvariant":138},[45,98754,98755],{"encoding":47},"\\frac{d\\,\\mathrm{TPR}}{d\\,\\mathrm{FPR}}\n=\\frac{\\pi_0\\,c_{\\mathrm{FP}}}{\\pi_1\\,c_{\\mathrm{FN}}}.",[15,98757,98759,98854],{"className":98758,"ariaHidden":53},[52],[15,98760,98762,98765,98845,98848,98851],{"className":98761},[57],[15,98763],{"className":98764,"style":38382},[61],[15,98766,98768,98771,98842],{"className":98767},[66],[15,98769],{"className":98770},[162,23412],[15,98772,98774],{"className":98773},[6465],[15,98775,98777,98834],{"className":98776},[202,1090],[15,98778,98780,98831],{"className":98779},[206],[15,98781,98783,98803,98811],{"className":98782,"style":27430},[210],[15,98784,98785,98788],{"style":23428},[15,98786],{"className":98787,"style":1368},[217],[15,98789,98791,98794,98797],{"className":98790},[66],[15,98792,458],{"className":98793},[66,67],[15,98795],{"className":98796,"style":697},[71],[15,98798,98800],{"className":98799},[66],[15,98801,98242],{"className":98802},[66,4427],[15,98804,98805,98808],{"style":23440},[15,98806],{"className":98807,"style":1368},[217],[15,98809],{"className":98810,"style":4438},[23447],[15,98812,98813,98816],{"style":23450},[15,98814],{"className":98815,"style":1368},[217],[15,98817,98819,98822,98825],{"className":98818},[66],[15,98820,458],{"className":98821},[66,67],[15,98823],{"className":98824,"style":697},[71],[15,98826,98828],{"className":98827},[66],[15,98829,98187],{"className":98830},[66,4427],[15,98832,1114],{"className":98833},[1113],[15,98835,98837],{"className":98836},[206],[15,98838,98840],{"className":98839,"style":23469},[210],[15,98841],{},[15,98843],{"className":98844},[169,23412],[15,98846],{"className":98847,"style":173},[71],[15,98849,129],{"className":98850},[177],[15,98852],{"className":98853,"style":173},[71],[15,98855,98857,98860,99094],{"className":98856},[57],[15,98858],{"className":98859,"style":52418},[61],[15,98861,98863,98866,99091],{"className":98862},[66],[15,98864],{"className":98865},[162,23412],[15,98867,98869],{"className":98868},[6465],[15,98870,98872,99083],{"className":98871},[202,1090],[15,98873,98875,99080],{"className":98874},[206],[15,98876,98878,98975,98983],{"className":98877,"style":39128},[210],[15,98879,98880,98883],{"style":23428},[15,98881],{"className":98882,"style":1368},[217],[15,98884,98886,98926,98929],{"className":98885},[66],[15,98887,98889,98892],{"className":98888},[66],[15,98890,33489],{"className":98891,"style":1515},[66,67],[15,98893,98895],{"className":98894},[198],[15,98896,98898,98918],{"className":98897},[202,1090],[15,98899,98901,98915],{"className":98900},[206],[15,98902,98904],{"className":98903,"style":2139},[210],[15,98905,98906,98909],{"style":1531},[15,98907],{"className":98908,"style":218},[217],[15,98910,98912],{"className":98911},[222,223,224,225],[15,98913,1211],{"className":98914},[66,225],[15,98916,1114],{"className":98917},[1113],[15,98919,98921],{"className":98920},[206],[15,98922,98924],{"className":98923,"style":1121},[210],[15,98925],{},[15,98927],{"className":98928,"style":697},[71],[15,98930,98932,98935],{"className":98931},[66],[15,98933,2256],{"className":98934},[66,67],[15,98936,98938],{"className":98937},[198],[15,98939,98941,98967],{"className":98940},[202,1090],[15,98942,98944,98964],{"className":98943},[206],[15,98945,98947],{"className":98946,"style":3636},[210],[15,98948,98949,98952],{"style":1584},[15,98950],{"className":98951,"style":218},[217],[15,98953,98955],{"className":98954},[222,223,224,225],[15,98956,98958],{"className":98957},[66,225],[15,98959,98961],{"className":98960},[66,225],[15,98962,74169],{"className":98963},[66,4427,225],[15,98965,1114],{"className":98966},[1113],[15,98968,98970],{"className":98969},[206],[15,98971,98973],{"className":98972,"style":1121},[210],[15,98974],{},[15,98976,98977,98980],{"style":23440},[15,98978],{"className":98979,"style":1368},[217],[15,98981],{"className":98982,"style":4438},[23447],[15,98984,98985,98988],{"style":23450},[15,98986],{"className":98987,"style":1368},[217],[15,98989,98991,99031,99034],{"className":98990},[66],[15,98992,98994,98997],{"className":98993},[66],[15,98995,33489],{"className":98996,"style":1515},[66,67],[15,98998,99000],{"className":98999},[198],[15,99001,99003,99023],{"className":99002},[202,1090],[15,99004,99006,99020],{"className":99005},[206],[15,99007,99009],{"className":99008,"style":2139},[210],[15,99010,99011,99014],{"style":1531},[15,99012],{"className":99013,"style":218},[217],[15,99015,99017],{"className":99016},[222,223,224,225],[15,99018,1804],{"className":99019},[66,225],[15,99021,1114],{"className":99022},[1113],[15,99024,99026],{"className":99025},[206],[15,99027,99029],{"className":99028,"style":1121},[210],[15,99030],{},[15,99032],{"className":99033,"style":697},[71],[15,99035,99037,99040],{"className":99036},[66],[15,99038,2256],{"className":99039},[66,67],[15,99041,99043],{"className":99042},[198],[15,99044,99046,99072],{"className":99045},[202,1090],[15,99047,99049,99069],{"className":99048},[206],[15,99050,99052],{"className":99051,"style":3636},[210],[15,99053,99054,99057],{"style":1584},[15,99055],{"className":99056,"style":218},[217],[15,99058,99060],{"className":99059},[222,223,224,225],[15,99061,99063],{"className":99062},[66,225],[15,99064,99066],{"className":99065},[66,225],[15,99067,74087],{"className":99068},[66,4427,225],[15,99070,1114],{"className":99071},[1113],[15,99073,99075],{"className":99074},[206],[15,99076,99078],{"className":99077,"style":1121},[210],[15,99079],{},[15,99081,1114],{"className":99082},[1113],[15,99084,99086],{"className":99085},[206],[15,99087,99089],{"className":99088,"style":52615},[210],[15,99090],{},[15,99092],{"className":99093},[169,23412],[15,99095,3571],{"className":99096},[66],[11,99098,99099,99100,1124],{},"这也说明 AUC 只是一个与阈值无关的排序指标。两个模型的 AUC 可以一个高一个低，但在某个特定的代价结构下，AUC 低的模型反而更优。",[2040,99101,99102],{},"脱离代价谈「最好的分类器」是没有意义的",[1904,99104,99106],{"id":99105},"_54-拒绝选项与选择性分类","5.4 拒绝选项与选择性分类",[11,99108,99109,99110,99160,99161,99164],{},"3.5 节已经导出了 Chow 规则：当最大后验低于 ",[15,99111,99113,99130],{"className":99112,"translate":19},[18],[15,99114,99116],{"className":99115},[23],[25,99117,99118],{"xmlns":27},[29,99119,99120,99128],{},[32,99121,99122,99124,99126],{},[1209,99123,1211],{},[39,99125,6600],{},[35,99127,2256],{},[45,99129,75389],{"encoding":47},[15,99131,99133,99151],{"className":99132,"ariaHidden":53},[52],[15,99134,99136,99139,99142,99145,99148],{"className":99135},[57],[15,99137],{"className":99138,"style":30463},[61],[15,99140,1211],{"className":99141},[66],[15,99143],{"className":99144,"style":72},[71],[15,99146,6600],{"className":99147},[76],[15,99149],{"className":99150,"style":72},[71],[15,99152,99154,99157],{"className":99153},[57],[15,99155],{"className":99156,"style":86},[61],[15,99158,2256],{"className":99159},[66,67]," 时拒绝。把这句话放到整个数据集上，就得到一条",[2040,99162,99163],{},"风险-覆盖率曲线","：随着拒绝阈值升高，被拒绝的样本增多，覆盖率下降，但在被接受的样本上错误率也下降。",[11,99166,99167],{},"选择性分类的理论目标不是最大化准确率，而是最小化",[15,99169,99171],{"className":99170,"translate":19},[294],[15,99172,99174,99201],{"className":99173,"translate":19},[18],[15,99175,99177],{"className":99176},[23],[25,99178,99179],{"xmlns":27,"display":303},[29,99180,99181,99198],{},[32,99182,99183,99186,99188,99196],{},[1274,99184,99185],{},"选择性风险",[39,99187,129],{},[6465,99189,99190,99193],{},[1274,99191,99192],{},"被接受样本上的错误数",[1274,99194,99195],{},"被接受样本数",[39,99197,335],{"separator":53},[45,99199,99200],{"encoding":47},"\\text{选择性风险}=\\frac{\\text{被接受样本上的错误数}}{\\text{被接受样本数}},",[15,99202,99204,99225],{"className":99203,"ariaHidden":53},[52],[15,99205,99207,99210,99216,99219,99222],{"className":99206},[57],[15,99208],{"className":99209,"style":490},[61],[15,99211,99213],{"className":99212},[66,1711],[15,99214,99185],{"className":99215},[66,68610],[15,99217],{"className":99218,"style":173},[71],[15,99220,129],{"className":99221},[177],[15,99223],{"className":99224,"style":173},[71],[15,99226,99228,99232,99300],{"className":99227},[57],[15,99229],{"className":99230,"style":99231},[61],"height:2.0463em;vertical-align:-0.686em;",[15,99233,99235,99238,99297],{"className":99234},[66],[15,99236],{"className":99237},[162,23412],[15,99239,99241],{"className":99240},[6465],[15,99242,99244,99289],{"className":99243},[202,1090],[15,99245,99247,99286],{"className":99246},[206],[15,99248,99250,99264,99272],{"className":99249,"style":85558},[210],[15,99251,99252,99255],{"style":23428},[15,99253],{"className":99254,"style":1368},[217],[15,99256,99258],{"className":99257},[66],[15,99259,99261],{"className":99260},[66,1711],[15,99262,99195],{"className":99263},[66,68610],[15,99265,99266,99269],{"style":23440},[15,99267],{"className":99268,"style":1368},[217],[15,99270],{"className":99271,"style":4438},[23447],[15,99273,99274,99277],{"style":23450},[15,99275],{"className":99276,"style":1368},[217],[15,99278,99280],{"className":99279},[66],[15,99281,99283],{"className":99282},[66,1711],[15,99284,99192],{"className":99285},[66,68610],[15,99287,1114],{"className":99288},[1113],[15,99290,99292],{"className":99291},[206],[15,99293,99295],{"className":99294,"style":23469},[210],[15,99296],{},[15,99298],{"className":99299},[169,23412],[15,99301,335],{"className":99302},[422],[11,99304,99305,99306,99334,99335,99363],{},"同时满足覆盖率约束。这两个量之间的权衡，仍然由损失函数决定：拒绝的代价 ",[15,99307,99309,99322],{"className":99308,"translate":19},[18],[15,99310,99312],{"className":99311},[23],[25,99313,99314],{"xmlns":27},[29,99315,99316,99320],{},[32,99317,99318],{},[35,99319,2256],{},[45,99321,2256],{"encoding":47},[15,99323,99325],{"className":99324,"ariaHidden":53},[52],[15,99326,99328,99331],{"className":99327},[57],[15,99329],{"className":99330,"style":86},[61],[15,99332,2256],{"className":99333},[66,67]," 越小，模型越倾向于拒绝；",[15,99336,99338,99351],{"className":99337,"translate":19},[18],[15,99339,99341],{"className":99340},[23],[25,99342,99343],{"xmlns":27},[29,99344,99345,99349],{},[32,99346,99347],{},[35,99348,2256],{},[45,99350,2256],{"encoding":47},[15,99352,99354],{"className":99353,"ariaHidden":53},[52],[15,99355,99357,99360],{"className":99356},[57],[15,99358],{"className":99359,"style":86},[61],[15,99361,2256],{"className":99362},[66,67]," 越大，模型越倾向于冒险。",[11,99365,99366],{},"在医疗、自动驾驶、内容审核这些场景里，拒绝选项往往比「提高模型容量」更有效。一个知道自己不知道的模型，比一个在所有样本上都自信地犯错的模型更有用。",[1904,99368,99370],{"id":99369},"_55-类别不平衡改损失还是改数据","5.5 类别不平衡：改损失，还是改数据",[11,99372,99373],{},"类别不平衡经常被当成一个数据处理问题，但在决策论的框架里，它首先是一个先验与损失的问题。",[11,99375,99376,99377,99501,99502,99660,99661,99812],{},"设原分布的类先验为 ",[15,99378,99380,99406],{"className":99379,"translate":19},[18],[15,99381,99383],{"className":99382},[23],[25,99384,99385],{"xmlns":27},[29,99386,99387,99403],{},[32,99388,99389,99395,99397],{},[1057,99390,99391,99393],{},[35,99392,33489],{},[1209,99394,1804],{},[39,99396,335],{"separator":53},[1057,99398,99399,99401],{},[35,99400,33489],{},[1209,99402,1211],{},[45,99404,99405],{"encoding":47},"\\pi_0,\\pi_1",[15,99407,99409],{"className":99408,"ariaHidden":53},[52],[15,99410,99412,99415,99455,99458,99461],{"className":99411},[57],[15,99413],{"className":99414,"style":471},[61],[15,99416,99418,99421],{"className":99417},[66],[15,99419,33489],{"className":99420,"style":1515},[66,67],[15,99422,99424],{"className":99423},[198],[15,99425,99427,99447],{"className":99426},[202,1090],[15,99428,99430,99444],{"className":99429},[206],[15,99431,99433],{"className":99432,"style":2139},[210],[15,99434,99435,99438],{"style":1531},[15,99436],{"className":99437,"style":218},[217],[15,99439,99441],{"className":99440},[222,223,224,225],[15,99442,1804],{"className":99443},[66,225],[15,99445,1114],{"className":99446},[1113],[15,99448,99450],{"className":99449},[206],[15,99451,99453],{"className":99452,"style":1121},[210],[15,99454],{},[15,99456,335],{"className":99457},[422],[15,99459],{"className":99460,"style":697},[71],[15,99462,99464,99467],{"className":99463},[66],[15,99465,33489],{"className":99466,"style":1515},[66,67],[15,99468,99470],{"className":99469},[198],[15,99471,99473,99493],{"className":99472},[202,1090],[15,99474,99476,99490],{"className":99475},[206],[15,99477,99479],{"className":99478,"style":2139},[210],[15,99480,99481,99484],{"style":1531},[15,99482],{"className":99483,"style":218},[217],[15,99485,99487],{"className":99486},[222,223,224,225],[15,99488,1211],{"className":99489},[66,225],[15,99491,1114],{"className":99492},[1113],[15,99494,99496],{"className":99495},[206],[15,99497,99499],{"className":99498,"style":1121},[210],[15,99500],{},"，重采样后的先验为 ",[15,99503,99505,99535],{"className":99504,"translate":19},[18],[15,99506,99508],{"className":99507},[23],[25,99509,99510],{"xmlns":27},[29,99511,99512,99532],{},[32,99513,99514,99522,99524],{},[1255,99515,99516,99518,99520],{},[35,99517,33489],{},[1209,99519,1804],{},[39,99521,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,99523,335],{"separator":53},[1255,99525,99526,99528,99530],{},[35,99527,33489],{},[1209,99529,1211],{},[39,99531,635],{"mathvariant":138,"lspace":634,"rspace":634},[45,99533,99534],{"encoding":47},"\\pi'_0,\\pi'_1",[15,99536,99538],{"className":99537,"ariaHidden":53},[52],[15,99539,99541,99545,99600,99603,99606],{"className":99540},[57],[15,99542],{"className":99543,"style":99544},[61],"height:1em;vertical-align:-0.2481em;",[15,99546,99548,99551],{"className":99547},[66],[15,99549,33489],{"className":99550,"style":1515},[66,67],[15,99552,99554],{"className":99553},[198],[15,99555,99557,99592],{"className":99556},[202,1090],[15,99558,99560,99589],{"className":99559},[206],[15,99561,99563,99575],{"className":99562,"style":6003},[210],[15,99564,99566,99569],{"style":99565},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[15,99567],{"className":99568,"style":218},[217],[15,99570,99572],{"className":99571},[222,223,224,225],[15,99573,1804],{"className":99574},[66,225],[15,99576,99577,99580],{"style":213},[15,99578],{"className":99579,"style":218},[217],[15,99581,99583],{"className":99582},[222,223,224,225],[15,99584,99586],{"className":99585},[66,225],[15,99587,635],{"className":99588},[66,225],[15,99590,1114],{"className":99591},[1113],[15,99593,99595],{"className":99594},[206],[15,99596,99598],{"className":99597,"style":6771},[210],[15,99599],{},[15,99601,335],{"className":99602},[422],[15,99604],{"className":99605,"style":697},[71],[15,99607,99609,99612],{"className":99608},[66],[15,99610,33489],{"className":99611,"style":1515},[66,67],[15,99613,99615],{"className":99614},[198],[15,99616,99618,99652],{"className":99617},[202,1090],[15,99619,99621,99649],{"className":99620},[206],[15,99622,99624,99635],{"className":99623,"style":6003},[210],[15,99625,99626,99629],{"style":99565},[15,99627],{"className":99628,"style":218},[217],[15,99630,99632],{"className":99631},[222,223,224,225],[15,99633,1211],{"className":99634},[66,225],[15,99636,99637,99640],{"style":213},[15,99638],{"className":99639,"style":218},[217],[15,99641,99643],{"className":99642},[222,223,224,225],[15,99644,99646],{"className":99645},[66,225],[15,99647,635],{"className":99648},[66,225],[15,99650,1114],{"className":99651},[1113],[15,99653,99655],{"className":99654},[206],[15,99656,99658],{"className":99657,"style":6771},[210],[15,99659],{},"。同一个似然比 ",[15,99662,99664,99702],{"className":99663,"translate":19},[18],[15,99665,99667],{"className":99666},[23],[25,99668,99669],{"xmlns":27},[29,99670,99671,99699],{},[32,99672,99673,99679,99681,99683,99685,99687,99693,99695,99697],{},[1057,99674,99675,99677],{},[35,99676,11],{},[1209,99678,1211],{},[39,99680,120],{"stretchy":119},[35,99682,123],{},[39,99684,126],{"stretchy":119},[35,99686,6623],{"mathvariant":138},[1057,99688,99689,99691],{},[35,99690,11],{},[1209,99692,1804],{},[39,99694,120],{"stretchy":119},[35,99696,123],{},[39,99698,126],{"stretchy":119},[45,99700,99701],{"encoding":47},"p_1(x)\u002Fp_0(x)",[15,99703,99705],{"className":99704,"ariaHidden":53},[52],[15,99706,99708,99711,99751,99754,99757,99760,99763,99803,99806,99809],{"className":99707},[57],[15,99709],{"className":99710,"style":154},[61],[15,99712,99714,99717],{"className":99713},[66],[15,99715,11],{"className":99716},[66,67],[15,99718,99720],{"className":99719},[198],[15,99721,99723,99743],{"className":99722},[202,1090],[15,99724,99726,99740],{"className":99725},[206],[15,99727,99729],{"className":99728,"style":2139},[210],[15,99730,99731,99734],{"style":1584},[15,99732],{"className":99733,"style":218},[217],[15,99735,99737],{"className":99736},[222,223,224,225],[15,99738,1211],{"className":99739},[66,225],[15,99741,1114],{"className":99742},[1113],[15,99744,99746],{"className":99745},[206],[15,99747,99749],{"className":99748,"style":1121},[210],[15,99750],{},[15,99752,120],{"className":99753},[162],[15,99755,123],{"className":99756},[66,67],[15,99758,126],{"className":99759},[169],[15,99761,6623],{"className":99762},[66],[15,99764,99766,99769],{"className":99765},[66],[15,99767,11],{"className":99768},[66,67],[15,99770,99772],{"className":99771},[198],[15,99773,99775,99795],{"className":99774},[202,1090],[15,99776,99778,99792],{"className":99777},[206],[15,99779,99781],{"className":99780,"style":2139},[210],[15,99782,99783,99786],{"style":1584},[15,99784],{"className":99785,"style":218},[217],[15,99787,99789],{"className":99788},[222,223,224,225],[15,99790,1804],{"className":99791},[66,225],[15,99793,1114],{"className":99794},[1113],[15,99796,99798],{"className":99797},[206],[15,99799,99801],{"className":99800,"style":1121},[210],[15,99802],{},[15,99804,120],{"className":99805},[162],[15,99807,123],{"className":99808},[66,67],[15,99810,126],{"className":99811},[169]," 在两种分布下给出的后验几率不同：",[15,99814,99816],{"className":99815,"translate":19},[294],[15,99817,99819,99953],{"className":99818,"translate":19},[18],[15,99820,99822],{"className":99821},[23],[25,99823,99824],{"xmlns":27,"display":303},[29,99825,99826,99950],{},[32,99827,99828,99867,99869,99908,99910,99948],{},[6465,99829,99830,99849],{},[32,99831,99832,99839,99841,99843,99845,99847],{},[1057,99833,99834,99836],{},[35,99835,11],{},[1274,99837,99838],{},"orig",[39,99840,120],{"stretchy":119},[1209,99842,1211],{},[39,99844,34425],{},[35,99846,123],{},[39,99848,126],{"stretchy":119},[32,99850,99851,99857,99859,99861,99863,99865],{},[1057,99852,99853,99855],{},[35,99854,11],{},[1274,99856,99838],{},[39,99858,120],{"stretchy":119},[1209,99860,1804],{},[39,99862,34425],{},[35,99864,123],{},[39,99866,126],{"stretchy":119},[39,99868,129],{},[6465,99870,99871,99890],{},[32,99872,99873,99880,99882,99884,99886,99888],{},[1057,99874,99875,99877],{},[35,99876,11],{},[1274,99878,99879],{},"res",[39,99881,120],{"stretchy":119},[1209,99883,1211],{},[39,99885,34425],{},[35,99887,123],{},[39,99889,126],{"stretchy":119},[32,99891,99892,99898,99900,99902,99904,99906],{},[1057,99893,99894,99896],{},[35,99895,11],{},[1274,99897,99879],{},[39,99899,120],{"stretchy":119},[1209,99901,1804],{},[39,99903,34425],{},[35,99905,123],{},[39,99907,126],{"stretchy":119},[39,99909,3266],{},[6465,99911,99912,99930],{},[32,99913,99914,99920,99922],{},[1057,99915,99916,99918],{},[35,99917,33489],{},[1209,99919,1211],{},[1274,99921,2462],{},[1255,99923,99924,99926,99928],{},[35,99925,33489],{},[1209,99927,1804],{},[39,99929,635],{"mathvariant":138,"lspace":634,"rspace":634},[32,99931,99932,99940,99942],{},[1255,99933,99934,99936,99938],{},[35,99935,33489],{},[1209,99937,1211],{},[39,99939,635],{"mathvariant":138,"lspace":634,"rspace":634},[1274,99941,2462],{},[1057,99943,99944,99946],{},[35,99945,33489],{},[1209,99947,1804],{},[35,99949,3571],{"mathvariant":138},[45,99951,99952],{"encoding":47},"\\frac{p_{\\text{orig}}(1\\mid x)}{p_{\\text{orig}}(0\\mid x)}\n=\n\\frac{p_{\\text{res}}(1\\mid x)}{p_{\\text{res}}(0\\mid x)}\n\\cdot\n\\frac{\\pi_1\\,\\pi'_0}{\\pi'_1\\,\\pi_0}.",[15,99954,99956,100164,100369],{"className":99955,"ariaHidden":53},[52],[15,99957,99959,99963,100155,100158,100161],{"className":99958},[57],[15,99960],{"className":99961,"style":99962},[61],"height:2.3991em;vertical-align:-0.9721em;",[15,99964,99966,99969,100152],{"className":99965},[66],[15,99967],{"className":99968},[162,23412],[15,99970,99972],{"className":99971},[6465],[15,99973,99975,100143],{"className":99974},[202,1090],[15,99976,99978,100140],{"className":99977},[206],[15,99979,99981,100057,100065],{"className":99980,"style":41206},[210],[15,99982,99983,99986],{"style":23428},[15,99984],{"className":99985,"style":1368},[217],[15,99987,99989,100036,100039,100042,100045,100048,100051,100054],{"className":99988},[66],[15,99990,99992,99995],{"className":99991},[66],[15,99993,11],{"className":99994},[66,67],[15,99996,99998],{"className":99997},[198],[15,99999,100001,100028],{"className":100000},[202,1090],[15,100002,100004,100025],{"className":100003},[206],[15,100005,100008],{"className":100006,"style":100007},[210],"height:0.3175em;",[15,100009,100010,100013],{"style":1584},[15,100011],{"className":100012,"style":218},[217],[15,100014,100016],{"className":100015},[222,223,224,225],[15,100017,100019],{"className":100018},[66,225],[15,100020,100022],{"className":100021},[66,1711,225],[15,100023,99838],{"className":100024},[66,225],[15,100026,1114],{"className":100027},[1113],[15,100029,100031],{"className":100030},[206],[15,100032,100034],{"className":100033,"style":2715},[210],[15,100035],{},[15,100037,120],{"className":100038},[162],[15,100040,1804],{"className":100041},[66],[15,100043],{"className":100044,"style":173},[71],[15,100046,34425],{"className":100047},[177],[15,100049],{"className":100050,"style":173},[71],[15,100052,123],{"className":100053},[66,67],[15,100055,126],{"className":100056},[169],[15,100058,100059,100062],{"style":23440},[15,100060],{"className":100061,"style":1368},[217],[15,100063],{"className":100064,"style":4438},[23447],[15,100066,100067,100070],{"style":23450},[15,100068],{"className":100069,"style":1368},[217],[15,100071,100073,100119,100122,100125,100128,100131,100134,100137],{"className":100072},[66],[15,100074,100076,100079],{"className":100075},[66],[15,100077,11],{"className":100078},[66,67],[15,100080,100082],{"className":100081},[198],[15,100083,100085,100111],{"className":100084},[202,1090],[15,100086,100088,100108],{"className":100087},[206],[15,100089,100091],{"className":100090,"style":100007},[210],[15,100092,100093,100096],{"style":1584},[15,100094],{"className":100095,"style":218},[217],[15,100097,100099],{"className":100098},[222,223,224,225],[15,100100,100102],{"className":100101},[66,225],[15,100103,100105],{"className":100104},[66,1711,225],[15,100106,99838],{"className":100107},[66,225],[15,100109,1114],{"className":100110},[1113],[15,100112,100114],{"className":100113},[206],[15,100115,100117],{"className":100116,"style":2715},[210],[15,100118],{},[15,100120,120],{"className":100121},[162],[15,100123,1211],{"className":100124},[66],[15,100126],{"className":100127,"style":173},[71],[15,100129,34425],{"className":100130},[177],[15,100132],{"className":100133,"style":173},[71],[15,100135,123],{"className":100136},[66,67],[15,100138,126],{"className":100139},[169],[15,100141,1114],{"className":100142},[1113],[15,100144,100146],{"className":100145},[206],[15,100147,100150],{"className":100148,"style":100149},[210],"height:0.9721em;",[15,100151],{},[15,100153],{"className":100154},[169,23412],[15,100156],{"className":100157,"style":173},[71],[15,100159,129],{"className":100160},[177],[15,100162],{"className":100163,"style":173},[71],[15,100165,100167,100170,100360,100363,100366],{"className":100166},[57],[15,100168],{"className":100169,"style":41187},[61],[15,100171,100173,100176,100357],{"className":100172},[66],[15,100174],{"className":100175},[162,23412],[15,100177,100179],{"className":100178},[6465],[15,100180,100182,100349],{"className":100181},[202,1090],[15,100183,100185,100346],{"className":100184},[206],[15,100186,100188,100263,100271],{"className":100187,"style":41206},[210],[15,100189,100190,100193],{"style":23428},[15,100191],{"className":100192,"style":1368},[217],[15,100194,100196,100242,100245,100248,100251,100254,100257,100260],{"className":100195},[66],[15,100197,100199,100202],{"className":100198},[66],[15,100200,11],{"className":100201},[66,67],[15,100203,100205],{"className":100204},[198],[15,100206,100208,100234],{"className":100207},[202,1090],[15,100209,100211,100231],{"className":100210},[206],[15,100212,100214],{"className":100213,"style":2198},[210],[15,100215,100216,100219],{"style":1584},[15,100217],{"className":100218,"style":218},[217],[15,100220,100222],{"className":100221},[222,223,224,225],[15,100223,100225],{"className":100224},[66,225],[15,100226,100228],{"className":100227},[66,1711,225],[15,100229,99879],{"className":100230},[66,225],[15,100232,1114],{"className":100233},[1113],[15,100235,100237],{"className":100236},[206],[15,100238,100240],{"className":100239,"style":1121},[210],[15,100241],{},[15,100243,120],{"className":100244},[162],[15,100246,1804],{"className":100247},[66],[15,100249],{"className":100250,"style":173},[71],[15,100252,34425],{"className":100253},[177],[15,100255],{"className":100256,"style":173},[71],[15,100258,123],{"className":100259},[66,67],[15,100261,126],{"className":100262},[169],[15,100264,100265,100268],{"style":23440},[15,100266],{"className":100267,"style":1368},[217],[15,100269],{"className":100270,"style":4438},[23447],[15,100272,100273,100276],{"style":23450},[15,100274],{"className":100275,"style":1368},[217],[15,100277,100279,100325,100328,100331,100334,100337,100340,100343],{"className":100278},[66],[15,100280,100282,100285],{"className":100281},[66],[15,100283,11],{"className":100284},[66,67],[15,100286,100288],{"className":100287},[198],[15,100289,100291,100317],{"className":100290},[202,1090],[15,100292,100294,100314],{"className":100293},[206],[15,100295,100297],{"className":100296,"style":2198},[210],[15,100298,100299,100302],{"style":1584},[15,100300],{"className":100301,"style":218},[217],[15,100303,100305],{"className":100304},[222,223,224,225],[15,100306,100308],{"className":100307},[66,225],[15,100309,100311],{"className":100310},[66,1711,225],[15,100312,99879],{"className":100313},[66,225],[15,100315,1114],{"className":100316},[1113],[15,100318,100320],{"className":100319},[206],[15,100321,100323],{"className":100322,"style":1121},[210],[15,100324],{},[15,100326,120],{"className":100327},[162],[15,100329,1211],{"className":100330},[66],[15,100332],{"className":100333,"style":173},[71],[15,100335,34425],{"className":100336},[177],[15,100338],{"className":100339,"style":173},[71],[15,100341,123],{"className":100342},[66,67],[15,100344,126],{"className":100345},[169],[15,100347,1114],{"className":100348},[1113],[15,100350,100352],{"className":100351},[206],[15,100353,100355],{"className":100354,"style":41277},[210],[15,100356],{},[15,100358],{"className":100359},[169,23412],[15,100361],{"className":100362,"style":72},[71],[15,100364,3266],{"className":100365},[76],[15,100367],{"className":100368,"style":72},[71],[15,100370,100372,100376,100631],{"className":100371},[57],[15,100373],{"className":100374,"style":100375},[61],"height:2.3812em;vertical-align:-0.9523em;",[15,100377,100379,100382,100628],{"className":100378},[66],[15,100380],{"className":100381},[162,23412],[15,100383,100385],{"className":100384},[6465],[15,100386,100388,100619],{"className":100387},[202,1090],[15,100389,100391,100616],{"className":100390},[206],[15,100392,100395,100503,100511],{"className":100393,"style":100394},[210],"height:1.4289em;",[15,100396,100397,100400],{"style":23428},[15,100398],{"className":100399,"style":1368},[217],[15,100401,100403,100460,100463],{"className":100402},[66],[15,100404,100406,100409],{"className":100405},[66],[15,100407,33489],{"className":100408,"style":1515},[66,67],[15,100410,100412],{"className":100411},[198],[15,100413,100415,100452],{"className":100414},[202,1090],[15,100416,100418,100449],{"className":100417},[206],[15,100419,100422,100434],{"className":100420,"style":100421},[210],"height:0.7337em;",[15,100423,100425,100428],{"style":100424},"top:-2.4337em;margin-left:-0.0359em;margin-right:0.05em;",[15,100426],{"className":100427,"style":218},[217],[15,100429,100431],{"className":100430},[222,223,224,225],[15,100432,1211],{"className":100433},[66,225],[15,100435,100437,100440],{"style":100436},"top:-3.0448em;margin-right:0.05em;",[15,100438],{"className":100439,"style":218},[217],[15,100441,100443],{"className":100442},[222,223,224,225],[15,100444,100446],{"className":100445},[66,225],[15,100447,635],{"className":100448},[66,225],[15,100450,1114],{"className":100451},[1113],[15,100453,100455],{"className":100454},[206],[15,100456,100458],{"className":100457,"style":93597},[210],[15,100459],{},[15,100461],{"className":100462,"style":697},[71],[15,100464,100466,100469],{"className":100465},[66],[15,100467,33489],{"className":100468,"style":1515},[66,67],[15,100470,100472],{"className":100471},[198],[15,100473,100475,100495],{"className":100474},[202,1090],[15,100476,100478,100492],{"className":100477},[206],[15,100479,100481],{"className":100480,"style":2139},[210],[15,100482,100483,100486],{"style":1531},[15,100484],{"className":100485,"style":218},[217],[15,100487,100489],{"className":100488},[222,223,224,225],[15,100490,1804],{"className":100491},[66,225],[15,100493,1114],{"className":100494},[1113],[15,100496,100498],{"className":100497},[206],[15,100499,100501],{"className":100500,"style":1121},[210],[15,100502],{},[15,100504,100505,100508],{"style":23440},[15,100506],{"className":100507,"style":1368},[217],[15,100509],{"className":100510,"style":4438},[23447],[15,100512,100513,100516],{"style":23450},[15,100514],{"className":100515,"style":1368},[217],[15,100517,100519,100559,100562],{"className":1005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3.5 节的公式移动阈值。两条路在理论上等价，差别只在实现难度与估计方差。",[11,100638,100639],{},"真正不能省掉的是损失函数。若两类错误的代价确实不同，那么无论怎么采样，都必须把代价写进决策规则；否则得到的只是一个在改动后的分布上「看起来平衡」的分类器，而不是原问题的最优决策。",[95,100641,100643],{"id":100642},"_6-从参数到分布","6. 从参数到分布",[1904,100645,100647],{"id":100646},"_61-三个预测器","6.1 三个预测器",[11,100649,100650,100651,100679,100680,100708],{},"给定训练集 ",[15,100652,100654,100667],{"className":100653,"translate":19},[18],[15,100655,100657],{"className":100656},[23],[25,100658,100659],{"xmlns":27},[29,100660,100661,100665],{},[32,100662,100663],{},[35,100664,32453],{},[45,100666,32453],{"encoding":47},[15,100668,100670],{"className":100669,"ariaHidden":53},[52],[15,100671,100673,100676],{"className":100672},[57],[15,100674],{"className":100675,"style":490},[61],[15,100677,32453],{"className":100678,"style":23020},[66,67],"，预测 ",[15,100681,100683,100696],{"className":100682,"translate":19},[18],[15,100684,100686],{"className":100685},[23],[25,100687,100688],{"xmlns":27},[29,100689,100690,100694],{},[32,100691,100692],{},[35,100693,61881],{},[45,100695,61881],{"encoding":47},[15,100697,100699],{"className":100698,"ariaHidden":53},[52],[15,100700,100702,100705],{"className":100701},[57],[15,100703],{"className":100704,"style":490},[61],[15,100706,61881],{"className":100707,"style":72},[66,67]," 至少有三层做法：",[3189,100710,100711,100891,101158],{},[3192,100712,100713,100716,100717,100778,100779,100890],{},[2040,100714,100715],{},"代入式","：先估计一个参数 ",[15,100718,100720,100738],{"className":100719,"translate":19},[18],[15,100721,100723],{"className":100722},[23],[25,100724,100725],{"xmlns":27},[29,100726,100727,100735],{},[32,100728,100729],{},[4294,100730,100731,100733],{"accent":53},[35,100732,61614],{},[39,100734,78122],{},[45,100736,100737],{"encoding":47},"\\hat\\theta",[15,100739,100741],{"className":100740,"ariaHidden":53},[52],[15,100742,100744,100747],{"className":100743},[57],[15,100745],{"className":100746,"style":78149},[61],[15,100748,100750],{"className":100749},[66,78139],[15,100751,100753],{"className":100752},[202],[15,100754,100756],{"className":100755},[206],[15,100757,100759,100767],{"className":100758,"style":78149},[210],[15,100760,100761,100764],{"style":1386},[15,100762],{"className":100763,"style":1368},[217],[15,100765,61614],{"className":100766,"style":23020},[66,67],[15,100768,100769,100772],{"style":78160},[15,100770],{"className":100771,"style":1368},[217],[15,100773,100775],{"className":100774,"style":96102},[78167],[15,100776,78122],{"className":100777},[66],"，再用 ",[15,100780,100782,100814],{"className":100781,"translate":19},[18],[15,100783,100785],{"className":100784},[23],[25,100786,100787],{"xmlns":27},[29,100788,100789,100811],{},[32,100790,100791,100793,100795,100797,100799,100801,100803,100809],{},[35,100792,11],{},[39,100794,120],{"stretchy":119},[35,100796,1224],{},[39,100798,34425],{},[35,100800,123],{},[39,100802,335],{"separator":53},[4294,100804,100805,100807],{"accent":53},[35,100806,61614],{},[39,100808,78122],{},[39,100810,126],{"stretchy":119},[45,100812,100813],{"encoding":47},"p(y\\mid x,\\hat\\theta)",[15,100815,100817,100841],{"className":100816,"ariaHidden":53},[52],[15,100818,100820,100823,100826,100829,100832,100835,100838],{"className":100819},[57],[15,100821],{"className":100822,"style":154},[61],[15,100824,11],{"className":100825},[66,67],[15,100827,120],{"className":100828},[162],[15,100830,1224],{"className":100831,"style":1515},[66,67],[15,100833],{"className":100834,"style":173},[71],[15,100836,34425],{"className":100837},[177],[15,100839],{"className":100840,"style":173},[71],[15,100842,100844,100847,100850,100853,100856,100887],{"className":100843},[57],[15,100845],{"className":100846,"style":78274},[61],[15,100848,123],{"className":100849},[66,67],[15,100851,335],{"className":100852},[422],[15,100854],{"className":100855,"style":697},[71],[15,100857,100859],{"className":100858},[66,78139],[15,100860,100862],{"className":100861},[202],[15,100863,100865],{"className":100864},[206],[15,100866,100868,100876],{"className":100867,"style":78149},[210],[15,100869,100870,100873],{"style":1386},[15,100871],{"className":100872,"style":1368},[217],[15,100874,61614],{"className":100875,"style":23020},[66,67],[15,100877,100878,100881],{"style":78160},[15,100879],{"className":100880,"style":1368},[217],[15,100882,100884],{"className":100883,"style":96102},[78167],[15,100885,78122],{"className":100886},[66],[15,100888,126],{"className":100889},[169]," 预测；",[3192,100892,100893,100896,100897],{},[2040,100894,100895],{},"后验预测","：对参数积分，\n",[15,100898,100900],{"className":100899,"translate":19},[294],[15,100901,100903,100974],{"className":100902,"translate":19},[18],[15,100904,100906],{"className":100905},[23],[25,100907,100908],{"xmlns":27,"display":303},[29,100909,100910,100971],{},[32,100911,100912,100919,100921,100923,100925,100927,100929,100931,100933,100935,100937,100939,100941,100943,100945,100947,100949,100951,100953,100955,100957,100959,100961,100963,100965,100967,100969],{},[4294,100913,100914,100916],{"accent":53},[35,100915,11],{},[39,100917,100918],{},"ˉ",[39,100920,120],{"stretchy":119},[35,100922,1224],{},[39,100924,34425],{},[35,100926,123],{},[39,100928,126],{"stretchy":119},[39,100930,129],{},[39,100932,29727],{},[35,100934,11],{},[39,100936,120],{"stretchy":119},[35,100938,1224],{},[39,100940,34425],{},[35,100942,123],{},[39,100944,335],{"separator":53},[35,100946,61614],{},[39,100948,126],{"stretchy":119},[1274,100950,2462],{},[35,100952,11],{},[39,100954,120],{"stretchy":119},[35,100956,61614],{},[39,100958,34425],{},[35,100960,32453],{},[39,100962,126],{"stretchy":119},[1274,100964,2462],{},[35,100966,458],{},[35,100968,61614],{},[39,100970,8408],{"separator":53},[45,100972,100973],{"encoding":47},"\\bar p(y\\mid x)=\\int p(y\\mid x,\\theta)\\,p(\\theta\\mid D)\\,d\\theta;",[15,100975,100977,101041,101062,101092,101134],{"className":100976,"ariaHidden":53},[52],[15,100978,100980,100983,101026,101029,101032,101035,101038],{"className":100979},[57],[15,100981],{"className":100982,"style":154},[61],[15,100984,100986],{"className":100985},[66,78139],[15,100987,100989,101018],{"className":100988},[202,1090],[15,100990,100992,101015],{"className":100991},[206],[15,100993,100996,101004],{"className":100994,"style":100995},[210],"height:0.5678em;",[15,100997,100998,101001],{"style":1386},[15,100999],{"className":101000,"style":1368},[217],[15,101002,11],{"className":101003},[66,67],[15,101005,101006,101009],{"style":1386},[15,101007],{"className":101008,"style":1368},[217],[15,101010,101012],{"className":101011,"style":96102},[78167],[15,101013,100918],{"className":101014},[66],[15,101016,1114],{"className":101017},[1113],[15,101019,101021],{"className":101020},[206],[15,101022,101024],{"className":101023,"style":4448},[210],[15,101025],{},[15,101027,120],{"className":101028},[162],[15,101030,1224],{"className":101031,"style":1515},[66,67],[15,101033],{"className":101034,"style":173},[71],[15,101036,34425],{"className":101037},[177],[15,101039],{"className":101040,"style":173},[71],[15,101042,101044,101047,101050,101053,101056,101059],{"className":101043},[57],[15,101045],{"className":101046,"style":154},[61],[15,101048,123],{"className":101049},[66,67],[15,101051,126],{"className":101052},[169],[15,101054],{"className":101055,"style":173},[71],[15,101057,129],{"className":101058},[177],[15,101060],{"className":101061,"style":173},[71],[15,101063,101065,101068,101071,101074,101077,101080,101083,101086,101089],{"className":101064},[57],[15,101066],{"className":101067,"style":65126},[61],[15,101069,29727],{"className":101070,"style":29791},[1339,1470,1471],[15,101072],{"className":101073,"style":697},[71],[15,101075,11],{"className":101076},[66,67],[15,101078,120],{"className":101079},[162],[15,101081,1224],{"className":101082,"style":1515},[66,67],[15,101084],{"className":101085,"style":173},[71],[15,101087,34425],{"className":101088},[177],[15,101090],{"className":101091,"style":173},[71],[15,101093,101095,101098,101101,101104,101107,101110,101113,101116,101119,101122,101125,101128,101131],{"className":101094},[57],[15,101096],{"className":101097,"style":154},[61],[15,101099,123],{"className":101100},[66,67],[15,101102,335],{"className":101103},[422],[15,101105],{"className":101106,"style":697},[71],[15,101108,61614],{"className":101109,"style":23020},[66,67],[15,101111,126],{"className":101112},[169],[15,101114],{"className":101115,"style":697},[71],[15,101117,11],{"className":101118},[66,67],[15,101120,120],{"className":101121},[162],[15,101123,61614],{"className":101124,"style":23020},[66,67],[15,101126],{"className":101127,"style":173},[71],[15,101129,34425],{"className":101130},[177],[15,101132],{"className":101133,"style":173},[71],[15,101135,101137,101140,101143,101146,101149,101152,101155],{"className":101136},[57],[15,101138],{"className":101139,"style":154},[61],[15,101141,32453],{"className":101142,"style":23020},[66,67],[15,101144,126],{"className":101145},[169],[15,101147],{"className":101148,"style":697},[71],[15,101150,458],{"className":101151},[66,67],[15,101153,61614],{"className":101154,"style":23020},[66,67],[15,101156,8408],{"className":101157},[422],[3192,101159,101160,101162,101163],{},[2040,101161,75103],{},"：在预测分布上再做一次决策，\n",[15,101164,101166],{"className":101165,"translate":19},[294],[15,101167,101169,101249],{"className":101168,"translate":19},[18],[15,101170,101172],{"className":101171},[23],[25,101173,101174],{"xmlns":27,"display":303},[29,101175,101176,101246],{},[32,101177,101178,101184,101186,101188,101190,101192,101194,101196,101206,101208,101210,101212,101214,101216,101218,101220,101222,101228,101230,101232,101234,101236,101238,101240,101242,101244],{},[131,101179,101180,101182],{},[35,101181,4554],{},[39,101183,1162],{},[39,101185,120],{"stretchy":119},[35,101187,123],{},[39,101189,126],{"stretchy":119},[39,101191,129],{},[35,101193,1167],{},[39,101195,1170],{},[1172,101197,101198,101204],{},[32,101199,101200,101202],{},[35,101201,1178],{},[39,101203,1170],{},[35,101205,4554],{},[39,101207,29727],{},[35,101209,1216],{},[39,101211,120],{"stretchy":119},[35,101213,1224],{},[39,101215,335],{"separator":53},[35,101217,4554],{},[39,101219,126],{"stretchy":119},[1274,101221,2462],{},[4294,101223,101224,101226],{"accent":53},[35,101225,11],{},[39,101227,100918],{},[39,101229,120],{"stretchy":119},[35,101231,1224],{},[39,101233,34425],{},[35,101235,123],{},[39,101237,126],{"stretchy":119},[1274,101239,2462],{},[35,101241,458],{},[35,101243,1224],{},[35,101245,3571],{"mathvariant":138},[45,101247,101248],{"encoding":47},"a^*(x)=\\arg\\min_a\\int L(y,a)\\,\\bar p(y\\mid x)\\,dy.",[15,101250,101252,101305,101453],{"className":101251,"ariaHidden":53},[52],[15,101253,101255,101258,101287,101290,101293,101296,101299,101302],{"className":101254},[57],[15,101256],{"className":101257,"style":154},[61],[15,101259,101261,101264],{"className":101260},[66],[15,101262,4554],{"className":101263},[66,67],[15,101265,101267],{"className":101266},[198],[15,101268,101270],{"className":101269},[202],[15,101271,101273],{"className":101272},[206],[15,101274,101276],{"className":101275,"style":1308},[210],[15,101277,101278,101281],{"style":397},[15,101279],{"className":101280,"style":218},[217],[15,101282,101284],{"className":101283},[222,223,224,225],[15,101285,1162],{"className":101286},[76,225],[15,101288,120],{"className":101289},[162],[15,101291,123],{"className":101292},[66,67],[15,101294,126],{"className":101295},[169],[15,101297],{"className":101298,"style":173},[71],[15,101300,129],{"className":101301},[177],[15,101303],{"className":101304,"style":173},[71],[15,101306,101308,101311,101316,101319,101363,101366,101369,101372,101375,101378,101381,101384,101387,101390,101393,101396,101438,101441,101444,101447,101450],{"className":101307},[57],[15,101309],{"className":101310,"style":65126},[61],[15,101312,1340,101314],{"className":101313},[1339],[15,101315,1344],{"style":1343},[15,101317],{"className":101318,"style":697},[71],[15,101320,101322],{"className":101321},[1339,1351],[15,101323,101325,101355],{"className":101324},[202,1090],[15,101326,101328,101352],{"className":101327},[206],[15,101329,101331,101342],{"className":101330,"style":1361},[210],[15,101332,101333,101336],{"style":48819},[15,101334],{"className":101335,"style":1368},[217],[15,101337,101339],{"className":101338},[222,223,224,225],[15,101340,4554],{"className":101341},[66,67,225],[15,101343,101344,101347],{"style":1386},[15,101345],{"className":101346,"style":1368},[217],[15,101348,101349],{},[15,101350,1178],{"className":101351},[1339],[15,101353,1114],{"className":101354},[1113],[15,101356,101358],{"className":101357},[206],[15,101359,101361],{"className":101360,"style":48857},[210],[15,101362],{},[15,101364],{"className":101365,"style":697},[71],[15,101367,29727],{"className":101368,"style":29791},[1339,1470,1471],[15,101370],{"className":101371,"style":697},[71],[15,101373,1216],{"className":101374},[66,67],[15,101376,120],{"className":101377},[162],[15,101379,1224],{"className":101380,"style":1515},[66,67],[15,101382,335],{"className":101383},[422],[15,101385],{"className":101386,"style":697},[71],[15,101388,4554],{"className":101389},[66,67],[15,101391,126],{"className":101392},[169],[15,101394],{"className":101395,"style":697},[71],[15,101397,101399],{"className":101398},[66,78139],[15,101400,101402,101430],{"className":101401},[202,1090],[15,101403,101405,101427],{"className":101404},[206],[15,101406,101408,101416],{"className":101407,"style":100995},[210],[15,101409,101410,101413],{"style":1386},[15,101411],{"className":101412,"style":1368},[217],[15,101414,11],{"className":101415},[66,67],[15,101417,101418,101421],{"style":1386},[15,101419],{"className":101420,"style":1368},[217],[15,101422,101424],{"className":101423,"style":96102},[78167],[15,101425,100918],{"className":101426},[66],[15,101428,1114],{"className":101429},[1113],[15,101431,101433],{"className":101432},[206],[15,101434,101436],{"className":101435,"style":4448},[210],[15,101437],{},[15,101439,120],{"className":101440},[162],[15,101442,1224],{"className":101443,"style":1515},[66,67],[15,101445],{"className":101446,"style":173},[71],[15,101448,34425],{"className":101449},[177],[15,101451],{"className":101452,"style":173},[71],[15,101454,101456,101459,101462,101465,101468,101471,101474],{"className":101455},[57],[15,101457],{"className":101458,"style":154},[61],[15,101460,123],{"className":101461},[66,67],[15,101463,126],{"className":101464},[169],[15,101466],{"className":101467,"style":697},[71],[15,101469,458],{"className":101470},[66,67],[15,101472,1224],{"className":101473,"style":1515},[66,67],[15,101475,3571],{"className":101476},[66],[11,101478,101479],{},"第二层是贝叶斯学习的核心，第三层才是完整的贝叶斯决策。很多讨论停在第二层，把「后验预测」直接当成答案；但在有明确损失的问题里，真正的答案是第三层。",[11,101481,101482,101483,101562,101563,101591],{},"后验预测是一个混合分布，不是「把后验均值代进去」。由于 ",[15,101484,101486,101514],{"className":101485,"translate":19},[18],[15,101487,101489],{"className":101488},[23],[25,101490,101491],{"xmlns":27},[29,101492,101493,101511],{},[32,101494,101495,101497,101499,101501,101503,101505,101507,101509],{},[35,101496,11],{},[39,101498,120],{"stretchy":119},[35,101500,1224],{},[39,101502,34425],{},[35,101504,123],{},[39,101506,335],{"separator":53},[35,101508,61614],{},[39,101510,126],{"stretchy":119},[45,101512,101513],{"encoding":47},"p(y\\mid x,\\theta)",[15,101515,101517,101541],{"className":101516,"ariaHidden":53},[52],[15,101518,101520,101523,101526,101529,101532,101535,101538],{"className":101519},[57],[15,101521],{"className":101522,"style":154},[61],[15,101524,11],{"className":101525},[66,67],[15,101527,120],{"className":101528},[162],[15,101530,1224],{"className":101531,"style":1515},[66,67],[15,101533],{"className":101534,"style":173},[71],[15,101536,34425],{"className":101537},[177],[15,101539],{"className":101540,"style":173},[71],[15,101542,101544,101547,101550,101553,101556,101559],{"className":101543},[57],[15,101545],{"className":101546,"style":154},[61],[15,101548,123],{"className":101549},[66,67],[15,101551,335],{"className":101552},[422],[15,101554],{"className":101555,"style":697},[71],[15,101557,61614],{"className":101558,"style":23020},[66,67],[15,101560,126],{"className":101561},[169]," 通常对 ",[15,101564,101566,101579],{"className":101565,"translate":19},[18],[15,101567,101569],{"className":101568},[23],[25,101570,101571],{"xmlns":27},[29,101572,101573,101577],{},[32,101574,101575],{},[35,101576,61614],{},[45,101578,64801],{"encoding":47},[15,101580,101582],{"className":101581,"ariaHidden":53},[52],[15,101583,101585,101588],{"className":101584},[57],[15,101586],{"className":101587,"style":840},[61],[15,101589,61614],{"className":101590,"style":23020},[66,67]," 非线性，",[15,101593,101595],{"className":101594,"translate":19},[294],[15,101596,101598,101678],{"className":101597,"translate":19},[18],[15,101599,101601],{"className":101600},[23],[25,101602,101603],{"xmlns":27,"display":303},[29,101604,101605,101675],{},[32,101606,101607,101609,101611,101613,101615,101617,101619,101621,101623,101625,101627,101629,101631,101633,101635,101637,101639,101641,101643,101645,101647,101649,101651,101653,101655,101657,101659,101661,101663,101665,101667,101669,101671,101673],{},[39,101608,29727],{},[35,101610,11],{},[39,101612,120],{"stretchy":119},[35,101614,1224],{},[39,101616,34425],{},[35,101618,123],{},[39,101620,335],{"separator":53},[35,101622,61614],{},[39,101624,126],{"stretchy":119},[1274,101626,2462],{},[35,101628,11],{},[39,101630,120],{"stretchy":119},[35,101632,61614],{},[39,101634,34425],{},[35,101636,32453],{},[39,101638,126],{"stretchy":119},[1274,101640,2462],{},[35,101642,458],{},[35,101644,61614],{},[39,101646,13014],{"mathvariant":138},[35,101648,11],{},[39,101650,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,101652,1224],{},[39,101654,34425],{},[35,101656,123],{},[39,101658,335],{"separator":53},[35,101660,62377],{"mathvariant":454},[39,101662,1194],{"stretchy":119},[35,101664,61614],{},[39,101666,34425],{},[35,101668,32453],{},[39,101670,1266],{"stretchy":119},[39,101672,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,101674,3571],{"mathvariant":138},[45,101676,101677],{"encoding":47},"\\int p(y\\mid x,\\theta)\\,p(\\theta\\mid D)\\,d\\theta\n\\ne\np\\big(y\\mid x,\\mathbb E[\\theta\\mid D]\\big).",[15,101679,101681,101711,101753,101816,101843,101876],{"className":101680,"ariaHidden":53},[52],[15,101682,101684,101687,101690,101693,101696,101699,101702,101705,101708],{"className":101683},[57],[15,101685],{"className":101686,"style":65126},[61],[15,101688,29727],{"className":101689,"style":29791},[1339,1470,1471],[15,101691],{"className":101692,"style":697},[71],[15,101694,11],{"className":101695},[66,67],[15,101697,120],{"className":101698},[162],[15,101700,1224],{"className":101701,"style":1515},[66,67],[15,101703],{"className":101704,"style":173},[71],[15,101706,34425],{"className":101707},[177],[15,101709],{"className":101710,"style":173},[71],[15,101712,101714,101717,101720,101723,101726,101729,101732,101735,101738,101741,101744,101747,101750],{"className":101713},[57],[15,101715],{"className":101716,"style":154},[61],[15,101718,123],{"className":101719},[66,67],[15,101721,335],{"className":101722},[422],[15,101724],{"className":101725,"style":697},[71],[15,101727,61614],{"className":101728,"style":23020},[66,67],[15,101730,126],{"className":101731},[169],[15,101733],{"className":101734,"style":697},[71],[15,101736,11],{"className":101737},[66,67],[15,101739,120],{"className":101740},[162],[15,101742,61614],{"className":101743,"style":23020},[66,67],[15,101745],{"className":101746,"style":173},[71],[15,101748,34425],{"className":101749},[177],[15,101751],{"className":101752,"style":173},[71],[15,101754,101756,101759,101762,101765,101768,101771,101774,101777,101813],{"className":101755},[57],[15,101757],{"className":101758,"style":154},[61],[15,101760,32453],{"className":101761,"style":23020},[66,67],[15,101763,126],{"className":101764},[169],[15,101766],{"className":101767,"style":697},[71],[15,101769,458],{"className":101770},[66,67],[15,101772,61614],{"className":101773,"style":23020},[66,67],[15,101775],{"className":101776,"style":173},[71],[15,101778,101780,101807,101810],{"className":101779},[177],[15,101781,101783],{"className":101782},[177],[15,101784,101786],{"className":101785},[66,13081],[15,101787,101789],{"className":101788},[13085],[15,101790,101792,101795,101804],{"className":101791},[13089],[15,101793],{"className":101794,"style":3406},[61],[15,101796,101798],{"className":101797},[13096],[15,101799,101801],{"className":101800},[66],[15,101802,13103],{"className":101803},[177],[15,101805],{"className":101806},[13107],[15,101808],{"className":101809},[71,13111],[15,101811,129],{"className":101812},[177],[15,101814],{"className":101815,"style":173},[71],[15,101817,101819,101822,101825,101831,101834,101837,101840],{"className":101818},[57],[15,101820],{"className":101821,"style":4398},[61],[15,101823,11],{"className":101824},[66,67],[15,101826,101828],{"className":101827},[66],[15,101829,120],{"className":101830},[1422,1508],[15,101832,1224],{"className":101833,"style":1515},[66,67],[15,101835],{"className":101836,"style":173},[71],[15,101838,34425],{"className":101839},[177],[15,101841],{"className":101842,"style":173},[71],[15,101844,101846,101849,101852,101855,101858,101861,101864,101867,101870,101873],{"className":101845},[57],[15,101847],{"className":101848,"style":154},[61],[15,101850,123],{"className":101851},[66,67],[15,101853,335],{"className":101854},[422],[15,101856],{"className":101857,"style":697},[71],[15,101859,62377],{"className":101860},[66,517],[15,101862,1194],{"className":101863},[162],[15,101865,61614],{"className":101866,"style":23020},[66,67],[15,101868],{"className":101869,"style":173},[71],[15,101871,34425],{"className":101872},[177],[15,101874],{"className":101875,"style":173},[71],[15,101877,101879,101882,101885,101888,101894],{"className":101878},[57],[15,101880],{"className":101881,"style":4398},[61],[15,101883,32453],{"className":101884,"style":23020},[66,67],[15,101886,1266],{"className":101887},[169],[15,101889,101891],{"className":101890},[66],[15,101892,126],{"className":101893},[1422,1508],[15,101895,3571],{"className":101896},[66],[11,101898,101899],{},"差别在不确定性大的区域尤其明显。设想后验在两个相距很远的参数值上各有一个峰，代入均值会得到一个位于两峰之间、几乎没有概率质量的预测；后验预测则会把两个峰的概率都保留下来。前者过度自信，后者诚实地表达了「我不知道是哪一个」。",[1904,101901,101903],{"id":101902},"_62-两种不确定性","6.2 两种不确定性",[11,101905,101906,101907,101910,101911,101939,101940,101968,101969,101972],{},"预测的不确定性可以分成两类。",[2040,101908,101909],{},"偶然不确定性","（aleatoric uncertainty）来自数据生成过程本身的噪声：同样的 ",[15,101912,101914,101927],{"className":101913,"translate":19},[18],[15,101915,101917],{"className":101916},[23],[25,101918,101919],{"xmlns":27},[29,101920,101921,101925],{},[32,101922,101923],{},[35,101924,123],{},[45,101926,123],{"encoding":47},[15,101928,101930],{"className":101929,"ariaHidden":53},[52],[15,101931,101933,101936],{"className":101932},[57],[15,101934],{"className":101935,"style":86},[61],[15,101937,123],{"className":101938},[66,67]," 可能对应不同的 ",[15,101941,101943,101956],{"className":101942,"translate":19},[18],[15,101944,101946],{"className":101945},[23],[25,101947,101948],{"xmlns":27},[29,101949,101950,101954],{},[32,101951,101952],{},[35,101953,1224],{},[45,101955,1224],{"encoding":47},[15,101957,101959],{"className":101958,"ariaHidden":53},[52],[15,101960,101962,101965],{"className":101961},[57],[15,101963],{"className":101964,"style":471},[61],[15,101966,1224],{"className":101967,"style":1515},[66,67],"，即使知道真实参数也无法消除。",[2040,101970,101971],{},"认知不确定性","（epistemic uncertainty）来自知识不足：我们不知道哪个参数是对的，但随着数据增多，它会收缩。",[11,101974,101975],{},"对平方损失，这两种不确定性由全方差公式精确分解：",[15,101977,101979],{"className":101978,"translate":19},[294],[15,101980,101982,102120],{"className":101981,"translate":19},[18],[15,101983,101985],{"className":101984},[23],[25,101986,101987],{"xmlns":27,"display":303},[29,101988,101989,102117],{},[32,101990,101991,101999,102001,102003,102005,102007,102009,102011,102013,102015,102064,102066,102115],{},[32,101992,101993,101995,101997],{},[35,101994,31727],{"mathvariant":138},[35,101996,4554],{"mathvariant":138},[35,101998,63103],{"mathvariant":138},[39,102000,120],{"stretchy":119},[35,102002,61881],{},[39,102004,34425],{},[35,102006,123],{},[39,102008,335],{"separator":53},[35,102010,32453],{},[39,102012,126],{"stretchy":119},[39,102014,129],{},[1172,102016,102017,102061],{},[1172,102018,102019,102059],{},[32,102020,102021,102033,102035,102043,102045,102047,102049,102051,102053,102055,102057],{},[1057,102022,102023,102025],{},[35,102024,62377],{"mathvariant":454},[32,102026,102027,102029,102031],{},[35,102028,61614],{},[39,102030,34425],{},[35,102032,32453],{},[39,102034,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[32,102036,102037,102039,102041],{},[35,102038,31727],{"mathvariant":138},[35,102040,4554],{"mathvariant":138},[35,102042,63103],{"mathvariant":138},[39,102044,120],{"stretchy":119},[35,102046,61881],{},[39,102048,34425],{},[35,102050,123],{},[39,102052,335],{"separator":53},[35,102054,61614],{},[39,102056,126],{"stretchy":119},[39,102058,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,102060,69838],{"stretchy":53},[1274,102062,102063],{},"偶然",[39,102065,1245],{},[1172,102067,102068,102112],{},[1172,102069,102070,102110],{},[32,102071,102072,102090,102092,102094,102096,102098,102100,102102,102104,102106,102108],{},[1057,102073,102074,102082],{},[32,102075,102076,102078,102080],{},[35,102077,31727],{"mathvariant":138},[35,102079,4554],{"mathvariant":138},[35,102081,63103],{"mathvariant":138},[32,102083,102084,102086,102088],{},[35,102085,61614],{},[39,102087,34425],{},[35,102089,32453],{},[39,102091,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,102093,62377],{"mathvariant":454},[39,102095,1194],{"stretchy":119},[35,102097,61881],{},[39,102099,34425],{},[35,102101,123],{},[39,102103,335],{"separator":53},[35,102105,61614],{},[39,102107,1266],{"stretchy":119},[39,102109,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,102111,69838],{"stretchy":53},[1274,102113,102114],{},"认知",[35,102116,3571],{"mathvariant":138},[45,102118,102119],{"encoding":47},"\\mathrm{Var}(Y\\mid x,D)\n=\n\\underbrace{\\mathbb E_{\\theta\\mid D}\\big[\\mathrm{Var}(Y\\mid x,\\theta)\\big]}_{\\text{偶然}}\n+\n\\underbrace{\\mathrm{Var}_{\\theta\\mid D}\\big(\\mathbb E[Y\\mid x,\\theta]\\big)}_{\\text{认知}}.",[15,102121,102123,102150,102180,102404],{"className":102122,"ariaHidden":53},[52],[15,102124,102126,102129,102135,102138,102141,102144,102147],{"className":102125},[57],[15,102127],{"className":102128,"style":154},[61],[15,102130,102132],{"className":102131},[66],[15,102133,72399],{"className":102134},[66,4427],[15,102136,120],{"className":102137},[162],[15,102139,61881],{"className":102140,"style":72},[66,67],[15,102142],{"className":102143,"style":173},[71],[15,102145,34425],{"className":102146},[177],[15,102148],{"className":102149,"style":173},[71],[15,102151,102153,102156,102159,102162,102165,102168,102171,102174,102177],{"className":102152},[57],[15,102154],{"className":102155,"style":154},[61],[15,102157,123],{"className":102158},[66,67],[15,102160,335],{"className":102161},[422],[15,102163],{"className":102164,"style":697},[71],[15,102166,32453],{"className":102167,"style":23020},[66,67],[15,102169,126],{"className":102170},[169],[15,102172],{"className":102173,"style":173},[71],[15,102175,129],{"className":102176},[177],[15,102178],{"className":102179,"style":173},[71],[15,102181,102183,102187,102395,102398,102401],{"className":102182},[57],[15,102184],{"className":102185,"style":102186},[61],"height:2.5315em;vertical-align:-1.6815em;",[15,102188,102190],{"className":102189},[1413,1172],[15,102191,102193,102386],{"className":102192},[202,1090],[15,102194,102196,102383],{"className":102195},[206],[15,102197,102200,102218],{"className":102198,"style":102199},[210],"height:0.85em;",[15,102201,102203,102206],{"style":102202},"top:-1.3185em;",[15,102204],{"className":102205,"style":1368},[217],[15,102207,102209],{"className":102208},[222,223,224,225],[15,102210,102212],{"className":102211},[66,225],[15,102213,102215],{"className":102214},[66,1711,225],[15,102216,102063],{"className":102217},[66,68610,225],[15,102219,102220,102223],{"style":1386},[15,102221],{"className":102222,"style":1368},[217],[15,102224,102226],{"className":102225},[1413,1172],[15,102227,102229,102374],{"className":102228},[202,1090],[15,102230,102232,102371],{"className":102231},[206],[15,102233,102235,102266],{"className":102234,"style":102199},[210],[15,102236,102239,102242],{"className":102237,"style":102238},[48560],"top:-1.9968em;",[15,102240],{"className":102241,"style":1368},[217],[15,102243,102245,102252,102259],{"className":102244,"style":70041},[70040],[15,102246,102248],{"className":102247,"style":70046},[70045],[34823,102249,102250],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":48593},[34829,102251],{"d":70053},[15,102253,102255],{"className":102254,"style":70046},[70057],[34823,102256,102257],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":70060},[34829,102258],{"d":70063},[15,102260,102262],{"className":102261,"style":70046},[70067],[34823,102263,102264],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":70070},[34829,102265],{"d":70073},[15,102267,102268,102271],{"style":1386},[15,102269],{"className":102270,"style":1368},[217],[15,102272,102274,102323,102329,102335,102338,102341,102344,102347,102350,102353,102356,102359,102362,102365],{"className":102273},[66],[15,102275,102277,102280],{"className":102276},[66],[15,102278,62377],{"className":102279},[66,517],[15,102281,102283],{"className":102282},[198],[15,102284,102286,102315],{"className":102285},[202,1090],[15,102287,102289,102312],{"className":102288},[206],[15,102290,102292],{"className":102291,"style":3666},[210],[15,102293,102294,102297],{"style":62478},[15,102295],{"className":102296,"style":218},[217],[15,102298,102300],{"className":102299},[222,223,224,225],[15,102301,102303,102306,102309],{"className":102302},[66,225],[15,102304,61614],{"className":102305,"style":23020},[66,67,225],[15,102307,34425],{"className":102308},[177,225],[15,102310,32453],{"className":102311,"style":23020},[66,67,225],[15,102313,1114],{"className":102314},[1113],[15,102316,102318],{"className":102317},[206],[15,102319,102321],{"className":102320,"style":9875},[210],[15,102322],{},[15,102324,102326],{"className":102325},[66],[15,102327,1194],{"className":102328},[1422,1508],[15,102330,102332],{"className":102331},[66],[15,102333,72399],{"className":102334},[66,4427],[15,102336,120],{"className":102337},[162],[15,102339,61881],{"className":102340,"style":72},[66,67],[15,102342],{"className":102343,"style":173},[71],[15,102345,34425],{"className":102346},[177],[15,102348],{"className":102349,"style":173},[71],[15,102351,123],{"className":102352},[66,67],[15,102354,335],{"className":102355},[422],[15,102357],{"className":102358,"style":697},[71],[15,102360,61614],{"className":102361,"style":23020},[66,67],[15,102363,126],{"className":102364},[169],[15,102366,102368],{"className":102367},[66],[15,102369,1266],{"className":102370},[1422,1508],[15,102372,1114],{"className":102373},[1113],[15,102375,102377],{"className":102376},[206],[15,102378,102381],{"className":102379,"style":102380},[210],"height:1.0032em;",[15,102382],{},[15,102384,1114],{"className":102385},[1113],[15,102387,102389],{"className":102388},[206],[15,102390,102393],{"className":102391,"style":102392},[210],"height:1.6815em;",[15,102394],{},[15,102396],{"className":102397,"style":72},[71],[15,102399,1245],{"className":102400},[76],[15,102402],{"className":102403,"style":72},[71],[15,102405,102407,102410,102614,102617],{"className":102406},[57],[15,102408],{"className":102409,"style":102186},[61],[15,102411,102413],{"className":102412},[1413,1172],[15,102414,102416,102606],{"className":102415},[202,1090],[15,102417,102419,102603],{"className":102418},[206],[15,102420,102422,102439],{"className":102421,"style":102199},[210],[15,102423,102424,102427],{"style":102202},[15,102425],{"className":102426,"style":1368},[217],[15,102428,102430],{"className":102429},[222,223,224,225],[15,102431,102433],{"className":102432},[66,225],[15,102434,102436],{"className":102435},[66,1711,225],[15,102437,102114],{"className":102438},[66,68610,225],[15,102440,102441,102444],{"style":1386},[15,102442],{"className":102443,"style":1368},[217],[15,102445,102447],{"className":102446},[1413,1172],[15,102448,102450,102595],{"className":102449},[202,1090],[15,102451,102453,102592],{"className":102452},[206],[15,102454,102456,102486],{"className":102455,"style":102199},[210],[15,102457,102459,102462],{"className":102458,"style":102238},[48560],[15,102460],{"className":102461,"style":1368},[217],[15,102463,102465,102472,102479],{"className":102464,"style":70041},[70040],[15,102466,102468],{"className":102467,"style":70046},[70045],[34823,102469,102470],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":48593},[34829,102471],{"d":70053},[15,102473,102475],{"className":102474,"style":70046},[70057],[34823,102476,102477],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":70060},[34829,102478],{"d":70063},[15,102480,102482],{"className":102481,"style":70046},[70067],[34823,102483,102484],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":70070},[34829,102485],{"d":70073},[15,102487,102488,102491],{"style":1386},[15,102489],{"className":102490,"style":1368},[217],[15,102492,102494,102547,102553,102556,102559,102562,102565,102568,102571,102574,102577,102580,102583,102586],{"className":102493},[66],[15,102495,102497,102503],{"className":102496},[66],[15,102498,102500],{"className":102499},[66],[15,102501,72399],{"className":102502},[66,4427],[15,102504,102506],{"className":102505},[198],[15,102507,102509,102539],{"className":102508},[202,1090],[15,102510,102512,102536],{"className":102511},[206],[15,102513,102515],{"className":102514,"style":3666},[210],[15,102516,102518,102521],{"style":102517},"top:-2.5198em;margin-right:0.05em;",[15,102519],{"className":102520,"style":218},[217],[15,102522,102524],{"className":102523},[222,223,224,225],[15,102525,102527,102530,102533],{"className":102526},[66,225],[15,102528,61614],{"className":102529,"style":23020},[66,67,225],[15,102531,34425],{"className":102532},[177,225],[15,102534,32453],{"className":102535,"style":23020},[66,67,225],[15,102537,1114],{"className":102538},[1113],[15,102540,102542],{"className":102541},[206],[15,102543,102545],{"className":102544,"style":9875},[210],[15,102546],{},[15,102548,102550],{"className":102549},[66],[15,102551,120],{"className":102552},[1422,1508],[15,102554,62377],{"className":102555},[66,517],[15,102557,1194],{"className":102558},[162],[15,102560,61881],{"className":102561,"style":72},[66,67],[15,102563],{"className":102564,"style":173},[71],[15,102566,34425],{"className":102567},[177],[15,102569],{"className":102570,"style":173},[71],[15,102572,123],{"className":102573},[66,67],[15,102575,335],{"className":102576},[422],[15,102578],{"className":102579,"style":697},[71],[15,102581,61614],{"className":102582,"style":23020},[66,67],[15,102584,1266],{"className":102585},[169],[15,102587,102589],{"className":102588},[66],[15,102590,126],{"className":102591},[1422,1508],[15,102593,1114],{"className":102594},[1113],[15,102596,102598],{"className":102597},[206],[15,102599,102601],{"className":102600,"style":102380},[210],[15,102602],{},[15,102604,1114],{"className":102605},[1113],[15,102607,102609],{"className":102608},[206],[15,102610,102612],{"className":102611,"style":102392},[210],[15,102613],{},[15,102615],{"className":102616,"style":697},[71],[15,102618,3571],{"className":102619},[66],[11,102621,102622],{},"第一项是「即使知道模型参数，预测仍然会波动的部分」；第二项是「不同参数给出的预测均值之间的差异」。在数据稀疏的区域，后验宽，第二项大；随着数据增加，后验收缩，第二项趋于零。第一项则不会消失。",[11,102624,102625],{},"这个分解也是主动学习的理论基础：我们想选的样本，是那些能最大程度降低认知不确定性的样本，而不是那些本身就带噪声的样本。",[1904,102627,102629],{"id":102628},"_63-推断本身也是决策","6.3 推断本身也是决策",[11,102631,102632],{},"在参数空间上定义一个损失，报一个点估计，本质上也是一次决策：",[6150,102634,102635,102645],{},[6153,102636,102637],{},[6156,102638,102639,102642],{},[6159,102640,102641],{},"参数空间上的损失",[6159,102643,102644],{},"最优估计",[6169,102646,102647,102653,102659],{},[6156,102648,102649,102651],{},[6174,102650,75118],{},[6174,102652,72658],{},[6156,102654,102655,102657],{},[6174,102656,75125],{},[6174,102658,75128],{},[6156,102660,102661,102663],{},[6174,102662,75110],{},[6174,102664,102665],{},"后验众数（MAP）",[11,102667,102668],{},"所以贝叶斯估计不是与决策论并列的东西，而是决策论在参数空间上的应用。后验均值与 MAP 会打架，是因为它们对应不同的损失，没有谁更「正确」。若你关心的是平方误差，均值是对的；若你关心的是「最可能的那个值」，众数是对的；若你关心的是「误差不超过某个范围」，分位数才是对的。",[11,102670,102671],{},"同样的逻辑适用于区间估计。一个可信区间是一个「集合值」的行动，它的损失函数衡量区间是否覆盖真值、以及区间有多宽。置信区间与可信区间的差异当然有概率哲学的成分；但从决策论看，它们也对应不同的损失：一个关心重复抽样下的覆盖率，一个关心给定数据下的后验覆盖概率。",[1904,102673,102675],{"id":102674},"_64-后验的集中","6.4 后验的集中",[11,102677,102678,102679,102682],{},"在正则条件下，后验会随着数据增多而集中到真实参数附近。",[2040,102680,102681],{},"Bernstein–von Mises 定理","给出了更精确的结论：后验在适当的缩放与中心化之后，趋近于一个正态分布，",[15,102684,102686],{"className":102685,"translate":19},[294],[15,102687,102689,102770],{"className":102688,"translate":19},[18],[15,102690,102692],{"className":102691},[23],[25,102693,102694],{"xmlns":27,"display":303},[29,102695,102696,102767],{},[32,102697,102698,102702,102704,102706,102708,102710,102720,102722,102724,102726,102728,102731,102733,102735,102737,102739,102741,102743,102745,102747,102753,102763,102765],{},[48504,102699,102700],{},[35,102701,37],{},[1274,102703,2462],{},[39,102705,120],{"stretchy":119},[35,102707,61614],{},[39,102709,6600],{},[1057,102711,102712,102718],{},[4294,102713,102714,102716],{"accent":53},[35,102715,61614],{},[39,102717,78122],{},[35,102719,37],{},[39,102721,126],{"stretchy":119},[39,102723,34425],{},[35,102725,32453],{},[1274,102727,24228],{},[39,102729,102730],{},"⇒",[1274,102732,24228],{},[35,102734,28530],{"mathvariant":445},[39,102736,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,102738,1804],{},[39,102740,335],{"separator":53},[1274,102742,24228],{},[35,102744,16422],{},[39,102746,120],{"stretchy":119},[1057,102748,102749,102751],{},[35,102750,61614],{},[1209,102752,1804],{},[131,102754,102755,102757],{},[39,102756,126],{"stretchy":119},[32,102758,102759,102761],{},[39,102760,6600],{},[1209,102762,1211],{},[39,102764,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,102766,335],{"separator":53},[45,102768,102769],{"encoding":47},"\\sqrt n\\,(\\theta-\\hat\\theta_n)\\mid D\n\\ \\Rightarrow\\\n\\mathcal N\\big(0,\\ I(\\theta_0)^{-1}\\big),",[15,102771,102773,102845,102931,102955],{"className":102772,"ariaHidden":53},[52],[15,102774,102776,102780,102827,102830,102833,102836,102839,102842],{"className":102775},[57],[15,102777],{"className":102778,"style":102779},[61],"height:1.0992em;vertical-align:-0.25em;",[15,102781,102783],{"className":102782},[66,48546],[15,102784,102786,102818],{"className":102785},[202,1090],[15,102787,102789,102815],{"className":102788},[206],[15,102790,102793,102802],{"className":102791,"style":102792},[210],"height:0.8492em;",[15,102794,102796,102799],{"className":102795,"style":1386},[48560],[15,102797],{"className":102798,"style":1368},[217],[15,102800,37],{"className":102801,"style":48567},[66,67],[15,102803,102805,102808],{"style":102804},"top:-2.8092em;",[15,102806],{"className":102807,"style":1368},[217],[15,102809,102811],{"className":102810,"style":48587},[48586],[34823,102812,102813],{"xmlns":34825,"width":48590,"height":48591,"viewBox":48592,"preserveAspectRatio":48593},[34829,102814],{"d":48596},[15,102816,1114],{"className":102817},[1113],[15,102819,102821],{"className":102820},[206],[15,102822,102825],{"className":102823,"style":102824},[210],"height:0.1908em;",[15,102826],{},[15,102828],{"className":102829,"style":697},[71],[15,102831,120],{"className":102832},[162],[15,102834,61614],{"className":102835,"style":23020},[66,67],[15,102837],{"className":102838,"style":72},[71],[15,102840,6600],{"className":102841},[76],[15,102843],{"className":102844,"style":72},[71],[15,102846,102848,102851,102919,102922,102925,102928],{"className":102847},[57],[15,102849],{"className":102850,"style":78274},[61],[15,102852,102854,102885],{"className":102853},[66],[15,102855,102857],{"className":102856},[66,78139],[15,102858,102860],{"className":102859},[202],[15,102861,102863],{"className":102862},[206],[15,102864,102866,102874],{"className":102865,"style":78149},[210],[15,102867,102868,102871],{"style":1386},[15,102869],{"className":102870,"style":1368},[217],[15,102872,61614],{"className":102873,"style":23020},[66,67],[15,102875,102876,102879],{"style":78160},[15,102877],{"className":102878,"style":1368},[217],[15,102880,102882],{"className":102881,"style":96102},[78167],[15,102883,78122],{"className":102884},[66],[15,102886,102888],{"className":102887},[198],[15,102889,102891,102911],{"className":102890},[202,1090],[15,102892,102894,102908],{"className":102893},[206],[15,102895,102897],{"className":102896,"style":2198},[210],[15,102898,102899,102902],{"style":88433},[15,102900],{"className":102901,"style":218},[217],[15,102903,102905],{"className":102904},[222,223,224,225],[15,102906,37],{"className":102907},[66,67,225],[15,102909,1114],{"className":102910},[1113],[15,102912,102914],{"className":102913},[206],[15,102915,102917],{"className":102916,"style":1121},[210],[15,102918],{},[15,102920,126],{"className":102921},[169],[15,102923],{"className":102924,"style":173},[71],[15,102926,34425],{"className":102927},[177],[15,102929],{"className":102930,"style":173},[71],[15,102932,102934,102937,102940,102943,102946,102949,102952],{"className":102933},[57],[15,102935],{"className":102936,"style":490},[61],[15,102938,32453],{"className":102939,"style":23020},[66,67],[15,102941,24228],{"className":102942},[71],[15,102944],{"className":102945,"style":173},[71],[15,102947,102730],{"className":102948},[177],[15,102950,24228],{"className":102951},[71],[15,102953],{"className":102954,"style":173},[71],[15,102956,102958,102961,102964,102970,102973,102976,102979,102982,102985,102988,103028,103063,103069],{"className":102957},[57],[15,102959],{"className":102960,"style":11713},[61],[15,102962,28530],{"className":102963,"style":83507},[66,494],[15,102965,102967],{"className":102966},[66],[15,102968,120],{"className":102969},[1422,1508],[15,102971,1804],{"className":102972},[66],[15,102974,335],{"className":102975},[422],[15,102977,24228],{"className":102978},[71],[15,102980],{"className":102981,"style":697},[71],[15,102983,16422],{"className":102984,"style":16492},[66,67],[15,102986,120],{"className":102987},[162],[15,102989,102991,102994],{"className":102990},[66],[15,102992,61614],{"className":102993,"style":23020},[66,67],[15,102995,102997],{"className":102996},[198],[15,102998,103000,103020],{"className":102999},[202,1090],[15,103001,103003,103017],{"className":103002},[206],[15,103004,103006],{"className":103005,"style":2139},[210],[15,103007,103008,103011],{"style":88433},[15,103009],{"className":103010,"style":218},[217],[15,103012,103014],{"className":103013},[222,223,224,225],[15,103015,1804],{"className":103016},[66,225],[15,103018,1114],{"className":103019},[1113],[15,103021,103023],{"className":103022},[206],[15,103024,103026],{"className":103025,"style":1121},[210],[15,103027],{},[15,103029,103031,103034],{"className":103030},[169],[15,103032,126],{"className":103033},[169],[15,103035,103037],{"className":103036},[198],[15,103038,103040],{"className":103039},[202],[15,103041,103043],{"className":103042},[206],[15,103044,103046],{"className":103045,"style":1650},[210],[15,103047,103048,103051],{"style":397},[15,103049],{"className":103050,"style":218},[217],[15,103052,103054],{"className":103053},[222,223,224,225],[15,103055,103057,103060],{"className":103056},[66,225],[15,103058,6600],{"className":103059},[66,225],[15,103061,1211],{"className":103062},[66,225],[15,103064,103066],{"className":103065},[66],[15,103067,126],{"className":103068},[1422,1508],[15,103070,335],{"className":103071},[422],[11,103073,425,103074,103159],{},[15,103075,103077,103101],{"className":103076,"translate":19},[18],[15,103078,103080],{"className":103079},[23],[25,103081,103082],{"xmlns":27},[29,103083,103084,103098],{},[32,103085,103086,103088,103090,103096],{},[35,103087,16422],{},[39,103089,120],{"stretchy":119},[1057,103091,103092,103094],{},[35,103093,61614],{},[1209,103095,1804],{},[39,103097,126],{"stretchy":119},[45,103099,103100],{"encoding":47},"I(\\theta_0)",[15,103102,103104],{"className":103103,"ariaHidden":53},[52],[15,103105,103107,103110,103113,103116,103156],{"className":103106},[57],[15,103108],{"className":103109,"style":154},[61],[15,103111,16422],{"className":103112,"style":16492},[66,67],[15,103114,120],{"className":103115},[162],[15,103117,103119,103122],{"className":103118},[66],[15,103120,61614],{"className":103121,"style":23020},[66,67],[15,103123,103125],{"className":103124},[198],[15,103126,103128,103148],{"className":103127},[202,1090],[15,103129,103131,103145],{"className":103130},[206],[15,103132,103134],{"className":103133,"style":2139},[210],[15,103135,103136,103139],{"style":88433},[15,103137],{"className":103138,"style":218},[217],[15,103140,103142],{"className":103141},[222,223,224,225],[15,103143,1804],{"className":103144},[66,225],[15,103146,1114],{"className":103147},[1113],[15,103149,103151],{"className":103150},[206],[15,103152,103154],{"className":103153,"style":1121},[210],[15,103155],{},[15,103157,126],{"className":103158},[169]," 是 Fisher 信息。等价地，后验近似为",[15,103161,103163],{"className":103162,"translate":19},[294],[15,103164,103166,103242],{"className":103165,"translate":19},[18],[15,103167,103169],{"className":103168},[23],[25,103170,103171],{"xmlns":27,"display":303},[29,103172,103173,103239],{},[32,103174,103175,103177,103179,103181,103183,103185,103187,103189,103191,103237],{},[35,103176,11],{},[39,103178,120],{"stretchy":119},[35,103180,61614],{},[39,103182,34425],{},[35,103184,32453],{},[39,103186,126],{"stretchy":119},[39,103188,31988],{},[35,103190,28530],{"mathvariant":445},[32,103192,103193,103195,103205,103207,103209,103215,103217,103219,103225,103235],{},[39,103194,120],{"fence":53},[1057,103196,103197,103203],{},[4294,103198,103199,103201],{"accent":53},[35,103200,61614],{},[39,103202,78122],{},[35,103204,37],{},[39,103206,335],{"separator":53},[1274,103208,24228],{},[6465,103210,103211,103213],{},[1209,103212,1211],{},[35,103214,37],{},[35,103216,16422],{},[39,103218,120],{"stretchy":119},[1057,103220,103221,103223],{},[35,103222,61614],{},[1209,103224,1804],{},[131,103226,103227,103229],{},[39,103228,126],{"stretchy":119},[32,103230,103231,103233],{},[39,103232,6600],{},[1209,103234,1211],{},[39,103236,126],{"fence":53},[35,103238,3571],{"mathvariant":138},[45,103240,103241],{"encoding":47},"p(\\theta\\mid D)\\approx\n\\mathcal N\\left(\\hat\\theta_n,\\ \\frac{1}{n}I(\\theta_0)^{-1}\\right).",[15,103243,103245,103269,103290],{"className":103244,"ariaHidden":53},[52],[15,103246,103248,103251,103254,103257,103260,103263,103266],{"className":103247},[57],[15,103249],{"className":103250,"style":154},[61],[15,103252,11],{"className":103253},[66,67],[15,103255,120],{"className":103256},[162],[15,103258,61614],{"className":103259,"style":23020},[66,67],[15,103261],{"className":103262,"style":173},[71],[15,103264,34425],{"className":103265},[177],[15,103267],{"className":103268,"style":173},[71],[15,103270,103272,103275,103278,103281,103284,103287],{"className":103271},[57],[15,103273],{"className":103274,"style":154},[61],[15,103276,32453],{"className":103277,"style":23020},[66,67],[15,103279,126],{"className":103280},[169],[15,103282],{"className":103283,"style":173},[71],[15,103285,31988],{"className":103286},[177],[15,103288],{"className":103289,"style":173},[71],[15,103291,103293,103296,103299,103302,103537,103540],{"className":103292},[57],[15,103294],{"className":103295,"style":43968},[61],[15,103297,28530],{"className":103298,"style":83507},[66,494],[15,103300],{"className":103301,"style":697},[71],[15,103303,103305,103311,103379,103382,103385,103388,103450,103453,103456,103496,103531],{"className":103304},[1413],[15,103306,103308],{"className":103307,"style":1418},[162,1417],[15,103309,120],{"className":103310},[1422,224],[15,103312,103314,103345],{"className":103313},[66],[15,103315,103317],{"className":103316},[66,78139],[15,103318,103320],{"className":103319},[202],[15,103321,103323],{"className":103322},[206],[15,103324,103326,103334],{"className":103325,"style":78149},[210],[15,103327,103328,103331],{"style":1386},[15,103329],{"className":103330,"style":1368},[217],[15,103332,61614],{"className":103333,"style":23020},[66,67],[15,103335,103336,103339],{"style":78160},[15,103337],{"className":103338,"style":1368},[217],[15,103340,103342],{"className":103341,"style":96102},[78167],[15,103343,78122],{"className":103344},[66],[15,103346,103348],{"className":103347},[198],[15,103349,103351,103371],{"className":103350},[202,1090],[15,103352,103354,103368],{"className":103353},[206],[15,103355,103357],{"className":103356,"style":2198},[210],[15,103358,103359,103362],{"style":88433},[15,103360],{"className":103361,"style":218},[217],[15,103363,103365],{"className":103364},[222,223,224,225],[15,103366,37],{"className":103367},[66,67,225],[15,103369,1114],{"className":103370},[1113],[15,103372,103374],{"className":103373},[206],[15,103375,103377],{"className":103376,"style":1121},[210],[15,103378],{},[15,103380,335],{"className":103381},[422],[15,103383,24228],{"className":103384},[71],[15,103386],{"className":103387,"style":697},[71],[15,103389,103391,103394,103447],{"className":103390},[66],[15,103392],{"className":103393},[162,23412],[15,103395,103397],{"className":103396},[6465],[15,103398,103400,103439],{"className":103399},[202,1090],[15,103401,103403,103436],{"className":103402},[206],[15,103404,103406,103417,103425],{"className":103405,"style":23425},[210],[15,103407,103408,103411],{"style":23428},[15,103409],{"className":103410,"style":1368},[217],[15,103412,103414],{"className":103413},[66],[15,103415,37],{"className":103416},[66,67],[15,103418,103419,103422],{"style":23440},[15,103420],{"className":103421,"style":1368},[217],[15,103423],{"className":103424,"style":4438},[23447],[15,103426,103427,103430],{"style":23450},[15,103428],{"className":103429,"style":1368},[217],[15,103431,103433],{"className":103432},[66],[15,103434,1211],{"className":103435},[66],[15,103437,1114],{"className":103438},[1113],[15,103440,103442],{"className":103441},[206],[15,103443,103445],{"className":103444,"style":23469},[210],[15,103446],{},[15,103448],{"className":103449},[169,23412],[15,103451,16422],{"className":103452,"style":16492},[66,67],[15,103454,120],{"className":103455},[162],[15,103457,103459,103462],{"className":103458},[66],[15,103460,61614],{"className":103461,"style":23020},[66,67],[15,103463,103465],{"className":103464},[198],[15,103466,103468,103488],{"className":103467},[202,1090],[15,103469,103471,103485],{"className":103470},[206],[15,103472,103474],{"className":103473,"style":2139},[210],[15,103475,103476,103479],{"style":88433},[15,103477],{"className":103478,"style":218},[217],[15,103480,103482],{"className":103481},[222,223,224,225],[15,103483,1804],{"className":103484},[66,225],[15,103486,1114],{"className":103487},[1113],[15,103489,103491],{"className":103490},[206],[15,103492,103494],{"className":103493,"style":1121},[210],[15,103495],{},[15,103497,103499,103502],{"className":103498},[169],[15,103500,126],{"className":103501},[169],[15,103503,103505],{"className":103504},[198],[15,103506,103508],{"className":103507},[202],[15,103509,103511],{"className":103510},[206],[15,103512,103514],{"className":103513,"style":1650},[210],[15,103515,103516,103519],{"style":397},[15,103517],{"className":103518,"style":218},[217],[15,103520,103522],{"className":103521},[222,223,224,225],[15,103523,103525,103528],{"className":103524},[66,225],[15,103526,6600],{"className":103527},[66,225],[15,103529,1211],{"className":103530},[66,225],[15,103532,103534],{"className":103533,"style":1418},[169,1417],[15,103535,126],{"className":103536},[1422,224],[15,103538],{"className":103539,"style":697},[71],[15,103541,3571],{"className":103542},[66],[11,103544,103545,103548],{},[2040,103546,103547],{},"Doob 定理","则从另一个角度保证一致性：对先验几乎所有的参数值，后验都会收敛到真值附近的任意邻域。注意「先验几乎处处」这个限定：先验可能把质量放在一个坏集合上，只是那个集合的概率为零。",[11,103550,103551,103552,103651],{},"这两个定理解释了为什么在大样本下，贝叶斯与频率派的答案往往一致。后验的宽度以 ",[15,103553,103555,103581],{"className":103554,"translate":19},[18],[15,103556,103558],{"className":103557},[23],[25,103559,103560],{"xmlns":27},[29,103561,103562,103578],{},[32,103563,103564,103566,103568,103570,103572,103576],{},[35,103565,22994],{},[39,103567,120],{"stretchy":119},[1209,103569,1211],{},[35,103571,6623],{"mathvariant":138},[48504,103573,103574],{},[35,103575,37],{},[39,103577,126],{"stretchy":119},[45,103579,103580],{"encoding":47},"O(1\u002F\\sqrt n)",[15,103582,103584],{"className":103583,"ariaHidden":53},[52],[15,103585,103587,103591,103594,103597,103601,103648],{"className":103586},[57],[15,103588],{"className":103589,"style":103590},[61],"height:1.0503em;vertical-align:-0.25em;",[15,103592,22994],{"className":103593,"style":23020},[66,67],[15,103595,120],{"className":103596},[162],[15,103598,103600],{"className":103599},[66],"1\u002F",[15,103602,103604],{"className":103603},[66,48546],[15,103605,103607,103639],{"className":103606},[202,1090],[15,103608,103610,103636],{"className":103609},[206],[15,103611,103614,103623],{"className":103612,"style":103613},[210],"height:0.8003em;",[15,103615,103617,103620],{"className":103616,"style":1386},[48560],[15,103618],{"className":103619,"style":1368},[217],[15,103621,37],{"className":103622,"style":48567},[66,67],[15,103624,103626,103629],{"style":103625},"top:-2.7603em;",[15,103627],{"className":103628,"style":1368},[217],[15,103630,103632],{"className":103631,"style":48587},[48586],[34823,103633,103634],{"xmlns":34825,"width":48590,"height":48591,"viewBox":48592,"preserveAspectRatio":48593},[34829,103635],{"d":48596},[15,103637,1114],{"className":103638},[1113],[15,103640,103642],{"className":103641},[206],[15,103643,103646],{"className":103644,"style":103645},[210],"height:0.2397em;",[15,103647],{},[15,103649,126],{"className":103650},[169]," 收缩，先验对形状的影响随之减弱，后验主要由似然决定。但这也划出了边界：当模型奇异（参数不可识别、网络层数过多、对称性导致 Fisher 信息退化）时，Bernstein–von Mises 失效，后验的形状本身携带信息，不能用一个高斯替代。这正是奇异学习理论要处理的情形。",[1904,103653,103655],{"id":103654},"_65-边缘似然与奥卡姆剃刀","6.5 边缘似然与奥卡姆剃刀",[11,103657,103658,103659,103662],{},"贝叶斯模型比较的核心量是",[2040,103660,103661],{},"边缘似然","（marginal likelihood），也就是贝叶斯公式的分母：",[15,103664,103666],{"className":103665,"translate":19},[294],[15,103667,103669,103735],{"className":103668,"translate":19},[18],[15,103670,103672],{"className":103671},[23],[25,103673,103674],{"xmlns":27,"display":303},[29,103675,103676,103732],{},[32,103677,103678,103680,103682,103684,103686,103688,103690,103692,103694,103696,103698,103700,103702,103704,103706,103708,103710,103712,103714,103716,103718,103720,103722,103724,103726,103728,103730],{},[35,103679,11],{},[39,103681,120],{"stretchy":119},[35,103683,32453],{},[39,103685,34425],{},[35,103687,95969],{},[39,103689,126],{"stretchy":119},[39,103691,129],{},[39,103693,29727],{},[35,103695,11],{},[39,103697,120],{"stretchy":119},[35,103699,32453],{},[39,103701,34425],{},[35,103703,61614],{},[39,103705,335],{"separator":53},[35,103707,95969],{},[39,103709,126],{"stretchy":119},[1274,103711,2462],{},[35,103713,11],{},[39,103715,120],{"stretchy":119},[35,103717,61614],{},[39,103719,34425],{},[35,103721,95969],{},[39,103723,126],{"stretchy":119},[1274,103725,2462],{},[35,103727,458],{},[35,103729,61614],{},[35,103731,3571],{"mathvariant":138},[45,103733,103734],{"encoding":47},"p(D\\mid M)=\\int p(D\\mid\\theta,M)\\,p(\\theta\\mid M)\\,d\\theta .",[15,103736,103738,103762,103783,103813,103855],{"className":103737,"ariaHidden":53},[52],[15,103739,103741,103744,103747,103750,103753,103756,103759],{"className":103740},[57],[15,103742],{"className":103743,"style":154},[61],[15,103745,11],{"className":103746},[66,67],[15,103748,120],{"className":103749},[162],[15,103751,32453],{"className":103752,"style":23020},[66,67],[15,103754],{"className":103755,"style":173},[71],[15,103757,34425],{"className":103758},[177],[15,103760],{"className":103761,"style":173},[71],[15,103763,103765,103768,103771,103774,103777,103780],{"className":103764},[57],[15,103766],{"className":103767,"style":154},[61],[15,103769,95969],{"className":103770,"style":28785},[66,67],[15,103772,126],{"className":103773},[169],[15,103775],{"className":103776,"style":173},[71],[15,103778,129],{"className":103779},[177],[15,103781],{"className":103782,"style":173},[71],[15,103784,103786,103789,103792,103795,103798,103801,103804,103807,103810],{"className":103785},[57],[15,103787],{"className":103788,"style":65126},[61],[15,103790,29727],{"className":103791,"style":29791},[1339,1470,1471],[15,103793],{"className":103794,"style":697},[71],[15,103796,11],{"className":103797},[66,67],[15,103799,120],{"className":103800},[162],[15,103802,32453],{"className":103803,"style":23020},[66,67],[15,103805],{"className":103806,"style":173},[71],[15,103808,34425],{"className":103809},[177],[15,103811],{"className":103812,"style":173},[71],[15,103814,103816,103819,103822,103825,103828,103831,103834,103837,103840,103843,103846,103849,103852],{"className":103815},[57],[15,103817],{"className":103818,"style":154},[61],[15,103820,61614],{"className":103821,"style":23020},[66,67],[15,103823,335],{"className":103824},[422],[15,103826],{"className":103827,"style":697},[71],[15,103829,95969],{"className":103830,"style":28785},[66,67],[15,103832,126],{"className":103833},[169],[15,103835],{"className":103836,"style":697},[71],[15,103838,11],{"className":103839},[66,67],[15,103841,120],{"className":103842},[162],[15,103844,61614],{"className":103845,"style":23020},[66,67],[15,103847],{"className":103848,"style":173},[71],[15,103850,34425],{"className":103851},[177],[15,103853],{"className":103854,"style":173},[71],[15,103856,103858,103861,103864,103867,103870,103873,103876],{"className":103857},[57],[15,103859],{"className":103860,"style":154},[61],[15,103862,95969],{"className":103863,"style":28785},[66,67],[15,103865,126],{"className":103866},[169],[15,103868],{"className":103869,"style":697},[71],[15,103871,458],{"className":103872},[66,67],[15,103874,61614],{"className":103875,"style":23020},[66,67],[15,103877,3571],{"className":103878},[66],[11,103880,103881,103882,37639],{},"两个模型的比较用",[2040,103883,103884],{},"贝叶斯因子",[15,103886,103888],{"className":103887,"translate":19},[294],[15,103889,103891,104042],{"className":103890,"translate":19},[18],[15,103892,103894],{"className":103893},[23],[25,103895,103896],{"xmlns":27,"display":303},[29,103897,103898,104039],{},[32,103899,103900,103911,103913,103951,103953,103955,103993,103995,104005,104007,104037],{},[1057,103901,103902,103908],{},[32,103903,103904,103906],{},[35,103905,63968],{"mathvariant":138},[35,103907,74037],{"mathvariant":138},[1209,103909,103910],{},"12",[39,103912,129],{},[6465,103914,103915,103933],{},[32,103916,103917,103919,103921,103923,103925,103931],{},[35,103918,11],{},[39,103920,120],{"stretchy":119},[35,103922,32453],{},[39,103924,34425],{},[1057,103926,103927,103929],{},[35,103928,95969],{},[1209,103930,1211],{},[39,103932,126],{"stretchy":119},[32,103934,103935,103937,103939,103941,103943,103949],{},[35,103936,11],{},[39,103938,120],{"stretchy":119},[35,103940,32453],{},[39,103942,34425],{},[1057,103944,103945,103947],{},[35,103946,95969],{},[1209,103948,1263],{},[39,103950,126],{"stretchy":119},[39,103952,335],{"separator":53},[71,103954],{"width":3902},[6465,103956,103957,103975],{},[32,103958,103959,103961,103963,103969,103971,103973],{},[35,103960,11],{},[39,103962,120],{"stretchy":119},[1057,103964,103965,103967],{},[35,103966,95969],{},[1209,103968,1211],{},[39,103970,34425],{},[35,103972,32453],{},[39,103974,126],{"stretchy":119},[32,103976,103977,103979,103981,103987,103989,103991],{},[35,103978,11],{},[39,103980,120],{"stretchy":119},[1057,103982,103983,103985],{},[35,103984,95969],{},[1209,103986,1263],{},[39,103988,34425],{},[35,103990,32453],{},[39,103992,126],{"stretchy":119},[39,103994,129],{},[1057,103996,103997,104003],{},[32,103998,103999,104001],{},[35,104000,63968],{"mathvariant":138},[35,104002,74037],{"mathvariant":138},[1209,104004,103910],{},[39,104006,3266],{},[6465,104008,104009,104023],{},[32,104010,104011,104013,104015,104021],{},[35,104012,11],{},[39,104014,120],{"stretchy":119},[1057,104016,104017,104019],{},[35,104018,95969],{},[1209,104020,1211],{},[39,104022,126],{"stretchy":119},[32,104024,104025,104027,104029,104035],{},[35,104026,11],{},[39,104028,120],{"stretchy":119},[1057,104030,104031,104033],{},[35,104032,95969],{},[1209,104034,1263],{},[39,104036,126],{"stretchy":119},[35,104038,3571],{"mathvariant":138},[45,104040,104041],{"encoding":47},"\\mathrm{BF}_{12}=\\frac{p(D\\mid M_1)}{p(D\\mid M_2)},\n\\qquad\n\\frac{p(M_1\\mid D)}{p(M_2\\mid D)}\n=\\mathrm{BF}_{12}\\cdot\\frac{p(M_1)}{p(M_2)}.",[15,104043,104045,104107,104488,104549],{"className":104044,"ariaHidden":53},[52],[15,104046,104048,104051,104098,104101,104104],{"className":104047},[57],[15,104049],{"className":104050,"style":1076},[61],[15,104052,104054,104061],{"className":104053},[66],[15,104055,104057],{"className":104056},[66],[15,104058,104060],{"className":104059},[66,4427],"BF",[15,104062,104064],{"className":104063},[198],[15,104065,104067,104090],{"className":104066},[202,1090],[15,104068,104070,104087],{"className":104069},[206],[15,104071,104073],{"className":104072,"style":2139},[210],[15,104074,104075,104078],{"style":69553},[15,104076],{"className":104077,"style":218},[217],[15,104079,104081],{"className":104080},[222,223,224,225],[15,104082,104084],{"className":104083},[66,225],[15,104085,103910],{"className":104086},[66,225],[15,104088,1114],{"className":104089},[1113],[15,104091,104093],{"className":104092},[206],[15,104094,104096],{"className":104095,"style":1121},[210],[15,104097],{},[15,104099],{"className":104100,"style":173},[71],[15,104102,129],{"className":104103},[177],[15,104105],{"className":104106,"style":173},[71],[15,104108,104110,104113,104292,104295,104298,104301,104479,104482,104485],{"className":104109},[57],[15,104111],{"className":104112,"style":41187},[61],[15,104114,104116,104119,104289],{"className":104115},[66],[15,104117],{"className":104118},[162,23412],[15,104120,104122],{"className":104121},[6465],[15,104123,104125,104281],{"className":104124},[202,1090],[15,104126,104128,104278],{"className":104127},[206],[15,104129,104131,104201,104209],{"className":104130,"style":41206},[210],[15,104132,104133,104136],{"style":23428},[15,104134],{"className":104135,"style":1368},[217],[15,104137,104139,104142,104145,104148,104151,104154,104157,104198],{"className":104138},[66],[15,104140,11],{"className":104141},[66,67],[15,104143,120],{"className":104144},[162],[15,104146,32453],{"className":104147,"style":23020},[66,67],[15,104149],{"className":104150,"style":173},[71],[15,104152,34425],{"className":104153},[177],[15,104155],{"className":104156,"style":173},[71],[15,104158,104160,104163],{"className":104159},[66],[15,104161,95969],{"className":104162,"style":28785},[66,67],[15,104164,104166],{"className":104165},[198],[15,104167,104169,104190],{"className":104168},[202,1090],[15,104170,104172,104187],{"className":104171},[206],[15,104173,104175],{"className":104174,"style":2139},[210],[15,104176,104178,104181],{"style":104177},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[15,104179],{"className":104180,"style":218},[217],[15,104182,104184],{"className":104183},[222,223,224,225],[15,104185,1263],{"className":104186},[66,225],[15,104188,1114],{"className":104189},[1113],[15,104191,104193],{"className":104192},[206],[15,104194,104196],{"className":104195,"style":1121},[210],[15,104197],{},[15,104199,126],{"className":104200},[169],[15,104202,104203,104206],{"style":23440},[15,104204],{"className":104205,"style":1368},[217],[15,104207],{"className":104208,"style":4438},[23447],[15,104210,104211,104214],{"style":23450},[15,104212],{"className":104213,"style":1368},[217],[15,104215,104217,104220,104223,104226,104229,104232,104235,104275],{"className":104216},[66],[15,104218,11],{"className":104219},[66,67],[15,104221,120],{"className":104222},[162],[15,104224,32453],{"className":104225,"style":23020},[66,67],[15,104227],{"className":104228,"style":173},[71],[15,104230,34425],{"className":104231},[177],[15,104233],{"className":104234,"style":173},[71],[15,104236,104238,104241],{"className":104237},[66],[15,104239,95969],{"className":104240,"style":28785},[66,67],[15,104242,104244],{"className":104243},[198],[15,104245,104247,104267],{"className":104246},[202,1090],[15,104248,104250,104264],{"className":104249},[206],[15,104251,104253],{"className":104252,"style":2139},[210],[15,104254,104255,104258],{"style":104177},[15,104256],{"className":104257,"style":218},[217],[15,104259,104261],{"className":104260},[222,223,224,225],[15,104262,1211],{"className":104263},[66,225],[15,104265,1114],{"className":104266},[1113],[15,104268,104270],{"className":104269},[206],[15,104271,104273],{"className":104272,"style":1121},[210],[15,104274],{},[15,104276,126],{"className":104277},[169],[15,104279,1114],{"className":104280},[1113],[15,104282,104284],{"className":104283},[206],[15,104285,104287],{"className":104286,"style":41277},[210],[15,104288],{},[15,104290],{"className":104291},[169,23412],[15,104293,335],{"className":104294},[422],[15,104296],{"className":104297,"style":4059},[71],[15,104299],{"className":104300,"style":697},[71],[15,104302,104304,104307,104476],{"className":104303},[66],[15,104305],{"className":104306},[162,23412],[15,104308,104310],{"className":104309},[6465],[15,104311,104313,104468],{"className":104312},[202,1090],[15,104314,104316,104465],{"className":104315},[206],[15,104317,104319,104388,104396],{"className":104318,"style":41206},[21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Laplace 近似可以看出这一点。设 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的主项。参数越多，惩罚越重；但更准确地说，惩罚的不是维度本身，而是参数空间中「能给出不同预测」的有效体积。一个参数如果对预测没有影响，它就不会贡献体积，也就不会被惩罚。",[11,106199,106200,106201,1124],{},"边缘似然也有它自己的问题：它对先验敏感。一个过分分散的先验会把概率质量摊到大量「几乎不预测数据」的区域，导致模型被过度惩罚。这就是 Lindley 悖论的一个侧面：当原假设是一个点、备择假设是一个很宽的分布时，贝叶斯因子可能强烈支持原假设，即使频率派的检验已经拒绝了它。",[2040,106202,106203],{},"贝叶斯方法不会自动给出「正确」的模型选择，它只是把选择的标准写清楚了",[95,106205,106207],{"id":106206},"_7-机器学习理论中的位置","7. 机器学习理论中的位置",[1904,106209,106211],{"id":106210},"_71-风险分解","7.1 风险分解",[11,106213,1910,106214,106242,106243,3247,106425,106487],{},[15,106215,106217,106230],{"className":106216,"translate":19},[18],[15,106218,106220],{"className":106219},[23],[25,106221,106222],{"xmlns":27},[29,106223,106224,106228],{},[32,106225,106226],{},[35,106227,1061],{"mathvariant":445},[45,106229,1742],{"encoding":47},[15,106231,106233],{"className":106232,"ariaHidden":53},[52],[15,106234,106236,106239],{"className":106235},[57],[15,106237],{"className":106238,"style":490},[61],[15,106240,1061],{"className":106241,"style":1083},[66,494]," 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是从数据中学到的模型。则",[15,106489,106491],{"className":106490,"translate":19},[294],[15,106492,106494,106596],{"className":106493,"translate":19},[18],[15,106495,106497],{"className":106496},[23],[25,106498,106499],{"xmlns":27,"display":303},[29,106500,106501,106593],{},[32,106502,106503,106505,106507,106513,106515,106517,106523,106525,106554,106556,106591],{},[35,106504,455],{},[39,106506,120],{"stretchy":119},[4294,106508,106509,106511],{"accent":53},[35,106510,20177],{},[39,106512,78122],{},[39,106514,126],{"stretchy":119},[39,106516,6600],{},[131,106518,106519,106521],{},[35,106520,455],{},[39,106522,1162],{},[39,106524,129],{},[1172,106526,106527,106551],{},[1172,106528,106529,106549],{},[32,106530,106531,106533,106539,106541,106547],{},[39,106532,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,106534,106535,106537],{},[35,106536,455],{},[35,106538,1061],{"mathvariant":445},[39,106540,6600],{},[131,106542,106543,106545],{},[35,106544,455],{},[39,106546,1162],{},[39,106548,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,106550,69838],{"stretchy":53},[1274,106552,106553],{},"近似误差",[39,106555,1245],{},[1172,106557,106558,106588],{},[1172,106559,106560,106586],{},[32,106561,106562,106564,106566,106568,106574,106576,106578,106584],{},[39,106563,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,106565,455],{},[39,106567,120],{"stretchy":119},[4294,106569,106570,106572],{"accent":53},[35,106571,20177],{},[39,106573,78122],{},[39,106575,126],{"stretchy":119},[39,106577,6600],{},[1057,106579,106580,106582],{},[35,106581,455],{},[35,106583,1061],{"mathvariant":445},[39,106585,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,106587,69838],{"stretchy":53},[1274,106589,106590],{},"估计误差",[35,106592,3571],{"mathvariant":138},[45,106594,106595],{"encoding":47},"R(\\hat h)-R^*\n=\n\\underbrace{\\big(R_{\\mathcal H}-R^*\\big)}_{\\text{近似误差}}\n+\n\\underbrace{\\big(R(\\hat h)-R_{\\mathcal H}\\big)}_{\\text{估计误差}}.",[15,106597,106599,106654,106698,106913],{"className":106598,"ariaHidden":53},[52],[15,106600,106602,106605,106608,106611,106642,106645,106648,106651],{"className":106601},[57],[15,106603],{"className":106604,"style":78274},[61],[15,106606,455],{"className":106607,"style":62433},[66,67],[15,106609,120],{"className":106610},[162],[15,106612,106614],{"className":106613},[66,78139],[15,106615,106617],{"className":106616},[202],[15,106618,106620],{"className":106619},[206],[15,106621,106623,106631],{"className":106622,"style":78149},[210],[15,106624,106625,106628],{"style":1386},[15,106626],{"className":106627,"style":1368},[217],[15,106629,20177],{"className":106630},[66,67],[15,106632,106633,106636],{"style":78160},[15,106634],{"className":106635,"style":1368},[217],[15,106637,106639],{"className":106638,"style":106483},[78167],[15,106640,78122],{"className":106641},[66],[15,106643,126],{"className":106644},[169],[15,106646],{"className":106647,"style":72},[71],[15,106649,6600],{"className":106650},[76],[15,106652],{"className":106653,"style":72},[71],[15,106655,106657,106660,106689,106692,106695],{"className":106656},[57],[15,106658],{"className":106659,"style":1308},[61],[15,106661,106663,106666],{"className":106662},[66],[15,106664,455],{"className":106665,"style":62433},[66,67],[15,106667,106669],{"className":106668},[198],[15,106670,106672],{"className":106671},[202],[15,106673,106675],{"className":106674},[206],[15,106676,106678],{"className":106677,"style":1308},[210],[15,106679,106680,106683],{"style":397},[15,106681],{"className":106682,"style":218},[217],[15,106684,106686],{"className":106685},[222,223,224,225],[15,106687,1162],{"className":106688},[76,225],[15,106690],{"className":106691,"style":173},[71],[15,106693,129],{"className":106694},[177],[15,106696],{"className":106697,"style":173},[71],[15,106699,106701,106705,106904,106907,106910],{"className":106700},[57],[15,106702],{"className":106703,"style":106704},[61],"height:2.5263em;vertical-align:-1.6763em;",[15,106706,106708],{"className":106707},[1413,1172],[15,106709,106711,106896],{"className":106710},[202,1090],[15,106712,106714,106893],{"className":106713},[206],[15,106715,106717,106734],{"className":106716,"style":102199},[210],[15,106718,106719,106722],{"style":91786},[15,106720],{"className":106721,"style":1368},[217],[15,106723,106725],{"className":106724},[222,223,224,225],[15,106726,106728],{"className":106727},[66,225],[15,106729,106731],{"className":106730},[66,1711,225],[15,106732,106553],{"className":106733},[66,68610,225],[15,106735,106736,106739],{"style":1386},[15,106737],{"className":106738,"style":1368},[217],[15,106740,106742],{"className":106741},[1413,1172],[15,106743,106745,106885],{"className":106744},[202,1090],[15,106746,106748,106882],{"className":106747},[206],[15,106749,106751,106781],{"className":106750,"style":102199},[210],[15,106752,106754,106757],{"className":106753,"style":91822},[48560],[15,106755],{"className":106756,"style":1368},[217],[15,106758,106760,106767,106774],{"className":106759,"style":70041},[70040],[15,106761,106763],{"className":106762,"style":70046},[70045],[34823,106764,106765],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":48593},[34829,106766],{"d":70053},[15,106768,106770],{"className":106769,"style":70046},[70057],[34823,106771,106772],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":70060},[34829,106773],{"d":70063},[15,106775,106777],{"className":106776,"style":70046},[70067],[34823,106778,106779],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":70070},[34829,106780],{"d":70073},[15,106782,106783,106786],{"style":1386},[15,106784],{"className":106785,"style":1368},[217],[15,106787,106789,106795,106838,106841,106844,106847,106876],{"className":106788},[66],[15,106790,106792],{"className":106791},[66],[15,106793,120],{"className":106794},[1422,1508],[15,106796,106798,106801],{"className":106797},[66],[15,106799,455],{"className":106800,"style":62433},[66,67],[15,106802,106804],{"className":106803},[198],[15,106805,106807,106830],{"className":106806},[202,1090],[15,106808,106810,106827],{"className":106809},[206],[15,106811,106813],{"className":106812,"style":3636},[210],[15,106814,106815,106818],{"style":106319},[15,106816],{"className":106817,"style":218},[217],[15,106819,106821],{"className":106820},[222,223,224,225],[15,106822,106824],{"className":106823},[66,225],[15,106825,1061],{"className":106826,"style":1083},[66,494,225],[15,106828,1114],{"className":106829},[1113],[15,106831,106833],{"className":106832},[206],[15,106834,106836],{"className":106835,"style":1121},[210],[15,106837],{},[15,106839],{"className":106840,"style":72},[71],[15,106842,6600],{"className":106843},[76],[15,106845],{"className":106846,"style":72},[71],[15,106848,106850,106853],{"className":106849},[66],[15,106851,455],{"className":106852,"style":62433},[66,67],[15,106854,106856],{"c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e":92021},[210],[15,107128],{},[15,107130],{"className":107131,"style":697},[71],[15,107133,3571],{"className":107134},[66],[11,107136,107137],{},"第一项由模型类决定：模型类太小，无论如何训练都逼近不了贝叶斯风险。第二项由样本量与模型复杂度决定：模型类太大，有限样本上的经验风险会偏离真实风险。",[11,107139,107140,107141,107199,107200,107258],{},"贝叶斯风险 ",[15,107142,107144,107161],{"className":107143,"translate":19},[18],[15,107145,107147],{"className":107146},[23],[25,107148,107149],{"xmlns":27},[29,107150,107151,107159],{},[32,107152,107153],{},[131,107154,107155,107157],{},[35,107156,455],{},[39,107158,1162],{},[45,107160,76275],{"encoding":47},[15,107162,107164],{"className":107163,"ariaHidden":53},[52],[15,107165,107167,107170],{"className":107166},[57],[15,107168],{"className":107169,"style":16244},[61],[15,107171,107173,107176],{"className":107172},[66],[15,107174,455],{"className":107175,"style":62433},[66,67],[15,107177,107179],{"className":107178},[198],[15,107180,107182],{"className":107181},[202],[15,107183,107185],{"className":107184},[206],[15,107186,107188],{"className":107187,"style":16244},[210],[15,107189,107190,107193],{"style":213},[15,107191],{"className":107192,"style":218},[217],[15,107194,107196],{"className":107195},[222,223,224,225],[15,107197,1162],{"className":107198},[76,225]," 是这两项的共同起点。它既不属于模型，也不属于算法，而是问题本身的下限。学习理论的大部分内容，都可以理解为在给定 ",[15,107201,107203,107220],{"className":107202,"translate":19},[18],[15,107204,107206],{"className":107205},[23],[25,107207,107208],{"xmlns":27},[29,107209,107210,107218],{},[32,107211,107212],{},[131,107213,107214,107216],{},[35,107215,455],{},[39,107217,1162],{},[45,107219,76275],{"encoding":47},[15,107221,107223],{"className":107222,"ariaHidden":53},[52],[15,107224,107226,107229],{"className":107225},[57],[15,107227],{"className":107228,"style":16244},[61],[15,107230,107232,107235],{"className":107231},[66],[15,107233,455],{"className":107234,"style":62433},[66,67],[15,107236,107238],{"className":107237},[198],[15,107239,107241],{"className":107240},[202],[15,107242,107244],{"className":107243},[206],[15,107245,107247],{"className":107246,"style":16244},[210],[15,107248,107249,107252],{"style":213},[15,107250],{"className":107251,"style":218},[217],[15,107253,107255],{"className":107254},[222,223,224,225],[15,107256,1162],{"className":107257},[76,225]," 的情况下，分别研究这两项能压到多低。",[11,107260,107261],{},"这也解释了为什么「更复杂的模型」有时有用、有时有害：它降低近似误差，却可能提高估计误差。深度学习成功的部分原因，是它在大规模数据上把估计误差压得足够低，同时用极宽的模型类把近似误差压得很小。这不是对偏差-方差权衡的否定，而是权衡曲线在数据量增大后的重新定位。",[1904,107263,107265],{"id":107264},"_72-一致性","7.2 一致性",[11,107267,107268,107269,2038,107339,107342,107343,3247],{},"一个决策规则序列 ",[15,107270,107272,107290],{"className":107271,"translate":19},[18],[15,107273,107275],{"className":107274},[23],[25,107276,107277],{"xmlns":27},[29,107278,107279,107287],{},[32,107280,107281],{},[1057,107282,107283,107285],{},[35,107284,3883],{},[35,107286,37],{},[45,107288,107289],{"encoding":47},"\\delta_n",[15,107291,107293],{"className":107292,"ariaHidden":53},[52],[15,107294,107296,107299],{"className":107295},[57],[15,107297],{"className":107298,"style":3937},[61],[15,107300,107302,107305],{"className":107301},[66],[15,107303,3883],{"className":107304,"style":3944},[66,67],[15,107306,107308],{"className":107307},[198],[15,107309,107311,107331],{"className":107310},[202,1090],[15,107312,107314,107328],{"className":107313},[206],[15,107315,107317],{"className":107316,"style":2198},[210],[15,107318,107319,107322],{"style":3959},[15,107320],{"className":107321,"style":218},[217],[15,107323,107325],{"className":107324},[222,223,224,225],[15,107326,37],{"className":107327},[66,67,225],[15,107329,1114],{"className":107330},[1113],[15,107332,107334],{"className":107333},[206],[15,107335,107337],{"className":107336,"style":1121},[210],[15,107338],{},[2040,107340,107341],{},"普遍一致的","（universally consistent），若对任意分布 ",[15,107344,107346,107359],{"className":107345,"translate":19},[18],[15,107347,107349],{"className":107348},[23],[25,107350,107351],{"xmlns":27},[29,107352,107353,107357],{},[32,107354,107355],{},[35,107356,28394],{},[45,107358,28394],{"encoding":47},[15,107360,107362],{"className":107361,"ariaHidden":53},[52],[15,107363,107365,107368],{"className":107364},[57],[15,107366],{"className":107367,"style":490},[61],[15,107369,28394],{"className":107370,"style":62509},[66,67],[15,107372,107374],{"className":107373,"translate":19},[294],[15,107375,107377,107429],{"className":107376,"translate":19},[18],[15,107378,107380],{"className":107379},[23],[25,107381,107382],{"xmlns":27,"display":303},[29,107383,107384,107426],{},[32,107385,107386,107388,107390,107396,107398,107400,107418,107420],{},[35,107387,455],{},[39,107389,120],{"stretchy":119},[1057,107391,107392,107394],{},[35,107393,3883],{},[35,107395,37],{},[39,107397,126],{"stretchy":119},[1274,107399,24228],{},[4294,107401,107402,107405],{},[39,107403,449],{"stretchy":53,"minsize":107404},"3.0em",[107406,107407,107410],"mpadded",{"width":107408,"lspace":107409},"+0.6em","0.3em",[32,107411,107412,107414,107416],{},[1274,107413,24228],{},[35,107415,28394],{},[1274,107417,24228],{},[1274,107419,24228],{},[131,107421,107422,107424],{},[35,107423,455],{},[39,107425,1162],{},[45,107427,107428],{"encoding":47},"R(\\delta_n)\\ \\xrightarrow{\\ P\\ }\\ R^*",[15,107430,107432,107572],{"className":107431,"ariaHidden":53},[52],[15,107433,107435,107439,107442,107445,107485,107488,107491,107494,107566,107569],{"className":107434},[57],[15,107436],{"className":107437,"style":107438},[61],"height:1.3503em;vertical-align:-0.25em;",[15,107440,455],{"className":107441,"style":62433},[66,67],[15,107443,120],{"className":107444},[162],[15,107446,107448,107451],{"className":107447},[66],[15,107449,3883],{"className":107450,"style":3944},[66,67],[15,107452,107454],{"className":107453},[198],[15,107455,107457,107477],{"className":107456},[202,1090],[15,107458,107460,107474],{"className":107459},[206],[15,107461,107463],{"className":107462,"style":2198},[210],[15,107464,107465,107468],{"style":3959},[15,107466],{"className":107467,"style":218},[217],[15,107469,107471],{"className":107470},[222,223,224,225],[15,107472,37],{"className":107473},[66,67,225],[15,107475,1114],{"className":107476},[1113],[15,107478,107480],{"className":107479},[206],[15,107481,107483],{"className":107482,"style":1121},[210],[15,107484],{},[15,107486,126],{"className":107487},[169],[15,107489,24228],{"className":107490},[71],[15,107492],{"className":107493,"style":173},[71],[15,107495,107498],{"className":107496},[177,107497],"x-arrow",[15,107499,107501,107557],{"className":107500},[202,1090],[15,107502,107504,107554],{"className":107503},[206],[15,107505,107508,107536],{"className":107506,"style":107507},[210],"height:1.1003em;",[15,107509,107511,107514],{"style":107510},"top:-3.322em;",[15,107512],{"className":107513,"style":218},[217],[15,107515,107518],{"className":107516},[222,223,224,225,107517],"x-arrow-pad",[15,107519,107521,107527,107530],{"className":107520},[66,225],[15,107522,107524],{"className":107523},[71,225],[15,107525,24228],{"className":107526},[225],[15,107528,28394],{"className":107529,"style":62509},[66,67,225],[15,107531,107533],{"className":107532},[71,225],[15,107534,24228],{"className":107535},[225],[15,107537,107540,107543],{"className":107538,"style":107539},[48560],"top:-2.689em;",[15,107541],{"className":107542,"style":218},[217],[15,107544,107547],{"className":107545,"style":107546},[48586],"height:0.522em;min-width:1.469em;",[34823,107548,107551],{"xmlns":34825,"width":48590,"height":107549,"viewBox":107550,"preserveAspectRatio":70070},"0.522em","0 0 400000 522",[34829,107552],{"d":107553},"M0 241v40h399891c-47.3 35.3-84 78-110 128\n-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20\n 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7\n 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85\n-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5\n-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67\n 151.7 139 205zm0 0v40h399900v-40z",[15,107555,1114],{"className":107556},[1113],[15,107558,107560],{"className":107559},[206],[15,107561,107564],{"className":107562,"style":107563},[210],"height:0.011em;",[15,107565],{},[15,107567,24228],{"className":107568},[71],[15,107570],{"className":107571,"style":173},[71],[15,107573,107575,107578],{"className":107574},[57],[15,107576],{"className":107577,"style":1308},[61],[15,107579,107581,107584],{"className":107580},[66],[15,107582,455],{"className":107583,"style":62433},[66,67],[15,107585,107587],{"className":107586},[198],[15,107588,107590],{"className":107589},[202],[15,107591,107593],{"className":107592},[206],[15,107594,107596],{"className":107595,"style":1308},[210],[15,107597,107598,107601],{"style":397},[15,107599],{"className":107600,"style":218},[217],[15,107602,107604],{"className":107603},[222,223,224,225],[15,107605,1162],{"className":107606},[76,225],[11,107608,107609,107610,107612],{},"依概率成立。注意这里收敛的是",[2040,107611,62346],{},"，不是后验估计。一个模型可以是一致的，但给出的概率完全不校准；反过来，一个校准很好的模型也可能因为模型类太小而无法逼近贝叶斯风险。",[11,107614,107615,107643,107644,107695,107696,13127,107757,107785,107786,107814,107815,107879],{},[15,107616,107618,107631],{"className":107617,"translate":19},[18],[15,107619,107621],{"className":107620},[23],[25,107622,107623],{"xmlns":27},[29,107624,107625,107629],{},[32,107626,107627],{},[35,107628,621],{},[45,107630,621],{"encoding":47},[15,107632,107634],{"className":107633,"ariaHidden":53},[52],[15,107635,107637,107640],{"className":107636},[57],[15,107638],{"className":107639,"style":840},[61],[15,107641,621],{"className":107642,"style":684},[66,67]," 近邻是最经典的例子。Stone 定理说：若 ",[15,107645,107647,107665],{"className":107646,"translate":19},[18],[15,107648,107650],{"className":107649},[23],[25,107651,107652],{"xmlns":27},[29,107653,107654,107662],{},[32,107655,107656,107658,107660],{},[35,107657,621],{},[39,107659,449],{},[35,107661,7615],{"mathvariant":138},[45,107663,107664],{"encoding":47},"k\\to\\infty",[15,107666,107668,107686],{"className":107667,"ariaHidden":53},[52],[15,107669,107671,107674,107677,107680,107683],{"className":107670},[57],[15,107672],{"className":107673,"style":840},[61],[15,107675,621],{"className":107676,"style":684},[66,67],[15,107678],{"className":107679,"style":173},[71],[15,107681,449],{"className":107682},[177],[15,107684],{"className":107685,"style":173},[71],[15,107687,107689,107692],{"className":107688},[57],[15,107690],{"className":107691,"style":86},[61],[15,107693,7615],{"className":107694},[66]," 且 ",[15,107697,107699,107721],{"className":107698,"translate":19},[18],[15,107700,107702],{"className":107701},[23],[25,107703,107704],{"xmlns":27},[29,107705,107706,107718],{},[32,107707,107708,107710,107712,107714,107716],{},[35,107709,621],{},[35,107711,6623],{"mathvariant":138},[35,107713,37],{},[39,107715,449],{},[1209,107717,1804],{},[45,107719,107720],{"encoding":47},"k\u002Fn\\to0",[15,107722,107724,107748],{"className":107723,"ariaHidden":53},[52],[15,107725,107727,107730,107733,107736,107739,107742,107745],{"className":107726},[57],[15,107728],{"className":107729,"style":154},[61],[15,107731,621],{"className":107732,"style":684},[66,67],[15,107734,6623],{"className":107735},[66],[15,107737,37],{"className":107738},[66,67],[15,107740],{"className":107741,"style":173},[71],[15,107743,449],{"className":107744},[177],[15,107746],{"className":107747,"style":173},[71],[15,107749,107751,107754],{"className":107750},[57],[15,107752],{"className":107753,"style":1836},[61],[15,107755,1804],{"className":107756},[66],[15,107758,107760,107773],{"className":107759,"translate":19},[18],[15,107761,107763],{"className":107762},[23],[25,107764,107765],{"xmlns":27},[29,107766,107767,107771],{},[32,107768,107769],{},[35,107770,621],{},[45,107772,621],{"encoding":47},[15,107774,107776],{"className":107775,"ariaHidden":53},[52],[15,107777,107779,107782],{"className":107778},[57],[15,107780],{"className":107781,"style":840},[61],[15,107783,621],{"className":107784,"style":684},[66,67],"-NN 是普遍一致的。",[15,107787,107789,107802],{"className":107788,"translate":19},[18],[15,107790,107792],{"className":107791},[23],[25,107793,107794],{"xmlns":27},[29,107795,107796,107800],{},[32,107797,107798],{},[1209,107799,1211],{},[45,107801,1211],{"encoding":47},[15,107803,107805],{"className":107804,"ariaHidden":53},[52],[15,107806,107808,107811],{"className":107807},[57],[15,107809],{"className":107810,"style":1836},[61],[15,107812,1211],{"className":107813},[66],"-NN 更极端：它的渐近错误率不超过 ",[15,107816,107818,107838],{"className":107817,"translate":19},[18],[15,107819,107821],{"className":107820},[23],[25,107822,107823],{"xmlns":27},[29,107824,107825,107835],{},[32,107826,107827,107829],{},[1209,107828,1263],{},[131,107830,107831,107833],{},[35,107832,455],{},[39,107834,1162],{},[45,107836,107837],{"encoding":47},"2R^*",[15,107839,107841],{"className":107840,"ariaHidden":53},[52],[15,107842,107844,107847,107850],{"className":107843},[57],[15,107845],{"className":107846,"style":16244},[61],[15,107848,1263],{"className":107849},[66],[15,107851,107853,107856],{"className":107852},[66],[15,107854,455],{"className":107855,"style":62433},[66,67],[15,107857,107859],{"className":107858},[198],[15,107860,107862],{"className":107861},[202],[15,107863,107865],{"className":107864},[206],[15,107866,107868],{"className":107867,"style":16244},[210],[15,107869,107870,107873],{"style":213},[15,107871],{"className":107872,"style":218},[217],[15,107874,107876],{"className":107875},[222,223,224,225],[15,107877,1162],{"className":107878},[76,225],"。这个「二倍」的常数很说明问题——即使是最简单的局部方法，也能在渐近意义下接近贝叶斯风险，代价是收敛速度很慢，而且在高维里被维数灾难拖垮。",[11,107881,107882,107883,107968],{},"一致性是一个渐近性质，它不告诉你有限样本下会发生什么。一个普遍一致的估计器可能在 ",[15,107884,107886,107909],{"className":107885,"translate":19},[18],[15,107887,107889],{"className":107888},[23],[25,107890,107891],{"xmlns":27},[29,107892,107893,107906],{},[32,107894,107895,107897,107899],{},[35,107896,37],{},[39,107898,129],{},[131,107900,107901,107904],{},[1209,107902,107903],{},"10",[1209,107905,59939],{},[45,107907,107908],{"encoding":47},"n=10^6",[15,107910,107912,107930],{"className":107911,"ariaHidden":53},[52],[15,107913,107915,107918,107921,107924,107927],{"className":107914},[57],[15,107916],{"className":107917,"style":86},[61],[15,107919,37],{"className":107920},[66,67],[15,107922],{"className":107923,"style":173},[71],[15,107925,129],{"className":107926},[177],[15,107928],{"className":107929,"style":173},[71],[15,107931,107933,107936,107939],{"className":107932},[57],[15,107934],{"className":107935,"style":6738},[61],[15,107937,1211],{"className":107938},[66],[15,107940,107942,107945],{"className":107941},[66],[15,107943,1804],{"className":107944},[66],[15,107946,107948],{"className":107947},[198],[15,107949,107951],{"className":107950},[202],[15,107952,107954],{"className":107953},[206],[15,107955,107957],{"className":107956,"style":6738},[210],[15,107958,107959,107962],{"style":213},[15,107960],{"className":107961,"style":218},[217],[15,107963,107965],{"className":107964},[222,223,224,225],[15,107966,59939],{"className":107967},[66,225]," 时仍然表现很差；一个不一致的估计器可能在某个特定问题上表现得很好。贝叶斯风险给出的是终点，不是路线。",[1904,107970,107972],{"id":107971},"_73-pac-bayes把信念放进泛化界","7.3 PAC-Bayes：把「信念」放进泛化界",[11,107974,107975,107976,108018],{},"PAC-Bayes 把贝叶斯式的分布对象与频率式的泛化保证放在同一个不等式里。这里的 ",[15,107977,107979,107997],{"className":107978,"translate":19},[18],[15,107980,107982],{"className":107981},[23],[25,107983,107984],{"xmlns":27},[29,107985,107986,107994],{},[32,107987,107988,107990,107992],{},[35,107989,28394],{},[39,107991,335],{"separator":53},[35,107993,83340],{},[45,107995,107996],{"encoding":47},"P,Q",[15,107998,108000],{"className":107999,"ariaHidden":53},[52],[15,108001,108003,108006,108009,108012,108015],{"className":108002},[57],[15,108004],{"className":108005,"style":17374},[61],[15,108007,28394],{"className":108008,"style":62509},[66,67],[15,108010,335],{"className":108011},[422],[15,108013],{"className":108014,"style":697},[71],[15,108016,83340],{"className":108017},[66,67]," 是假设空间上的分布，不要与数据分布混淆。",[3054,108020,108022,108262,108851],{"title":108021},"定理 7.1（McAllester 界）",[11,108023,108024,108025,108053,108054,108105,108106,108180,108181,108232,108233,108261],{},"设先验 ",[15,108026,108028,108041],{"className":108027,"translate":19},[18],[15,108029,108031],{"className":108030},[23],[25,108032,108033],{"xmlns":27},[29,108034,108035,108039],{},[32,108036,108037],{},[35,108038,28394],{},[45,108040,28394],{"encoding":47},[15,108042,108044],{"className":108043,"ariaHidden":53},[52],[15,108045,108047,108050],{"className":108046},[57],[15,108048],{"className":108049,"style":490},[61],[15,108051,28394],{"className":108052,"style":62509},[66,67]," 在看到数据前固定，损失取值于 ",[15,108055,108057,108078],{"className":108056,"translate":19},[18],[15,108058,108060],{"className":108059},[23],[25,108061,108062],{"xmlns":27},[29,108063,108064,108076],{},[32,108065,108066,108068,108070,108072,108074],{},[39,108067,1194],{"stretchy":119},[1209,108069,1804],{},[39,108071,335],{"separator":53},[1209,108073,1211],{},[39,108075,1266],{"stretchy":119},[45,108077,19036],{"encoding":47},[15,108079,108081],{"className":108080,"ariaHidden":53},[52],[15,108082,108084,108087,108090,108093,108096,108099,108102],{"className":108083},[57],[15,108085],{"className":108086,"style":154},[61],[15,108088,1194],{"className":108089},[162],[15,108091,1804],{"className":108092},[66],[15,108094,335],{"className":108095},[422],[15,108097],{"className":108098,"style":697},[71],[15,108100,1211],{"className":108101},[66],[15,108103,1266],{"className":108104},[169],"。对任意 ",[15,108107,108109,108135],{"className":108108,"translate":19},[18],[15,108110,108112],{"className":108111},[23],[25,108113,108114],{"xmlns":27},[29,108115,108116,108132],{},[32,108117,108118,108120,108122,108124,108126,108128,108130],{},[35,108119,3883],{},[39,108121,1187],{},[39,108123,120],{"stretchy":119},[1209,108125,1804],{},[39,108127,335],{"separator":53},[1209,108129,1211],{},[39,108131,126],{"stretchy":119},[45,108133,108134],{"encoding":47},"\\delta\\in(0,1)",[15,108136,108138,108156],{"className":108137,"ariaHidden":53},[52],[15,108139,108141,108144,108147,108150,108153],{"className":108140},[57],[15,108142],{"className":108143,"style":9293},[61],[15,108145,3883],{"className":108146,"style":3944},[66,67],[15,108148],{"className":108149,"style":173},[71],[15,108151,1187],{"className":108152},[177],[15,108154],{"className":108155,"style":173},[71],[15,108157,108159,108162,108165,108168,108171,108174,108177],{"className":108158},[57],[15,108160],{"className":108161,"style":154},[61],[15,108163,120],{"className":108164},[162],[15,108166,1804],{"className":108167},[66],[15,108169,335],{"className":108170},[422],[15,108172],{"className":108173,"style":697},[71],[15,108175,1211],{"className":108176},[66],[15,108178,126],{"className":108179},[169],"，以至少 ",[15,108182,108184,108202],{"className":108183,"translate":19},[18],[15,108185,108187],{"className":108186},[23],[25,108188,108189],{"xmlns":27},[29,108190,108191,108199],{},[32,108192,108193,108195,108197],{},[1209,108194,1211],{},[39,108196,6600],{},[35,108198,3883],{},[45,108200,108201],{"encoding":47},"1-\\delta",[15,108203,108205,108223],{"className":108204,"ariaHidden":53},[52],[15,108206,108208,108211,108214,108217,108220],{"className":108207},[57],[15,108209],{"className":108210,"style":30463},[61],[15,108212,1211],{"className":108213},[66],[15,108215],{"className":108216,"style":72},[71],[15,108218,6600],{"className":108219},[76],[15,108221],{"className":108222,"style":72},[71],[15,108224,108226,108229],{"className":108225},[57],[15,108227],{"className":108228,"style":840},[61],[15,108230,3883],{"className":108231,"style":3944},[66,67]," 的概率，对所有后验分布 ",[15,108234,108236,108249],{"className":108235,"translate":19},[18],[15,108237,108239],{"className":108238},[23],[25,108240,108241],{"xmlns":27},[29,108242,108243,108247],{},[32,108244,108245],{},[35,108246,83340],{},[45,108248,83340],{"encoding":47},[15,108250,108252],{"className":108251,"ariaHidden":53},[52],[15,108253,108255,108258],{"className":108254},[57],[15,108256],{"className":108257,"style":17374},[61],[15,108259,83340],{"className":108260},[66,67]," 同时成立：",[15,108263,108265],{"className":108264,"translate":19},[294],[15,108266,108268,108390],{"className":108267,"translate":19},[18],[15,108269,108271],{"className":108270},[23],[25,108272,108273],{"xmlns":27,"display":303},[29,108274,108275,108387],{},[32,108276,108277,108289,108291,108293,108295,108297,108299,108301,108303,108315,108317,108327,108329,108331,108333,108335,108337,108385],{},[1057,108278,108279,108281],{},[35,108280,62377],{"mathvariant":454},[32,108282,108283,108285,108287],{},[35,108284,20177],{},[39,108286,28386],{},[35,108288,83340],{},[39,108290,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,108292,455],{},[39,108294,120],{"stretchy":119},[35,108296,20177],{},[39,108298,126],{"stretchy":119},[39,108300,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,108302,24215],{},[1057,108304,108305,108307],{},[35,108306,62377],{"mathvariant":454},[32,108308,108309,108311,108313],{},[35,108310,20177],{},[39,108312,28386],{},[35,108314,83340],{},[39,108316,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,108318,108319,108325],{},[4294,108320,108321,108323],{"accent":53},[35,108322,455],{},[39,108324,78122],{},[35,108326,37],{},[39,108328,120],{"stretchy":119},[35,108330,20177],{},[39,108332,126],{"stretchy":119},[39,108334,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,108336,1245],{},[48504,108338,108339],{},[6465,108340,108341,108379],{},[32,108342,108343,108349,108351,108353,108355,108357,108359,108361,108363,108365,108367,108369,108373,108375,108377],{},[32,108344,108345,108347],{},[35,108346,2853],{"mathvariant":138},[35,108348,1216],{"mathvariant":138},[39,108350,120],{"stretchy":119},[35,108352,83340],{},[35,108354,1251],{"mathvariant":138},[35,108356,28394],{},[39,108358,126],{"stretchy":119},[39,108360,1245],{},[35,108362,27068],{},[39,108364,1170],{},[39,108366,120],{"stretchy":119},[1209,108368,1263],{},[48504,108370,108371],{},[35,108372,37],{},[35,108374,6623],{"mathvariant":138},[35,108376,3883],{},[39,108378,126],{"stretchy":119},[32,108380,108381,108383],{},[1209,108382,1263],{},[35,108384,37],{},[39,108386,335],{"separator":53},[45,108388,108389],{"encoding":47},"\\mathbb E_{h\\sim Q}\\big[R(h)\\big]\n\\le\n\\mathbb E_{h\\sim Q}\\big[\\hat R_n(h)\\big]\n+\n\\sqrt{\\frac{\\mathrm{KL}(Q\\|P)+\\log(2\\sqrt n\u002F\\delta)}{2n}},",[15,108391,108393,108481,108637],{"className":108392,"ariaHidden":53},[52],[15,108394,108396,108399,108448,108454,108457,108460,108463,108466,108472,108475,108478],{"className":108395},[57],[15,108397],{"className":108398,"style":4398},[61],[15,108400,108402,108405],{"className":108401},[66],[15,108403,62377],{"className":108404},[66,517],[15,108406,108408],{"className":108407},[198],[15,108409,108411,108440],{"className":108410},[202,1090],[15,108412,108414,108437],{"className":108413},[206],[15,108415,108417],{"className":108416,"style":1097},[210],[15,108418,108419,108422],{"style":1584},[15,108420],{"className":108421,"style":218},[217],[15,108423,108425],{"className":108424},[222,223,224,225],[15,108426,108428,108431,108434],{"className":108427},[66,225],[15,108429,20177],{"className":108430},[66,67,225],[15,108432,28386],{"className":108433},[177,225],[15,108435,83340],{"className":108436},[66,67,225],[15,108438,1114],{"className":108439},[1113],[15,108441,108443],{"className":108442},[206],[15,108444,108446],{"className":108445,"style":2715},[210],[15,108447],{},[15,108449,108451],{"className":108450},[66],[15,108452,1194],{"className":108453},[1422,1508],[15,108455,455],{"className":108456,"style":62433},[66,67],[15,108458,120],{"className":108459},[162],[15,108461,20177],{"className":108462},[66,67],[15,108464,126],{"className":108465},[169],[15,108467,108469],{"className":108468},[66],[15,108470,1266],{"className":108471},[1422,1508],[15,108473],{"className":108474,"style":173},[71],[15,108476,24215],{"className":108477},[177],[15,108479],{"className":108480,"style":173},[71],[15,108482,108484,108488,108537,108543,108613,108616,108619,108622,108628,108631,108634],{"className":108483},[57],[15,108485],{"className":108486,"style":108487},[61],"height:1.2968em;vertical-align:-0.35em;",[15,108489,108491,108494],{"className":108490},[66],[15,108492,62377],{"className":108493},[66,517],[15,108495,108497],{"className":108496},[198],[15,108498,108500,108529],{"className":108499},[202,1090],[15,108501,108503,108526],{"className":108502},[206],[15,108504,108506],{"className":108505,"style":1097},[210],[15,108507,108508,108511],{"style":1584},[15,108509],{"className":108510,"style":218},[217],[15,108512,108514],{"className":108513},[222,223,224,225],[15,108515,108517,108520,108523],{"className":108516},[66,225],[15,108518,20177],{"className":108519},[66,67,225],[15,108521,28386],{"className":108522},[177,225],[15,108524,83340],{"className":108525},[66,67,225],[15,108527,1114],{"className":108528},[1113],[15,108530,108532],{"className":108531},[206],[15,108533,108535],{"className":108534,"style":2715},[210],[15,108536],{},[15,108538,108540],{"className":108539},[66],[15,108541,1194],{"className":108542},[1422,1508],[15,108544,108546,108579],{"className":108545},[66],[15,108547,108549],{"className":108548},[66,78139],[15,108550,108552],{"className":108551},[202],[15,108553,108555],{"className":108554},[206],[15,108556,108559,108567],{"className":108557,"style":108558},[210],"height:0.9468em;",[15,108560,108561,108564],{"style":1386},[15,108562],{"className":108563,"style":1368},[217],[15,108565,455],{"className":108566,"style":62433},[66,67],[15,108568,108570,108573],{"style":108569},"top:-3.2523em;",[15,108571],{"className":108572,"style":1368},[217],[15,108574,108576],{"className":108575,"style":96102},[78167],[15,108577,78122],{"className":108578},[66],[15,108580,108582],{"className":108581},[198],[15,108583,108585,108605],{"className":108584},[202,1090],[15,108586,108588,108602],{"className":108587},[206],[15,108589,108591],{"className":108590,"style":2198},[210],[15,108592,108593,108596],{"style":106319},[15,108594],{"className":108595,"style":218},[217],[15,108597,108599],{"className":108598},[222,223,224,225],[15,108600,37],{"className":108601},[66,67,225],[15,108603,1114],{"className":108604},[1113],[15,108606,108608],{"className":108607},[206],[15,108609,108611],{"className":108610,"style":1121},[210],[15,108612],{},[15,108614,120],{"className":108615},[162],[15,108617,20177],{"className":108618},[66,67],[15,108620,126],{"className":108621},[169],[15,108623,108625],{"className":108624},[66],[15,108626,1266],{"className":108627},[1422,1508],[15,108629],{"className":108630,"style":72},[71],[15,108632,1245],{"className":108633},[76],[15,108635],{"className":108636,"style":72},[71],[15,108638,108640,108644,108848],{"className":108639},[57],[15,108641],{"className":108642,"style":108643},[61],"height:2.44em;vertical-align:-0.7105em;",[15,108645,108647],{"className":108646},[66,48546],[15,108648,108650,108839],{"className":108649},[202,1090],[15,108651,108653,108836],{"className":108652},[206],[15,108654,108657,108823],{"className":108655,"style":108656},[210],"height:1.7295em;",[15,108658,108660,108663],{"className":108659,"style":89047},[48560],[15,108661],{"className":108662,"style":89051},[217],[15,108664,108666],{"className":108665,"style":89055},[66],[15,108667,108669,108672,10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R_n",[15,108879,108881],{"className":108880,"ariaHidden":53},[52],[15,108882,108884,108888],{"className":108883},[57],[15,108885],{"className":108886,"style":108887},[61],"height:1.0968em;vertical-align:-0.15em;",[15,108889,108891,108922],{"className":108890},[66],[15,108892,108894],{"className":108893},[66,78139],[15,108895,108897],{"className":108896},[202],[15,108898,108900],{"className":108899},[206],[15,108901,108903,108911],{"className":108902,"style":108558},[210],[15,108904,108905,108908],{"style":1386},[15,108906],{"className":108907,"style":1368},[217],[15,108909,455],{"className":108910,"style":62433},[66,67],[15,108912,108913,108916],{"style":108569},[15,108914],{"className":108915,"style":1368},[217],[15,108917,108919],{"className":108918,"style":96102},[78167],[15,108920,78122],{"className":108921},[66],[15,108923,108925],{"className":108924},[198],[15,108926,108928,108948],{"className":108927},[202,1090],[15,108929,108931,108945],{"className":108930},[206],[15,108932,108934],{"className":108933,"style":2198},[210],[15,108935,108936,108939],{"style":106319},[15,108937],{"className":108938,"style":218},[217],[15,108940,108942],{"className":108941},[222,223,224,225],[15,108943,37],{"className":108944},[66,67,225],[15,108946,1114],{"className":108947},[1113],[15,108949,108951],{"className":108950},[206],[15,108952,108954],{"className":108953,"style":1121},[210],[15,108955],{}," 是经验风险。",[11,108958,108959,108960,108988,108989,109017],{},"这个界的美妙之处在于：",[15,108961,108963,108976],{"className":108962,"translate":19},[18],[15,108964,108966],{"className":108965},[23],[25,108967,108968],{"xmlns":27},[29,108969,108970,108974],{},[32,108971,108972],{},[35,108973,28394],{},[45,108975,28394],{"encoding":47},[15,108977,108979],{"className":108978,"ariaHidden":53},[52],[15,108980,108982,108985],{"className":108981},[57],[15,108983],{"className":108984,"style":490},[61],[15,108986,28394],{"className":108987,"style":62509},[66,67]," 必须在看到数据前固定，",[15,108990,108992,109005],{"className":108991,"translate":19},[18],[15,108993,108995],{"className":108994},[23],[25,108996,108997],{"xmlns":27},[29,108998,108999,109003],{},[32,109000,109001],{},[35,109002,83340],{},[45,109004,83340],{"encoding":47},[15,109006,109008],{"className":109007,"ariaHidden":53},[52],[15,109009,109011,109014],{"className":109010},[57],[15,109012],{"className":109013,"style":17374},[61],[15,109015,83340],{"className":109016},[66,67]," 可以是任何依赖于数据的分布。KL 项衡量「为了解释数据，你把信念从先验挪了多远」。先验越集中、后验越贴近先验，界就越紧；后验越远离先验，界就越松。",[11,109019,109020,109021,109024,109025,109053,109054,109082],{},"这正是奥卡姆剃刀的概率版本，只不过它作用在",[2040,109022,109023],{},"分布","而不是单个假设上。一个在大量假设上分散的 ",[15,109026,109028,109041],{"className":109027,"translate":19},[18],[15,109029,109031],{"className":109030},[23],[25,109032,109033],{"xmlns":27},[29,109034,109035,109039],{},[32,109036,109037],{},[35,109038,83340],{},[45,109040,83340],{"encoding":47},[15,109042,109044],{"className":109043,"ariaHidden":53},[52],[15,109045,109047,109050],{"className":109046},[57],[15,109048],{"className":109049,"style":17374},[61],[15,109051,83340],{"className":109052},[66,67]," 会付出很大的 KL，即使它的经验风险很低；一个集中在少数假设上的 ",[15,109055,109057,109070],{"className":109056,"translate":19},[18],[15,109058,109060],{"className":109059},[23],[25,109061,109062],{"xmlns":27},[29,109063,109064,109068],{},[32,109065,109066],{},[35,109067,83340],{},[45,109069,83340],{"encoding":47},[15,109071,109073],{"className":109072,"ariaHidden":53},[52],[15,109074,109076,109079],{"className":109075},[57],[15,109077],{"className":109078,"style":17374},[61],[15,109080,83340],{"className":109081},[66,67]," 则可以用较小的 KL 换取较好的界。",[11,109084,109085],{},"PAC-Bayes 也有它的局限：界在深度网络里常常很松，常数项可能大到没有实际意义。但它提供了一个正确的概念框架：泛化不是「参数少」的同义词，而是「后验没有比先验复杂太多」。后来的 flatness、sharpness、信息论泛化界，都可以在这个框架里找到位置。",[1904,109087,109089],{"id":109088},"_74-变分推断当后验收敛不了","7.4 变分推断：当后验收敛不了",[11,109091,109092,109093,109136,109137,109166],{},"精确后验通常不可行，因为归一化常数 ",[15,109094,109096,109115],{"className":109095,"translate":19},[18],[15,109097,109099],{"className":109098},[23],[25,109100,109101],{"xmlns":27},[29,109102,109103,109113],{},[32,109104,109105,109107,109109,109111],{},[35,109106,11],{},[39,109108,120],{"stretchy":119},[35,109110,32453],{},[39,109112,126],{"stretchy":119},[45,109114,65369],{"encoding":47},[15,109116,109118],{"className":109117,"ariaHidden":53},[52],[15,109119,109121,109124,109127,109130,109133],{"className":109120},[57],[15,109122],{"className":109123,"style":154},[61],[15,109125,11],{"className":109126},[66,67],[15,109128,120],{"className":109129},[162],[15,109131,32453],{"className":109132,"style":23020},[66,67],[15,109134,126],{"className":109135},[169]," 是一个高维积分。变分推断用一个受限制的分布族 ",[15,109138,109140,109154],{"className":109139,"translate":19},[18],[15,109141,109143],{"className":109142},[23],[25,109144,109145],{"xmlns":27},[29,109146,109147,109151],{},[32,109148,109149],{},[35,109150,83340],{"mathvariant":445},[45,109152,109153],{"encoding":47},"\\mathcal Q",[15,109155,109157],{"className":109156,"ariaHidden":53},[52],[15,109158,109160,109163],{"className":109159},[57],[15,109161],{"className":109162,"style":61916},[61],[15,109164,83340],{"className":109165},[66,494]," 去逼近后验，目标是",[15,109168,109170],{"className":109169,"translate":19},[294],[15,109171,109173,109253],{"className":109172,"translate":19},[18],[15,109174,109176],{"className":109175},[23],[25,109177,109178],{"xmlns":27,"display":303},[29,109179,109180,109250],{},[32,109181,109182,109188,109190,109192,109194,109210,109212,109218,109220,109222,109224,109226,109228,109230,109232,109234,109236,109238,109240,109242,109244,109246,109248],{},[131,109183,109184,109186],{},[35,109185,94268],{},[39,109187,1162],{},[39,109189,129],{},[35,109191,1167],{},[39,109193,1170],{},[1172,109195,109196,109202],{},[32,109197,109198,109200],{},[35,109199,1178],{},[39,109201,1170],{},[32,109203,109204,109206,109208],{},[35,109205,94268],{},[39,109207,1187],{},[35,109209,83340],{"mathvariant":445},[1274,109211,24228],{},[32,109213,109214,109216],{},[35,109215,2853],{"mathvariant":138},[35,109217,1216],{"mathvariant":138},[39,109219,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,109221,94268],{},[39,109223,120],{"stretchy":119},[35,109225,61614],{},[39,109227,126],{"stretchy":119},[1274,109229,24228],{},[35,109231,1251],{"mathvariant":138},[1274,109233,24228],{},[35,109235,11],{},[39,109237,120],{"stretchy":119},[35,109239,61614],{},[39,109241,34425],{},[35,109243,32453],{},[39,109245,126],{"stretchy":119},[39,109247,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,109249,3571],{"mathvariant":138},[45,109251,109252],{"encoding":47},"q^*=\\arg\\min_{q\\in\\mathcal Q}\\ \\mathrm{KL}\\big(q(\\theta)\\ \\|\\ p(\\theta\\mid D)\\big).",[15,109254,109256,109300,109425],{"className":109255,"ariaHidden":53},[52],[15,109257,109259,109262,109291,109294,109297],{"className":109258},[57],[15,109260],{"className":109261,"style":1289},[61],[15,109263,109265,109268],{"className":109264},[66],[15,109266,94268],{"className":109267,"style":1515},[66,67],[15,109269,109271],{"className":109270},[198],[15,109272,109274],{"className":109273},[202],[15,109275,109277],{"className":109276},[206],[15,109278,109280],{"className":109279,"style":1308},[210],[15,109281,109282,109285],{"style":397},[15,109283],{"className":109284,"style":218},[217],[15,109286,109288],{"className":109287},[222,223,224,225],[15,109289,1162],{"className":109290},[76,225],[15,109292],{"className":109293,"style":173},[71],[15,109295,129],{"className":109296},[177],[15,109298],{"className":109299,"style":173},[71],[15,109301,109303,109307,109312,109315,109368,109371,109374,109380,109386,109389,109392,109395,109398,109401,109404,109407,109410,109413,109416,109419,109422],{"className":109302},[57],[15,109304],{"className":109305,"style":109306},[61],"height:1.7304em;vertical-align:-0.8804em;",[15,109308,1340,109310],{"className":109309},[1339],[15,109311,1344],{"style":1343},[15,109313],{"className":109314,"style":697},[71],[15,109316,109318],{"className":109317},[1339,1351],[15,109319,109321,109360],{"className":109320},[202,1090],[15,109322,109324,109357],{"className":109323},[206],[15,109325,109327,109347],{"className":109326,"style":1361},[210],[15,109328,109329,109332],{"style":24349},[15,109330],{"className":109331,"style":1368},[217],[15,109333,109335],{"className":109334},[222,223,224,225],[15,109336,109338,109341,109344],{"className":109337},[66,225],[15,109339,94268],{"className":109340,"style":1515},[66,67,225],[15,109342,1187],{"className":109343},[177,225],[15,109345,83340],{"className":109346},[66,494,225],[15,109348,109349,109352],{"style":1386},[15,109350],{"className":109351,"style":1368},[217],[15,109353,109354],{},[15,109355,1178],{"className":109356},[1339],[15,109358,1114],{"className":109359},[1113],[15,109361,109363],{"className":109362},[206],[15,109364,109366],{"className":109365,"style":67076},[210],[15,109367],{},[15,109369,24228],{"className":109370},[71],[15,109372],{"className":109373,"style":697},[71],[15,109375,109377],{"className":109376},[66],[15,109378,88449],{"className":109379},[66,4427],[15,109381,109383],{"className":109382},[66],[15,109384,120],{"className":109385},[1422,1508],[15,109387,94268],{"className":109388,"style":1515},[66,67],[15,109390,120],{"className":109391},[162],[15,109393,61614],{"className":109394,"style":23020},[66,67],[15,109396,126],{"className":109397},[169],[15,109399,24228],{"className":109400},[71],[15,109402,1251],{"className":109403},[66],[15,109405,24228],{"className":109406},[71],[15,109408,11],{"className":109409},[66,67],[15,109411,120],{"className":109412},[162],[15,109414,61614],{"className":109415,"style":23020},[66,67],[15,109417],{"className":109418,"style":173},[71],[15,109420,34425],{"className":109421},[177],[15,109423],{"className":109424,"style":173},[71],[15,109426,109428,109431,109434,109437,109443],{"className":109427},[57],[15,109429],{"className":109430,"style":4398},[61],[15,109432,32453],{"className":109433,"style":23020},[66,67],[15,109435,126],{"className":109436},[169],[15,109438,109440],{"className":109439},[66],[15,109441,126],{"className":109442},[1422,1508],[15,109444,3571],{"className":109445},[66],[11,109447,109448,109449,109492],{},"直接最小化这个 KL 仍然需要 ",[15,109450,109452,109471],{"className":109451,"translate":19},[18],[15,109453,109455],{"className":109454},[23],[25,109456,109457],{"xmlns":27},[29,109458,109459,109469],{},[32,109460,109461,109463,109465,109467],{},[35,109462,11],{},[39,109464,120],{"stretchy":119},[35,109466,32453],{},[39,109468,126],{"stretchy":119},[45,109470,65369],{"encoding":47},[15,109472,109474],{"className":109473,"ariaHidden":53},[52],[15,109475,109477,109480,109483,109486,109489],{"className":109476},[57],[15,109478],{"className":109479,"style":154},[61],[15,109481,11],{"className":109482},[66,67],[15,109484,120],{"className":109485},[162],[15,109487,32453],{"className":109488,"style":23020},[66,67],[15,109490,126],{"className":109491},[169],"，于是利用恒等式",[15,109494,109496],{"className":109495,"translate":19},[294],[15,109497,109499,109633],{"className":109498,"translate":19},[18],[15,109500,109502],{"className":109501},[23],[25,109503,109504],{"xmlns":27,"display":303},[29,109505,109506,109630],{},[32,109507,109508,109510,109512,109514,109516,109518,109520,109522,109590,109592,109598,109600,109602,109604,109606,109608,109610,109612,109614,109616,109618,109620,109622,109624,109626,109628],{},[35,109509,27068],{},[39,109511,1170],{},[35,109513,11],{},[39,109515,120],{"stretchy":119},[35,109517,32453],{},[39,109519,126],{"stretchy":119},[39,109521,129],{},[1172,109523,109524,109572],{},[1172,109525,109526,109570],{},[32,109527,109528,109534],{},[1057,109529,109530,109532],{},[35,109531,62377],{"mathvariant":454},[35,109533,94268],{},[32,109535,109536,109538,109540,109542,109568],{},[39,109537,1194],{"fence":53},[35,109539,27068],{},[39,109541,1170],{},[6465,109543,109544,109558],{},[32,109545,109546,109548,109550,109552,109554,109556],{},[35,109547,11],{},[39,109549,120],{"stretchy":119},[35,109551,32453],{},[39,109553,335],{"separator":53},[35,109555,61614],{},[39,109557,126],{"stretchy":119},[32,109559,109560,109562,109564,109566],{},[35,109561,94268],{},[39,109563,120],{"stretchy":119},[35,109565,61614],{},[39,109567,126],{"stretchy":119},[39,109569,1266],{"fence":53},[39,109571,69838],{"stretchy":53},[32,109573,109574,109584,109586,109588],{},[32,109575,109576,109578,109580,109582],{},[35,109577,62377],{"mathvariant":138},[35,109579,1216],{"mathvariant":138},[35,109581,63968],{"mathvariant":138},[35,109583,22994],{"mathvariant":138},[39,109585,120],{"stretchy":119},[35,109587,94268],{},[39,109589,126],{"stretchy":119},[39,109591,1245],{},[32,109593,109594,109596],{},[35,109595,2853],{"mathvariant":138},[35,109597,1216],{"mathvariant":138},[39,109599,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,109601,94268],{},[39,109603,120],{"stretchy":119},[35,109605,61614],{},[39,109607,126],{"stretchy":119},[1274,109609,24228],{},[35,109611,1251],{"mathvariant":138},[1274,109613,24228],{},[35,109615,11],{},[39,109617,120],{"stretchy":119},[35,109619,61614],{},[39,109621,34425],{},[35,109623,32453],{},[39,109625,126],{"stretchy":119},[39,109627,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,109629,335],{"separator":53},[45,109631,109632],{"encoding":47},"\\log p(D)\n=\n\\underbrace{\\mathbb E_{q}\\left[\\log\\frac{p(D,\\theta)}{q(\\theta)}\\right]}_{\\mathrm{ELBO}(q)}\n+\n\\mathrm{KL}\\big(q(\\theta)\\ \\|\\ p(\\theta\\mid D)\\big),",[15,109634,109636,109671,109967,110024],{"className":109635,"ariaHidden":53},[52],[15,109637,109639,109642,109647,109650,109653,109656,109659,109662,109665,109668],{"className":109638},[57],[15,109640],{"className":109641,"style":154},[61],[15,109643,27278,109645],{"className":109644},[1339],[15,109646,1344],{"style":1343},[15,109648],{"className":109649,"style":697},[71],[15,109651,11],{"className":109652},[66,67],[15,109654,120],{"className":109655},[162],[15,109657,32453],{"className":109658,"style":23020},[66,67],[15,109660,126],{"className":109661},[169],[15,109663],{"className":109664,"style":173},[71],[15,109666,129],{"className":109667},[177],[15,109669],{"className":109670,"style":173},[71],[15,109672,109674,109678,109958,109961,109964],{"className":109673},[57],[15,109675],{"className":109676,"style":109677},[61],"height:3.948em;vertical-align:-2.498em;",[15,109679,109681],{"className":109680},[1413,1172],[15,109682,109684,109949],{"className":109683},[202,1090],[15,109685,109687,109946],{"className":109686},[206],[15,109688,109691,109719],{"className":109689,"style":109690},[210],"height:1.45em;",[15,109692,109694,109697],{"style":109693},"top:-1.127em;",[15,109695],{"className":109696,"style":63163},[217],[15,109698,109700],{"className":109699},[222,223,224,225],[15,109701,109703,109710,109713,109716],{"className":109702},[66,225],[15,109704,109706],{"className":109705},[66,225],[15,109707,109709],{"className":109708},[66,4427,225],"ELBO",[15,109711,120],{"className":109712},[162,225],[15,109714,94268],{"className":109715,"style":1515},[66,67,225],[15,109717,126],{"className":109718},[169,225],[15,109720,109722,109725],{"style":109721},"top:-3.45em;",[15,109723],{"className":109724,"style":63163},[217],[15,109726,109728],{"className":109727},[1413,1172],[15,109729,109731,109937],{"className":109730},[202,1090],[15,109732,109734,109934],{"className":109733},[206],[15,109735,109737,109768],{"className":109736,"style":109690},[210],[15,109738,109741,109744],{"className":109739,"style":109740},[48560],"top:-1.852em;",[15,109742],{"className":109743,"style":63163},[217],[15,109745,109747,109754,109761],{"className":109746,"style":70041},[70040],[15,109748,109750],{"className":109749,"style":70046},[70045],[34823,109751,109752],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":48593},[34829,109753],{"d":70053},[15,109755,109757],{"className":109756,"style":70046},[70057],[34823,109758,109759],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":70060},[34829,109760],{"d":70063},[15,109762,109764],{"className":109763,"style":70046},[70067],[34823,109765,109766],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":70070},[34829,109767],{"d":70073},[15,109769,109770,109773],{"style":109721},[15,109771],{"className":109772,"style":63163},[217],[15,109774,109776,109819,109822],{"className":109775},[66],[15,109777,109779,109782],{"className":109778},[66],[15,109780,62377],{"className":109781},[66,517],[15,109783,109785],{"className":109784},[198],[15,109786,109788,109811],{"className":109787},[202,1090],[15,109789,109791,109808],{"className":109790},[206],[15,109792,109794],{"className":109793,"style":2198},[210],[15,109795,109796,109799],{"style":1584},[15,109797],{"className":109798,"style":218},[217],[15,109800,109802],{"className":109801},[222,223,224,225],[15,109803,109805],{"className":109804},[66,225],[15,109806,94268],{"className":109807,"style":1515},[66,67,225],[15,109809,1114],{"className":109810},[1113],[15,109812,109814],{"className":109813},[206],[15,109815,109817],{"className":109816,"style":2715},[210],[15,109818],{},[15,109820],{"className":109821,"style":697},[71],[15,109823,109825,109831,109836,109839,109928],{"className":109824},[1413],[15,109826,109828],{"className":109827,"style":1418},[162,1417],[15,109829,1194],{"className":109830},[1422,224],[15,109832,27278,109834],{"className":109833},[1339],[15,109835,1344],{"style":1343},[15,109837],{"className":109838,"style":697},[71],[15,109840,109842,109845,109925],{"className":109841},[66],[15,109843],{"className":109844},[162,23412],[15,109846,109848],{"className":109847},[6465],[15,109849,109851,109917],{"className":109850},[202,1090],[15,109852,109854,109914],{"className":109853},[206],[15,109855,109857,109877,109885],{"className":109856,"style":41206},[210],[15,109858,109859,109862],{"style":23428},[15,109860],{"className":109861,"style":1368},[217],[15,109863,109865,109868,109871,109874],{"className":109864},[66],[15,109866,94268],{"className":109867,"style":1515},[66,67],[15,109869,120],{"className":109870},[162],[15,109872,61614],{"className":109873,"style":23020},[66,67],[15,109875,126],{"className":109876},[169],[15,109878,109879,109882],{"style":23440},[15,109880],{"className":109881,"style":1368},[217],[15,109883],{"className":109884,"style":4438},[23447],[15,109886,109887,109890],{"style":23450},[15,109888],{"className":109889,"s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E_q\\big[\\log p(D\\mid\\theta)\\big]\n-\n\\mathrm{KL}\\big(q(\\theta)\\ \\|\\ p(\\theta)\\big).",[15,110151,110153,110183,110264,110291],{"className":110152,"ariaHidden":53},[52],[15,110154,110156,110159,110165,110168,110171,110174,110177,110180],{"className":110155},[57],[15,110157],{"className":110158,"style":154},[61],[15,110160,110162],{"className":110161},[66],[15,110163,109709],{"className":110164},[66,4427],[15,110166,120],{"className":110167},[162],[15,110169,94268],{"className":110170,"style":1515},[66,67],[15,110172,126],{"className":110173},[169],[15,110175],{"className":110176,"style":173},[71],[15,110178,129],{"className":110179},[177],[15,110181],{"className":110182,"style":173},[71],[15,110184,110186,110189,110229,110235,110238,110243,110246,110249,110252,110255,110258,110261],{"className":110185},[57],[15,110187],{"className":110188,"style":4398},[61],[15,110190,110192,110195],{"className":110191},[66],[15,110193,62377],{"className":110194},[66,517],[15,110196,110198],{"className":110197},[198],[15,110199,110201,110221],{"className":110200},[202,1090],[15,110202,110204,110218],{"className":110203},[206],[15,110205,110207],{"className":110206,"style":2198},[210],[15,110208,110209,110212],{"style":1584},[15,110210],{"className":110211,"style":218},[217],[15,110213,110215],{"className":110214},[222,223,224,225],[15,110216,94268],{"className":110217,"style":1515},[66,67,225],[15,110219,1114],{"className":110220},[1113],[15,110222,110224],{"className":110223},[206],[15,110225,110227],{"className":110226,"style":2715},[210],[15,110228],{},[15,110230,110232],{"className":110231},[66],[15,110233,1194],{"className":110234},[1422,1508],[15,110236],{"className":110237,"style":697},[71],[15,110239,27278,110241],{"className":110240},[1339],[15,110242,1344],{"style":1343},[15,110244],{"className":110245,"style":697},[71],[15,110247,11],{"className":110248},[66,67],[15,110250,120],{"className":110251},[162],[15,110253,32453],{"className":110254,"style":23020},[66,67],[15,110256],{"className":110257,"style":173},[71],[15,110259,34425],{"className":110260},[177],[15,110262],{"className":110263,"style":173},[71],[15,110265,110267,110270,110273,110276,110282,110285,110288],{"className":110266},[57],[15,110268],{"className":110269,"style":4398},[61],[15,110271,61614],{"className":110272,"style":23020},[66,67],[15,110274,126],{"className":110275},[169],[15,110277,110279],{"className":110278},[66],[15,110280,1266],{"className":110281},[1422,1508],[15,110283],{"className":110284,"style":72},[71],[15,110286,6600],{"className":110287},[76],[15,110289],{"className":110290,"style":72},[71],[15,110292,110294,110297,110303,110309,110312,110315,110318,110321,110324,110327,110330,110333,110336,110339,110342,110348],{"className":110293},[57],[15,110295],{"className":110296,"style":4398},[61],[15,110298,110300],{"className":110299},[66],[15,110301,88449],{"className":110302},[66,4427],[15,110304,110306],{"className":110305},[66],[15,110307,120],{"className":110308},[1422,1508],[15,110310,94268],{"className":110311,"style":1515},[66,67],[15,110313,120],{"className":110314},[162],[15,110316,61614],{"className":110317,"style":23020},[66,67],[15,110319,126],{"className":110320},[169],[15,110322,24228],{"className":110323},[71],[15,110325,1251],{"className":110326},[66],[15,110328,24228],{"className":110329},[71],[15,110331,11],{"className":110332},[66,67],[15,110334,120],{"className":110335},[162],[15,110337,61614],{"className":110338,"style":23020},[66,67],[15,110340,126],{"className":110341},[169],[15,110343,110345],{"className":110344},[66],[15,110346,126],{"className":110347},[1422,1508],[15,110349,3571],{"className":110350},[66],[11,110352,110353,110354,110382,110383,110411,110412,10233,110507,48361,110563,110591],{},"第一项鼓励 ",[15,110355,110357,110370],{"className":110356,"translate":19},[18],[15,110358,110360],{"className":110359},[23],[25,110361,110362],{"xmlns":27},[29,110363,110364,110368],{},[32,110365,110366],{},[35,110367,94268],{},[45,110369,94268],{"encoding":47},[15,110371,110373],{"className":110372,"ariaHidden":53},[52],[15,110374,110376,110379],{"className":110375},[57],[15,110377],{"className":110378,"style":471},[61],[15,110380,94268],{"className":110381,"style":1515},[66,67]," 解释数据，第二项惩罚 ",[15,110384,110386,110399],{"className":110385,"translate":19},[18],[15,110387,110389],{"className":110388},[23],[25,110390,110391],{"xmlns":27},[29,110392,110393,110397],{},[32,110394,110395],{},[35,110396,94268],{},[45,110398,94268],{"encoding":47},[15,110400,110402],{"className":110401,"ariaHidden":53},[52],[15,110403,110405,110408],{"className":110404},[57],[15,110406],{"className":110407,"style":471},[61],[15,110409,94268],{"className":110410,"style":1515},[66,67]," 偏离先验。最大化 ELBO 等价于最小化 ",[15,110413,110415,110453],{"className":110414,"translate":19},[18],[15,110416,110418],{"className":110417},[23],[25,110419,110420],{"xmlns":27},[29,110421,110422,110450],{},[32,110423,110424,110430,110432,110434,110436,110438,110440,110442,110444,110446,110448],{},[32,110425,110426,110428],{},[35,110427,2853],{"mathvariant":138},[35,110429,1216],{"mathvariant":138},[39,110431,120],{"stretchy":119},[35,110433,94268],{},[35,110435,1251],{"mathvariant":138},[35,110437,11],{},[39,110439,120],{"stretchy":119},[35,110441,61614],{},[39,110443,34425],{},[35,110445,32453],{},[39,110447,126],{"stretchy":119},[39,110449,126],{"stretchy":119},[45,110451,110452],{"encoding":47},"\\mathrm{KL}(q\\|p(\\theta\\mid D))",[15,110454,110456,110495],{"className":110455,"ariaHidden":53},[52],[15,110457,110459,110462,110468,110471,110474,110477,110480,110483,110486,110489,110492],{"className":110458},[57],[15,110460],{"className":110461,"style":154},[61],[15,110463,110465],{"className":110464},[66],[15,110466,88449],{"className":110467},[66,4427],[15,110469,120],{"className":110470},[162],[15,110472,94268],{"className":110473,"style":1515},[66,67],[15,110475,1251],{"className":110476},[66],[15,110478,11],{"className":110479},[66,67],[15,110481,120],{"className":110482},[162],[15,110484,61614],{"className":110485,"style":23020},[66,67],[15,110487],{"className":110488,"style":173},[71],[15,110490,34425],{"className":110491},[177],[15,110493],{"className":110494,"style":173},[71],[15,110496,110498,110501,110504],{"className":110497},[57],[15,110499],{"className":110500,"style":154},[61],[15,110502,32453],{"className":110503,"style":23020},[66,67],[15,110505,18766],{"className":110506},[169],[15,110508,110510,110534],{"className":110509,"translate":19},[18],[15,110511,110513],{"className":110512},[23],[25,110514,110515],{"xmlns":27},[29,110516,110517,110531],{},[32,110518,110519,110521,110523,110525,110527,110529],{},[35,110520,27068],{},[39,110522,1170],{},[35,110524,11],{},[39,110526,120],{"stretchy":119},[35,110528,32453],{},[39,110530,126],{"stretchy":119},[45,110532,110533],{"encoding":47},"\\log p(D)",[15,110535,110537],{"className":110536,"ariaHidden":53},[52],[15,110538,110540,110543,110548,110551,110554,110557,110560],{"className":110539},[57],[15,110541],{"className":110542,"style":154},[61],[15,110544,27278,110546],{"className":110545},[1339],[15,110547,1344],{"style":1343},[15,110549],{"className":110550,"style":697},[71],[15,110552,11],{"className":110553},[66,67],[15,110555,120],{"className":110556},[162],[15,110558,32453],{"className":110559,"style":23020},[66,67],[15,110561,126],{"className":110562},[169],[15,110564,110566,110579],{"className":110565,"translate":19},[18],[15,110567,110569],{"className":110568},[23],[25,110570,110571],{"xmlns":27},[29,110572,110573,110577],{},[32,110574,110575],{},[35,110576,94268],{},[45,110578,94268],{"encoding":47},[15,110580,110582],{"className":110581,"ariaHidden":53},[52],[15,110583,110585,110588],{"className":110584},[57],[15,110586],{"className":110587,"style":471},[61],[15,110589,94268],{"className":110590,"style":1515},[66,67]," 无关。",[11,110593,110594,110595,110656,110657,110660,110661,110689,110690,110656,110751,110754,110755,110820],{},"KL 的方向很重要。",[15,110596,110598,110626],{"className":110597,"translate":19},[18],[15,110599,110601],{"className":110600},[23],[25,110602,110603],{"xmlns":27},[29,110604,110605,110623],{},[32,110606,110607,110613,110615,110617,110619,110621],{},[32,110608,110609,110611],{},[35,110610,2853],{"mathvariant":138},[35,110612,1216],{"mathvariant":138},[39,110614,120],{"stretchy":119},[35,110616,94268],{},[35,110618,1251],{"mathvariant":138},[35,110620,11],{},[39,110622,126],{"stretchy":119},[45,110624,110625],{"encoding":47},"\\mathrm{KL}(q\\|p)",[15,110627,110629],{"className":110628,"ariaHidden":53},[52],[15,110630,110632,110635,110641,110644,110647,110650,110653],{"className":110631},[57],[15,110633],{"className":110634,"style":154},[61],[15,110636,110638],{"className":110637},[66],[15,110639,88449],{"className":110640},[66,4427],[15,110642,120],{"className":110643},[162],[15,110645,94268],{"className":110646,"style":1515},[66,67],[15,110648,1251],{"className":110649},[66],[15,110651,11],{"className":110652},[66,67],[15,110654,126],{"className":110655},[169]," 倾向",[2040,110658,110659],{},"寻模","（mode-seeking）：只要 ",[15,110662,110664,110677],{"className":110663,"translate":19},[18],[15,110665,110667],{"className":110666},[23],[25,110668,110669],{"xmlns":27},[29,110670,110671,110675],{},[32,110672,110673],{},[35,110674,94268],{},[45,110676,94268],{"encoding":47},[15,110678,110680],{"className":110679,"ariaHidden":53},[52],[15,110681,110683,110686],{"className":110682},[57],[15,110684],{"className":110685,"style":471},[61],[15,110687,94268],{"className":110688,"style":1515},[66,67]," 覆盖了后验的一个峰，它就不太在意漏掉其他峰，因此常常低估方差。反方向 ",[15,110691,110693,110721],{"className":110692,"translate":19},[18],[15,110694,110696],{"className":110695},[23],[25,110697,110698],{"xmlns":27},[29,110699,110700,110718],{},[32,110701,110702,110708,110710,110712,110714,110716],{},[32,110703,110704,110706],{},[35,110705,2853],{"mathvariant":138},[35,110707,1216],{"mathvariant":138},[39,110709,120],{"stretchy":119},[35,110711,11],{},[35,110713,1251],{"mathvariant":138},[35,110715,94268],{},[39,110717,126],{"stretchy":119},[45,110719,110720],{"encoding":47},"\\mathrm{KL}(p\\|q)",[15,110722,110724],{"className":110723,"ariaHidden":53},[52],[15,110725,110727,110730,110736,110739,110742,110745,110748],{"className":110726},[57],[15,110728],{"className":110729,"style":154},[61],[15,110731,110733],{"className":110732},[66],[15,110734,88449],{"className":110735},[66,4427],[15,110737,120],{"className":110738},[162],[15,110740,11],{"className":110741},[66,67],[15,110743,1251],{"className":110744},[66],[15,110746,94268],{"className":110747,"style":1515},[66,67],[15,110749,126],{"className":110750},[169],[2040,110752,110753],{},"覆盖质量","（mass-covering），但它的计算需要 ",[15,110756,110758,110781],{"className":110757,"translate":19},[18],[15,110759,110761],{"className":110760},[23],[25,110762,110763],{"xmlns":27},[29,110764,110765,110779],{},[32,110766,110767,110769,110771,110773,110775,110777],{},[35,110768,11],{},[39,110770,120],{"stretchy":119},[35,110772,61614],{},[39,110774,34425],{},[35,110776,32453],{},[39,110778,126],{"stretchy":119},[45,110780,65456],{"encoding":47},[15,110782,110784,110808],{"className":110783,"ariaHidden":53},[52],[15,110785,110787,110790,110793,110796,110799,110802,110805],{"className":110786},[57],[15,110788],{"className":110789,"style":154},[61],[15,110791,11],{"className":110792},[66,67],[15,110794,120],{"className":110795},[162],[15,110797,61614],{"className":110798,"style":23020},[66,67],[15,110800],{"className":110801,"style":173},[71],[15,110803,34425],{"className":110804},[177],[15,110806],{"className":110807,"style":173},[71],[15,110809,110811,110814,110817],{"className":110810},[57],[15,110812],{"className":110813,"style":154},[61],[15,110815,32453],{"className":110816,"style":23020},[66,67],[15,110818,126],{"className":110819},[169]," 的归一化常数。变分推断通常用前者，因为 ELBO 只需要未归一化的联合分布。",[11,110822,110823,110824,110853,110854,1124],{},"从决策论的角度看，变分推断本身也是一个决策问题：在受限的分布族里，选一个使 KL 最小的近似。这里的「损失」是 KL，行动是一个分布，而不是一个点。贝叶斯决策理论的框架在这里再次出现，只是行动空间从 ",[15,110825,110827,110841],{"className":110826,"translate":19},[18],[15,110828,110830],{"className":110829},[23],[25,110831,110832],{"xmlns":27},[29,110833,110834,110838],{},[32,110835,110836],{},[35,110837,62001],{"mathvariant":445},[45,110839,110840],{"encoding":47},"\\mathcal A",[15,110842,110844],{"className":110843,"ariaHidden":53},[52],[15,110845,110847,110850],{"className":110846},[57],[15,110848],{"className":110849,"style":490},[61],[15,110851,62001],{"className":110852},[66,494]," 变成了 ",[15,110855,110857,110870],{"className":110856,"translate":19},[18],[15,110858,110860],{"className":110859},[23],[25,110861,110862],{"xmlns":27},[29,110863,110864,110868],{},[32,110865,110866],{},[35,110867,83340],{"mathvariant":445},[45,110869,109153],{"encoding":47},[15,110871,110873],{"className":110872,"ariaHidden":53},[52],[15,110874,110876,110879],{"className":110875},[57],[15,110877],{"className":110878,"style":61916},[61],[15,110880,83340],{"className":110881},[66,494],[1904,110883,110885],{"id":110884},"_75-贝叶斯神经网络与不确定性","7.5 贝叶斯神经网络与不确定性",[11,110887,110888,110889,110917,110918,110984],{},"把参数换成网络权重 ",[15,110890,110892,110905],{"className":110891,"translate":19},[18],[15,110893,110895],{"className":110894},[23],[25,110896,110897],{"xmlns":27},[29,110898,110899,110903],{},[32,110900,110901],{},[35,110902,135],{},[45,110904,135],{"encoding":47},[15,110906,110908],{"className":110907,"ariaHidden":53},[52],[15,110909,110911,110914],{"className":110910},[57],[15,110912],{"className":110913,"style":86},[61],[15,110915,135],{"className":110916,"style":194},[66,67],"，后验 ",[15,110919,110921,110945],{"className":110920,"translate":19},[18],[15,110922,110924],{"className":110923},[23],[25,110925,110926],{"xmlns":27},[29,110927,110928,110942],{},[32,110929,110930,110932,110934,110936,110938,110940],{},[35,110931,11],{},[39,110933,120],{"stretchy":119},[35,110935,135],{},[39,110937,34425],{},[35,110939,32453],{},[39,110941,126],{"stretchy":119},[45,110943,110944],{"encoding":47},"p(w\\mid D)",[15,110946,110948,110972],{"className":110947,"ariaHidden":53},[52],[15,110949,110951,110954,110957,110960,110963,110966,110969],{"className":110950},[57],[15,110952],{"className":110953,"style":154},[61],[15,110955,11],{"className":110956},[66,67],[15,110958,120],{"className":110959},[162],[15,110961,135],{"className":110962,"style":194},[66,67],[15,110964],{"className":110965,"style":173},[71],[15,110967,34425],{"className":110968},[177],[15,110970],{"className":110971,"style":173},[71],[15,110973,110975,110978,110981],{"className":110974},[57],[15,110976],{"className":110977,"style":154},[61],[15,110979,32453],{"className":110980,"style":23020},[66,67],[15,110982,126],{"className":110983},[169]," 就是贝叶斯神经网络。精确后验不可行，实践中用 MC dropout、Laplace 近似、变分推断或深度集成来近似。",[11,110986,110987],{},"对分类问题，预测熵可以分解为",[15,110989,110991],{"className":110990,"translate":19},[294],[15,110992,110994,111080],{"className":110993,"translate":19},[18],[15,110995,110997],{"className":110996},[23],[25,110998,110999],{"xmlns":27,"display":303},[29,111000,111001,111077],{},[32,111002,111003,111005,111007,111009,111011,111013,111015,111017,111019,111021,111033,111035,111037,111039,111041,111043,111045,111047,111049,111051,111053,111055,111057,111059,111061,111063,111065,111067,111069,111071,111073,111075],{},[35,111004,1061],{"mathvariant":454},[39,111006,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,111008,61881],{},[39,111010,34425],{},[35,111012,123],{},[39,111014,335],{"separator":53},[35,111016,32453],{},[39,111018,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,111020,129],{},[1057,111022,111023,111025],{},[35,111024,62377],{"mathvariant":454},[32,111026,111027,111029,111031],{},[35,111028,135],{},[39,111030,34425],{},[35,111032,32453],{},[39,111034,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,111036,1061],{"mathvariant":454},[39,111038,1194],{"stretchy":119},[35,111040,61881],{},[39,111042,34425],{},[35,111044,123],{},[39,111046,335],{"separator":53},[35,111048,135],{},[39,111050,1266],{"stretchy":119},[39,111052,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,111054,1245],{},[35,111056,16422],{"mathvariant":454},[39,111058,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,111060,61881],{},[39,111062,8408],{"separator":53},[35,111064,135],{},[39,111066,34425],{},[35,111068,123],{},[39,111070,335],{"separator":53},[35,111072,32453],{},[39,111074,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,111076,3571],{"mathvariant":138},[45,111078,111079],{"encoding":47},"\\mathbb H\\big[Y\\mid x,D\\big]\n=\n\\mathbb E_{w\\mid D}\\big[\\mathbb H[Y\\mid x,w]\\big]\n+\n\\mathbb I\\big(Y;w\\mid x,D\\big).",[15,111081,111083,111110,111143,111222,111258,111294],{"className":111082,"ariaHidden":53},[52],[15,111084,111086,111089,111092,111098,111101,111104,111107],{"className":111085},[57],[15,111087],{"className":111088,"style":4398},[61],[15,111090,1061],{"className":111091},[66,517],[15,111093,111095],{"className":111094},[66],[15,111096,1194],{"className":111097},[1422,1508],[15,111099,61881],{"className":111100,"style":72},[66,67],[15,111102],{"className":111103,"style":173},[71],[15,111105,34425],{"className":111106},[177],[15,111108],{"className":111109,"style":173},[71],[15,111111,111113,111116,111119,111122,111125,111128,111134,111137,111140],{"className":111112},[57],[15,111114],{"className":111115,"style":4398},[61],[15,111117,123],{"className":111118},[66,67],[15,111120,335],{"className":111121},[422],[15,111123],{"className":111124,"style":697},[71],[15,111126,32453],{"className":111127,"style":23020},[66,67],[15,111129,111131],{"className":111130},[66],[15,111132,1266],{"className":111133},[1422,1508],[15,111135],{"className":111136,"style":173},[71],[15,111138,129],{"className":111139},[177],[15,111141],{"className":111142,"style":173},[71],[15,111144,111146,111149,111198,111204,111207,111210,111213,111216,111219],{"className":111145},[57],[15,111147],{"className":111148,"style":12286},[61],[15,111150,111152,111155],{"className":111151},[66],[15,111153,62377],{"className":111154},[66,517],[15,111156,111158],{"className":111157},[198],[15,111159,111161,111190],{"className":111160},[202,1090],[15,111162,111164,111187],{"className":111163},[206],[15,111165,111167],{"className":111166,"style":3666},[210],[15,111168,111169,111172],{"style":62478},[15,111170],{"className":111171,"style":218},[217],[15,111173,111175],{"className":111174},[222,223,224,225],[15,111176,111178,111181,111184],{"className":111177},[66,225],[15,111179,135],{"className":111180,"style":194},[66,67,225],[15,111182,34425],{"className":111183},[177,225],[15,111185,32453],{"className":111186,"style":23020},[66,67,225],[15,111188,1114],{"className":111189},[1113],[15,111191,111193],{"className":111192},[206],[15,111194,111196],{"className":111195,"style":9875},[210],[15,111197],{},[15,111199,111201],{"className":111200},[66],[15,111202,1194],{"className":111203},[1422,1508],[15,111205,1061],{"className":111206},[66,517],[15,111208,1194],{"className":111209},[162],[15,111211,61881],{"className":111212,"style":72},[66,67],[15,111214],{"className":111215,"style":173},[71],[15,111217,34425],{"className":111218},[177],[15,111220],{"className":111221,"style":173},[71],[15,111223,111225,111228,111231,111234,111237,111240,111243,111249,111252,111255],{"className":111224},[57],[15,111226],{"className":111227,"style":4398},[61],[15,111229,123],{"className":111230},[66,67],[15,111232,335],{"className":111233},[422],[15,111235],{"className":111236,"style":697},[71],[15,111238,135],{"className":111239,"style":194},[66,67],[15,111241,1266],{"className":111242},[169],[15,111244,111246],{"className":111245},[66],[15,111247,1266],{"className":111248},[1422,1508],[15,111250],{"className":111251,"style":72},[71],[15,111253,1245],{"className":111254},[76],[15,111256],{"className":111257,"style":72},[71],[15,111259,111261,111264,111267,111273,111276,111279,111282,111285,111288,111291],{"className":111260},[57],[15,111262],{"className":111263,"style":4398},[61],[15,111265,16422],{"className":111266},[66,517],[15,111268,111270],{"className":111269},[66],[15,111271,120],{"className":111272},[1422,1508],[15,111274,61881],{"className":111275,"style":72},[66,67],[15,111277,8408],{"className":111278},[422],[15,111280],{"className":111281,"style":697},[71],[15,111283,135],{"className":111284,"style":194},[66,67],[15,111286],{"className":111287,"style":173},[71],[15,111289,34425],{"className":111290},[177],[15,111292],{"className":111293,"style":173},[71],[15,111295,111297,111300,111303,111306,111309,111312,111318],{"className":111296},[57],[15,111298],{"className":111299,"style":4398},[61],[15,111301,123],{"className":111302},[66,67],[15,111304,335],{"className":111305},[422],[15,111307],{"className":111308,"style":697},[71],[15,111310,32453],{"className":111311,"style":23020},[66,67],[15,111313,111315],{"className":111314},[66],[15,111316,126],{"className":111317},[1422,1508],[15,111319,3571],{"className":111320},[66],[11,111322,111323],{},"第一项是偶然不确定性：即使固定权重，类别仍然有噪声。第二项是认知不确定性：不同权重给出的预测分布之间的差异。BALD 等主动学习准则，就是用它来挑选「最能减少认知不确定性」的样本。",[11,111325,111326],{},"但「贝叶斯」这个标签不保证校准。一个近似的后验可能给出过度自信的预测；深度集成不是严格的贝叶斯方法，却常常校准得更好。判断的标准应该是决策质量与校准，而不是方法的名字。一个模型如果能在高损失的区域拒绝预测，或者把不确定性传递给下游决策，它就已经在做贝叶斯决策了，哪怕它没有显式地计算后验。",[1904,111328,111330],{"id":111329},"_76-序贯决策后验采样与探索","7.6 序贯决策：后验采样与探索",[11,111332,111333,111334,111337,111338,111503],{},"在序贯决策里，后验不只是用来「估计」，还用来「探索」。",[2040,111335,111336],{},"Thompson 采样","的规则很简单：每一步从后验中采样一个环境参数 ",[15,111339,111341,111378],{"className":111340,"translate":19},[18],[15,111342,111344],{"className":111343},[23],[25,111345,111346],{"xmlns":27},[29,111347,111348,111375],{},[32,111349,111350,111357,111359,111361,111363,111365,111367,111373],{},[4294,111351,111352,111354],{"accent":53},[35,111353,61614],{},[39,111355,111356],{},"~",[39,111358,28386],{},[35,111360,11],{},[39,111362,120],{"stretchy":119},[35,111364,61614],{},[39,111366,34425],{},[1057,111368,111369,111371],{},[35,111370,32453],{},[35,111372,9207],{},[39,111374,126],{"stretchy":119},[45,111376,111377],{"encoding":47},"\\tilde\\theta\\sim p(\\theta\\mid D_t)",[15,111379,111381,111429,111453],{"className":111380,"ariaHidden":53},[52],[15,111382,111384,111388,111420,111423,111426],{"className":111383},[57],[15,111385],{"className":111386,"style":111387},[61],"height:0.9313em;",[15,111389,111391],{"className":111390},[66,78139],[15,111392,111394],{"className":111393},[202],[15,111395,111397],{"className":111396},[206],[15,111398,111400,111408],{"className":111399,"style":111387},[210],[15,111401,111402,111405],{"style":1386},[15,111403],{"className":111404,"style":1368},[217],[15,111406,61614],{"className":111407,"style":23020},[66,67],[15,111409,111411,111414],{"style":111410},"top:-3.6134em;",[15,111412],{"className":111413,"style":1368},[217],[15,111415,111417],{"className":111416,"style":96102},[78167],[15,111418,111356],{"className":111419},[66],[15,111421],{"className":111422,"style":173},[71],[15,111424,28386],{"className":111425},[177],[15,111427],{"className":111428,"style":173},[71],[15,111430,111432,111435,111438,111441,111444,111447,111450],{"className":111431},[57],[15,111433],{"className":111434,"style":154},[61],[15,111436,11],{"className":111437},[66,67],[15,111439,120],{"className":111440},[162],[15,111442,61614],{"className":111443,"style":23020},[66,67],[15,111445],{"className":111446,"style":173},[71],[15,111448,34425],{"className":111449},[177],[15,111451],{"className":111452,"style":173},[71],[15,111454,111456,111459,111500],{"className":111455},[57],[15,111457],{"className":111458,"style":154},[61],[15,111460,111462,111465],{"className":111461},[66],[15,111463,32453],{"className":111464,"style":23020},[66,67],[15,111466,111468],{"className":111467},[198],[15,111469,111471,111492],{"className":111470},[202,1090],[15,111472,111474,111489],{"className":111473},[206],[15,111475,111478],{"className":111476,"style":111477},[210],"height:0.2806em;",[15,111479,111480,111483],{"style":88433},[15,111481],{"className":111482,"style":218},[217],[15,111484,111486],{"className":111485},[222,223,224,225],[15,111487,9207],{"className":111488},[66,67,225],[15,111490,1114],{"className":111491},[1113],[15,111493,111495],{"className":111494},[206],[15,111496,111498],{"className":111497,"style":1121},[210],[15,111499],{},[15,111501,126],{"className":111502},[169],"，然后按这个采样出的环境做最优动作。",[11,111505,111506],{},"这个规则自动平衡了利用与探索。不确定性大的动作在采样中更可能显得最优，因此会被更频繁地尝试；随着数据增多，后验收窄，采样出的环境越来越接近真实环境，策略逐渐转向利用。它的贝叶斯遗憾有理论保证，而且实现起来只需要一个后验采样器。",[11,111508,111509],{},"从决策论的角度看，Thompson 采样把「信息」当成了决策的一部分：当前动作的收益不只是它带来的即时回报，还包括它揭示的信息。完整的贝叶斯解决方案是动态规划，在信念空间（belief space）上求解；Thompson 采样是它的一个可计算的近似。",[1904,111511,111513],{"id":111512},"_77-主动学习与贝叶斯优化","7.7 主动学习与贝叶斯优化",[11,111515,111516],{},"主动学习问的是：下一个标签应该买哪一个？从决策论看，这是一个两级决策：先选一个样本，再在拿到标签后做预测。理想的采集函数是「期望后验风险降低」：",[15,111518,111520],{"className":111519,"translate":19},[294],[15,111521,111523,111622],{"className":111522,"translate":19},[18],[15,111524,111526],{"className":111525},[23],[25,111527,111528],{"xmlns":27,"display":303},[29,111529,111530,111619],{},[32,111531,111532,111539,111541,111543,111545,111555,111557,111583,111617],{},[1057,111533,111534,111536],{},[35,111535,123],{},[1274,111537,111538],{},"next",[39,111540,129],{},[35,111542,1167],{},[39,111544,1170],{},[1172,111546,111547,111553],{},[32,111548,111549,111551],{},[35,111550,1178],{},[39,111552,1170],{},[35,111554,123],{},[1274,111556,24228],{},[1057,111558,111559,111561],{},[35,111560,62377],{"mathvariant":454},[32,111562,111563,111565,111567,111569,111571,111573,111575,111577,111579,111581],{},[35,111564,1224],{},[39,111566,28386],{},[35,111568,11],{},[39,111570,120],{"stretchy":119},[35,111572,1224],{},[39,111574,34425],{},[35,111576,123],{},[39,111578,335],{"separator":53},[35,111580,32453],{},[39,111582,126],{"stretchy":119},[32,111584,111585,111587,111593,111595,111597,111599,111601,111603,111605,111607,111609,111611,111613,111615],{},[39,111586,1194],{"fence":53},[131,111588,111589,111591],{},[35,111590,455],{},[39,111592,1162],{},[39,111594,120],{"stretchy":119},[35,111596,32453],{},[39,111598,7608],{},[39,111600,4305],{"stretchy":119},[39,111602,120],{"stretchy":119},[35,111604,123],{},[39,111606,335],{"separator":53},[35,111608,1224],{},[39,111610,126],{"stretchy":119},[39,111612,4328],{"stretchy":119},[39,111614,126],{"stretchy":119},[39,111616,1266],{"fence":53},[39,111618,335],{"separator":53},[45,111620,111621],{"encoding":47},"x_{\\text{next}}\n=\n\\arg\\min_{x}\\ \n\\mathbb E_{y\\sim p(y\\mid x,D)}\n\\left[\nR^*(D\\cup\\{(x,y)\\})\n\\right],",[15,111623,111625,111686],{"className":111624,"ariaHidden":53},[52],[15,111626,111628,111631,111677,111680,111683],{"className":111627},[57],[15,111629],{"className":111630,"style":10192},[61],[15,111632,111634,111637],{"className":111633},[66],[15,111635,123],{"className":111636},[66,67],[15,111638,111640],{"className":111639},[198],[15,111641,111643,111669],{"className":111642},[202,1090],[15,111644,111646,111666],{"className":111645},[206],[15,111647,111649],{"className":111648,"style":111477},[210],[15,111650,111651,111654],{"style":1584},[15,111652],{"className":111653,"style":218},[217],[15,111655,111657],{"className":111656},[222,223,224,225],[15,111658,111660],{"className":111659},[66,225],[15,111661,111663],{"className":111662},[66,1711,225],[15,111664,111538],{"className":111665},[66,225],[15,111667,1114],{"className":111668},[1113],[15,111670,111672],{"className":111671},[206],[15,111673,111675],{"className":111674,"style":1121},[210],[15,111676],{},[15,111678],{"className":111679,"style":173},[71],[15,111681,129],{"className":111682},[177],[15,111684],{"className":111685,"style":173},[71],[15,111687,111689,111693,111698,111701,111748,111751,111754,111824,111827,111899,111902],{"className":111688},[57],[15,111690],{"className":111691,"style":111692},[61],"height:1.45em;vertical-align:-0.7em;",[15,111694,1340,111696],{"className":111695},[1339],[15,111697,1344],{"style":1343},[15,111699],{"className":111700,"style":697},[71],[15,111702,111704],{"className":111703},[1339,1351],[15,111705,111707,111740],{"className":111706},[202,1090],[15,111708,111710,111737],{"className":111709},[206],[15,111711,111713,111727],{"className":111712,"style":1361},[210],[15,111714,111715,111718],{"style":48819},[15,111716],{"className":111717,"style":1368},[217],[15,111719,111721],{"className":111720},[222,223,224,225],[15,111722,111724],{"className":111723},[66,225],[15,111725,123],{"className":111726},[66,67,225],[15,111728,111729,111732],{"style":1386},[15,111730],{"className":111731,"style":1368},[217],[15,111733,111734],{},[15,111735,1178],{"className":111736},[1339],[15,111738,1114],{"className":111739},[1113],[15,111741,111743],{"className":111742},[206],[15,111744,111746],{"className":111745,"style":48857},[210],[15,111747],{},[15,111749,24228],{"className":111750},[71],[15,111752],{"className":111753,"style":697},[71],[15,111755,111757,111760],{"className":111756},[66],[15,111758,62377],{"className":111759},[66,517],[15,111761,111763],{"className":111762},[198],[15,111764,111766,111816],{"className":111765},[202,1090],[15,111767,111769,111813],{"className":111768},[206],[15,111770,111772],{"className":111771,"style":3666},[210],[15,111773,111774,111777],{"style":62478},[15,111775],{"className":111776,"style":218},[217],[15,111778,111780],{"className":111779},[222,223,224,225],[15,111781,111783,111786,111789,111792,111795,111798,111801,111804,111807,111810],{"className":111782},[66,225],[15,111784,1224],{"className":111785,"style":1515},[66,67,225],[15,111787,28386],{"className":111788},[177,225],[15,111790,11],{"className":111791},[66,67,225],[15,111793,120],{"className":111794},[162,225],[15,111796,1224],{"className":111797,"style":1515},[66,67,225],[15,111799,34425],{"className":111800},[177,225],[15,111802,123],{"className":111803},[66,67,225],[15,111805,335],{"className":111806},[422,225],[15,111808,32453],{"className":111809,"style":23020},[66,67,225],[15,111811,126],{"className":111812},[169,225],[15,111814,1114],{"className":111815},[1113],[15,111817,111819],{"className":111818},[206],[15,111820,111822],{"className":111821,"style":9875},[210],[15,111823],{},[15,111825],{"className":111826,"style":697},[71],[15,111828,111830,111833,111862,111865,111868,111871,111874,111877,111880,111883,111886,111889,111892,111896],{"className":111829},[1413],[15,111831,1194],{"className":111832,"style":1418},[162,1417],[15,111834,111836,111839],{"className":111835},[66],[15,111837,455],{"className":111838,"style":62433},[66,67],[15,111840,111842],{"className":111841},[198],[15,111843,111845],{"className":111844},[202],[15,111846,111848],{"className":111847},[206],[15,111849,111851],{"className":111850,"style":1308},[210],[15,111852,111853,111856],{"style":397},[15,111854],{"className":111855,"style":218},[217],[15,111857,111859],{"className":111858},[222,223,224,225],[15,111860,1162],{"className":111861},[76,225],[15,111863,120],{"className":111864},[162],[15,111866,32453],{"className":111867,"style":23020},[66,67],[15,111869],{"className":111870,"style":72},[71],[15,111872,7608],{"className":111873},[76],[15,111875],{"className":111876,"style":72},[71],[15,111878,19294],{"className":111879},[162],[15,111881,123],{"className":111882},[66,67],[15,111884,335],{"className":111885},[422],[15,111887],{"className":111888,"style":697},[71],[15,111890,1224],{"className":111891,"style":1515},[66,67],[15,111893,111895],{"className":111894},[169],")})",[15,111897,1266],{"className":111898,"style":1418},[169,1417],[15,111900],{"className":111901,"style":697},[71],[15,111903,335],{"className":111904},[422],[11,111906,425,111907,111981],{},[15,111908,111910,111934],{"className":111909,"translate":19},[18],[15,111911,111913],{"className":111912},[23],[25,111914,111915],{"xmlns":27},[29,111916,111917,111931],{},[32,111918,111919,111925,111927,111929],{},[131,111920,111921,111923],{},[35,111922,455],{},[39,111924,1162],{},[39,111926,120],{"stretchy":119},[35,111928,32453],{},[39,111930,126],{"stretchy":119},[45,111932,111933],{"encoding":47},"R^*(D)",[15,111935,111937],{"className":111936,"ariaHidden":53},[52],[15,111938,111940,111943,111972,111975,111978],{"className":111939},[57],[15,111941],{"className":111942,"style":154},[61],[15,111944,111946,111949],{"className":111945},[66],[15,111947,455],{"className":111948,"style":62433},[66,67],[15,111950,111952],{"className":111951},[198],[15,111953,111955],{"className":111954},[202],[15,111956,111958],{"className":111957},[206],[15,111959,111961],{"className":111960,"style":16244},[210],[15,111962,111963,111966],{"style":213},[15,111964],{"className":111965,"style":218},[217],[15,111967,111969],{"className":111968},[222,223,224,225],[15,111970,1162],{"className":111971},[76,225],[15,111973,120],{"className":111974},[162],[15,111976,32453],{"className":111977,"style":23020},[66,67],[15,111979,126],{"className":111980},[169]," 是当前数据下的贝叶斯风险。这个量通常算不出来，于是有了各种近似：熵、互信息、BALD、期望模型变化。",[11,111983,111984],{},"贝叶斯优化把目标函数本身当成随机变量，用后验选择下一个评估点。Expected Improvement 就是",[15,111986,111988],{"className":111987,"translate":19},[294],[15,111989,111991,112061],{"className":111990,"translate":19},[18],[15,111992,111994],{"className":111993},[23],[25,111995,111996],{"xmlns":27,"display":303},[29,111997,111998,112058],{},[32,111999,112000,112006,112008,112010,112012,112014,112016,112018,112020,112022,112024,112026,112028,112030,112038,112040,112042,112044,112046,112048,112050,112052,112054,112056],{},[32,112001,112002,112004],{},[35,112003,62377],{"mathvariant":138},[35,112005,16422],{"mathvariant":138},[39,112007,120],{"stretchy":119},[35,112009,123],{},[39,112011,126],{"stretchy":119},[39,112013,129],{},[35,112015,62377],{"mathvariant":454},[39,112017,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,112019,23268],{},[39,112021,1170],{},[39,112023,4305],{"stretchy":119},[1209,112025,1804],{},[39,112027,335],{"separator":53},[1274,112029,24228],{},[1057,112031,112032,112034,112036],{},[35,112033,116],{},[35,112035,1178],{},[39,112037,1170],{},[39,112039,6600],{},[35,112041,116],{},[39,112043,120],{"stretchy":119},[35,112045,123],{},[39,112047,126],{"stretchy":119},[39,112049,4328],{"stretchy":119},[39,112051,34425],{},[35,112053,32453],{},[39,112055,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,112057,335],{"separator":53},[45,112059,112060],{"encoding":47},"\\mathrm{EI}(x)=\\mathbb E\\big[\\max\\{0,\\ f_{\\min}-f(x)\\}\\mid D\\big],",[15,112062,112064,112095,112192,112219],{"className":112063,"ariaHidden":53},[52],[15,112065,112067,112070,112077,112080,112083,112086,112089,112092],{"className":112066},[57],[15,112068],{"className":112069,"style":154},[61],[15,112071,112073],{"className":112072},[66],[15,112074,112076],{"className":112075},[66,4427],"EI",[15,112078,120],{"className":112079},[162],[15,112081,123],{"className":112082},[66,67],[15,112084,126],{"className":112085},[169],[15,112087],{"className":112088,"style":173},[71],[15,112090,129],{"className":112091},[177],[15,112093],{"className":112094,"style":173},[71],[15,112096,112098,112101,112104,112110,112113,112116,112119,112122,112125,112128,112131,112183,112186,112189],{"className":112097},[57],[15,112099],{"className":112100,"style":4398},[61],[15,112102,62377],{"className":112103},[66,517],[15,112105,112107],{"className":112106},[66],[15,112108,1194],{"className":112109},[1422,1508],[15,112111],{"className":112112,"style":697},[71],[15,112114,23268],{"className":112115},[1339],[15,112117,4305],{"className":112118},[162],[15,112120,1804],{"className":112121},[66],[15,112123,335],{"className":112124},[422],[15,112126,24228],{"className":112127},[71],[15,112129],{"className":112130,"style":697},[71],[15,112132,112134,112137],{"className":112133},[66],[15,112135,116],{"className":112136,"style":158},[66,67],[15,112138,112140],{"className":112139},[198],[15,112141,112143,112175],{"className":112142},[202,1090],[15,112144,112146,112172],{"className":112145},[206],[15,112147,112149],{"className":112148,"style":100007},[210],[15,112150,112151,112154],{"style":9913},[15,112152],{"className":112153,"style":218},[217],[15,112155,112157],{"className":112156},[222,223,224,225],[15,112158,112160],{"className":112159},[66,225],[15,112161,112163,112166,112169],{"className":112162},[1339,225],[15,112164,28853],{"className":112165},[225],[15,112167,1205],{"className":112168},[225],[15,112170,37],{"className":112171},[225],[15,112173,1114],{"className":112174},[1113],[15,112176,112178],{"className":112177},[206],[15,112179,112181],{"className":112180,"style":1121},[210],[15,112182],{},[15,112184],{"className":112185,"style":72},[71],[15,112187,6600],{"className":112188},[76],[15,112190],{"className":112191,"style":72},[71],[15,112193,112195,112198,112201,112204,112207,112210,112213,112216],{"className":112194},[57],[15,112196],{"className":112197,"style":154},[61],[15,112199,116],{"className":112200,"style":158},[66,67],[15,112202,120],{"className":112203},[162],[15,112205,123],{"className":112206},[66,67],[15,112208,19275],{"className":112209},[169],[15,112211],{"className":112212,"style":173},[71],[15,112214,34425],{"className":112215},[177],[15,112217],{"className":112218,"style":173},[71],[15,112220,112222,112225,112228,112234],{"className":112221},[57],[15,112223],{"className":112224,"style":4398},[61],[15,112226,32453],{"className":112227,"style":23020},[66,67],[15,112229,112231],{"className":112230},[66],[15,112232,1266],{"className":112233},[1422,1508],[15,112235,335],{"className":112236},[422],[11,112238,112239],{},"它把「可能变好多少」与「变好的概率」乘在一起。探索与利用的权衡，在这里被写成了一个一维积分的最大化问题。",[1904,112241,112243],{"id":112242},"_78-可容许性与完整类定理","7.8 可容许性与完整类定理",[11,112245,112246,112247,2038,112275,112278,112279,112341],{},"最后回到 2.4 节留下的那个说法。一个决策规则 ",[15,112248,112250,112263],{"className":112249,"translate":19},[18],[15,112251,112253],{"className":112252},[23],[25,112254,112255],{"xmlns":27},[29,112256,112257,112261],{},[32,112258,112259],{},[35,112260,3883],{},[45,112262,68709],{"encoding":47},[15,112264,112266],{"className":112265,"ariaHidden":53},[52],[15,112267,112269,112272],{"className":112268},[57],[15,112270],{"className":112271,"style":840},[61],[15,112273,3883],{"className":112274,"style":3944},[66,67],[2040,112276,112277],{},"可容许的","（admissible），若不存在另一个规则 ",[15,112280,112282,112300],{"className":112281,"translate":19},[18],[15,112283,112285],{"className":112284},[23],[25,112286,112287],{"xmlns":27},[29,112288,112289,112297],{},[32,112290,112291],{},[131,112292,112293,112295],{},[35,112294,3883],{},[39,112296,635],{"mathvariant":138,"lspace":634,"rspace":634},[45,112298,112299],{"encoding":47},"\\delta'",[15,112301,112303],{"className":112302,"ariaHidden":53},[52],[15,112304,112306,112309],{"className":112305},[57],[15,112307],{"className":112308,"style":6003},[61],[15,112310,112312,112315],{"className":112311},[66],[15,112313,3883],{"className":112314,"style":3944},[66,67],[15,112316,112318],{"className":112317},[198],[15,112319,112321],{"className":112320},[202],[15,112322,112324],{"className":112323},[206],[15,112325,112327],{"className":112326,"style":6003},[210],[15,112328,112329,112332],{"style":213},[15,112330],{"className":112331,"style":218},[217],[15,112333,112335],{"className":112334},[222,223,224,225],[15,112336,112338],{"className":112337},[66,225],[15,112339,635],{"className":112340},[66,225],"，使得",[15,112343,112345],{"className":112344,"translate":19},[294],[15,112346,112348,112391],{"className":112347,"translate":19},[18],[15,112349,112351],{"className":112350},[23],[25,112352,112353],{"xmlns":27,"display":303},[29,112354,112355,112388],{},[32,112356,112357,112359,112361,112367,112369,112371,112373,112375,112377,112379,112381,112384,112386],{},[35,112358,455],{},[39,112360,120],{"stretchy":119},[131,112362,112363,112365],{},[35,112364,3883],{},[39,112366,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,112368,126],{"stretchy":119},[39,112370,24215],{},[35,112372,455],{},[39,112374,120],{"stretchy":119},[35,112376,3883],{},[39,112378,126],{"stretchy":119},[71,112380],{"width":9786},[1274,112382,112383],{},"对所有分布 ",[35,112385,28394],{},[39,112387,335],{"separator":53},[45,112389,112390],{"encoding":47},"R(\\delta')\\le R(\\delta)\\quad\\text{对所有分布 }P,",[15,112392,112394,112450],{"className":112393,"ariaHidden":53},[52],[15,112395,112397,112400,112403,112406,112438,112441,112444,112447],{"className":112396},[57],[15,112398],{"className":112399,"style":680},[61],[15,112401,455],{"className":112402,"style":62433},[66,67],[15,112404,120],{"className":112405},[162],[15,112407,112409,112412],{"className":112408},[66],[15,112410,3883],{"className":112411,"style":3944},[66,67],[15,112413,112415],{"className":112414},[198],[15,112416,112418],{"className":112417},[202],[15,112419,112421],{"className":112420},[206],[15,112422,112424],{"className":112423,"style":716},[210],[15,112425,112426,112429],{"style":397},[15,112427],{"className":112428,"style":218},[217],[15,112430,112432],{"className":112431},[222,223,224,225],[15,112433,112435],{"className":112434},[66,225],[15,112436,635],{"className":112437},[66,225],[15,112439,126],{"className":112440},[169],[15,112442],{"className":112443,"style":173},[71],[15,112445,24215],{"className":112446},[177],[15,112448],{"className":112449,"style":173},[71],[15,112451,112453,112456,112459,112462,112465,112468,112471,112481,112484],{"className":112452},[57],[15,112454],{"className":112455,"style":154},[61],[15,112457,455],{"className":112458,"style":62433},[66,67],[15,112460,120],{"className":112461},[162],[15,112463,3883],{"className":112464,"style":3944},[66,67],[15,112466,126],{"className":112467},[169],[15,112469],{"className":112470,"style":10044},[71],[15,112472,112474,112478],{"className":112473},[66,1711],[15,112475,112477],{"className":112476},[66,68610],"对所有分布",[15,112479,24228],{"className":112480},[66],[15,112482,28394],{"className":112483,"style":62509},[66,67],[15,112485,335],{"className":112486},[422],[11,112488,112489,112490,112518],{},"且至少对一个 ",[15,112491,112493,112506],{"className":112492,"translate":19},[18],[15,112494,112496],{"className":112495},[23],[25,112497,112498],{"xmlns":27},[29,112499,112500,112504],{},[32,112501,112502],{},[35,112503,28394],{},[45,112505,28394],{"encoding":47},[15,112507,112509],{"className":112508,"ariaHidden":53},[52],[15,112510,112512,112515],{"className":112511},[57],[15,112513],{"className":112514,"style":490},[61],[15,112516,28394],{"className":112517,"style":62509},[66,67]," 严格小于。换句话说，可容许就是「没有被另一个规则全面支配」。一个处处比别的规则差的决策规则，理应被淘汰；可容许性把这个直觉形式化了。",[3054,112520,112522],{"title":112521},"定理 7.2（Wald 完整类定理）",[11,112523,112524,112525,112554],{},"在损失函数凸、行动空间紧、实验族满足适当正则条件（例如被某个 ",[15,112526,112528,112542],{"className":112527,"translate":19},[18],[15,112529,112531],{"className":112530},[23],[25,112532,112533],{"xmlns":27},[29,112534,112535,112539],{},[32,112536,112537],{},[35,112538,6632],{},[45,112540,112541],{"encoding":47},"\\sigma",[15,112543,112545],{"className":112544,"ariaHidden":53},[52],[15,112546,112548,112551],{"className":112547},[57],[15,112549],{"className":112550,"style":86},[61],[15,112552,6632],{"className":112553,"style":1515},[66,67],"-有限测度支配）时，每一个可容许的决策规则都是某个先验（或先验序列的极限）下的贝叶斯规则。",[11,112556,112557],{},"这个定理的意义在于：在适当条件下，它把「频率派的极大极小规则」「看起来没有先验的规则」也纳入了贝叶斯框架。先验不再是一个可选的信仰，而是可容许性这个要求所对应的对偶变量。你可以拒绝某个具体的先验，但很难拒绝「存在某个先验」这件事。",[11,112559,112560,112561,1124],{},"定理的条件不能省。损失不凸、行动空间不紧，或者问题是非参数的，结论可能失效；此时需要广义先验、极限贝叶斯规则，或者放弃完整类定理的结论。这些技术细节不改变核心信息：",[2040,112562,112563],{},"贝叶斯规则不是许多决策规则中的一种，而是所有不被支配的决策规则的一般形式",[11,112565,112566],{},"从实践的角度看，这个定理不告诉你该选哪个先验，只告诉你：如果你想要一个不被全面支配的规则，你总可以把它写成某个先验下的贝叶斯规则。先验的选择仍然是建模的一部分，但「要不要用先验」已经不再是一个真正的选项。",[95,112568,112570],{"id":112569},"_8-几个可以手算的例子","8. 几个可以手算的例子",[1904,112572,112574],{"id":112573},"_81-betabernoulli-的完整更新","8.1 Beta–Bernoulli 的完整更新",[11,112576,112577,112578,112646,112647,66227,112675,112704],{},"取先验 ",[15,112579,112581,112613],{"className":112580,"translate":19},[18],[15,112582,112584],{"className":112583},[23],[25,112585,112586],{"xmlns":27},[29,112587,112588,112610],{},[32,112589,112590,112600,112602,112604,112606,112608],{},[32,112591,112592,112594,112596,112598],{},[35,112593,63968],{"mathvariant":138},[35,112595,48378],{"mathvariant":138},[35,112597,9207],{"mathvariant":138},[35,112599,4554],{"mathvariant":138},[39,112601,120],{"stretchy":119},[1209,112603,8673],{},[39,112605,335],{"separator":53},[1209,112607,8673],{},[39,112609,126],{"stretchy":119},[45,112611,112612],{"encoding":47},"\\mathrm{Beta}(3,3)",[15,112614,112616],{"className":112615,"ariaHidden":53},[52],[15,112617,112619,112622,112628,112631,112634,112637,112640,112643],{"className":112618},[57],[15,112620],{"className":112621,"style":154},[61],[15,112623,112625],{"className":112624},[66],[15,112626,66788],{"className":112627},[66,4427],[15,112629,120],{"className":112630},[162],[15,112632,8673],{"className":112633},[66],[15,112635,335],{"className":112636},[422],[15,112638],{"className":112639,"style":697},[71],[15,112641,8673],{"className":112642},[66],[15,112644,126],{"className":112645},[169],"，观测到 ",[15,112648,112650,112663],{"className":112649,"translate":19},[18],[15,112651,112653],{"className":112652},[23],[25,112654,112655],{"xmlns":27},[29,112656,112657,112661],{},[32,112658,112659],{},[1209,112660,107903],{},[45,112662,107903],{"encoding":47},[15,112664,112666],{"className":112665,"ariaHidden":53},[52],[15,112667,112669,112672],{"className":112668},[57],[15,112670],{"className":112671,"style":1836},[61],[15,112673,107903],{"className":112674},[66],[15,112676,112678,112692],{"className":112677,"translate":19},[18],[15,112679,112681],{"className":112680},[23],[25,112682,112683],{"xmlns":27},[29,112684,112685,112690],{},[32,112686,112687],{},[1209,112688,112689],{},"7",[45,112691,112689],{"encoding":47},[15,112693,112695],{"className":112694,"ariaHidden":53},[52],[15,112696,112698,112701],{"className":112697},[57],[15,112699],{"className":112700,"style":1836},[61],[15,112702,112689],{"className":112703},[66]," 次正面。后验为",[15,112706,112708],{"className":112707,"translate":19},[294],[15,112709,112711,112777],{"className":112710,"translate":19},[18],[15,112712,112714],{"className":112713},[23],[25,112715,112716],{"xmlns":27,"display":303},[29,112717,112718,112774],{},[32,112719,112720,112730,112732,112734,112736,112738,112740,112742,112744,112746,112748,112750,112752,112762,112764,112766,112768,112770,112772],{},[32,112721,112722,112724,112726,112728],{},[35,112723,63968],{"mathvariant":138},[35,112725,48378],{"mathvariant":138},[35,112727,9207],{"mathvariant":138},[35,112729,4554],{"mathvariant":138},[39,112731,120],{"stretchy":119},[1209,112733,8673],{},[39,112735,1245],{},[1209,112737,112689],{},[39,112739,335],{"separator":53},[1274,112741,24228],{},[1209,112743,8673],{},[39,112745,1245],{},[1209,112747,8673],{},[39,112749,126],{"stretchy":119},[39,112751,129],{},[32,112753,112754,112756,112758,112760],{},[35,112755,63968],{"mathvariant":138},[35,112757,48378],{"mathvariant":138},[35,112759,9207],{"mathvariant":138},[35,112761,4554],{"mathvariant":138},[39,112763,120],{"stretchy":119},[1209,112765,107903],{},[39,112767,335],{"separator":53},[1209,112769,59939],{},[39,112771,126],{"stretchy":119},[35,112773,3571],{"mathvariant":138},[45,112775,112776],{"encoding":47},"\\mathrm{Beta}(3+7,\\ 3+3)=\\mathrm{Beta}(10,6).",[15,112778,112780,112807,112838,112859],{"className":112779,"ariaHidden":53},[52],[15,112781,112783,112786,112792,112795,112798,112801,112804],{"className":112782},[57],[15,112784],{"className":112785,"style":154},[61],[15,112787,112789],{"className":112788},[66],[15,112790,66788],{"className":112791},[66,4427],[15,112793,120],{"className":112794},[162],[15,112796,8673],{"className":112797},[66],[15,112799],{"className":112800,"style":72},[71],[15,112802,1245],{"className":112803},[76],[15,112805],{"className":112806,"style":72},[71],[15,112808,112810,112814,112817,112820,112823,112826,112829,112832,112835],{"className":112809},[57],[15,112811],{"className":112812,"style":112813},[61],"height:0.8389em;vertical-align:-0.1944em;",[15,112815,112689],{"className":112816},[66],[15,112818,335],{"className":112819},[422],[15,112821,24228],{"className":112822},[71],[15,112824],{"className":112825,"style":697},[71],[15,112827,8673],{"className":112828},[66],[15,112830],{"className":112831,"style":72},[71],[15,112833,1245],{"className":112834},[76],[15,112836],{"className":112837,"style":72},[71],[15,112839,112841,112844,112847,112850,112853,112856],{"className":112840},[57],[15,112842],{"className":112843,"style":154},[61],[15,112845,8673],{"className":112846},[66],[15,112848,126],{"className":112849},[169],[15,112851],{"className":112852,"style":173},[71],[15,112854,129],{"className":112855},[177],[15,112857],{"className":112858,"style":173},[71],[15,112860,112862,112865,112871,112874,112877,112880,112883,112886,112889],{"className":112861},[57],[15,112863],{"className":112864,"style":154},[61],[15,112866,112868],{"className":112867},[66],[15,112869,66788],{"className":112870},[66,4427],[15,112872,120],{"className":112873},[162],[15,112875,107903],{"className":112876},[66],[15,112878,335],{"className":112879},[422],[15,112881],{"className":112882,"style":697},[71],[15,112884,59939],{"className":112885},[66],[15,112887,126],{"className":112888},[169],[15,112890,3571],{"className":112891},[66],[11,112893,112894],{},"把几个估计量放在一起：",[6150,112896,112897,112907],{},[6153,112898,112899],{},[6156,112900,112901,112904],{},[6159,112902,112903],{},"量",[6159,112905,112906],{},"值",[6169,112908,112909,112973,113037,113100,113165],{},[6156,112910,112911,112914],{},[6174,112912,112913],{},"先验均值",[6174,112915,112916],{},[15,112917,112919,112942],{"className":112918,"translate":19},[18],[15,112920,112922],{"className":112921},[23],[25,112923,112924],{"xmlns":27},[29,112925,112926,112939],{},[32,112927,112928,112930,112932,112934,112936],{},[1209,112929,8673],{},[35,112931,6623],{"mathvariant":138},[1209,112933,59939],{},[39,112935,129],{},[1209,112937,112938],{},"0.5",[45,112940,112941],{"encoding":47},"3\u002F6=0.5",[15,112943,112945,112964],{"className":112944,"ariaHidden":53},[52],[15,112946,112948,112951,112955,112958,112961],{"className":112947},[57],[15,112949],{"className":112950,"style":154},[61],[15,112952,112954],{"className":112953},[66],"3\u002F6",[15,112956],{"className":112957,"style":173},[71],[15,112959,129],{"className":112960},[177],[15,112962],{"className":112963,"style":173},[71],[15,112965,112967,112970],{"className":112966},[57],[15,112968],{"className":112969,"style":1836},[61],[15,112971,112938],{"className":112972},[66],[6156,112974,112975,112978],{},[6174,112976,112977],{},"最大似然估计",[6174,112979,112980],{},[15,112981,112983,113006],{"className":112982,"translate":19},[18],[15,112984,112986],{"className":112985},[23],[25,112987,112988],{"xmlns":27},[29,112989,112990,113003],{},[32,112991,112992,112994,112996,112998,113000],{},[1209,112993,112689],{},[35,112995,6623],{"mathvariant":138},[1209,112997,107903],{},[39,112999,129],{},[1209,113001,113002],{},"0.7",[45,113004,113005],{"encoding":47},"7\u002F10=0.7",[15,113007,113009,113028],{"className":113008,"ariaHidden":53},[52],[15,113010,113012,113015,113019,113022,113025],{"className":113011},[57],[15,113013],{"className":113014,"style":154},[61],[15,113016,113018],{"className":113017},[66],"7\u002F10",[15,113020],{"className":113021,"style":173},[71],[15,113023,129],{"className":113024},[177],[15,113026],{"className":113027,"style":173},[71],[15,113029,113031,113034],{"className":113030},[57],[15,113032],{"className":113033,"style":1836},[61],[15,113035,113002],{"className":113036},[66],[6156,113038,113039,113041],{},[6174,113040,72658],{},[6174,113042,113043],{},[15,113044,113046,113069],{"className":113045,"translate":19},[18],[15,113047,113049],{"className":113048},[23],[25,113050,113051],{"xmlns":27},[29,113052,113053,113066],{},[32,113054,113055,113057,113059,113061,113063],{},[1209,113056,107903],{},[35,113058,6623],{"mathvariant":138},[1209,113060,59723],{},[39,113062,129],{},[1209,113064,113065],{},"0.625",[45,113067,113068],{"encoding":47},"10\u002F16=0.625",[15,113070,113072,113091],{"className":113071,"ariaHidden":53},[52],[15,113073,113075,113078,113082,113085,113088],{"className":113074},[57],[15,113076],{"className":113077,"style":154},[61],[15,113079,113081],{"className":113080},[66],"10\u002F16",[15,113083],{"className":113084,"style":173},[71],[15,113086,129],{"className":113087},[177],[15,113089],{"className":113090,"style":173},[71],[15,113092,113094,113097],{"className":113093},[57],[15,113095],{"className":113096,"style":1836},[61],[15,113098,113065],{"className":113099},[66],[6156,113101,113102,113104],{},[6174,113103,75113],{},[6174,113105,113106],{},[15,113107,113109,113134],{"className":113108,"translate":19},[18],[15,113110,113112],{"className":113111},[23],[25,113113,113114],{"xmlns":27},[29,113115,113116,113131],{},[32,113117,113118,113121,113123,113126,113128],{},[1209,113119,113120],{},"9",[35,113122,6623],{"mathvariant":138},[1209,113124,113125],{},"14",[39,113127,31988],{},[1209,113129,113130],{},"0.643",[45,113132,113133],{"encoding":47},"9\u002F14\\approx0.643",[15,113135,113137,113156],{"className":113136,"ariaHidden":53},[52],[15,113138,113140,113143,113147,113150,113153],{"className":113139},[57],[15,113141],{"className":113142,"style":154},[61],[15,113144,113146],{"className":113145},[66],"9\u002F14",[15,113148],{"className":113149,"style":173},[71],[15,113151,31988],{"className":113152},[177],[15,113154],{"className":113155,"style":173},[71],[15,113157,113159,113162],{"className":113158},[57],[15,113160],{"className":113161,"style":1836},[61],[15,113163,113130],{"className":113164},[66],[6156,113166,113167,113305],{},[6174,113168,113169,113170],{},"后验预测 ",[15,113171,113173,113205],{"className":113172,"translate":19},[18],[15,113174,113176],{"className":113175},[23],[25,113177,113178],{"xmlns":27},[29,113179,113180,113202],{},[32,113181,113182,113184,113186,113192,113194,113196,113198,113200],{},[35,113183,11],{},[39,113185,120],{"stretchy":119},[1057,113187,113188,113190],{},[35,113189,123],{},[1274,113191,67801],{},[39,113193,129],{},[1209,113195,1211],{},[39,113197,34425],{},[35,113199,32453],{},[39,113201,126],{"stretchy":119},[45,113203,113204],{"encoding":47},"p(x_{\\text{new}}=1\\mid D)",[15,113206,113208,113275,113293],{"className":113207,"ariaHidden":53},[52],[15,113209,113211,113214,113217,113220,113266,113269,113272],{"className":113210},[57],[15,113212],{"className":113213,"style":154},[61],[15,113215,11],{"className":113216},[66,67],[15,113218,120],{"className":113219},[162],[15,113221,113223,113226],{"className":113222},[66],[15,113224,123],{"className":113225},[66,67],[15,113227,113229],{"className":113228},[198],[15,113230,113232,113258],{"className":113231},[202,1090],[15,113233,113235,113255],{"className":113234},[206],[15,113236,113238],{"className":113237,"style":2198},[210],[15,113239,113240,113243],{"style":1584},[15,113241],{"className":113242,"style":218},[217],[15,113244,113246],{"className":113245},[222,223,224,225],[15,113247,113249],{"className":113248},[66,225],[15,113250,113252],{"className":113251},[66,1711,225],[15,113253,67801],{"className":113254},[66,225],[15,113256,1114],{"className":113257},[1113],[15,113259,113261],{"className":113260},[206],[15,113262,113264],{"className":113263,"style":1121},[210],[15,113265],{},[15,113267],{"className":113268,"style":173},[71],[15,113270,129],{"className":113271},[177],[15,113273],{"className":113274,"style":173},[71],[15,113276,113278,113281,113284,113287,113290],{"className":113277},[57],[15,113279],{"className":113280,"style":154},[61],[15,113282,1211],{"className":113283},[66],[15,113285],{"className":113286,"style":173},[71],[15,113288,34425],{"className":113289},[177],[15,113291],{"className":113292,"style":173},[71],[15,113294,113296,113299,113302],{"className":113295},[57],[15,113297],{"className":113298,"style":154},[61],[15,113300,32453],{"className":113301,"style":23020},[66,67],[15,113303,126],{"className":113304},[169],[6174,113306,113307],{},[15,113308,113310,113323],{"className":113309,"translate":19},[18],[15,113311,113313],{"className":113312},[23],[25,113314,113315],{"xmlns":27},[29,113316,113317,113321],{},[32,113318,113319],{},[1209,113320,113065],{},[45,113322,113065],{"encoding":47},[15,113324,113326],{"className":113325,"ariaHidden":53},[52],[15,113327,113329,113332],{"className":113328},[57],[15,113330],{"className":113331,"style":1836},[61],[15,113333,113065],{"className":113334},[66],[11,113336,113337,113338,113366,113367,113395,113396,113425,113426,113455,113456,113515,113516,1124],{},"后验均值被先验从 ",[15,113339,113341,113354],{"className":113340,"translate":19},[18],[15,113342,113344],{"className":113343},[23],[25,113345,113346],{"xmlns":27},[29,113347,113348,113352],{},[32,113349,113350],{},[1209,113351,113002],{},[45,113353,113002],{"encoding":47},[15,113355,113357],{"className":113356,"ariaHidden":53},[52],[15,113358,113360,113363],{"className":113359},[57],[15,113361],{"className":113362,"style":1836},[61],[15,113364,113002],{"className":113365},[66]," 拉回到 ",[15,113368,113370,113383],{"className":113369,"translate":19},[18],[15,113371,113373],{"className":113372},[23],[25,113374,113375],{"xmlns":27},[29,113376,113377,113381],{},[32,113378,113379],{},[1209,113380,113065],{},[45,113382,113065],{"encoding":47},[15,113384,113386],{"className":113385,"ariaHidden":53},[52],[15,113387,113389,113392],{"className":113388},[57],[15,113390],{"className":113391,"style":1836},[61],[15,113393,113065],{"className":113394},[66],"，这就是收缩。数据越多，拉力越弱：若观测到 ",[15,113397,113399,113413],{"className":113398,"translate":19},[18],[15,113400,113402],{"className":113401},[23],[25,113403,113404],{"xmlns":27},[29,113405,113406,113411],{},[32,113407,113408],{},[1209,113409,113410],{},"100",[45,113412,113410],{"encoding":47},[15,113414,113416],{"className":113415,"ariaHidden":53},[52],[15,113417,113419,113422],{"className":113418},[57],[15,113420],{"className":113421,"style":1836},[61],[15,113423,113410],{"className":113424},[66]," 次中 ",[15,113427,113429,113443],{"className":113428,"translate":19},[18],[15,113430,113432],{"className":113431},[23],[25,113433,113434],{"xmlns":27},[29,113435,113436,113441],{},[32,113437,113438],{},[1209,113439,113440],{},"70",[45,113442,113440],{"encoding":47},[15,113444,113446],{"className":113445,"ariaHidden":53},[52],[15,113447,113449,113452],{"className":113448},[57],[15,113450],{"className":113451,"style":1836},[61],[15,113453,113440],{"className":113454},[66]," 次正面，后验均值变为 ",[15,113457,113459,113484],{"className":113458,"translate":19},[18],[15,113460,113462],{"className":113461},[23],[25,113463,113464],{"xmlns":27},[29,113465,113466,113481],{},[32,113467,113468,113471,113473,113476,113478],{},[1209,113469,113470],{},"73",[35,113472,6623],{"mathvariant":138},[1209,113474,113475],{},"106",[39,113477,31988],{},[1209,113479,113480],{},"0.689",[45,113482,113483],{"encoding":47},"73\u002F106\\approx0.689",[15,113485,113487,113506],{"className":113486,"ariaHidden":53},[52],[15,113488,113490,113493,113497,113500,113503],{"className":113489},[57],[15,113491],{"className":113492,"style":154},[61],[15,113494,113496],{"className":113495},[66],"73\u002F106",[15,113498],{"className":113499,"style":173},[71],[15,113501,31988],{"className":113502},[177],[15,113504],{"className":113505,"style":173},[71],[15,113507,113509,113512],{"className":113508},[57],[15,113510],{"className":113511,"style":1836},[61],[15,113513,113480],{"className":113514},[66],"，已经非常接近 ",[15,113517,113519,113532],{"className":113518,"translate":19},[18],[15,113520,113522],{"className":113521},[23],[25,113523,113524],{"xmlns":27},[29,113525,113526,113530],{},[32,113527,113528],{},[1209,113529,113002],{},[45,113531,113002],{"encoding":47},[15,113533,113535],{"className":113534,"ariaHidden":53},[52],[15,113536,113538,113541],{"className":113537},[57],[15,113539],{"className":113540,"style":1836},[61],[15,113542,113002],{"className":113543},[66],[11,113545,113546,113547,113579,113580,113608,113609,113637,113638,113666],{},"如果决策是「猜下一次是否为正面」，且两类错误的代价相同，那么只要后验预测大于 ",[15,113548,113550,113567],{"className":113549,"translate":19},[18],[15,113551,113553],{"className":113552},[23],[25,113554,113555],{"xmlns":27},[29,113556,113557,113565],{},[32,113558,113559,113561,113563],{},[1209,113560,1211],{},[35,113562,6623],{"mathvariant":138},[1209,113564,1263],{},[45,113566,71801],{"encoding":47},[15,113568,113570],{"className":113569,"ariaHidden":53},[52],[15,113571,113573,113576],{"className":113572},[57],[15,113574],{"className":113575,"style":154},[61],[15,113577,71801],{"className":113578},[66]," 就猜正面，",[15,113581,113583,113596],{"className":113582,"translate":19},[18],[15,113584,113586],{"className":113585},[23],[25,113587,113588],{"xmlns":27},[29,113589,113590,113594],{},[32,113591,113592],{},[1209,113593,113065],{},[45,113595,113065],{"encoding":47},[15,113597,113599],{"className":113598,"ariaHidden":53},[52],[15,113600,113602,113605],{"className":113601},[57],[15,113603],{"className":113604,"style":1836},[61],[15,113606,113065],{"className":113607},[66]," 满足条件。若两类错误的代价不同：把反面误判为正面的代价为 ",[15,113610,113612,113625],{"className":113611,"translate":19},[18],[15,113613,113615],{"className":113614},[23],[25,113616,113617],{"xmlns":27},[29,113618,113619,113623],{},[32,113620,113621],{},[1209,113622,1263],{},[45,113624,1263],{"encoding":47},[15,113626,113628],{"className":113627,"ariaHidden":53},[52],[15,113629,113631,113634],{"className":113630},[57],[15,113632],{"className":113633,"style":1836},[61],[15,113635,1263],{"className":113636},[66],"，把正面误判为反面的代价为 ",[15,113639,113641,113654],{"className":113640,"translate":19},[18],[15,113642,113644],{"className":113643},[23],[25,113645,113646],{"xmlns":27},[29,113647,113648,113652],{},[32,113649,113650],{},[1209,113651,1211],{},[45,113653,1211],{"encoding":47},[15,113655,113657],{"className":113656,"ariaHidden":53},[52],[15,113658,113660,113663],{"className":113659},[57],[15,113661],{"className":113662,"style":1836},[61],[15,113664,1211],{"className":113665},[66],"，则阈值变为",[15,113668,113670],{"className":113669,"translate":19},[294],[15,113671,113673,113745],{"className":113672,"translate":19},[18],[15,113674,113676],{"className":113675},[23],[25,113677,113678],{"xmlns":27,"display":303},[29,113679,113680,113742],{},[32,113681,113682,113718,113720,113732,113734,113740],{},[6465,113683,113684,113694],{},[1057,113685,113686,113688],{},[35,113687,2256],{},[32,113689,113690,113692],{},[35,113691,74037],{"mathvariant":138},[35,113693,28394],{"mathvariant":138},[32,113695,113696,113706,113708],{},[1057,113697,113698,113700],{},[35,113699,2256],{},[32,113701,113702,113704],{},[35,113703,74037],{"mathvariant":138},[35,113705,28394],{"mathvariant":138},[39,113707,1245],{},[1057,113709,113710,113712],{},[35,113711,2256],{},[32,113713,113714,113716],{},[35,113715,74037],{"mathvariant":138},[35,113717,28530],{"mathvariant":138},[39,113719,129],{},[6465,113721,113722,113724],{},[1209,113723,1263],{},[32,113725,113726,113728,113730],{},[1209,113727,1263],{},[39,113729,1245],{},[1209,113731,1211],{},[39,113733,129],{},[6465,113735,113736,113738],{},[1209,113737,1263],{},[1209,113739,8673],{},[39,113741,335],{"separator":53},[45,113743,113744],{"encoding":47},"\\frac{c_{\\mathrm{FP}}}{c_{\\mathrm{FP}}+c_{\\mathrm{FN}}}\n=\\frac{2}{2+1}=\\frac23,",[15,113746,113748,113966,114055],{"className":113747,"ariaHidden":53},[52],[15,113749,113751,113754,113957,113960,113963],{"className":113750},[57],[15,113752],{"className":113753,"style":52418},[61],[15,113755,113757,113760,113954],{"className":113756},[66],[15,113758],{"className":113759},[162,23412],[15,113761,113763],{"className":113762},[6465],[15,113764,113766,113946],{"className":113765},[202,1090],[15,113767,113769,113943],{"className":113768},[206],[15,113770,113772,113881,113889],{"className":113771,"style":39128},[210],[15,113773,113774,113777],{"style":23428},[15,113775],{"className":113776,"style":1368},[217],[15,113778,113780,113826,113829,113832,113835],{"className":113779},[66],[15,113781,113783,113786],{"className":113782},[66],[15,113784,2256],{"className":113785},[66,67],[15,113787,113789],{"className":113788},[198],[15,113790,113792,113818],{"className":113791},[202,1090],[15,113793,113795,113815],{"className":113794},[206],[15,113796,113798],{"className":113797,"style":3636},[210],[15,113799,113800,113803],{"style":1584},[15,113801],{"className":113802,"style":218},[217],[15,113804,113806],{"className":113805},[222,223,224,225],[15,113807,113809],{"className":113808},[66,225],[15,113810,113812],{"className":113811},[66,225],[15,113813,74087],{"className":113814},[66,4427,225],[15,113816,1114],{"className":113817},[1113],[15,113819,113821],{"className":113820},[206],[15,113822,113824],{"className":113823,"style":1121},[210],[15,113825],{},[15,113827],{"className":113828,"style":72},[71],[15,113830,1245],{"className":113831},[76],[15,113833],{"className":113834,"style":72},[71],[15,113836,113838,113841],{"className":113837},[66],[15,113839,2256],{"className":113840},[66,67],[15,113842,113844],{"className":113843},[198],[15,113845,113847,113873],{"className":113846},[202,1090],[15,113848,113850,113870],{"className":113849},[206],[15,113851,113853],{"className":113852,"style":3636},[210],[15,113854,113855,113858],{"style":1584},[15,113856],{"className":113857,"style":218},[217],[15,113859,113861],{"className":113860},[222,223,224,225],[15,113862,113864],{"className":113863},[66,225],[15,113865,113867],{"className":113866},[66,225],[15,113868,74169],{"className":113869},[66,4427,225],[15,113871,1114],{"className":113872},[1113],[15,113874,113876],{"className":113875},[206],[15,113877,113879],{"className":113878,"style":1121},[210],[15,113880],{},[15,113882,113883,113886],{"style":23440},[15,113884],{"className":113885,"style":1368},[217],[15,113887],{"className":113888,"style":4438},[23447],[15,113890,113891,113894],{"style":23450},[15,113892],{"className":113893,"style":1368},[217],[15,113895,113897],{"className":113896},[66],[15,113898,113900,113903],{"className":113899},[66],[15,113901,2256],{"className":113902},[66,67],[15,113904,113906],{"className":113905},[198],[15,113907,113909,113935],{"className":113908},[202,1090],[15,113910,113912,113932],{"className":113911},[206],[15,113913,113915],{"className":113914,"style":3636},[210],[15,113916,113917,113920],{"style":1584},[15,113918],{"className":113919,"style":218},[217],[15,113921,113923],{"className":113922},[222,223,224,225],[15,113924,113926],{"className":113925},[66,225],[15,113927,113929],{"className":113928},[66,225],[15,113930,74087],{"className":113931},[66,4427,225],[15,113933,1114],{"className":113934},[1113],[15,113936,113938],{"className":113937},[206],[15,113939,113941],{"className":113940,"style":1121},[210],[15,113942],{},[15,113944,1114],{"className":113945},[1113],[15,113947,113949],{"className":113948},[206],[15,113950,113952],{"className":113951,"style":52615},[210],[15,113953],{},[15,113955],{"className":113956},[169,23412],[15,113958],{"className":113959,"style":173},[71],[15,113961,129],{"className":113962},[177],[15,113964],{"className":113965,"style":173},[71],[15,113967,113969,113972,114046,114049,114052],{"className":113968},[57],[15,113970],{"className":113971,"style":44665},[61],[15,113973,113975,113978,114043],{"className":113974},[66],[15,113976],{"className":113977},[162,23412],[15,113979,113981],{"className":113980},[6465],[15,113982,113984,114035],{"className":113983},[202,1090],[15,113985,113987,114032],{"className":113986},[206],[15,113988,113990,114013,114021],{"className":113989,"style":23425},[210],[15,113991,113992,113995],{"style":23428},[15,113993],{"className":113994,"style":1368},[217],[15,113996,113998,114001,114004,114007,114010],{"className":113997},[66],[15,113999,1263],{"className":114000},[66],[15,114002],{"className":114003,"style":72},[71],[15,114005,1245],{"className":114006},[76],[15,114008],{"className":114009,"style":72},[71],[15,114011,1211],{"className":114012},[66],[15,114014,114015,114018],{"style":23440},[15,114016],{"className":114017,"style":1368},[217],[15,114019],{"className":114020,"style":4438},[23447],[15,114022,114023,114026],{"style":23450},[15,114024],{"className":114025,"style":1368},[217],[15,114027,114029],{"className":114028},[66],[15,114030,1263],{"className":114031},[66],[15,114033,1114],{"className":114034},[1113],[15,114036,114038],{"className":114037},[206],[15,114039,114041],{"className":114040,"style":38565},[210],[15,114042],{},[15,114044],{"className":114045},[169,23412],[15,114047],{"className":114048,"style":173},[71],[15,114050,129],{"className":114051},[177],[15,114053],{"className":114054,"style":173},[71],[15,114056,114058,114061,114123],{"className":114057},[57],[15,114059],{"className":114060,"style":49082},[61],[15,114062,114064,114067,114120],{"className":114063},[66],[15,114065],{"className":114066},[162,23412],[15,114068,114070],{"className":114069},[6465],[15,114071,114073,114112],{"className":114072},[202,1090],[15,114074,114076,114109],{"className":114075},[206],[15,114077,114079,114090,114098],{"className":114078,"style":23425},[210],[15,114080,114081,114084],{"style":23428},[15,114082],{"className":114083,"style":1368},[217],[15,114085,114087],{"className":114086},[66],[15,114088,8673],{"className":114089},[66],[15,114091,114092,114095],{"style":23440},[15,114093],{"className":114094,"style":1368},[217],[15,114096],{"className":114097,"style":4438},[23447],[15,114099,114100,114103],{"style":23450},[15,114101],{"className":114102,"style":1368},[217],[15,114104,114106],{"className":114105},[66],[15,114107,1263],{"className":114108},[66],[15,114110,1114],{"className":114111},[1113],[15,114113,114115],{"className":114114},[206],[15,114116,114118],{"className":114117,"style":23469},[210],[15,114119],{},[15,114121],{"className":114122},[169,23412],[15,114124,335],{"className":114125},[422],[11,114127,956,114128,114185],{},[15,114129,114131,114153],{"className":114130,"translate":19},[18],[15,114132,114134],{"className":114133},[23],[25,114135,114136],{"xmlns":27},[29,114137,114138,114150],{},[32,114139,114140,114142,114144,114146,114148],{},[1209,114141,113065],{},[39,114143,25767],{},[1209,114145,1263],{},[35,114147,6623],{"mathvariant":138},[1209,114149,8673],{},[45,114151,114152],{"encoding":47},"0.625\u003C2\u002F3",[15,114154,114156,114175],{"className":114155,"ariaHidden":53},[52],[15,114157,114159,114163,114166,114169,114172],{"className":114158},[57],[15,114160],{"className":114161,"style":114162},[61],"height:0.6835em;vertical-align:-0.0391em;",[15,114164,113065],{"className":114165},[66],[15,114167],{"className":114168,"style":173},[71],[15,114170,25767],{"className":114171},[177],[15,114173],{"className":114174,"style":173},[71],[15,114176,114178,114181],{"className":114177},[57],[15,114179],{"className":114180,"style":154},[61],[15,114182,114184],{"className":114183},[66],"2\u002F3","，最优行动变成猜反面。同一个后验，换一个损失，结论就变了。这正是贝叶斯决策理论与「最大后验」的分水岭：后验只有一个，行动却取决于代价。",[1904,114187,114189],{"id":114188},"_82-高斯先验与收缩","8.2 高斯先验与收缩",[11,114191,114192,114193,114343,114344,114494,114495,114557,114558,114586],{},"设观测模型 ",[15,114194,114196,114232],{"className":114195,"translate":19},[18],[15,114197,114199],{"className":114198},[23],[25,114200,114201],{"xmlns":27},[29,114202,114203,114229],{},[32,114204,114205,114211,114213,114215,114217,114219,114221,114227],{},[1057,114206,114207,114209],{},[35,114208,1224],{},[35,114210,1205],{},[39,114212,28386],{},[35,114214,28530],{"mathvariant":445},[39,114216,120],{"stretchy":119},[35,114218,61614],{},[39,114220,335],{"separator":53},[131,114222,114223,114225],{},[35,114224,6632],{},[1209,114226,1263],{},[39,114228,126],{"stretchy":119},[45,114230,114231],{"encoding":47},"y_i\\sim\\mathcal N(\\theta,\\sigma^2)",[15,114233,114235,114290],{"className":114234,"ariaHidden":53},[52],[15,114236,114238,114241,114281,114284,114287],{"className":114237},[57],[15,114239],{"className":114240,"style":471},[61],[15,114242,114244,114247],{"className":114243},[66],[15,114245,1224],{"className":114246,"style":1515},[66,67],[15,114248,114250],{"className":114249},[198],[15,114251,114253,114273],{"className":114252},[202,1090],[15,114254,114256,114270],{"className":114255},[206],[15,114257,114259],{"className":114258,"style":1528},[210],[15,114260,114261,114264],{"style":1531},[15,114262],{"className":114263,"style":218},[217],[15,114265,114267],{"className":114266},[222,223,224,225],[15,114268,1205],{"className":114269},[66,67,225],[15,114271,1114],{"className":114272},[1113],[15,114274,114276],{"className":114275},[206],[15,114277,114279],{"className":114278,"style":1121},[210],[15,114280],{},[15,114282],{"className":114283,"style":173},[71],[15,114285,28386],{"className":114286},[177],[15,114288],{"className":114289,"style":173},[71],[15,114291,114293,114296,114299,114302,114305,114308,114311,114340],{"className":114292},[57],[15,114294],{"className":114295,"style":9149},[61],[15,114297,28530],{"className":114298,"style":83507},[66,494],[15,114300,120],{"className":114301},[162],[15,114303,61614],{"className":114304,"style":23020},[66,67],[15,114306,335],{"className":114307},[422],[15,114309],{"className":114310,"style":697},[71],[15,114312,114314,114317],{"className":114313},[66],[15,114315,6632],{"className":114316,"style":1515},[66,67],[15,114318,114320],{"className":114319},[198],[15,114321,114323],{"className":114322},[202],[15,114324,114326],{"className":114325},[206],[15,114327,114329],{"className":114328,"style":6738},[210],[15,114330,114331,114334],{"style":213},[15,114332],{"className":114333,"style":218},[217],[15,114335,114337],{"className":114336},[222,223,224,225],[15,114338,1263],{"className":114339},[66,225],[15,114341,126],{"className":114342},[169],"，先验 ",[15,114345,114347,114383],{"className":114346,"translate":19},[18],[15,114348,114350],{"className":114349},[23],[25,114351,114352],{"xmlns":27},[29,114353,114354,114380],{},[32,114355,114356,114358,114360,114362,114364,114370,114372,114378],{},[35,114357,61614],{},[39,114359,28386],{},[35,114361,28530],{"mathvariant":445},[39,114363,120],{"stretchy":119},[1057,114365,114366,114368],{},[35,114367,83394],{},[1209,114369,1804],{},[39,114371,335],{"separator":53},[131,114373,114374,114376],{},[35,114375,95586],{},[1209,114377,1263],{},[39,114379,126],{"stretchy":119},[45,114381,114382],{"encoding":47},"\\theta\\sim\\mathcal N(\\mu_0,\\tau^2)",[15,114384,114386,114404],{"className":114385,"ariaHidden":53},[52],[15,114387,114389,114392,114395,114398,114401],{"className":114388},[57],[15,114390],{"className":114391,"style":840},[61],[15,114393,61614],{"className":114394,"style":23020},[66,67],[15,114396],{"className":114397,"style":173},[71],[15,114399,28386],{"className":114400},[177],[15,114402],{"className":114403,"style":173},[71],[15,114405,114407,114410,114413,114416,114456,114459,114462,114491],{"className":114406},[57],[15,114408],{"className":114409,"style":9149},[61],[15,114411,28530],{"className":114412,"style":83507},[66,494],[15,114414,120],{"className":114415},[162],[15,114417,114419,114422],{"className":114418},[66],[15,114420,83394],{"className":114421},[66,67],[15,114423,114425],{"className":114424},[198],[15,114426,114428,114448],{"className":114427},[202,1090],[15,114429,114431,114445],{"className":114430},[206],[15,114432,114434],{"className":114433,"style":2139},[210],[15,114435,114436,114439],{"style":1584},[15,114437],{"className":114438,"style":218},[217],[15,114440,114442],{"className":114441},[222,223,224,225],[15,114443,1804],{"className":114444},[66,225],[15,114446,1114],{"className":114447},[1113],[15,114449,114451],{"className":114450},[206],[15,114452,114454],{"className":114453,"style":1121},[210],[15,114455],{},[15,114457,335],{"className":114458},[422],[15,114460],{"className":114461,"style":697},[71],[15,114463,114465,114468],{"className":114464},[66],[15,114466,95586],{"className":114467,"style":95644},[66,67],[15,114469,114471],{"className":114470},[198],[15,114472,114474],{"className":114473},[202],[15,114475,114477],{"className":114476},[206],[15,114478,114480],{"className":114479,"style":6738},[210],[15,114481,114482,114485],{"style":213},[15,114483],{"className":114484,"style":218},[217],[15,114486,114488],{"className":114487},[222,223,224,225],[15,114489,1263],{"className":114490},[66,225],[15,114492,126],{"className":114493},[169],"，样本均值为 ",[15,114496,114498,114516],{"className":114497,"translate":19},[18],[15,114499,114501],{"className":114500},[23],[25,114502,114503],{"xmlns":27},[29,114504,114505,114513],{},[32,114506,114507],{},[4294,114508,114509,114511],{"accent":53},[35,114510,123],{},[39,114512,100918],{},[45,114514,114515],{"encoding":47},"\\bar x",[15,114517,114519],{"className":114518,"ariaHidden":53},[52],[15,114520,114522,114525],{"className":114521},[57],[15,114523],{"className":114524,"style":100995},[61],[15,114526,114528],{"className":114527},[66,78139],[15,114529,114531],{"className":114530},[202],[15,114532,114534],{"className":114533},[206],[15,114535,114537,114545],{"className":114536,"style":100995},[210],[15,114538,114539,114542],{"style":1386},[15,114540],{"className":114541,"style":1368},[217],[15,114543,123],{"className":114544},[66,67],[15,114546,114547,114550],{"style":1386},[15,114548],{"className":114549,"style":1368},[217],[15,114551,114554],{"className":114552,"style":114553},[78167],"left:-0.2222em;",[15,114555,100918],{"className":114556},[66],"，样本量为 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x\u002F\\sigma^2}{1\u002F\\tau^2+n\u002F\\sigma^2},\n\\qquad\n\\frac{1}{\\tau_n^2}=\\frac{1}{\\tau^2}+\\frac{n}{\\sigma^2}.",[15,114923,114925,114980,115394,115497],{"className":114924,"ariaHidden":53},[52],[15,114926,114928,114931,114971,114974,114977],{"className":114927},[57],[15,114929],{"className":114930,"style":471},[61],[15,114932,114934,114937],{"className":114933},[66],[15,114935,83394],{"className":114936},[66,67],[15,114938,114940],{"className":114939},[198],[15,114941,114943,114963],{"className":114942},[202,1090],[15,114944,114946,114960],{"className":114945},[206],[15,114947,114949],{"className":114948,"style":2198},[210],[15,114950,114951,114954],{"style":1584},[15,114952],{"className":114953,"style":218},[217],[15,114955,114957],{"className":114956},[222,223,224,225],[15,114958,37],{"className":114959},[66,67,225],[15,114961,1114],{"className":114962},[1113],[15,114964,114966],{"className":114965},[206],[15,114967,114969],{"className":114968,"style":1121},[210],[15,114970],{},[15,114972],{"className":114973,"style":173},[71],[15,114975,129],{"className":114976},[177],[15,114978],{"className":114979,"style":173},[71],[15,114981,114983,114986,115265,115268,115271,115274,115385,115388,115391],{"className":114982},[57],[15,114984],{"className":114985,"style":45939},[61],[15,114987,114989,114992,115262],{"className":114988},[66],[15,114990],{"className":114991},[162,23412],[15,114993,114995],{"className":114994},[6465],[15,114996,114998,115254],{"className":114997},[202,1090],[15,114999,115001,115251],{"className":115000},[206],[15,115002,115004,115088,115096],{"className":115003,"style":45958},[210],[15,115005,115006,115009],{"style":23428},[15,115007],{"className":115008,"style":1368},[217],[15,115010,115012,115015,115044,115047,115050,115053,115056,115059],{"className":115011},[66],[15,115013,103600],{"className":115014},[66],[15,115016,115018,115021],{"className":115017},[66],[15,115019,95586],{"className":115020,"style":95644},[66,67],[15,115022,115024],{"className":115023},[198],[15,115025,115027],{"className":115026},[202],[15,115028,115030],{"className":115029},[206],[15,115031,115033],{"className":115032,"style":46247},[210],[15,115034,115035,115038],{"style":46250},[15,115036],{"className":115037,"style":218},[217],[15,115039,115041],{"className":115040},[222,223,224,225],[15,115042,1263],{"className":115043},[66,225],[15,115045],{"className":115046,"style":72},[71],[15,115048,1245],{"className":115049},[76],[15,115051],{"className":115052,"style":72},[71],[15,115054,37],{"className":115055},[66,67],[15,115057,6623],{"className":115058},[66],[15,115060,115062,115065],{"className":115061},[66],[15,115063,6632],{"className":115064,"style":1515},[66,67],[15,115066,115068],{"className":115067},[198],[15,115069,115071],{"className":115070},[202],[15,115072,115074],{"className":115073},[206],[15,115075,115077],{"className":115076,"style":46247},[210],[15,115078,115079,115082],{"style":46250},[15,115080],{"className":115081,"style":218},[217],[15,115083,115085],{"className":115084},[222,223,224,225],[15,115086,1263],{"className":115087},[66,225],[15,115089,115090,115093],{"style":23440},[15,115091],{"className":115092,"style":1368},[217],[15,115094],{"className":115095,"style":4438},[23447],[15,115097,115098,115101],{"style":23450},[15,115099],{"className":115100,"style":1368},[217],[15,115102,115104,115144,115147,115176,115179,115182,115185,115188,115219,115222],{"className":115103},[66],[15,115105,115107,115110],{"className":115106},[66],[15,115108,83394],{"className":115109},[66,67],[15,115111,115113],{"className":115112},[198],[15,115114,115116,115136],{"className":115115},[202,1090],[15,115117,115119,115133],{"className":115118},[206],[15,115120,115122],{"className":115121,"style":2139},[210],[15,115123,115124,115127],{"style":1584},[15,115125],{"className":115126,"style":218},[217],[15,115128,115130],{"className":115129},[222,223,224,225],[15,115131,1804],{"className":115132},[66,225],[15,115134,1114],{"className":115135},[1113],[15,115137,115139],{"className":115138},[206],[15,115140,115142],{"className":115141,"style":1121},[210],[15,115143],{},[15,115145,6623],{"className":115146},[66],[15,115148,115150,115153],{"className":115149},[66],[15,115151,95586],{"className":115152,"style":95644},[66,67],[15,115154,115156],{"className":115155},[198],[15,115157,115159],{"className":115158},[202],[15,115160,115162],{"className":115161},[206],[15,115163,115165],{"className":115164,"style":6738},[210],[15,115166,115167,115170],{"style":213},[15,115168],{"className":115169,"style":218},[217],[15,115171,115173],{"className":115172},[222,223,224,225],[15,115174,1263],{"className":115175},[66,225],[15,115177],{"className":115178,"style":72},[71],[15,115180,1245],{"className":115181},[76],[15,115183],{"className":115184,"style":72},[71],[15,115186,37],{"className":115187},[66,67],[15,115189,115191],{"className":115190},[66,78139],[15,115192,115194],{"className":115193},[202],[15,115195,115197],{"clas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写成加权平均：",[15,115668,115670],{"className":115669,"translate":19},[294],[15,115671,115673,115775],{"className":115672,"translate":19},[18],[15,115674,115676],{"className":115675},[23],[25,115677,115678],{"xmlns":27,"display":303},[29,115679,115680,115772],{},[32,115681,115682,115688,115690,115724,115726,115732,115734,115762,115764,115770],{},[1057,115683,115684,115686],{},[35,115685,83394],{},[35,115687,37],{},[39,115689,129],{},[6465,115691,115692,115704],{},[32,115693,115694,115700,115702],{},[131,115695,115696,115698],{},[35,115697,6632],{},[1209,115699,1263],{},[35,115701,6623],{"mathvariant":138},[35,115703,37],{},[32,115705,115706,115712,115714,115720,115722],{},[131,115707,115708,115710],{},[35,115709,95586],{},[1209,115711,1263],{},[39,115713,1245],{},[131,115715,115716,115718],{},[35,115717,6632],{},[1209,115719,1263],{},[35,115721,6623],{"mathvariant":138},[35,115723,37],{},[1274,115725,2462],{},[1057,115727,115728,115730],{},[35,115729,83394],{},[1209,115731,1804],{},[39,115733,1245],{},[6465,115735,115736,115742],{},[131,115737,115738,115740],{},[35,115739,95586],{},[1209,115741,1263],{},[32,115743,115744,115750,115752,115758,115760],{},[131,115745,115746,115748],{},[35,115747,95586],{},[1209,115749,1263],{},[39,115751,1245],{},[131,115753,115754,115756],{},[35,115755,6632],{},[1209,115757,1263],{},[35,115759,6623],{"mathvariant":138},[35,115761,37],{},[1274,115763,2462],{},[4294,115765,115766,115768],{"accent":53},[35,115767,123],{},[39,115769,100918],{},[35,115771,3571],{"mathvariant":138},[45,115773,115774],{"encoding":47},"\\mu_n\n=\n\\frac{\\sigma^2\u002Fn}{\\tau^2+\\sigma^2\u002Fn}\\,\\mu_0\n+\n\\frac{\\tau^2}{\\tau^2+\\sigma^2\u002Fn}\\,\\bar x 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x=1",[15,116479,116481,116527],{"className":116480,"ariaHidden":53},[52],[15,116482,116484,116487,116518,116521,116524],{"className":116483},[57],[15,116485],{"className":116486,"style":100995},[61],[15,116488,116490],{"className":116489},[66,78139],[15,116491,116493],{"className":116492},[202],[15,116494,116496],{"className":116495},[206],[15,116497,116499,116507],{"className":116498,"style":100995},[210],[15,116500,116501,116504],{"style":1386},[15,116502],{"className":116503,"style":1368},[217],[15,116505,123],{"className":116506},[66,67],[15,116508,116509,116512],{"style":1386},[15,116510],{"className":116511,"style":1368},[217],[15,116513,116515],{"className":116514,"style":114553},[78167],[15,116516,100918],{"className":116517},[66],[15,116519],{"className":116520,"style":173},[71],[15,116522,129],{"className":116523},[177],[15,116525],{"className":116526,"style":173},[71],[15,116528,116530,116533],{"className":116529},[57],[15,116531],{"className":116532,"style":1836},[61],[15,116534,1211],{"className":116535},[66],[6150,116537,116538,116731],{},[6153,116539,116540],{},[6156,116541,116542,116572,116644],{},[6159,116543,116544],{},[15,116545,116547,116560],{"className":116546,"translate":19},[18],[15,116548,116550],{"className":116549},[23],[25,116551,116552],{"xmlns":27},[29,116553,116554,116558],{},[32,116555,116556],{},[35,116557,37],{},[45,116559,37],{"encoding":47},[15,116561,116563],{"className":116562,"ariaHidden":53},[52],[15,116564,116566,116569],{"className":116565},[57],[15,116567],{"className":116568,"style":86},[61],[15,116570,37],{"className":116571},[66,67],[6159,116573,116574,116575],{},"后验均值 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",[15,116648,116650,116670],{"className":116649,"translate":19},[18],[15,116651,116653],{"className":116652},[23],[25,116654,116655],{"xmlns":27},[29,116656,116657,116667],{},[32,116658,116659],{},[1255,116660,116661,116663,116665],{},[35,116662,95586],{},[35,116664,37],{},[1209,116666,1263],{},[45,116668,116669],{"encoding":47},"\\tau_n^2",[15,116671,116673],{"className":116672,"ariaHidden":53},[52],[15,116674,116676,116680],{"className":116675},[57],[15,116677],{"className":116678,"style":116679},[61],"height:1.0611em;vertical-align:-0.247em;",[15,116681,116683,116686],{"className":116682},[66],[15,116684,95586],{"className":116685,"style":95644},[66,67],[15,116687,116689],{"className":116688},[198],[15,116690,116692,116723],{"className":116691},[202,1090],[15,116693,116695,116720],{"className":116694},[206],[15,116696,116698,116709],{"className":116697,"style":6738},[210],[15,116699,116700,116703],{"style":114754},[15,116701],{"className":116702,"style":218},[217],[15,116704,116706],{"className":116705},[222,223,224,225],[15,116707,37],{"className":116708},[66,67,225],[15,116710,116711,116714],{"style":213},[15,116712],{"className":116713,"style":218},[217],[15,116715,116717],{"className":116716},[222,223,224,225],[15,116718,1263],{"className":116719},[66,225],[15,116721,1114],{"className":116722},[1113],[15,116724,116726],{"className":116725},[206],[15,116727,116729],{"className":116728,"style":1686},[210],[15,116730],{},[6169,116732,116733,116850,116973,117097],{},[6156,116734,116735,116737,116794],{},[6174,116736,1211],{},[6174,116738,116739],{},[15,116740,116742,116764],{"className":116741,"translate":19},[18],[15,116743,116745],{"className":116744},[23],[25,116746,116747],{"xmlns":27},[29,116748,116749,116761],{},[32,116750,116751,116753,116755,116757,116759],{},[1209,116752,1211],{},[35,116754,6623],{"mathvariant":138},[1209,116756,1263],{},[39,116758,129],{},[1209,116760,112938],{},[45,116762,116763],{"encoding":47},"1\u002F2=0.5",[15,116765,116767,116785],{"className":116766,"ariaHidden":53},[52],[15,116768,116770,116773,116776,116779,116782],{"className":116769},[57],[15,116771],{"className":116772,"style":154},[61],[15,116774,71801],{"className":116775},[66],[15,116777],{"className":116778,"style":173},[71],[15,116780,129],{"className":116781},[177],[15,116783],{"className":116784,"style":173},[71],[15,116786,116788,116791],{"className":116787},[57],[15,116789],{"className":116790,"style":1836},[61],[15,116792,112938],{"className":116793},[66],[6174,116795,116796],{},[15,116797,116799,116820],{"className":116798,"translate":19},[18],[15,116800,116802],{"className":116801},[23],[25,116803,116804],{"xmlns":27},[29,116805,116806,116818],{},[32,116807,116808,116810,116812,116814,116816],{},[1209,116809,1211],{},[35,116811,6623],{"mathvariant":138},[1209,116813,1263],{},[39,116815,129],{},[1209,116817,112938],{},[45,116819,116763],{"encoding":47},[15,116821,116823,116841],{"className":116822,"ariaHidden":53},[52],[15,116824,116826,116829,116832,116835,116838],{"className":116825},[57],[15,116827],{"className":116828,"style":154},[61],[15,116830,71801],{"className":116831},[66],[15,116833],{"className":116834,"style":173},[71],[15,116836,129],{"className":116837},[177],[15,116839],{"className":116840,"style":173},[71],[15,116842,116844,116847],{"className":116843},[57],[15,116845],{"className":116846,"style":1836},[61],[15,116848,112938],{"className":116849},[66],[6156,116851,116852,116854,116914],{},[6174,116853,15850],{},[6174,116855,116856],{},[15,116857,116859,116883],{"className":116858,"translate":19},[18],[15,116860,116862],{"className":116861},[23],[25,116863,116864],{"xmlns":27},[29,116865,116866,116880],{},[32,116867,116868,116870,116872,116875,116877],{},[1209,116869,15850],{},[35,116871,6623],{"mathvariant":138},[1209,116873,116874],{},"5",[39,116876,129],{},[1209,116878,116879],{},"0.8",[45,116881,116882],{"encoding":47},"4\u002F5=0.8",[15,116884,116886,116905],{"className":116885,"ariaHidden":53},[52],[15,116887,116889,116892,116896,116899,116902],{"className":116888},[57],[15,116890],{"className":116891,"style":154},[61],[15,116893,116895],{"className":116894},[66],"4\u002F5",[15,116897],{"className":116898,"style":173},[71],[15,116900,129],{"className":116901},[177],[15,116903],{"className":116904,"style":173},[71],[15,116906,116908,116911],{"className":116907},[57],[15,116909],{"className":116910,"style":1836},[61],[15,116912,116879],{"className":116913},[66],[6174,116915,116916],{},[15,116917,116919,116942],{"className":116918,"translate":19},[18],[15,116920,116922],{"className":116921},[23],[25,116923,116924],{"xmlns":27},[29,116925,116926,116939],{},[32,116927,116928,116930,116932,116934,116936],{},[1209,116929,1211],{},[35,116931,6623],{"mathvariant":138},[1209,116933,116874],{},[39,116935,129],{},[1209,116937,116938],{},"0.2",[45,116940,116941],{"encoding":47},"1\u002F5=0.2",[15,116943,116945,116964],{"className":116944,"ariaHidden":53},[52],[15,116946,116948,116951,116955,116958,116961],{"className":116947},[57],[15,116949],{"className":116950,"style":154},[61],[15,116952,116954],{"className":116953},[66],"1\u002F5",[15,116956],{"className":116957,"style":173},[71],[15,116959,129],{"className":116960},[177],[15,116962],{"className":116963,"style":173},[71],[15,116965,116967,116970],{"className":116966},[57],[15,116968],{"className":116969,"style":1836},[61],[15,116971,116938],{"className":116972},[66],[6156,116974,116975,116978,117038],{},[6174,116976,116977],{},"25",[6174,116979,116980],{},[15,116981,116983,117007],{"className":116982,"translate":19},[18],[15,116984,116986],{"className":116985},[23],[25,116987,116988],{"xmlns":27},[29,116989,116990,117004],{},[32,116991,116992,116994,116996,116999,117001],{},[1209,116993,116977],{},[35,116995,6623],{"mathvariant":138},[1209,116997,116998],{},"26",[39,117000,31988],{},[1209,117002,117003],{},"0.962",[45,117005,117006],{"encoding":47},"25\u002F26\\approx0.962",[15,117008,117010,117029],{"className":117009,"ariaHidden":53},[52],[15,117011,117013,117016,117020,117023,117026],{"className":117012},[57],[15,117014],{"className":117015,"style":154},[61],[15,117017,117019],{"className":117018},[66],"25\u002F26",[15,117021],{"className":117022,"style":173},[71],[15,117024,31988],{"className":117025},[177],[15,117027],{"className":117028,"style":173},[71],[15,117030,117032,117035],{"className":117031},[57],[15,117033],{"className":117034,"style":1836},[61],[15,117036,117003],{"className":117037},[66],[6174,117039,117040],{},[15,117041,117043,117066],{"className":117042,"translate":19},[18],[15,117044,117046],{"className":117045},[23],[25,117047,117048],{"xmlns":27},[29,117049,117050,117063],{},[32,117051,117052,117054,117056,117058,117060],{},[1209,117053,1211],{},[35,117055,6623],{"mathvariant":138},[1209,117057,116998],{},[39,117059,31988],{},[1209,117061,117062],{},"0.038",[45,117064,117065],{"encoding":47},"1\u002F26\\approx0.038",[15,117067,117069,117088],{"className":117068,"ariaHidden":53},[52],[15,117070,117072,117075,117079,117082,117085],{"className":117071},[57],[15,117073],{"className":117074,"style":154},[61],[15,117076,117078],{"className":117077},[66],"1\u002F26",[15,117080],{"className":117081,"style":173},[71],[15,117083,31988],{"className":117084},[177],[15,117086],{"className":117087,"style":173},[71],[15,117089,117091,117094],{"className":117090},[57],[15,117092],{"className":117093,"style":1836},[61],[15,117095,117062],{"className":117096},[66],[6156,117098,117099,117101,117161],{},[6174,117100,113410],{},[6174,117102,117103],{},[15,117104,117106,117130],{"className":117105,"translate":19},[18],[15,117107,117109],{"className":117108},[23],[25,117110,117111],{"xmlns":27},[29,117112,117113,117127],{},[32,117114,117115,117117,117119,117122,117124],{},[1209,117116,113410],{},[35,117118,6623],{"mathvariant":138},[1209,117120,117121],{},"101",[39,117123,31988],{},[1209,117125,117126],{},"0.990",[45,117128,117129],{"encoding":47},"100\u002F101\\approx0.990",[15,117131,117133,117152],{"className":117132,"ariaHidden":53},[52],[15,117134,117136,117139,117143,117146,117149],{"className":117135},[57],[15,117137],{"className":117138,"style":154},[61],[15,117140,117142],{"className":117141},[66],"100\u002F101",[15,117144],{"className":117145,"style":173},[71],[15,117147,31988],{"className":117148},[177],[15,117150],{"className":117151,"style":173},[71],[15,117153,117155,117158],{"className":117154},[57],[15,117156],{"className":117157,"style":1836},[61],[15,117159,117126],{"className":117160},[66],[6174,117162,117163],{},[15,117164,117166,117189],{"className":117165,"translate":19},[18],[15,117167,117169],{"className":117168},[23],[25,117170,117171],{"xmlns":27},[29,117172,117173,117186],{},[32,117174,117175,117177,117179,117181,117183],{},[1209,117176,1211],{},[35,117178,6623],{"mathvariant":138},[1209,117180,117121],{},[39,117182,31988],{},[1209,117184,117185],{},"0.0099",[45,117187,117188],{"encoding":47},"1\u002F101\\approx0.0099",[15,117190,117192,117211],{"className":117191,"ariaHidden":53},[52],[15,117193,117195,117198,117202,117205,117208],{"className":117194},[57],[15,117196],{"className":117197,"style":154},[61],[15,117199,117201],{"className":117200},[66],"1\u002F101",[15,117203],{"className":117204,"style":173},[71],[15,117206,31988],{"className":117207},[177],[15,117209],{"className":117210,"style":173},[71],[15,117212,117214,117217],{"className":117213},[57],[15,117215],{"className":117216,"style":1836},[61],[15,117218,117185],{"className":117219},[66],[11,117221,117222],{},"样本越多，后验越靠近样本均值，方差越小。先验的作用不是「永远拉住估计」，而是在小样本时提供一个稳定的起点，随着证据积累自动退场。",[1904,117224,117226],{"id":117225},"_83-医疗筛查先验似然与代价","8.3 医疗筛查：先验、似然与代价",[11,117228,117229,117230,117264,117265,117298,117299,117332],{},"设某疾病的患病率为 ",[15,117231,117233,117250],{"className":117232,"translate":19},[18],[15,117234,117236],{"className":117235},[23],[25,117237,117238],{"xmlns":27},[29,117239,117240,117247],{},[32,117241,117242,117244],{},[1209,117243,1211],{},[35,117245,117246],{"mathvariant":138},"%",[45,117248,117249],{"encoding":47},"1\\%",[15,117251,117253],{"className":117252,"ariaHidden":53},[52],[15,117254,117256,117260],{"className":117255},[57],[15,117257],{"className":117258,"style":117259},[61],"height:0.8056em;vertical-align:-0.0556em;",[15,117261,117263],{"className":117262},[66],"1%","，检测的灵敏度为 ",[15,117266,117268,117285],{"className":117267,"translate":19},[18],[15,117269,117271],{"className":117270},[23],[25,117272,117273],{"xmlns":27},[29,117274,117275,117282],{},[32,117276,117277,117280],{},[1209,117278,117279],{},"99",[35,117281,117246],{"mathvariant":138},[45,117283,117284],{"encoding":47},"99\\%",[15,117286,117288],{"className":117287,"ariaHidden":53},[52],[15,117289,117291,117294],{"className":117290},[57],[15,117292],{"className":117293,"style":117259},[61],[15,117295,117297],{"className":117296},[66],"99%","，特异度为 ",[15,117300,117302,117319],{"className":117301,"translate":19},[18],[15,117303,117305],{"className":117304},[23],[25,117306,117307],{"xmlns":27},[29,117308,117309,117316],{},[32,117310,117311,117314],{},[1209,117312,117313],{},"95",[35,117315,117246],{"mathvariant":138},[45,117317,117318],{"encoding":47},"95\\%",[15,117320,117322],{"className":117321,"ariaHidden":53},[52],[15,117323,117325,117328],{"className":117324},[57],[15,117326],{"className":117327,"style":117259},[61],[15,117329,117331],{"className":117330},[66],"95%","。一个人检测阳性，患病概率是多少？",[11,117334,117335],{},"用贝叶斯公式：",[15,117337,117339],{"className":117338,"translate":19},[294],[15,117340,117342,117540],{"className":117341,"translate":19},[18],[15,117343,117345],{"className":117344},[23],[25,117346,117347],{"xmlns":27,"display":303},[29,117348,117349,117537],{},[1142,117350,117351,117454,117501],{"rowspacing":62976,"columnalign":62977,"columnspacing":634},[1146,117352,117353,117371],{},[1149,117354,117355],{},[62983,117356,117357],{"scriptlevel":1804,"displaystyle":53},[32,117358,117359,117361,117363,117365,117367,117369],{},[35,117360,28394],{},[39,117362,120],{"stretchy":119},[35,117364,32453],{},[39,117366,34425],{},[39,117368,1245],{},[39,117370,126],{"stretchy":119},[1149,117372,117373],{},[62983,117374,117375],{"scriptlevel":1804,"displaystyle":53},[32,117376,117377,117379,117381],{},[32,117378],{},[39,117380,129],{},[6465,117382,117383,117405],{},[32,117384,117385,117387,117389,117391,117393,117395,117397,117399,117401,117403],{},[35,117386,28394],{},[39,117388,120],{"stretchy":119},[39,117390,1245],{},[39,117392,34425],{},[35,117394,32453],{},[39,117396,126],{"stretchy":119},[35,117398,28394],{},[39,117400,120],{"stretchy":119},[35,117402,32453],{},[39,117404,126],{"stretchy":119},[32,117406,117407,117409,117411,117413,117415,117417,117419,117421,117423,117425,117427,117429,117431,117433,117435,117437,117440,117442,117444,117446,117448,117450,117452],{},[35,117408,28394],{},[39,117410,120],{"stretchy":119},[39,117412,1245],{},[39,117414,34425],{},[35,117416,32453],{},[39,117418,126],{"stretchy":119},[35,117420,28394],{},[39,117422,120],{"stretchy":119},[35,117424,32453],{},[39,117426,126],{"stretchy":119},[39,117428,1245],{},[35,117430,28394],{},[39,117432,120],{"stretchy":119},[39,117434,1245],{},[39,117436,34425],{},[35,117438,117439],{"mathvariant":138},"¬",[35,117441,32453],{},[39,117443,126],{"stretchy":119},[35,117445,28394],{},[39,117447,120],{"stretchy":119},[35,117449,117439],{"mathvariant":138},[35,117451,32453],{},[39,117453,126],{"stretchy":119},[1146,117455,117456,117462],{},[1149,117457,117458],{},[62983,117459,117460],{"scriptlevel":1804,"displaystyle":53},[32,117461],{},[1149,117463,117464],{},[62983,117465,117466],{"scriptlevel":1804,"displaystyle":53},[32,117467,117468,117470,117472],{},[32,117469],{},[39,117471,129],{},[6465,117473,117474,117484],{},[32,117475,117476,117479,117481],{},[1209,117477,117478],{},"0.99",[39,117480,41],{},[1209,117482,117483],{},"0.01",[32,117485,117486,117488,117490,117492,117494,117497,117499],{},[1209,117487,117478],{},[39,117489,41],{},[1209,117491,117483],{},[39,117493,1245],{},[1209,117495,117496],{},"0.05",[39,117498,41],{},[1209,117500,117478],{},[1146,117502,117503,117509],{},[1149,117504,117505],{},[62983,117506,117507],{"scriptlevel":1804,"displaystyle":53},[32,117508],{},[1149,117510,117511],{},[62983,117512,117513],{"scriptlevel":1804,"displaystyle":53},[32,117514,117515,117517,117519,117532,117534],{},[32,117516],{},[39,117518,129],{},[6465,117520,117521,117523],{},[1209,117522,117185],{},[32,117524,117525,117527,117529],{},[1209,117526,117185],{},[39,117528,1245],{},[1209,117530,117531],{},"0.0495",[39,117533,31988],{},[1209,117535,117536],{},"0.167.",[45,117538,117539],{"encoding":47},"\\begin{aligned}\nP(D\\mid +)\n&=\\frac{P(+\\mid D)P(D)}{P(+\\mid D)P(D)+P(+\\mid \\neg D)P(\\neg D)}\\\\\n&=\\frac{0.99\\times0.01}{0.99\\times0.01+0.05\\times0.99}\\\\\n&=\\frac{0.0099}{0.0099+0.0495}\n\\approx0.167 .\n\\end{aligned}",[15,117541,117543],{"className":117542,"ariaHidden":53},[52],[15,117544,117546,117550],{"className":117545},[57],[15,117547],{"className":117548,"style":117549},[61],"height:7.1445em;vertical-align:-3.3223em;",[15,117551,117553],{"className":117552},[66],[15,117554,117556,117633],{"className":117555},[1142],[15,117557,117559],{"className":117558},[63146],[15,117560,117562,117624],{"className":117561},[202,1090],[15,117563,117565,117621],{"className":117564},[206],[15,117566,117569,117603,117612],{"className":117567,"style":117568},[210],"height:3.8223em;",[15,117570,117572,117576],{"style":117571},"top:-5.8223em;",[15,117573],{"className":117574,"style":117575},[217],"height:3.427em;",[15,117577,117579,117582,117585,117588,117591,117594,117597,117600],{"className":117578},[66],[15,117580,28394],{"className":117581,"style":62509},[66,67],[15,117583,120],{"className":117584},[162],[15,117586,32453],{"className":117587,"style":23020},[66,67],[15,117589],{"className":117590,"style":173},[71],[15,117592,34425],{"className":117593},[177],[15,117595],{"className":117596,"style":173},[71],[15,117598,1245],{"className":117599},[66],[15,117601,126],{"className":117602},[169],[15,117604,117606,117609],{"style":117605},"top:-3.2648em;",[15,117607],{"className":117608,"style":117575},[217],[15,117610],{"className":117611},[66],[15,117613,117615,117618],{"style":117614},"top:-0.8741em;",[15,117616],{"className":117617,"style":117575},[217],[15,117619],{"className":117620},[66],[15,117622,1114],{"className":117623},[1113],[15,117625,117627],{"className":117626},[206],[15,117628,117631],{"className":117629,"style":117630},[210],"height:3.3223em;",[15,117632],{},[15,117634,117636],{"className":117635},[63203],[15,117637,117639,118083],{"className":117638},[202,1090],[15,117640,117642,118080],{"className":117641},[206],[15,117643,117645,117844,117974],{"className":117644,"style":117568},[210],[15,117646,117647,117650],{"style":117571},[15,117648],{"className":117649,"style":117575},[217],[15,117651,117653,117656,117659,117662,117665],{"className":117652},[66],[15,117654],{"className":117655},[66],[15,117657],{"className":117658,"style":173},[71],[15,117660,129],{"className":117661},[177],[15,117663],{"className":117664,"style":173},[71],[15,117666,117668,117671,117841],{"className":117667},[66],[15,117669],{"className":117670},[162,23412],[15,117672,117674],{"className":117673},[6465],[15,117675,117677,117833],{"className":117676},[202,1090],[15,117678,117680,117830],{"className":117679},[206],[15,117681,117683,117778,117786],{"className":117682,"style":41206},[210],[15,117684,117685,117688],{"style":23428},[15,117686],{"className":117687,"style":1368},[217],[15,117689,117691,117694,117697,117700,117703,117706,117709,117712,117715,117718,117721,117724,117727,117730,117733,117736,117739,117742,117745,117748,117751,117754,117757,117760,117763,117766,117769,117772,117775],{"className":117690},[66],[15,117692,28394],{"className":117693,"style":62509},[66,67],[15,117695,120],{"className":117696},[162],[15,117698,1245],{"className":117699},[66],[15,117701],{"className":117702,"style":173},[71],[15,117704,34425],{"className":117705},[177],[15,117707],{"className":117708,"style":173},[71],[15,117710,32453],{"className":117711,"style":23020},[66,67],[15,117713,126],{"className":117714},[169],[15,117716,28394],{"className":117717,"style":62509},[66,67],[15,117719,120],{"className":117720},[162],[15,117722,32453],{"className":117723,"style":23020},[66,67],[15,117725,126],{"className":117726},[169],[15,117728],{"className":117729,"style":72},[71],[15,117731,1245],{"className":117732},[76],[15,117734],{"className":117735,"style":72},[71],[15,117737,28394],{"className":117738,"style":62509},[66,67],[15,117740,120],{"className":117741},[162],[15,117743,1245],{"className":117744},[66],[15,117746],{"className":117747,"style":173},[71],[15,117749,34425],{"className":117750},[177],[15,117752],{"className":117753,"style":173},[71],[15,117755,117439],{"className":117756},[66],[15,117758,32453],{"className":117759,"style":23020},[66,67],[15,117761,126],{"className":117762},[169],[15,117764,28394],{"className":117765,"style":62509},[66,67],[15,117767,120],{"className":117768},[162],[15,117770,117439],{"className":117771},[66],[15,117773,32453],{"className":117774,"style":23020},[66,67],[15,117776,126],{"className":117777},[169],[15,117779,117780,117783],{"style":23440},[15,117781],{"className":117782,"style":1368},[217],[15,117784],{"className":117785,"style":4438},[23447],[15,117787,117788,117791],{"style":23450},[15,117789],{"className":117790,"style":1368},[217],[15,117792,117794,117797,117800,117803,117806,117809,117812,117815,117818,117821,117824,117827],{"className":117793},[66],[15,117795,28394],{"className":117796,"style":62509},[66,67],[15,117798,120],{"className":117799},[162],[15,117801,1245],{"className":117802},[66],[15,117804],{"className":117805,"style":173},[71],[15,117807,34425],{"className":117808},[177],[15,11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",[15,118095,118097,118114],{"className":118096,"translate":19},[18],[15,118098,118100],{"className":118099},[23],[25,118101,118102],{"xmlns":27},[29,118103,118104,118111],{},[32,118105,118106,118109],{},[1209,118107,118108],{},"16.7",[35,118110,117246],{"mathvariant":138},[45,118112,118113],{"encoding":47},"16.7\\%",[15,118115,118117],{"className":118116,"ariaHidden":53},[52],[15,118118,118120,118123],{"className":118119},[57],[15,118121],{"className":118122,"style":117259},[61],[15,118124,118126],{"className":118125},[66],"16.7%","，因为健康人占了绝大多数，",[15,118129,118131,118147],{"className":118130,"translate":19},[18],[15,118132,118134],{"className":118133},[23],[25,118135,118136],{"xmlns":27},[29,118137,118138,118144],{},[32,118139,118140,118142],{},[1209,118141,116874],{},[35,118143,117246],{"mathvariant":138},[45,118145,118146],{"encoding":47},"5\\%",[15,118148,118150],{"className":118149,"ariaHidden":53},[52],[15,118151,118153,118156],{"className":118152},[57],[15,118154],{"className":118155,"style":117259},[61],[15,118157,118159],{"className":118158},[66],"5%"," 的假阳性率乘以 ",[15,118162,118164,118179],{"className":118163,"translate":19},[18],[15,118165,118167],{"className":118166},[23],[25,118168,118169],{"xmlns":27},[29,118170,118171,118177],{},[32,118172,118173,118175],{},[1209,118174,117279],{},[35,118176,117246],{"mathvariant":138},[45,118178,117284],{"encoding":47},[15,118180,118182],{"className":118181,"ariaHidden":53},[52],[15,118183,118185,118188],{"className":118184},[57],[15,118186],{"className":118187,"style":117259},[61],[15,118189,117297],{"className":118190},[66]," 的基础人数，产生了一个比真阳性更大的绝对数量。这就是基础率谬误：把 ",[15,118193,118195,118219],{"className":118194,"translate":19},[18],[15,118196,118198],{"className":118197},[23],[25,118199,118200],{"xmlns":27},[29,118201,118202,118216],{},[32,118203,118204,118206,118208,118210,118212,118214],{},[35,118205,28394],{},[39,118207,120],{"stretchy":119},[39,118209,1245],{},[39,118211,34425],{},[35,118213,32453],{},[39,118215,126],{"stretchy":119},[45,118217,118218],{"encoding":47},"P(+\\mid D)",[15,118220,118222,118246],{"className":118221,"ariaHidden":53},[52],[15,118223,118225,118228,118231,118234,118237,118240,118243],{"className":118224},[57],[15,118226],{"className":118227,"style":154},[61],[15,118229,28394],{"className":118230,"style":62509},[66,67],[15,118232,120],{"className":118233},[162],[15,118235,1245],{"className":118236},[66],[15,118238],{"className":118239,"style":173},[71],[15,118241,34425],{"className":118242},[177],[15,118244],{"className":118245,"style":173},[71],[15,118247,118249,118252,118255],{"className":118248},[57],[15,118250],{"className":118251,"style":154},[61],[15,118253,32453],{"className":118254,"style":23020},[66,67],[15,118256,126],{"className":118257},[169],[15,118259,118261,118285],{"className":118260,"translate":19},[18],[15,118262,118264],{"className":118263},[23],[25,118265,118266],{"xmlns":27},[29,118267,118268,118282],{},[32,118269,118270,118272,118274,118276,118278,118280],{},[35,118271,28394],{},[39,118273,120],{"stretchy":119},[35,118275,32453],{},[39,118277,34425],{},[39,118279,1245],{},[39,118281,126],{"stretchy":119},[45,118283,118284],{"encoding":47},"P(D\\mid +)",[15,118286,118288,118312],{"className":118287,"ariaHidden":53},[52],[15,118289,118291,118294,118297,118300,118303,118306,118309],{"className":118290},[57],[15,118292],{"className":118293,"style":154},[61],[15,118295,28394],{"className":118296,"style":62509},[66,67],[15,118298,120],{"className":118299},[162],[15,118301,32453],{"className":118302,"style":23020},[66,67],[15,118304],{"className":118305,"style":173},[71],[15,118307,34425],{"className":118308},[177],[15,118310],{"className":118311,"style":173},[71],[15,118313,118315,118318,118321],{"className":118314},[57],[15,118316],{"className":118317,"style":154},[61],[15,118319,1245],{"className":118320},[66],[15,118322,126],{"className":118323},[169]," 混为一谈。",[11,118326,118327],{},"患病率变化时，同一个检测给出的后验完全不同：",[6150,118329,118330,118404],{},[6153,118331,118332],{},[6156,118333,118334,118337],{},[6159,118335,118336],{},"患病率",[6159,118338,118339],{},[15,118340,118342,118365],{"className":118341,"translate":19},[18],[15,118343,118345],{"className":118344},[23],[25,118346,118347],{"xmlns":27},[29,118348,118349,118363],{},[32,118350,118351,118353,118355,118357,118359,118361],{},[35,118352,28394],{},[39,118354,120],{"stretchy":119},[35,118356,32453],{},[39,118358,34425],{},[39,118360,1245],{},[39,118362,126],{"stretchy":119},[45,118364,118284],{"encoding":47},[15,118366,118368,118392],{"className":118367,"ariaHidden":53},[52],[15,118369,118371,118374,118377,118380,118383,118386,118389],{"className":118370},[57],[15,118372],{"className":118373,"style":154},[61],[15,118375,28394],{"className":118376,"style":62509},[66,67],[15,118378,120],{"className":118379},[162],[15,118381,32453],{"className":118382,"style":23020},[66,67],[15,118384],{"className":118385,"style":173},[71],[15,118387,34425],{"className":118388},[177],[15,118390],{"className":118391,"style":173},[71],[15,118393,118395,118398,118401],{"className":118394},[57],[15,118396],{"className":118397,"style":154},[61],[15,118399,1245],{"className":118400},[66],[15,118402,126],{"className":118403},[169],[6169,118405,118406,118473,118538,118603],{},[6156,118407,118408,118442],{},[6174,118409,118410],{},[15,118411,118413,118429],{"className":118412,"translate":19},[18],[15,118414,118416],{"className":118415},[23],[25,118417,118418],{"xmlns":27},[29,118419,118420,118426],{},[32,118421,118422,118424],{},[1209,118423,112938],{},[35,118425,117246],{"mathvariant":138},[45,118427,118428],{"encoding":47},"0.5\\%",[15,118430,118432],{"className":118431,"ariaHidden":53},[52],[15,118433,118435,118438],{"className":118434},[57],[15,118436],{"className":118437,"style":117259},[61],[15,118439,118441],{"className":118440},[66],"0.5%",[6174,118443,118444],{},[15,118445,118447,118461],{"className":118446,"translate":19},[18],[15,118448,118450],{"className":118449},[23],[25,118451,118452],{"xmlns":27},[29,118453,118454,118459],{},[32,118455,118456],{},[1209,118457,118458],{},"0.090",[45,118460,118458],{"encoding":47},[15,118462,118464],{"className":118463,"ariaHidden":53},[52],[15,118465,118467,118470],{"className":118466},[57],[15,118468],{"className":118469,"style":1836},[61],[15,118471,118458],{"className":118472},[66],[6156,118474,118475,118507],{},[6174,118476,118477],{},[15,118478,118480,118495],{"className":118479,"translate":19},[18],[15,118481,118483],{"className":118482},[23],[25,118484,118485],{"xmlns":27},[29,118486,118487,118493],{},[32,118488,118489,118491],{},[1209,118490,1211],{},[35,118492,117246],{"mathvariant":138},[45,118494,117249],{"encoding":47},[15,118496,118498],{"className":118497,"ariaHidden":53},[52],[15,118499,118501,118504],{"className":118500},[57],[15,118502],{"className":118503,"style":117259},[61],[15,118505,117263],{"className":118506},[66],[6174,118508,118509],{},[15,118510,118512,118526],{"className":118511,"translate":19},[18],[15,118513,118515],{"className":118514},[23],[25,118516,118517],{"xmlns":27},[29,118518,118519,118524],{},[32,118520,118521],{},[1209,118522,118523],{},"0.167",[45,118525,118523],{"encoding":47},[15,118527,118529],{"className":118528,"ariaHidden":53},[52],[15,118530,118532,118535],{"className":118531},[57],[15,118533],{"className":118534,"style":1836},[61],[15,118536,118523],{"className":118537},[66],[6156,118539,118540,118572],{},[6174,118541,118542],{},[15,118543,118545,118560],{"className":118544,"translate":19},[18],[15,118546,118548],{"className":118547},[23],[25,118549,118550],{"xmlns":27},[29,118551,118552,118558],{},[32,118553,118554,118556],{},[1209,118555,116874],{},[35,118557,117246],{"mathvariant":138},[45,118559,118146],{"encoding":47},[15,118561,118563],{"className":118562,"ariaHidden":53},[52],[15,118564,118566,118569],{"className":118565},[57],[15,118567],{"className":118568,"style":117259},[61],[15,118570,118159],{"className":118571},[66],[6174,118573,118574],{},[15,118575,118577,118591],{"className":118576,"translate":19},[18],[15,118578,118580],{"className":118579},[23],[25,118581,118582],{"xmlns":27},[29,118583,118584,118589],{},[32,118585,118586],{},[1209,118587,118588],{},"0.510",[45,118590,118588],{"encoding":47},[15,118592,118594],{"className":118593,"ariaHidden":53},[52],[15,118595,118597,118600],{"className":118596},[57],[15,118598],{"className":118599,"style":1836},[61],[15,118601,118588],{"className":118602},[66],[6156,118604,118605,118639],{},[6174,118606,118607],{},[15,118608,118610,118626],{"className":118609,"translate":19},[18],[15,118611,118613],{"className":118612},[23],[25,118614,118615],{"xmlns":27},[29,118616,118617,118623],{},[32,118618,118619,118621],{},[1209,118620,107903],{},[35,118622,117246],{"mathvariant":138},[45,118624,118625],{"encoding":47},"10\\%",[15,118627,118629],{"className":118628,"ariaHidden":53},[52],[15,118630,118632,118635],{"className":118631},[57],[15,118633],{"className":118634,"style":117259},[61],[15,118636,118638],{"className":118637},[66],"10%",[6174,118640,118641],{},[15,118642,118644,118658],{"className":118643,"translate":19},[18],[15,118645,118647],{"className":118646},[23],[25,118648,118649],{"xmlns":27},[29,118650,118651,118656],{},[32,118652,118653],{},[1209,118654,118655],{},"0.688",[45,118657,118655],{"encoding":47},[15,118659,118661],{"className":118660,"ariaHidden":53},[52],[15,118662,118664,118667],{"className":118663},[57],[15,118665],{"className":118666,"style":1836},[61],[15,118668,118655],{"className":118669},[66],[11,118671,118672,118673,118701,118702,48301,118804,118906],{},"现在加入代价。设漏诊的代价是误诊的 ",[15,118674,118676,118689],{"className":118675,"translate":19},[18],[15,118677,118679],{"className":118678},[23],[25,118680,118681],{"xmlns":27},[29,118682,118683,118687],{},[32,118684,118685],{},[1209,118686,107903],{},[45,118688,107903],{"encoding":47},[15,118690,118692],{"className":118691,"ariaHidden":53},[52],[15,118693,118695,118698],{"className":118694},[57],[15,118696],{"className":118697,"style":1836},[61],[15,118699,107903],{"className":118700},[66]," 倍，即 ",[15,118703,118705,118731],{"className":118704,"translate":19},[18],[15,118706,118708],{"className":118707},[23],[25,118709,118710],{"xmlns":27},[29,118711,118712,118728],{},[32,118713,118714,118724,118726],{},[1057,118715,118716,118718],{},[35,118717,2256],{},[32,118719,118720,118722],{},[35,118721,74037],{"mathvariant":138},[35,118723,28530],{"mathvariant":138},[39,118725,129],{},[1209,118727,107903],{},[45,118729,118730],{"encoding":47},"c_{\\mathrm{FN}}=10",[15,118732,118734,118795],{"className":118733,"ariaHidden":53},[52],[15,118735,118737,118740,118786,118789,118792],{"className":118736},[57],[15,118738],{"className":118739,"style":10192},[61],[15,118741,118743,118746],{"className":118742},[66],[15,118744,2256],{"className":118745},[66,67],[15,118747,118749],{"className":118748},[198],[15,118750,118752,118778],{"className":118751},[202,1090],[15,118753,118755,118775],{"className":118754},[206],[15,118756,118758],{"className":118757,"style":3636},[210],[15,118759,118760,118763],{"style":1584},[15,118761],{"className":118762,"style":218},[217],[15,118764,118766],{"className":118765},[222,223,224,225],[15,118767,118769],{"className":118768},[66,225],[15,118770,118772],{"className":118771},[66,225],[15,118773,74169],{"className":118774},[66,4427,225],[15,118776,1114],{"className":118777},[1113],[15,118779,118781],{"className":118780},[206],[15,118782,118784],{"className":118783,"style":1121},[210],[15,118785],{},[15,118787],{"className":118788,"style":173},[71],[15,118790,129],{"className":118791},[177],[15,118793],{"className":118794,"style":173},[71],[15,118796,118798,118801],{"className":118797},[57],[15,118799],{"className":118800,"style":1836},[61],[15,118802,107903],{"className":118803},[66],[15,118805,118807,118833],{"className":118806,"translate":19},[18],[15,118808,118810],{"className":118809},[23],[25,118811,118812],{"xmlns":27},[29,118813,118814,118830],{},[32,118815,118816,118826,118828],{},[1057,118817,118818,118820],{},[35,118819,2256],{},[32,118821,118822,118824],{},[35,118823,74037],{"mathvariant":138},[35,118825,28394],{"mathvariant":138},[39,118827,129],{},[1209,118829,1211],{},[45,118831,118832],{"encoding":47},"c_{\\mathrm{FP}}=1",[15,118834,118836,118897],{"className":118835,"ariaHidden":53},[52],[15,118837,118839,118842,118888,118891,118894],{"className":118838},[57],[15,118840],{"className":118841,"style":10192},[61],[15,118843,118845,118848],{"className":118844},[66],[15,118846,2256],{"className":118847},[66,67],[15,118849,118851],{"className":118850},[198],[15,118852,118854,118880],{"className":118853},[202,1090],[15,118855,118857,118877],{"className":118856},[206],[15,118858,118860],{"className":118859,"style":3636},[210],[15,118861,118862,118865],{"style":1584},[15,118863],{"className":118864,"style":218},[217],[15,118866,118868],{"className":118867},[222,223,224,225],[15,118869,118871],{"className":118870},[66,225],[15,118872,118874],{"className":118873},[66,225],[15,118875,74087],{"className":118876},[66,4427,225],[15,118878,1114],{"className":118879},[1113],[15,118881,118883],{"className":118882},[206],[15,118884,118886],{"className":118885,"style":1121},[210],[15,118887],{},[15,118889],{"className":118890,"style":173},[71],[15,118892,129],{"className":118893},[177],[15,118895],{"className":118896,"style":173},[71],[15,118898,118900,118903],{"className":118899},[57],[15,118901],{"className":118902,"style":1836},[61],[15,118904,1211],{"className":118905},[66],"，则治疗阈值是",[15,118908,118910],{"className":118909,"translate":19},[294],[15,118911,118913,118974],{"className":118912,"translate":19},[18],[15,118914,118916],{"className":118915},[23],[25,118917,118918],{"xmlns":27,"display":303},[29,118919,118920,118971],{},[32,118921,118922,118958,118960,118966,118968],{},[6465,118923,118924,118934],{},[1057,118925,118926,118928],{},[35,118927,2256],{},[32,118929,118930,118932],{},[35,118931,74037],{"mathvariant":138},[35,118933,28394],{"mathvariant":138},[32,118935,118936,118946,118948],{},[1057,118937,118938,118940],{},[35,118939,2256],{},[32,118941,118942,118944],{},[35,118943,74037],{"mathvariant":138},[35,118945,28394],{"mathvariant":138},[39,118947,1245],{},[1057,118949,118950,118952],{},[35,118951,2256],{},[32,118953,118954,118956],{},[35,118955,74037],{"mathvariant":138},[35,118957,28530],{"mathvariant":138},[39,118959,129],{},[6465,118961,118962,118964],{},[1209,118963,1211],{},[1209,118965,75066],{},[39,118967,31988],{},[1209,118969,118970],{},"0.091.",[45,118972,118973],{"encoding":47},"\\frac{c_{\\mathrm{FP}}}{c_{\\mathrm{FP}}+c_{\\mathrm{FN}}}\n=\\frac{1}{11}\\approx0.091.",[15,118975,118977,119195,119272],{"className":118976,"ariaHidden":53},[52],[15,118978,118980,118983,119186,119189,119192],{"className":118979},[57],[15,118981],{"className":118982,"style":52418},[61],[15,118984,118986,118989,119183],{"className":118985},[66],[15,118987],{"className":118988},[162,23412],[15,118990,118992],{"className":118991},[6465],[15,118993,118995,119175],{"className":118994},[202,109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",[15,119285,119287,119302],{"className":119286,"translate":19},[18],[15,119288,119290],{"className":119289},[23],[25,119291,119292],{"xmlns":27},[29,119293,119294,119300],{},[32,119295,119296,119298],{},[1209,119297,1211],{},[35,119299,117246],{"mathvariant":138},[45,119301,117249],{"encoding":47},[15,119303,119305],{"className":119304,"ariaHidden":53},[52],[15,119306,119308,119311],{"className":119307},[57],[15,119309],{"className":119310,"style":117259},[61],[15,119312,117263],{"className":119313},[66]," 时，",[15,119316,119318,119351],{"className":119317,"translate":19},[18],[15,119319,119321],{"className":119320},[23],[25,119322,119323],{"xmlns":27},[29,119324,119325,119348],{},[32,119326,119327,119329,119331,119333,119335,119337,119339,119341,119343,119345],{},[35,119328,28394],{},[39,119330,120],{"stretchy":119},[35,119332,32453],{},[39,119334,34425],{},[39,119336,1245],{},[39,119338,126],{"stretchy":119},[39,119340,31988],{},[1209,119342,118523],{},[39,119344,9278],{},[1209,119346,119347],{},"0.091",[45,119349,119350],{"encoding":47},"P(D\\mid +)\\approx0.167>0.091",[15,119352,119354,119378,119399,119417],{"className":119353,"ariaHidden":53},[52],[15,119355,119357,119360,119363,119366,119369,119372,119375],{"className":119356},[57],[15,119358],{"className":119359,"style":154},[61],[15,119361,28394],{"className":119362,"style":62509},[66,67],[15,119364,120],{"className":119365},[162],[15,119367,32453],{"className":119368,"style":23020},[66,67],[15,119370],{"className":119371,"style":173},[71],[15,119373,34425],{"className":119374},[177],[15,119376],{"className":119377,"style":173},[71],[15,119379,119381,119384,119387,119390,119393,119396],{"className":119380},[57],[15,119382],{"className":119383,"style":154},[61],[15,119385,1245],{"className":119386},[66],[15,119388,126],{"className":119389},[169],[15,119391],{"className":119392,"style":173},[71],[15,119394,31988],{"className":119395},[177],[15,119397],{"className":119398,"style":173},[71],[15,119400,119402,119405,119408,119411,119414],{"className":119401},[57],[15,119403],{"className":119404,"style":114162},[61],[15,119406,118523],{"className":119407},[66],[15,119409],{"className":119410,"style":173},[71],[15,119412,9278],{"className":119413},[177],[15,119415],{"className":119416,"style":173},[71],[15,119418,119420,119423],{"className":119419},[57],[15,119421],{"className":119422,"style":1836},[61],[15,119424,119347],{"className":119425},[66],"，应当治疗；在患病率 ",[15,119428,119430,119445],{"className":119429,"translate":19},[18],[15,119431,119433],{"className":119432},[23],[25,119434,119435],{"xmlns":27},[29,119436,119437,119443],{},[32,119438,119439,119441],{},[1209,119440,112938],{},[35,119442,117246],{"mathvariant":138},[45,119444,118428],{"encoding":47},[15,119446,119448],{"className":119447,"ariaHidden":53},[52],[15,119449,119451,119454],{"className":119450},[57],[15,119452],{"className":119453,"style":117259},[61],[15,119455,118441],{"className":119456},[66],[15,119458,119460,119488],{"className":119459,"translate":19},[18],[15,119461,119463],{"className":119462},[23],[25,119464,119465],{"xmlns":27},[29,119466,119467,119485],{},[32,119468,119469,119471,119473,119475,119477,119479,119481,119483],{},[35,119470,28394],{},[39,119472,120],{"stretchy":119},[35,119474,32453],{},[39,119476,34425],{},[39,119478,1245],{},[39,119480,126],{"stretchy":119},[39,119482,31988],{},[1209,119484,118458],{},[45,119486,119487],{"encoding":47},"P(D\\mid +)\\approx0.090",[15,119489,119491,119515,119536],{"className":119490,"ariaHidden":53},[52],[15,119492,119494,119497,119500,119503,119506,119509,119512],{"className":119493},[57],[15,119495],{"className":119496,"style":154},[61],[15,119498,28394],{"className":119499,"style":62509},[66,67],[15,119501,120],{"className":119502},[162],[15,119504,32453],{"className":119505,"style":23020},[66,67],[15,119507],{"className":119508,"style":173},[71],[15,119510,34425],{"className":119511},[177],[15,119513],{"className":119514,"style":173},[71],[15,119516,119518,119521,119524,119527,119530,119533],{"className":119517},[57],[15,119519],{"className":119520,"style":154},[61],[15,119522,1245],{"className":119523},[66],[15,119525,126],{"className":119526},[169],[15,119528],{"className":119529,"style":173},[71],[15,119531,31988],{"className":119532},[177],[15,119534],{"className":119535,"style":173},[71],[15,119537,119539,119542],{"className":119538},[57],[15,119540],{"className":119541,"style":1836},[61],[15,119543,118458],{"className":119544},[66],"，已经落到阈值以下，治疗与不治疗的期望代价几乎相同。同一个阳性结果，在不同的先验与代价下导向不同的行动，这不是检测本身变了，而是决策问题变了。",[11,119547,119548],{},"这个例子也说明了为什么「准确率」不是一个好的汇报指标。检测的准确率是",[15,119550,119552],{"className":119551,"translate":19},[294],[15,119553,119555,119589],{"className":119554,"translate":19},[18],[15,119556,119558],{"className":119557},[23],[25,119559,119560],{"xmlns":27,"display":303},[29,119561,119562,119586],{},[32,119563,119564,119566,119568,119570,119572,119575,119577,119579,119581,119584],{},[1209,119565,117478],{},[39,119567,41],{},[1209,119569,117483],{},[39,119571,1245],{},[1209,119573,119574],{},"0.95",[39,119576,41],{},[1209,119578,117478],{},[39,119580,31988],{},[1209,119582,119583],{},"0.9504",[39,119585,335],{"separator":53},[45,119587,119588],{"encoding":47},"0.99\\times0.01+0.95\\times0.99\\approx0.9504,",[15,119590,119592,119610,119628,119646,119664],{"className":119591,"ariaHidden":53},[52],[15,119593,119595,119598,119601,119604,119607],{"className":119594},[57],[15,119596],{"className":119597,"style":30463},[61],[15,119599,117478],{"className":119600},[66],[15,119602],{"className":119603,"style":72},[71],[15,119605,41],{"className":119606},[76],[15,119608],{"className":119609,"style":72},[71],[15,119611,119613,119616,119619,119622,119625],{"className":119612},[57],[15,119614],{"className":119615,"style":30463},[61],[15,119617,117483],{"className":119618},[66],[15,119620],{"className":119621,"style":72},[71],[15,119623,1245],{"className":119624},[76],[15,119626],{"className":119627,"style":72},[71],[15,119629,119631,119634,119637,119640,119643],{"className":119630},[57],[15,119632],{"className":119633,"style":30463},[61],[15,119635,119574],{"className":119636},[66],[15,119638],{"className":119639,"style":72},[71],[15,119641,41],{"className":119642},[76],[15,119644],{"className":119645,"style":72},[71],[15,119647,119649,119652,119655,119658,119661],{"className":119648},[57],[15,119650],{"className":119651,"style":1836},[61],[15,119653,117478],{"className":119654},[66],[15,119656],{"className":119657,"style":173},[71],[15,119659,31988],{"className":119660},[177],[15,119662],{"className":119663,"style":173},[71],[15,119665,119667,119670,119673],{"className":119666},[57],[15,119668],{"className":119669,"style":112813},[61],[15,119671,119583],{"className":119672},[66],[15,119674,335],{"className":119675},[422],[11,119677,119678],{},"看起来很高，但它几乎完全由健康人贡献；在真正需要决策的那一小部分人身上，后验概率才是关键。",[1904,119680,119682],{"id":119681},"_84-高斯两类问题","8.4 高斯两类问题",[11,119684,119685,119686,119767,119768,48361,119796,119824,119825,119876],{},"设两类共享方差 ",[15,119687,119689,119711],{"className":119688,"translate":19},[18],[15,119690,119692],{"className":119691},[23],[25,119693,119694],{"xmlns":27},[29,119695,119696,119708],{},[32,119697,119698,119704,119706],{},[131,119699,119700,119702],{},[35,119701,6632],{},[1209,119703,1263],{},[39,119705,129],{},[1209,119707,1211],{},[45,119709,119710],{"encoding":47},"\\sigma^2=1",[15,119712,119714,119758],{"className":119713,"ariaHidden":53},[52],[15,119715,119717,119720,119749,119752,119755],{"className":119716},[57],[15,119718],{"className":119719,"style":6738},[61],[15,119721,119723,119726],{"className":119722},[66],[15,119724,6632],{"className":119725,"style":1515},[66,67],[15,119727,119729],{"className":119728},[198],[15,119730,119732],{"className":119731},[202],[15,119733,119735],{"className":119734},[206],[15,119736,119738],{"className":119737,"style":6738},[210],[15,119739,119740,119743],{"style":213},[15,119741],{"className":119742,"style":218},[217],[15,119744,119746],{"className":119745},[222,223,224,225],[15,119747,1263],{"className":119748},[66,225],[15,119750],{"className":119751,"style":173},[71],[15,119753,129],{"className":119754},[177],[15,119756],{"className":119757,"style":173},[71],[15,119759,119761,119764],{"className":119760},[57],[15,119762],{"className":119763,"style":1836},[61],[15,119765,1211],{"className":119766},[66],"，均值分别为 ",[15,119769,119771,119784],{"className":119770,"translate":19},[18],[15,119772,119774],{"className":119773},[23],[25,119775,119776],{"xmlns":27},[29,119777,119778,119782],{},[32,119779,119780],{},[1209,119781,1804],{},[45,119783,1804],{"encoding":47},[15,119785,119787],{"className":119786,"ariaHidden":53},[52],[15,119788,119790,119793],{"className":119789},[57],[15,119791],{"className":119792,"style":1836},[61],[15,119794,1804],{"className":119795},[66],[15,119797,119799,119812],{"className":119798,"translate":19},[18],[15,119800,119802],{"className":119801},[23],[25,119803,119804],{"xmlns":27},[29,119805,119806,119810],{},[32,119807,119808],{},[1209,119809,1263],{},[45,119811,1263],{"encoding":47},[15,119813,119815],{"className":119814,"ariaHidden":53},[52],[15,119816,119818,119821],{"className":119817},[57],[15,119819],{"className":119820,"style":1836},[61],[15,119822,1263],{"className":119823},[66],"，先验相等。则 ",[15,119826,119828,119846],{"className":119827,"translate":19},[18],[15,119829,119831],{"className":119830},[23],[25,119832,119833],{"xmlns":27},[29,119834,119835,119843],{},[32,119836,119837,119839,119841],{},[35,119838,84660],{"mathvariant":138},[39,119840,129],{},[1209,119842,1263],{},[45,119844,119845],{"encoding":47},"\\Delta=2",[15,119847,119849,119867],{"className":119848,"ariaHidden":53},[52],[15,119850,119852,119855,119858,119861,119864],{"className":119851},[57],[15,119853],{"className":119854,"style":490},[61],[15,119856,84660],{"className":119857},[66],[15,119859],{"className":119860,"style":173},[71],[15,119862,129],{"className":119863},[177],[15,119865],{"className":119866,"style":173},[71],[15,119868,119870,119873],{"className":119869},[57],[15,119871],{"className":119872,"style":1836},[61],[15,119874,1263],{"className":119875},[66],"，贝叶斯错误率为",[15,119878,119880],{"className":119879,"translate":19},[294],[15,119881,119883,119918],{"className":119882,"translate":19},[18],[15,119884,119886],{"className":119885},[23],[25,119887,119888],{"xmlns":27,"display":303},[29,119889,119890,119915],{},[32,119891,119892,119900,119902,119904,119906,119908,119910,119912],{},[1255,119893,119894,119896,119898],{},[35,119895,11],{},[35,119897,48378],{},[39,119899,1162],{},[39,119901,129],{},[35,119903,83340],{},[39,119905,120],{"stretchy":119},[1209,119907,1211],{},[39,119909,126],{"stretchy":119},[39,119911,31988],{},[1209,119913,119914],{},"0.1587.",[45,119916,119917],{"encoding":47},"p_e^*=Q(1)\\approx0.1587.",[15,119919,119921,119987,120014],{"className":119920,"ariaHidden":53},[52],[15,119922,119924,119927,119978,119981,119984],{"className":119923},[57],[15,119925],{"className":119926,"style":82015},[61],[15,119928,119930,119933],{"className":119929},[66],[15,119931,11],{"className":119932},[66,67],[15,119934,119936],{"className":119935},[198],[15,119937,119939,119970],{"className":119938},[202,1090],[15,119940,119942,119967],{"className":119941},[206],[15,119943,119945,119956],{"className":119944,"style":1308},[210],[15,119946,119947,119950],{"style":1653},[15,119948],{"className":119949,"style":218},[217],[15,119951,119953],{"className":119952},[222,223,224,225],[15,119954,48378],{"className":119955},[66,67,225],[15,119957,119958,119961],{"style":397},[15,119959],{"className":119960,"style":218},[217],[15,119962,119964],{"className":119963},[222,223,224,225],[15,119965,1162],{"className":119966},[76,225],[15,119968,1114],{"className":119969},[1113],[15,119971,119973],{"className":119972},[206],[15,119974,119976],{"className":119975,"style":1686},[210],[15,119977],{},[15,119979],{"className":119980,"style":173},[71],[15,119982,129],{"className":119983},[177],[15,119985],{"className":119986,"style":173},[71],[15,119988,119990,119993,119996,119999,120002,120005,120008,120011],{"className":119989},[57],[15,119991],{"className":119992,"style":154},[61],[15,119994,83340],{"className":119995},[66,67],[15,119997,120],{"className":119998},[162],[15,120000,1211],{"className":120001},[66],[15,120003,126],{"className":120004},[169],[15,120006],{"className":120007,"style":173},[71],[15,120009,31988],{"className":120010},[177],[15,120012],{"className":120013,"style":173},[71],[15,120015,120017,120020],{"className":120016},[57],[15,120018],{"className":120019,"style":1836},[61],[15,120021,119914],{"className":120022},[66],[11,120024,120025,120026,3247,120054,120105,120106,120172,120173,120279],{},"若把均值差扩大到 ",[15,120027,120029,120042],{"className":120028,"translate":19},[18],[15,120030,120032],{"className":120031},[23],[25,120033,120034],{"xmlns":27},[29,120035,120036,120040],{},[32,120037,120038],{},[1209,120039,15850],{},[45,120041,15850],{"encoding":47},[15,120043,120045],{"className":120044,"ariaHidden":53},[52],[15,120046,120048,120051],{"className":120047},[57],[15,120049],{"className":120050,"style":1836},[61],[15,120052,15850],{"className":120053},[66],[15,120055,120057,120075],{"className":120056,"translate":19},[18],[15,120058,120060],{"className":120059},[23],[25,120061,120062],{"xmlns":27},[29,120063,120064,120072],{},[32,120065,120066,120068,120070],{},[35,120067,84660],{"mathvariant":138},[39,120069,129],{},[1209,120071,15850],{},[45,120073,120074],{"encoding":47},"\\Delta=4",[15,120076,120078,120096],{"className":120077,"ariaHidden":53},[52],[15,120079,120081,120084,120087,120090,120093],{"className":120080},[57],[15,120082],{"className":120083,"style":490},[61],[15,120085,84660],{"className":120086},[66],[15,120088],{"className":120089,"style":173},[71],[15,120091,129],{"className":120092},[177],[15,120094],{"className":120095,"style":173},[71],[15,120097,120099,120102],{"className":120098},[57],[15,120100],{"className":120101,"style":1836},[61],[15,120103,15850],{"className":120104},[66],"，错误率降为 ",[15,120107,120109,120133],{"className":120108,"translate":19},[18],[15,120110,120112],{"className":120111},[23],[25,120113,120114],{"xmlns":27},[29,120115,120116,120130],{},[32,120117,120118,120120,120122,120124,120126,120128],{},[35,120119,83340],{},[39,120121,120],{"stretchy":119},[1209,120123,1263],{},[39,120125,126],{"stretchy":119},[39,120127,31988],{},[1209,120129,86214],{},[45,120131,120132],{"encoding":47},"Q(2)\\approx0.0228",[15,120134,120136,120163],{"className":120135,"ariaHidden":53},[52],[15,120137,120139,120142,120145,120148,120151,120154,120157,120160],{"className":120138},[57],[15,120140],{"className":120141,"style":154},[61],[15,120143,83340],{"className":120144},[66,67],[15,120146,120],{"className":120147},[162],[15,120149,1263],{"className":120150},[66],[15,120152,126],{"className":120153},[169],[15,120155],{"className":120156,"style":173},[71],[15,120158,31988],{"className":120159},[177],[15,120161],{"className":120162,"style":173},[71],[15,120164,120166,120169],{"className":120165},[57],[15,120167],{"className":120168,"style":1836},[61],[15,120170,86214],{"className":120171},[66],"。错误率随马氏距离以 ",[15,120174,120176,120206],{"className":120175,"translate":19},[18],[15,120177,120179],{"className":120178},[23],[25,120180,120181],{"xmlns":27},[29,120182,120183,120203],{},[32,120184,120185],{},[131,120186,120187,120189],{},[35,120188,48378],{},[32,120190,120191,120193,120199,120201],{},[39,120192,6600],{},[131,120194,120195,120197],{},[35,120196,84660],{"mathvariant":138},[1209,120198,1263],{},[35,120200,6623],{"mathvariant":138},[1209,120202,86026],{},[45,120204,120205],{"encoding":47},"e^{-\\Delta^2\u002F8}",[15,120207,120209],{"className":120208,"ariaHidden":53},[52],[15,120210,120212,120215],{"className":120211},[57],[15,120213],{"className":120214,"style":86126},[61],[15,120216,120218,120221],{"className":120217},[66],[15,120219,48378],{"className":120220},[66,67],[15,120222,120224],{"className":120223},[198],[15,120225,120227],{"className":120226},[202],[15,120228,120230],{"className":120229},[206],[15,120231,120233],{"className":120232,"style":86126},[210],[15,120234,120235,120238],{"style":213},[15,120236],{"className":120237,"style":218},[217],[15,120239,120241],{"className":120240},[222,223,224,225],[15,120242,120244,120247,120276],{"className":120243},[66,225],[15,120245,6600],{"className":120246},[66,225],[15,120248,120250,120253],{"className":120249},[66,225],[15,120251,84660],{"className":120252},[66,225],[15,120254,120256],{"className":120255},[198],[15,120257,120259],{"className":120258},[202],[15,120260,120262],{"className":120261},[206],[15,120263,120265],{"className":120264,"style":86159},[210],[15,120266,120267,120270],{"style":86162},[15,120268],{"className":120269,"style":3673},[217],[15,120271,120273],{"className":120272},[222,3677,1508,225],[15,120274,1263],{"className":120275},[66,225],[15,120277,86175],{"className":120278},[66,225]," 的速率衰减，但永远不为零，因为两个高斯的支撑集在整个实轴上重叠。",[11,120281,120282],{},"Bhattacharyya 上界给出",[15,120284,120286],{"className":120285,"translate":19},[294],[15,120287,120289,120337],{"className":120288,"translate":19},[18],[15,120290,120292],{"className":120291},[23],[25,120293,120294],{"xmlns":27,"display":303},[29,120295,120296,120334],{},[32,120297,120298,120306,120308,120314,120332],{},[1255,120299,120300,120302,120304],{},[35,120301,11],{},[35,120303,48378],{},[39,120305,1162],{},[39,120307,24215],{},[6465,120309,120310,120312],{},[1209,120311,1211],{},[1209,120313,1263],{},[131,120315,120316,120318],{},[35,120317,48378],{},[32,120319,120320,120322,120328,120330],{},[39,120321,6600],{},[131,120323,120324,120326],{},[35,120325,84660],{"mathvariant":138},[1209,120327,1263],{},[35,120329,6623],{"mathvariant":138},[1209,120331,86026],{},[39,120333,335],{"separator":53},[45,120335,120336],{"encoding":47},"p_e^*\\le\\frac12e^{-\\Delta^2\u002F8},",[15,120338,120340,120406],{"className":120339,"ariaHidden":53},[52],[15,120341,120343,120346,120397,120400,120403],{"className":120342},[57],[15,120344],{"className":120345,"style":82015},[61],[15,120347,120349,120352],{"className":120348},[66],[15,120350,11],{"className":120351},[66,67],[15,120353,120355],{"className":120354},[198],[15,120356,120358,120389],{"className":120357},[202,1090],[15,120359,120361,120386],{"className":120360},[206],[15,120362,120364,120375],{"className":120363,"style":1308},[210],[15,120365,120366,120369],{"style":1653},[15,120367],{"className":120368,"style":218},[217],[15,120370,120372],{"className":120371},[222,223,224,225],[15,120373,48378],{"className":120374},[66,67,225],[15,120376,120377,120380],{"style":397},[15,120378],{"className":120379,"style":218},[217],[15,120381,120383],{"className":120382},[222,223,224,225],[15,120384,1162],{"className":120385},[76,225],[15,120387,1114],{"className":120388},[1113],[15,120390,120392],{"className":120391},[206],[15,120393,120395],{"className":120394,"style":1686},[210],[15,120396],{},[15,120398],{"className":120399,"style":173},[71],[15,120401,24215],{"className":120402},[177],[15,120404],{"className":120405,"style":173},[71],[15,120407,120409,120412,120474,120538],{"className":120408},[57],[15,120410],{"className":120411,"style":49082},[61],[15,120413,120415,120418,120471],{"className":120414},[66],[15,120416],{"className":120417},[162,23412],[15,120419,120421],{"className":120420},[6465],[15,120422,120424,120463],{"className":120423},[202,1090],[15,120425,120427,120460],{"className":120426},[206],[15,120428,120430,120441,120449],{"className":120429,"style":23425},[210],[15,120431,120432,120435],{"style":23428},[15,120433],{"className":120434,"style":1368},[217],[15,120436,120438],{"className":120437},[66],[15,120439,1263],{"className":120440},[66],[15,120442,120443,120446],{"style":23440},[15,120444],{"className":120445,"style":1368},[217],[15,120447],{"className":120448,"style":4438},[23447],[15,120450,120451,120454],{"style":23450},[15,120452],{"className":120453,"style":1368},[217],[15,120455,120457],{"className":120456},[66],[15,120458,1211],{"className":120459},[66],[15,120461,1114],{"className":120462},[1113],[15,120464,120466],{"className":120465},[206],[15,120467,120469],{"className":120468,"style":23469},[210],[15,120470],{},[15,120472],{"className":120473},[169,23412],[15,120475,120477,120480],{"className":120476},[66],[15,120478,48378],{"className":120479},[66,67],[15,120481,120483],{"className":120482},[198],[15,120484,120486],{"className":120485},[202],[15,120487,120489],{"className":120488},[206],[15,120490,120492],{"className":120491,"style":87407},[210],[15,120493,120494,120497],{"style":397},[15,120495],{"className":120496,"style":218},[217],[15,120498,120500],{"className":120499},[222,223,224,225],[15,120501,120503,120506,120535],{"className":120502},[66,225],[15,120504,6600],{"className":120505},[66,225],[15,120507,120509,120512],{"className":120508},[66,225],[15,120510,84660],{"className":120511},[66,225],[15,120513,120515],{"className":120514},[198],[15,120516,120518],{"className":120517},[202],[15,120519,120521],{"className":120520},[206],[15,120522,120524],{"className":120523,"style":86159},[210],[15,120525,120526,120529],{"style":86162},[15,120527],{"className":120528,"style":3673},[217],[15,120530,120532],{"className":120531},[222,3677,1508,225],[15,120533,1263],{"className":120534},[66,225],[15,120536,86175],{"className":120537},[66,225],[15,120539,335],{"className":120540},[422],[11,120542,19354,120543,120593,120594,120623,120624,120593,120674,120702],{},[15,120544,120546,120563],{"className":120545,"translate":19},[18],[15,120547,120549],{"className":120548},[23],[25,120550,120551],{"xmlns":27},[29,120552,120553,120561],{},[32,120554,120555,120557,120559],{},[35,120556,84660],{"mathvariant":138},[39,120558,129],{},[1209,120560,1263],{},[45,120562,119845],{"encoding":47},[15,120564,120566,120584],{"className":120565,"ariaHidden":53},[52],[15,120567,120569,120572,120575,120578,120581],{"className":120568},[57],[15,120570],{"className":120571,"style":490},[61],[15,120573,84660],{"className":120574},[66],[15,120576],{"className":120577,"style":173},[71],[15,120579,129],{"className":120580},[177],[15,120582],{"className":120583,"style":173},[71],[15,120585,120587,120590],{"className":120586},[57],[15,120588],{"className":120589,"style":1836},[61],[15,120591,1263],{"className":120592},[66]," 时为 ",[15,120595,120597,120611],{"className":120596,"translate":19},[18],[15,120598,120600],{"className":120599},[23],[25,120601,120602],{"xmlns":27},[29,120603,120604,120609],{},[32,120605,120606],{},[1209,120607,120608],{},"0.303",[45,120610,120608],{"encoding":47},[15,120612,120614],{"className":120613,"ariaHidden":53},[52],[15,120615,120617,120620],{"className":120616},[57],[15,120618],{"className":120619,"style":1836},[61],[15,120621,120608],{"className":120622},[66],"，在 ",[15,120625,120627,120644],{"className":120626,"translate":19},[18],[15,120628,120630],{"className":120629},[23],[25,120631,120632],{"xmlns":27},[29,120633,120634,120642],{},[32,120635,120636,120638,120640],{},[35,120637,84660],{"mathvariant":138},[39,120639,129],{},[1209,120641,15850],{},[45,120643,120074],{"encoding":47},[15,120645,120647,120665],{"className":120646,"ariaHidden":53},[52],[15,120648,120650,120653,120656,120659,120662],{"className":120649},[57],[15,120651],{"className":120652,"style":490},[61],[15,120654,84660],{"className":120655},[66],[15,120657],{"className":120658,"style":173},[71],[15,120660,129],{"className":120661},[177],[15,120663],{"className":120664,"style":173},[71],[15,120666,120668,120671],{"className":120667},[57],[15,120669],{"className":120670,"style":1836},[61],[15,120672,15850],{"className":120673},[66],[15,120675,120677,120690],{"className":120676,"translate":19},[18],[15,120678,120680],{"className":120679},[23],[25,120681,120682],{"xmlns":27},[29,120683,120684,120688],{},[32,120685,120686],{},[1209,120687,86217],{},[45,120689,86217],{"encoding":47},[15,120691,120693],{"className":120692,"ariaHidden":53},[52],[15,120694,120696,120699],{"className":120695},[57],[15,120697],{"className":120698,"style":1836},[61],[15,120700,86217],{"className":120701},[66],"。上界不紧，但它的指数部分与真实错误率一致；这正是我们在 4.2 节强调的「速率」而非「常数」。",[95,120704,120705],{"id":120705},"结语",[11,120707,120708],{},"把全文的线索压缩成一行：",[15,120710,120712],{"className":120711,"translate":19},[294],[15,120713,120715,120881],{"className":120714,"translate":19},[18],[15,120716,120718],{"className":120717},[23],[25,120719,120720],{"xmlns":27,"display":303},[29,120721,120722,120878],{},[32,120723,120724,120742,120744,120766,120768,120770,120773,120775,120797,120799,120801,120803,120805,120853,120855,120857,120859,120861,120876],{},[1172,120725,120726,120740],{},[1172,120727,120728,120738],{},[32,120729,120730,120732,120734,120736],{},[35,120731,11],{},[39,120733,120],{"stretchy":119},[35,120735,61614],{},[39,120737,126],{"stretchy":119},[39,120739,69838],{"stretchy":53},[1274,120741,65214],{},[39,120743,41],{},[1172,120745,120746,120764],{},[1172,120747,120748,120762],{},[32,120749,120750,120752,120754,120756,120758,120760],{},[35,120751,11],{},[39,120753,120],{"stretchy":119},[35,120755,32453],{},[39,120757,34425],{},[35,120759,61614],{},[39,120761,126],{"stretchy":119},[39,120763,69838],{"stretchy":53},[1274,120765,65268],{},[1274,120767,24228],{},[71,120769],{"linebreak":10451},[39,120771,120772],{},"⟶",[1274,120774,24228],{},[1172,120776,120777,120795],{},[1172,120778,120779,120793],{},[32,120780,120781,120783,120785,120787,120789,120791],{},[35,120782,11],{},[39,120784,120],{"stretchy":119},[35,120786,61614],{},[39,120788,34425],{},[35,120790,32453],{},[39,120792,126],{"stretchy":119},[39,120794,69838],{"stretchy":53},[1274,120796,65427],{},[1274,120798,24228],{},[71,120800],{"linebreak":10451},[39,120802,120772],{},[1274,120804,24228],{},[1172,120806,120807,120851],{},[1172,120808,120809,120849],{},[32,120810,120811,120813,120815,120825,120827,120829,120831,120833,120835,120837,120839,120841,120843,120845,120847],{},[35,120812,1167],{},[39,120814,1170],{},[1172,120816,120817,120823],{},[32,120818,120819,120821],{},[35,120820,1178],{},[39,120822,1170],{},[35,120824,4554],{},[1274,120826,24228],{},[35,120828,62377],{"mathvariant":454},[39,120830,1194],{"stretchy":119},[35,120832,1216],{},[39,120834,120],{"stretchy":119},[35,120836,61881],{},[39,120838,335],{"separator":53},[35,120840,4554],{},[39,120842,126],{"stretchy":119},[39,120844,34425],{},[35,120846,32453],{},[39,120848,1266],{"stretchy":119},[39,120850,69838],{"stretchy":53},[1274,120852,61982],{},[1274,120854,24228],{},[71,120856],{"linebreak":10451},[39,120858,120772],{},[1274,120860,24228],{},[1172,120862,120863,120873],{},[1172,120864,120865,120871],{},[131,120866,120867,120869],{},[35,120868,455],{},[39,120870,1162],{},[39,120872,69838],{"stretchy":53},[1274,120874,120875],{},"下限",[35,120877,3571],{"mathvariant":138},[45,120879,120880],{"encoding":47},"\\underbrace{p(\\theta)}_{\\text{先验}}\n\\times\n\\underbrace{p(D\\mid\\theta)}_{\\text{似然}}\n\\ \\\\ \\longrightarrow\\\n\\underbrace{p(\\theta\\mid D)}_{\\text{后验}}\n\\ \\\\ \\longrightarrow\\\n\\underbrace{\\arg\\min_a\\ \\mathbb E[L(Y,a)\\mid D]}_{\\text{行动}}\n\\ \\\\ \\longrightarrow\\\n\\underbrace{R^*}_{\\text{下限}}.",[15,120882,120884,121022,121161,121164,121179,121318,121321,121336,121554,121557,121572],{"className":120883,"ariaHidden":53},[52],[15,120885,120887,120891,121013,121016,121019],{"className":120886},[57],[15,120888],{"className":120889,"style":120890},[61],"height:2.3263em;vertical-align:-1.5763em;",[15,120892,120894],{"className":120893},[1413,1172],[15,120895,120897,121004],{"className":120896},[202,1090],[15,120898,120900,121001],{"className":120899},[206],[15,120901,120903,120921],{"className":120902,"style":69994},[210],[15,120904,120906,120909],{"style":120905},"top:-1.4237em;",[15,120907],{"className":120908,"style":1368},[217],[15,120910,120912],{"className":120911},[222,223,224,225],[15,120913,120915],{"className":120914},[66,225],[15,120916,120918],{"className":120917},[66,1711,225],[15,120919,65214],{"className":120920},[66,68610,225],[15,120922,120923,120926],{"style":1386},[15,120924],{"className":120925,"style":1368},[217],[15,120927,120929],{"className":120928},[1413,1172],[15,120930,120932,120992],{"className":120931},[202,1090],[15,120933,120935,120989],{"className":120934},[206],[15,120936,120938,120969],{"className":120937,"style":69994},[210],[15,120939,120942,120945],{"className":120940,"style":120941},[48560],"top:-2.102em;",[15,120943],{"className":120944,"style":1368},[217],[15,120946,120948,120955,120962],{"className":120947,"style":70041},[70040],[15,120949,120951],{"className":120950,"style":70046},[70045],[34823,120952,120953],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":48593},[34829,120954],{"d":70053},[15,120956,120958],{"className":120957,"style":70046},[70057],[34823,120959,120960],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":70060},[34829,120961],{"d":70063},[15,120963,120965],{"className":120964,"style":70046},[70067],[34823,120966,120967],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":70070},[34829,120968],{"d":70073},[15,120970,120971,120974],{"style":1386},[15,120972],{"className":120973,"style":1368},[217],[15,120975,120977,120980,120983,120986],{"className":120976},[66],[15,120978,11],{"className":120979},[66,67],[15,120981,120],{"className":120982},[162],[15,120984,61614],{"className":120985,"style":23020},[66,67],[15,120987,126],{"className":120988},[169],[15,120990,1114],{"className":120991},[1113],[15,120993,120995],{"className":120994},[206],[15,120996,120999],{"className":120997,"style":120998},[210],"height:0.898em;",[15,121000],{},[15,121002,1114],{"className":121003},[1113],[15,121005,121007],{"className":121006},[206],[15,121008,121011],{"className":121009,"style":121010},[210],"height:1.5763em;",[15,121012],{},[15,121014],{"className":121015,"style":72},[71],[15,121017,41],{"className":121018},[76],[15,121020],{"className":121021,"style":72},[71],[15,121023,121025,121028,121158],{"className":121024},[57],[15,121026],{"className":121027,"style":120890},[61],[15,121029,121031],{"className":121030},[1413,1172],[15,121032,121034,121150],{"className":121033},[202,1090],[15,121035,121037,121147],{"className":121036},[206],[15,121038,121040,121057],{"className":121039,"style":69994},[210],[15,121041,121042,121045],{"style":120905},[15,121043],{"className":121044,"style":1368},[217],[15,121046,121048],{"className":121047},[222,223,224,225],[15,121049,121051],{"className":121050},[66,225],[15,121052,121054],{"className":121053},[66,1711,225],[15,121055,65268],{"className":121056},[66,68610,225],[15,121058,121059,121062],{"style":1386},[15,121060],{"className":121061,"style":1368},[217],[15,121063,121065],{"className":121064},[1413,1172],[15,121066,121068,121139],{"className":121067},[202,1090],[15,121069,121071,121136],{"className":121070},[206],[15,121072,121074,121104],{"className":121073,"style":69994},[210],[15,121075,121077,121080],{"className":121076,"style":120941},[48560],[15,121078],{"className":121079,"style":1368},[217],[15,121081,121083,121090,121097],{"className":121082,"style":70041},[70040],[15,121084,121086],{"className":121085,"style":70046},[70045],[34823,121087,121088],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":48593},[34829,121089],{"d":70053},[15,121091,121093],{"className":121092,"style":70046},[70057],[34823,121094,121095],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":70060},[34829,121096],{"d":70063},[15,121098,121100],{"className":121099,"style":70046},[70067],[34823,121101,121102],{"xmlns":34825,"width":48590,"height":70049,"viewBox":70050,"preserveAspectRatio":70070},[34829,121103],{"d":70073},[15,121105,121106,121109],{"style":1386},[15,121107],{"className":121108,"style":1368},[217],[15,121110,121112,121115,121118,121121,121124,121127,121130,121133],{"className":121111},[66],[15,121113,11],{"className":121114},[66,67],[15,121116,120],{"className":121117},[162],[15,121119,32453],{"className":121120,"style"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