[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki15\/archives\/321241#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki15\/archives\/321241","headline":"\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf – Wikipedia","name":"\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf – Wikipedia","description":"\u3053\u306e\u8a18\u4e8b\u306f\u691c\u8a3c\u53ef\u80fd\u306a\u53c2\u8003\u6587\u732e\u3084\u51fa\u5178\u304c\u5168\u304f\u793a\u3055\u308c\u3066\u3044\u306a\u3044\u304b\u3001\u4e0d\u5341\u5206\u3067\u3059\u3002\u51fa\u5178\u3092\u8ffd\u52a0\u3057\u3066\u8a18\u4e8b\u306e\u4fe1\u983c\u6027\u5411\u4e0a\u306b\u3054\u5354\u529b\u304f\u3060\u3055\u3044\u3002\u51fa\u5178\u691c\u7d22?:\u00a0“\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf”\u00a0\u2013\u00a0\u30cb\u30e5\u30fc\u30b9\u00a0\u00b7 \u66f8\u7c4d\u00a0\u00b7 \u30b9\u30ab\u30e9\u30fc\u00a0\u00b7 CiNii\u00a0\u00b7 J-STAGE\u00a0\u00b7 NDL\u00a0\u00b7 dlib.jp\u00a0\u00b7 \u30b8\u30e3\u30d1\u30f3\u30b5\u30fc\u30c1\u00a0\u00b7 TWL\uff082016\u5e7410\u6708\uff09 \u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\uff08\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u3058\u3087\u3046\u307b\u3046\u308a\u3087\u3046\u3001\u82f1: Fisher information \uff09 IX(\u03b8){displaystyle {mathcal {I}}_{X}(theta )} \u306f\u3001\u7d71\u8a08\u5b66\u3084\u60c5\u5831\u7406\u8ad6\u3067\u767b\u5834\u3059\u308b\u91cf\u3067\u3001\u78ba\u7387\u5909\u6570 X{displaystyle X}","datePublished":"2022-04-19","dateModified":"2022-04-19","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki15\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki15\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/6\/64\/Question_book-4.svg\/50px-Question_book-4.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/6\/64\/Question_book-4.svg\/50px-Question_book-4.svg.png","height":"39","width":"50"},"url":"https:\/\/wiki.edu.vn\/jp\/wiki15\/archives\/321241","about":["Wiki"],"wordCount":14919,"articleBody":"\u3053\u306e\u8a18\u4e8b\u306f\u691c\u8a3c\u53ef\u80fd\u306a\u53c2\u8003\u6587\u732e\u3084\u51fa\u5178\u304c\u5168\u304f\u793a\u3055\u308c\u3066\u3044\u306a\u3044\u304b\u3001\u4e0d\u5341\u5206\u3067\u3059\u3002\u51fa\u5178\u3092\u8ffd\u52a0\u3057\u3066\u8a18\u4e8b\u306e\u4fe1\u983c\u6027\u5411\u4e0a\u306b\u3054\u5354\u529b\u304f\u3060\u3055\u3044\u3002\u51fa\u5178\u691c\u7d22?:\u00a0“\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf”\u00a0\u2013\u00a0\u30cb\u30e5\u30fc\u30b9\u00a0\u00b7 \u66f8\u7c4d\u00a0\u00b7 \u30b9\u30ab\u30e9\u30fc\u00a0\u00b7 CiNii\u00a0\u00b7 J-STAGE\u00a0\u00b7 NDL\u00a0\u00b7 dlib.jp\u00a0\u00b7 \u30b8\u30e3\u30d1\u30f3\u30b5\u30fc\u30c1\u00a0\u00b7 TWL\uff082016\u5e7410\u6708\uff09\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\uff08\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u3058\u3087\u3046\u307b\u3046\u308a\u3087\u3046\u3001\u82f1: Fisher information\uff09 IX(\u03b8){displaystyle {mathcal {I}}_{X}(theta )}  \u306f\u3001\u7d71\u8a08\u5b66\u3084\u60c5\u5831\u7406\u8ad6\u3067\u767b\u5834\u3059\u308b\u91cf\u3067\u3001\u78ba\u7387\u5909\u6570X{displaystyle X}\u304c\u6bcd\u6570\u03b8{displaystyle theta }\u306b\u95a2\u3057\u3066\u6301\u3064\u300c\u60c5\u5831\u300d\u306e\u91cf\u3092\u8868\u3059\u3002\u7d71\u8a08\u5b66\u8005\u306e\u30ed\u30ca\u30eb\u30c9\u30fb\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u306b\u56e0\u3093\u3067\u540d\u4ed8\u3051\u3089\u308c\u305f\u3002  \u03b8{displaystyle theta }\u3092\u6bcd\u6570\u3068\u3057\u3001X{displaystyle X}\u3092\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304cf(x|\u03b8){displaystyle f(x|theta )}\u3067\u8868\u3055\u308c\u308b\u78ba\u7387\u5909\u6570\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\u3001  \u03b8{displaystyle theta }\u306e\u5c24\u5ea6\u95a2\u6570L(\u03b8|x){displaystyle L(theta |x)}\u306fL(\u03b8|x)=f(x|\u03b8){displaystyle L(theta |x)=f(x|theta ),}\u3067\u5b9a\u7fa9\u3055\u308c\u3001\u30b9\u30b3\u30a2\u95a2\u6570\u306f\u5bfe\u6570\u5c24\u5ea6\u95a2\u6570\u306e\u5fae\u5206V(x;\u03b8)=\u2202\u2202\u03b8ln\u2061L(\u03b8|x){displaystyle V(x;theta )={frac {partial }{partial theta }}ln L(theta |x)}\u306b\u3088\u308a\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u3053\u306e\u3068\u304d\u3001\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cfIX(\u03b8){displaystyle {mathcal {I}}_{X}(theta )}\u306f\u30b9\u30b3\u30a2\u95a2\u6570\u306e2\u6b21\u306e\u30e2\u30fc\u30e1\u30f3\u30c8IX(\u03b8)=E[V(x;\u03b8)2|\u03b8]=E[(\u2202\u2202\u03b8ln\u2061L(\u03b8|x))2|\u03b8]{displaystyle {begin{aligned}{mathcal {I}}_{X}(theta )&=mathrm {E} [V(x;theta )^{2}|theta ]\\&=mathrm {E} left[left.{biggl (}{frac {partial }{partial theta }}ln L(theta |x){biggr )}^{2}right|,theta right]end{aligned}}}\u306b\u3088\u308a\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u7d1b\u308c\u304c\u306a\u3051\u308c\u3070\u6dfb\u3048\u5b57\u306eX{displaystyle X}\u3092\u7701\u7565\u3057\u3001I(\u03b8){displaystyle {mathcal {I}}(theta )}\u3068\u3082\u8868\u8a18\u3059\u308b\u3002\u306a\u304a\u3001X{displaystyle X}\u306b\u95a2\u3057\u3066\u306f\u671f\u5f85\u5024\u304c\u53d6\u3089\u308c\u3066\u3044\u308b\u70ba\u3001\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\u306fX{displaystyle X}\u306e\u5f93\u3046\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570f(x|\u03b8){displaystyle f(x|theta )}\u306e\u307f\u306b\u4f9d\u5b58\u3057\u3066\u6c7a\u307e\u308b\u3002\u3088\u3063\u3066X{displaystyle X}\u3068Y{displaystyle Y}\u304c\u540c\u3058\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u6301\u3066\u3070\u3001\u305d\u308c\u3089\u306e\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\u306f\u540c\u4e00\u3067\u3042\u308b\u3002\u30b9\u30b3\u30a2\u95a2\u6570\u306fE[V(x;\u03b8)|\u03b8]=0{displaystyle mathrm {E} [V(x;theta )|theta ]=0,}\u3092\u6e80\u305f\u3059\u4e8b\u304c\u77e5\u3089\u308c\u3066\u3044\u308b\u306e\u3067\u3001IX(\u03b8)=var(V(x;\u03b8)){displaystyle {mathcal {I}}_{X}(theta )=mathrm {var} (V(x;theta ))}\u304c\u6210\u7acb\u3059\u308b\u3002\u3053\u3053\u3067 var{displaystyle mathrm {var} } \u306f\u5206\u6563\u3092\u8868\u3059\u3002\u307e\u305fln\u2061f(x|\u03b8){displaystyle ln f(x|theta )}\u304c\u4e8c\u56de\u5fae\u5206\u53ef\u80fd\u3067\u4ee5\u4e0b\u306e\u6a19\u6e96\u5316\u6761\u4ef6\u222b\u22022\u2202\u03b82f(X;\u03b8)dx=0,{displaystyle int {frac {partial ^{2}}{partial theta ^{2}}}f(X;theta ),dx=0,}\u3092\u6e80\u305f\u3059\u306a\u3089\u3001\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304d\u63db\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002I(\u03b8)=\u2212E[\u22022\u2202\u03b82ln\u2061f(X;\u03b8)].{displaystyle {mathcal {I}}(theta )=-mathrm {E} left[{frac {partial ^{2}}{partial theta ^{2}}}ln f(X;theta )right].}\u3053\u306e\u3068\u304d\u3001\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\u306f\u3001f{displaystyle f} \u306e\u5bfe\u6570\u306e\u03b8{displaystyle theta }\u306b\u3064\u3044\u3066\u306e2\u6b21\u306e\u5c0e\u95a2\u6570\u306b\u30de\u30a4\u30ca\u30b9\u3092\u4ed8\u3051\u305f\u3082\u306e\u306b\u306a\u308b\u3002\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\u306f\u3001\u03b8{displaystyle theta }\u306b\u3064\u3044\u3066\u306e\u6700\u5c24\u63a8\u5b9a\u91cf\u4ed8\u8fd1\u306e\u30b5\u30dd\u30fc\u30c8\u66f2\u7dda\u306e\u300c\u92ed\u3055\u300d\u3068\u3057\u3066\u3082\u3068\u3089\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u4f8b\u3048\u3070\u3001\u300c\u920d\u3044\u300d\uff08\u3064\u307e\u308a\u3001\u6d45\u3044\u6700\u5927\u5024\u3092\u6301\u3064\uff09\u30b5\u30dd\u30fc\u30c8\u66f2\u7dda\u306f\u30012\u6b21\u306e\u5c0e\u95a2\u6570\u3068\u3057\u3066\u5c0f\u3055\u306a\u5024\u3092\u6301\u3064\u305f\u3081\u3001\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\u3068\u3057\u3066\u3082\u5c0f\u3055\u306a\u5024\u3092\u6301\u3064\u3053\u3068\u306b\u306a\u308b\u3057\u3001\u92ed\u3044\u30b5\u30dd\u30fc\u30c8\u66f2\u7dda\u306f\u30012\u6b21\u5c0e\u95a2\u6570\u3068\u3057\u3066\u5927\u304d\u306a\u5024\u3092\u6301\u3064\u305f\u3081\u3001\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\u3082\u5927\u304d\u306a\u5024\u306b\u306a\u308b\u3002Table of Contents\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u884c\u5217[\u7de8\u96c6]\u57fa\u672c\u7684\u6027\u8cea[\u7de8\u96c6]Cram\u00e9r\u2013Rao\u306e\u4e0d\u7b49\u5f0f[\u7de8\u96c6]\u5341\u5206\u7d71\u8a08\u91cf\u3068\u306e\u95a2\u4fc2[\u7de8\u96c6]\u30ab\u30eb\u30d0\u30c3\u30af\u30fb\u30e9\u30a4\u30d6\u30e9\u30fc\u60c5\u5831\u91cf\u3068\u306e\u95a2\u4fc2[\u7de8\u96c6]\u30d9\u30eb\u30cc\u30fc\u30a4\u5206\u5e03[\u7de8\u96c6]\u30ac\u30f3\u30de\u5206\u5e03[\u7de8\u96c6]\u6b63\u898f\u5206\u5e03[\u7de8\u96c6]\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u884c\u5217[\u7de8\u96c6]\u30d1\u30e9\u30e1\u30fc\u30bf\u304cN\u500b\u306e\u5834\u5408\u3001\u3064\u307e\u308a\u3001\u03b8{displaystyle mathbf {theta } } \u304cN\u6b21\u306e\u30d9\u30af\u30c8\u30eb\u03b8=(\u03b81,\u03b82,\u22ef,\u03b8N)T{displaystyle theta =(theta _{1},theta _{2},cdots ,theta _{N})^{T}}\u3067\u3042\u308b\u3068\u304d\u3001\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\u306f\u3001\u4ee5\u4e0b\u3067\u5b9a\u7fa9\u3055\u308c\u308bNxN \u884c\u5217\u306b\u62e1\u5f35\u3055\u308c\u308b\u3002I(\u03b8)=E[\u2202\u2202\u03b8ln\u2061f(X;\u03b8)\u2202\u2202\u03b8Tln\u2061f(X;\u03b8)].{displaystyle {mathcal {I}}(mathbf {theta } )=mathrm {E} left[{frac {partial }{partial mathbf {theta } }}ln f(X;theta ){frac {partial }{partial mathbf {theta } ^{T}}}ln f(X;theta )right].}\u3053\u308c\u3092\u3001\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u884c\u5217(FIM, Fisher information matrix)\u3068\u547c\u3076\u3002\u6210\u5206\u8868\u793a\u3059\u308c\u3070\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002(I(\u03b8))i,j=E[\u2202\u2202\u03b8iln\u2061f(X;\u03b8)\u2202\u2202\u03b8jln\u2061f(X;\u03b8)].{displaystyle {left({mathcal {I}}left(theta right)right)}_{i,j}=mathrm {E} left[{frac {partial }{partial theta _{i}}}ln f(X;theta ){frac {partial }{partial theta _{j}}}ln f(X;theta )right].}\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u884c\u5217\u306f\u3001NxN \u306e\u6b63\u5b9a\u5024\u5bfe\u79f0\u884c\u5217\u3067\u3042\u308a\u3001\u305d\u306e\u6210\u5206\u306f\u3001N\u6b21\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u7a7a\u9593\u304b\u3089\u306a\u308b\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u8ddd\u96e2\u3092\u5b9a\u7fa9\u3059\u308b\u3002p{displaystyle p}\u500b\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u306b\u3088\u308b\u5c24\u5ea6\u304c\u3042\u308b\u3068\u304d\u3001\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u884c\u5217\u306ei\u756a\u76ee\u306e\u884c\u3068\u3001j\u756a\u76ee\u306e\u5217\u306e\u8981\u7d20\u304c\u30bc\u30ed\u3067\u3042\u308b\u306a\u3089\u30012\u3064\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u3001\u03b8i{displaystyle theta _{i}}\u3068\u03b8j{displaystyle theta _{j}}\u306f\u76f4\u4ea4\u3067\u3042\u308b\u3002\u30d1\u30e9\u30e1\u30fc\u30bf\u304c\u76f4\u4ea4\u3067\u3042\u308b\u3068\u304d\u3001\u6700\u5c24\u63a8\u5b9a\u91cf\u304c\u72ec\u7acb\u306b\u306a\u308a\u3001\u5225\u3005\u306b\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u305f\u3081\u3001\u6271\u3044\u3084\u3059\u304f\u306a\u308b\u3002\u3053\u306e\u305f\u3081\u3001\u7814\u7a76\u8005\u304c\u4f55\u3089\u304b\u306e\u7814\u7a76\u4e0a\u306e\u554f\u984c\u3092\u6271\u3046\u3068\u304d\u3001\u305d\u306e\u554f\u984c\u306b\u95a2\u308f\u308b\u78ba\u7387\u5bc6\u5ea6\u304c\u76f4\u4ea4\u306b\u306a\u308b\u3088\u3046\u306b\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u5316\u3059\u308b\u65b9\u6cd5\u3092\u63a2\u3059\u306e\u306b\u4e00\u5b9a\u306e\u6642\u9593\u3092\u8cbb\u3084\u3059\u306e\u304c\u666e\u901a\u3067\u3042\u308b\u3002\u57fa\u672c\u7684\u6027\u8cea[\u7de8\u96c6]\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\u306f0\u2264I(\u03b8))=IX(\u03b8)+IY(\u03b8){displaystyle {mathcal {I}}_{X,Y}(theta )={mathcal {I}}_{X}(theta )+{mathcal {I}}_{Y}(theta )}\u3000(\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\u306e\u52a0\u7b97\u6027\uff09\u304c\u6210\u7acb\u3059\u308b\u3002\u3059\u306a\u308f\u3061\u3001\u300c(X,Y){displaystyle (X,Y)}\u304c\u03b8{displaystyle theta }\u306b\u95a2\u3057\u3066\u6301\u3064\u60c5\u5831\u306e\u91cf\u300d\u306f\u300cX{displaystyle X}\u304c\u03b8{displaystyle theta }\u306b\u95a2\u3057\u3066\u6301\u3064\u60c5\u5831\u306e\u91cf\u300d\u3068\u300cY{displaystyle Y}\u304c\u03b8{displaystyle theta }\u306b\u95a2\u3057\u3066\u6301\u3064\u60c5\u5831\u306e\u91cf\u300d\u306e\u548c\u3067\u3042\u308b\u3002\u3088\u3063\u3066\u7279\u306b\u3001\u7121\u4f5c\u70ba\u306b\u53d6\u3089\u308c\u305fn\u500b\u306e\u6a19\u672c\u304c\u6301\u3064\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\u306f\u30011\u3064\u306e\u6a19\u672c\u304c\u6301\u3064\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\u306en\u500d\u3067\u3042\u308b\uff08\u89b3\u5bdf\u304c\u72ec\u7acb\u3067\u3042\u308b\u5834\u5408\uff09\u3002Cram\u00e9r\u2013Rao\u306e\u4e0d\u7b49\u5f0f[\u7de8\u96c6]\u03b8{displaystyle theta }\u306e\u4efb\u610f\u306e\u4e0d\u504f\u63a8\u5b9a\u91cf\u03b8^{displaystyle {hat {theta }}}\u306f\u4ee5\u4e0b\u306eCram\u00e9r\u2013Rao(\u30af\u30e9\u30e1\u30fc\u30eb-\u30e9\u30aa)\u306e\u4e0d\u7b49\u5f0f\u3092\u6e80\u305f\u3059\uff1avar(\u03b8^)\u22651I(\u03b8){displaystyle mathrm {var} ({hat {theta }})geq {frac {1}{{mathcal {I}}(theta )}},}\u3053\u306e\u4e0d\u7b49\u5f0f\u306e\u76f4\u89b3\u7684\u610f\u5473\u3092\u8aac\u660e\u3059\u308b\u70ba\u3001\u4e21\u8fba\u306e\u9006\u6570\u3092\u53d6\u3063\u305f\u4e0a\u3067\u78ba\u7387\u5909\u6570X{displaystyle X}\u3078\u306e\u4f9d\u5b58\u95a2\u4fc2\u3092\u660e\u793a\u3059\u308b\u3068\u3001IX(\u03b8)\u22651var(\u03b8^(X)){displaystyle {mathcal {I}}_{X}(theta )geq {frac {1}{mathrm {var} ({hat {theta }}(X))}},}\u3068\u306a\u308b\u3002\u4e00\u822c\u306b\u63a8\u5b9a\u91cf\u306f\u305d\u306e\u5206\u6563\u304c\u5c0f\u3055\u3044\u307b\u3069(\u3088\u3063\u3066\u5206\u6563\u306e\u9006\u6570\u304c\u5927\u304d\u3044\u307b\u3069)\u6bcd\u6570\u03b8{displaystyle theta }\u306b\u8fd1\u3044\u5024\u3092\u51fa\u3057\u3084\u3059\u3044\u306e\u3067\u3001\u300c\u3088\u3044\u300d\u63a8\u5b9a\u91cf\u3067\u3042\u308b\u3068\u8a00\u3048\u308b\u3002\u03b8{displaystyle theta }\u3092\u300c\u63a8\u5b9a\u3059\u308b\u300d\u3068\u3044\u3046\u884c\u70ba\u306f\u3001\u300c\u3088\u3044\u300d\u63a8\u5b9a\u91cf\u03b8^(X){displaystyle {hat {theta }}(X)}\u3092\u4f7f\u3063\u3066\u03b8{displaystyle theta }\u3092\u53ef\u80fd\u306a\u9650\u308a\u5fa9\u5143\u3059\u308b\u884c\u70ba\u306b\u4ed6\u306a\u3089\u306a\u3044\u304c\u3001\u4e0a\u306e\u4e0d\u7b49\u5f0f\u306fX{displaystyle X}\u304b\u3089\u7b97\u51fa\u3055\u308c\u305f\u3069\u3093\u306a\u4e0d\u504f\u63a8\u5b9a\u91cf\u3067\u3042\u3063\u3066\u3082X{displaystyle X}\u304c\u5143\u3005\u6301\u3063\u3066\u3044\u308b\u300c\u60c5\u5831\u300d\u4ee5\u4e0a\u306b\u300c\u3088\u3044\u300d\u63a8\u5b9a\u91cf\u306b\u306f\u306a\u308a\u3048\u306a\u3044\u4e8b\u3092\u610f\u5473\u3059\u308b\u3002\u5341\u5206\u7d71\u8a08\u91cf\u3068\u306e\u95a2\u4fc2[\u7de8\u96c6]\u4e00\u822c\u306bT=t(X){displaystyle T=t(X)}\u304c\u7d71\u8a08\u91cf\u3067\u3042\u308b\u306a\u3089\u3070\u3001IT(\u03b8)\u2264IX(\u03b8){displaystyle {mathcal {I}}_{T}(theta )leq {mathcal {I}}_{X}(theta )}\u304c\u6210\u7acb\u3059\u308b\u3002\u3059\u306a\u308f\u3061\u3001\u300cX{displaystyle X}\u304b\u3089\u8a08\u7b97\u3055\u308c\u308b\u5024T=t(X){displaystyle T=t(X)}\u304c\u6301\u3063\u3066\u3044\u308b\u03b8{displaystyle theta }\u306e\u60c5\u5831\u300d\u306f\u300cX{displaystyle X}\u81ea\u8eab\u304c\u6301\u3063\u3066\u3044\u308b\u03b8{displaystyle theta }\u306e\u60c5\u5831\u300d\u3088\u308a\u3082\u5927\u304d\u304f\u306a\u3044\u3002\u4e0a\u5f0f\u3067\u7b49\u53f7\u6210\u7acb\u3059\u308b\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306fT{displaystyle T}\u304c\u5341\u5206\u7d71\u8a08\u91cf\u3067\u3042\u308b\u3053\u3068\u3002\u3053\u308c\u306fT(X){displaystyle T(X)}\u304c \u03b8{displaystyle theta }\u306b\u5bfe\u3057\u3066\u5341\u5206\u7d71\u8a08\u91cf\u3067\u3042\u308b\u306a\u3089\u3070\u3001\u3042\u308b\u95a2\u6570f{displaystyle f}\u304a\u3088\u3073g{displaystyle g}\u304c\u5b58\u5728\u3057\u3066f(X;\u03b8)=g(T(X),\u03b8)h(X){displaystyle f(X;theta )=g(T(X),theta )h(X)}\u304c\u6210\u308a\u7acb\u3064\uff08\u30cd\u30a4\u30de\u30f3\u5206\u89e3\u57fa\u6e96\uff09\u4e8b\u3092\u4f7f\u3063\u3066\u8a3c\u660e\u3067\u304d\u308b\u3002\u30ab\u30eb\u30d0\u30c3\u30af\u30fb\u30e9\u30a4\u30d6\u30e9\u30fc\u60c5\u5831\u91cf\u3068\u306e\u95a2\u4fc2[\u7de8\u96c6]X\u03b8{displaystyle X_{theta }}\u3092\u6bcd\u6570\u03b8\u2192=(\u03b81,\u2026,\u03b8n){displaystyle {vec {theta }}=(theta _{1},ldots ,theta _{n})}\u3092\u6301\u3064\u78ba\u7387\u5909\u6570\u3068\u3059\u308b\u3068\u3001\u30ab\u30eb\u30d0\u30c3\u30af\u30fb\u30e9\u30a4\u30d6\u30e9\u30fc\u60c5\u5831\u91cf DKL{displaystyle D_{mathrm {KL} }}\u3068\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u884c\u5217\u306f\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3064\u3002DKL(X\u03b8\u2192+h\u2192\u2016X\u03b8\u2192)=th\u2192\u22c5I(\u03b8\u2192)\u22c5h\u21922+o(|h\u2192|2){displaystyle D_{mathrm {KL} }(X_{{vec {theta }}+{vec {h}}}|X_{vec {theta }})={frac {{}^{t}{vec {h}}cdot {mathcal {I}}({vec {theta }})cdot {vec {h}}}{2}}+o(|{vec {h}}|^{2})}\u3059\u306a\u308f\u3061\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u884c\u5217\u306f\u30ab\u30eb\u30d0\u30c3\u30af\u30fb\u30e9\u30a4\u30d6\u30e9\u30fc\u60c5\u5831\u91cf\u3092\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3057\u305f\u3068\u304d\u306e2\u6b21\u306e\u9805\u3068\u3057\u3066\u767b\u5834\u3059\u308b\u3002\uff080\u6b21\u30011\u6b21\u306e\u9805\u306f0\uff09\u3002\u30d9\u30eb\u30cc\u30fc\u30a4\u5206\u5e03[\u7de8\u96c6]\u30d9\u30eb\u30cc\u30fc\u30a4\u5206\u5e03\u306f\u3001\u78ba\u7387\u03b8 \u3067\u3082\u305f\u3089\u3055\u308c\u308b\u300c\u6210\u529f\u300d\u3068\u3001\u305d\u308c\u4ee5\u5916\u306e\u5834\u5408\u306b\u8d77\u304d\u308b\u300c\u5931\u6557\u300d\u3068\u3044\u30462\u3064\u306e\u7d50\u679c\u3092\u3082\u305f\u3089\u3059\u78ba\u7387\u5909\u6570\u304c\u5f93\u3046\u5206\u5e03\u3067\u3042\u308b\uff08\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\uff09\u3002\u4f8b\u3048\u3070\u3001\u8868\u304c\u51fa\u308b\u78ba\u7387\u304c\u03b8\u3001\u88cf\u304c\u51fa\u308b\u78ba\u7387\u304c1 – \u03b8\u3067\u3042\u308b\u3088\u3046\u306a\u3001\u30b3\u30a4\u30f3\u306e\u6295\u3052\u4e0a\u3052\u3092\u8003\u3048\u308c\u3070\u826f\u3044\u3002n\u56de\u306e\u72ec\u7acb\u306a\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u304c\u542b\u3080\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf\u306f\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u6c42\u3081\u3089\u308c\u308b\u3002\u306a\u304a\u3001\u4ee5\u4e0b\u306e\u5f0f\u4e2d\u3067\u3001A \u306f\u6210\u529f\u306e\u56de\u6570\u3001B \u306f\u5931\u6557\u306e\u56de\u6570\u3001n =A +B \u306f\u8a66\u884c\u306e\u5408\u8a08\u56de\u6570\u3092\u793a\u3057\u3066\u3044\u308b\u3002\u5bfe\u6570\u5c24\u5ea6\u95a2\u6570\u306e2\u968e\u5c0e\u95a2\u6570\u306f\u3001\u22022\u2202\u03b82ln\u2061f(A;\u03b8)=\u22022\u2202\u03b82ln\u2061[\u03b8A(1\u2212\u03b8)B(A+B)!A!B!]=\u22022\u2202\u03b82[Aln\u2061(\u03b8)+Bln\u2061(1\u2212\u03b8)]=\u2212A\u03b82\u2212B(1\u2212\u03b8)2{displaystyle {begin{aligned}{frac {partial ^{2}}{partial theta ^{2}}}ln {f(A;theta )}&={frac {partial ^{2}}{partial theta ^{2}}}ln left[theta ^{A}(1-theta )^{B}{frac {(A+B)!}{A!B!}}right]\\&={frac {partial ^{2}}{partial theta ^{2}}}left[Aln(theta )+Bln(1-theta )right]\\&=-{frac {A}{theta ^{2}}}-{frac {B}{(1-theta )^{2}}}end{aligned}}}\u3067\u3042\u308b\u304b\u3089\u3001I(\u03b8)=\u2212E[\u22022\u2202\u03b82ln\u2061(f(A;\u03b8))]=n\u03b8\u03b82+n(1\u2212\u03b8)(1\u2212\u03b8)2{displaystyle {begin{aligned}{mathcal {I}}(theta )&=-mathrm {E} left[{frac {partial ^{2}}{partial theta ^{2}}}ln(f(A;theta ))right]\\&={frac {ntheta }{theta ^{2}}}+{frac {n(1-theta )}{(1-theta )^{2}}}end{aligned}}}\u3068\u306a\u308b\u3002\u4f46\u3057\u3001A\u306e\u671f\u5f85\u5024\u306fn \u03b8\u3001B \u306e\u671f\u5f85\u5024\u306fn (1-\u03b8 )\u3067\u3042\u308b\u3053\u3068\u3092\u7528\u3044\u305f \u3002\u3064\u307e\u308a\u3001\u6700\u7d42\u7684\u306a\u7d50\u679c\u306f\u3001I(\u03b8)=n\u03b8(1\u2212\u03b8),{displaystyle {mathcal {I}}(theta )={frac {n}{theta (1-theta )}},}\u3067\u3042\u308b\u3002\u3053\u308c\u306f\u3001n\u56de\u306e\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u6210\u529f\u6570\u306e\u5e73\u5747\u306e\u5206\u6563\u306e\u9006\u6570\u306b\u7b49\u3057\u3044\u3002\u30ac\u30f3\u30de\u5206\u5e03[\u7de8\u96c6]\u5f62\u72b6\u30d1\u30e9\u30e1\u30fc\u30bf\u03b1\u3001\u5c3a\u5ea6\u30d1\u30e9\u30e1\u30fc\u30bf\u03b2\u306e\u30ac\u30f3\u30de\u5206\u5e03\u306b\u304a\u3044\u3066\u3001\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u884c\u5217\u306fI(\u03b1,\u03b2)=(\u03c8\u2032(\u03b1)1\u03b21\u03b2\u03b1\u03b22){displaystyle {mathcal {I}}(alpha ,beta )={begin{pmatrix}psi ‘(alpha )&{frac {1}{beta }}\\{frac {1}{beta }}&{frac {alpha }{beta ^{2}}}end{pmatrix}}}\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u4f46\u3057\u3001\u03c8(\u03b1)\u306f\u30c7\u30a3\u30ac\u30f3\u30de\u95a2\u6570\u3092\u8868\u3059\u3002\u6b63\u898f\u5206\u5e03[\u7de8\u96c6]\u5e73\u5747\u03bc\u3001\u5206\u6563\u03c32\u306e\u6b63\u898f\u5206\u5e03N(\u03bc, \u03c32)\u306b\u304a\u3044\u3066\u3001\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u884c\u5217\u306fI(\u03bc,\u03c32)=(1\u03c320012(\u03c32)2){displaystyle {mathcal {I}}(mu ,sigma ^{2})={begin{pmatrix}{frac {1}{sigma ^{2}}}&0\\0&{frac {1}{2(sigma ^{2})^{2}}}end{pmatrix}}}\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03[\u7de8\u96c6]N\u500b\u306e\u5909\u6570\u306e\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u3064\u3044\u3066\u306e\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u884c\u5217\u306f\u3001\u7279\u5225\u306a\u5f62\u5f0f\u3092\u6301\u3064\u3002\u03bc(\u03b8)=(\u03bc1(\u03b8),\u03bc2(\u03b8),\u22ef,\u03bcN(\u03b8)),{displaystyle mu (theta )={begin{pmatrix}mu _{1}(theta ),mu _{2}(theta ),cdots ,mu _{N}(theta )end{pmatrix}},}\u3067\u3042\u308b\u3068\u3057\u3001\u03a3(\u03b8){displaystyle Sigma (theta )}\u304c\u03bc(\u03b8){displaystyle mu (theta )}\u306e\u5171\u5206\u6563\u884c\u5217\u3067\u3042\u308b\u3068\u3059\u308b\u306a\u3089\u3001X{displaystyle X}\uff5eN(\u03bc(\u03b8),\u03a3(\u03b8)){displaystyle N(mu (theta ),Sigma (theta ))}\u306e\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u884c\u5217\u3001Im,n(0\u2264;m,n\u2202\u03b8m\u03a3\u22121\u2202\u03bc\u22a4\u2202\u03b8n+12tr(\u03a3\u22121\u2202\u03a3\u2202\u03b8m\u03a3\u22121\u2202\u03a3\u2202\u03b8n),{displaystyle {mathcal {I}}_{m,n}={frac {partial mu }{partial theta _{m}}}Sigma ^{-1}{frac {partial mu ^{top }}{partial theta _{n}}}+{frac {1}{2}}mathrm {tr} left(Sigma ^{-1}{frac {partial Sigma }{partial theta _{m}}}Sigma ^{-1}{frac {partial Sigma }{partial theta _{n}}}right),}\u3053\u3053\u3067\u3001(..)\u22a4{displaystyle (..)^{top }}\u306f\u30d9\u30af\u30c8\u30eb\u306e\u8ee2\u7f6e\u3092\u793a\u3059\u8a18\u53f7\u3067\u3042\u308a\u3001tr(..){displaystyle mathrm {tr} (..)}\u306f\u3001\u5e73\u65b9\u884c\u5217\u306e\u30c8\u30ec\u30fc\u30b9\u3092\u8868\u3059\u8a18\u53f7\u3067\u3042\u308b\u3002\u307e\u305f\u3001\u5fae\u5206\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u2202\u03bc\u2202\u03b8m=(\u2202\u03bc1\u2202\u03b8m,\u2202\u03bc2\u2202\u03b8m,\u22ef,\u2202\u03bcN\u2202\u03b8m){displaystyle {frac {partial mu }{partial theta _{m}}}={begin{pmatrix}{frac {partial mu _{1}}{partial theta _{m}}},&{frac {partial mu _{2}}{partial theta _{m}}},&cdots ,&{frac {partial mu _{N}}{partial theta _{m}}}end{pmatrix}}}\u2202\u03a3\u2202\u03b8m=(\u2202\u03a31,1\u2202\u03b8m\u2202\u03a31,2\u2202\u03b8m\u22ef\u2202\u03a31,N\u2202\u03b8m\u2202\u03a32,1\u2202\u03b8m\u2202\u03a32,2\u2202\u03b8m\u22ef\u2202\u03a32,N\u2202\u03b8m\u22ee\u22ee\u22f1\u22ee\u2202\u03a3N,1\u2202\u03b8m\u2202\u03a3N,2\u2202\u03b8m\u22ef\u2202\u03a3N,N\u2202\u03b8m).{displaystyle {frac {partial Sigma }{partial theta _{m}}}={begin{pmatrix}{frac {partial Sigma _{1,1}}{partial theta _{m}}}&{frac {partial Sigma _{1,2}}{partial theta _{m}}}&cdots &{frac {partial Sigma _{1,N}}{partial theta _{m}}}\\\\{frac {partial Sigma _{2,1}}{partial theta _{m}}}&{frac {partial Sigma _{2,2}}{partial theta _{m}}}&cdots &{frac {partial Sigma _{2,N}}{partial theta _{m}}}\\\\vdots &vdots &ddots &vdots \\\\{frac {partial Sigma _{N,1}}{partial theta _{m}}}&{frac {partial Sigma _{N,2}}{partial theta _{m}}}&cdots &{frac {partial Sigma _{N,N}}{partial theta _{m}}}end{pmatrix}}.}\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki15\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki15\/archives\/321241#breadcrumbitem","name":"\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u91cf – Wikipedia"}}]}]