{"id":935,"date":"2020-06-05T11:00:00","date_gmt":"2020-06-05T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=935"},"modified":"2020-06-24T23:34:18","modified_gmt":"2020-06-24T14:34:18","slug":"bayes-theorm","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/bayes-theorm","title":{"rendered":"\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u3010\u30d9\u30a4\u30ba\u7d71\u8a08\u5b66\u3078\u306e\u4e00\u6b69\u3011"},"content":{"rendered":"\n<p class=\"has-text-align-right\"><span style=\"text-decoration: underline;\">\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u9ad8\u6821\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606<\/span><\/p>\n\n\n\n\n\n\n\n<p>\u3000\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u3068\u95a2\u9023\u4ed8\u3051\u3066\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u3068\u3044\u3046\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u3002\u30a2\u30bf\u30ea\u3068\u30cf\u30ba\u30ec\u304c\u5165\u3063\u3066\u3044\u308b\u304f\u3058\u3092\u8003\u3048\u3066\u307f\u307e\u3059\u3002\u81ea\u5206\u304c\u304f\u3058\u3092\u5f15\u304f\u524d\u306b\u304f\u3058\u304c\u5f15\u304b\u308c\u3066\u3044\u305f\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u81ea\u5206\u304c\u5f15\u304f\u524d\u306b\u3069\u306e\u304f\u3089\u3044\u30a2\u30bf\u30ea\u30fb\u30cf\u30ba\u30ec\u304c\u5f15\u304b\u308c\u3066\u3044\u305f\u306e\u304b\u306f\u5206\u304b\u3089\u306a\u3044\u3082\u306e\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u72b6\u614b\u306e\u3068\u304d\u3001\u81ea\u5206\u304c\u304f\u3058\u3092\u5f15\u3044\u305f\u7d50\u679c\u304b\u3089\u3001\u524d\u306e\u4eba\u304c\u3069\u306e\u304f\u3089\u3044\u304f\u3058\u3092\u5f15\u304b\u308c\u3066\u3044\u305f\u306e\u304b\u300c\u539f\u56e0\u300d\u3092\u63a8\u5b9a\u3057\u305f\u3044\u3068\u304d\u304c\u3042\u308a\u307e\u3059\u3002<br>\u3000\u3053\u306e\u3088\u3046\u306b\u3001\u7d50\u679c\u304b\u3089\u539f\u56e0\u3092\u63a8\u5b9a\u3059\u308b\u65b9\u6cd5\u3092\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u3068\u3044\u3044\u307e\u3059\u3002\u3053\u306e\u8a18\u4e8b\u3067\u306f\u3001\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u8aac\u660e\u3057\u3066\u3044\u304d\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<div id=\"toc_container\" class=\"toc_light_blue no_bullets\"><p class=\"toc_title\">\u76ee\u6b21<\/p><ul class=\"toc_list\"><li><a href=\"#i\"><span class=\"toc_number toc_depth_1\">1<\/span> \u30d9\u30a4\u30ba\u306e\u5b9a\u7406<\/a><ul><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_2\">1.1<\/span> \u5177\u4f53\u4f8b<\/a><\/li><\/ul><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_1\">2<\/span> \u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u306e\u8a3c\u660e<\/a><\/li><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_1\">3<\/span> \u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u306e\u307e\u3068\u3081<\/a><\/li><\/ul><\/div>\n<h3><span id=\"i\">\u30d9\u30a4\u30ba\u306e\u5b9a\u7406<\/span><\/h3>\n\n\n\n<p>\u3000\u7d50\u679c\u3068\u306a\u308b\u4e8b\u8c61\u3092\\(A\\)\u3001\u539f\u56e0\u3068\u306a\u308b\u4e8b\u8c61\u3092\\(H_{1},\\cdots,H_{k}\\)\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"wp-block-liquid-speech-balloon liquid-speech-balloon-wrap liquid-speech-balloon-00  liquid-speech-balloon-short\"><div class=\"liquid-speech-balloon-avatar\"><\/div><div class=\"liquid-speech-balloon-text\" style=\"border-color:#d6ebf8;background-color:#d6ebf8;color:#000000\"><p>\u4eca\u3001\u79c1\u305f\u3061\u304c\u8003\u3048\u305f\u3044\u306e\u306f\u4e8b\u8c61\\(A\\)\u3068\u3044\u3046\u7d50\u679c\u304c\u5f97\u3089\u308c\u305f\u3068\u304d\u3001\u4e8b\u8c61\\(H_{i}\\)\u304c\u8d77\u3053\u308b\u78ba\u7387\u3067\u3059\u3002\u3053\u306e\u78ba\u7387\u3092\u8868\u3059\u3082\u306e\u304c\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u3067\u3059\u3002\u65e9\u901f\u3001\u5b9a\u7406\u3092\u898b\u3066\u307f\u307e\u3057\u3087\u3046\uff01<\/p><div class=\"liquid-speech-balloon-arrow\" style=\"border-color:transparent #d6ebf8 transparent transparent\"><\/div><\/div><\/div>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u30d9\u30a4\u30ba\u306e\u5b9a\u7406(Bayes&#8217; theorm)<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\u539f\u56e0\u3068\u306a\u308b\u4e8b\u8c61\\(H_{1},\\cdots,H_{k}\\)\u304c\u304a\u4e92\u3044\u306b\u3059\u3079\u3066\u6392\u53cd\u3060\u3068\u3057\u3001\\(H_{1}\\cup\\cdots\\cup H_{k}=\\Omega\\)\u304c\u6210\u308a\u7acb\u3064\u3082\u306e\u3060\u3068\u4eee\u5b9a\u3057\u307e\u3059\uff08\u305f\u3060\u3057\\(\\Omega\\)\u306f\u5168\u4e8b\u8c61\uff09\u3002\u3053\u306e\u3068\u304d\u7d50\u679c\\(A\\)\u304c\u5f97\u3089\u308c\u305f\u4e0b\u3067\u3001\u539f\u56e0\\(H_{i}\\)\u304c\u5f97\u3089\u308c\u308b\u78ba\u7387\u306f$$\\mathrm{P}(H_{i}|A)=\\frac{\\mathrm{P}(H_{i})\\cdot\\mathrm{P}(A|H_{i})}{\\sum_{j=1}^{k}\\mathrm{P}(H_{j})\\mathrm{P}(A|H_{j})}$$\u3067\u4e0e\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u306e\u5b9a\u7406\u3092<strong>\u30d9\u30a4\u30ba\u306e\u5b9a\u7406<\/strong>\u3068\u3044\u3044\u307e\u3059\u3002\u3053\u3053\u3067\u3001\u78ba\u7387\\(\\mathrm{P}(H_{i})\\)\u306f<strong>\u4e8b\u524d\u78ba\u7387<\/strong>\u3001\u78ba\u7387\\(\\mathrm{P}(H_{i}|A)\\)\u306f<strong>\u4e8b\u5f8c\u78ba\u7387<\/strong>\u3068\u547c\u3070\u308c\u3066\u3044\u307e\u3059\u3002\u300c\u4e8b\u524d\u300d\u300c\u4e8b\u5f8c\u300d\u306f\u7d50\u679c\u306e\u4e8b\u8c61\\(A\\)\u3092\u57fa\u6e96\u306b\u8003\u3048\u3066\u3044\u307e\u3059\u3002<\/div>\n<p>&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-liquid-speech-balloon liquid-speech-balloon-wrap liquid-speech-balloon-00  liquid-speech-balloon-short\"><div class=\"liquid-speech-balloon-avatar\"><\/div><div class=\"liquid-speech-balloon-text\" style=\"border-color:#d6ebf8;background-color:#d6ebf8;color:#000000\"><p>\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u306e\u8a3c\u660e\u306f\u4e0b\u306e\u65b9\u306b\u7d39\u4ecb\u3057\u3066\u3044\u307e\u3059\u3002<br>\u3053\u3053\u304b\u3089\u306f\u5177\u4f53\u4f8b\u3067\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u3092\u4f7f\u3063\u3066\u307f\u307e\u3057\u3087\u3046\uff01<\/p><div class=\"liquid-speech-balloon-arrow\" style=\"border-color:transparent #d6ebf8 transparent transparent\"><\/div><\/div><\/div>\n\n\n\n<h4><span id=\"i-2\">\u5177\u4f53\u4f8b<\/span><\/h4>\n\n\n\n<div style=\"background: #e6ffe6; padding: 15px; border-radius: 10px; border: 0px solid #99cc00; word-break: break-all;\">\u30002\u3064\u306e\u5de5\u5834A\u3001B\u3067\u4f5c\u3089\u308c\u308b\u88fd\u54c1\u306b\u3064\u3044\u3066\u4e0d\u826f\u54c1\u304c\u51fa\u308b\u78ba\u7387\u304c\u305d\u308c\u305e\u308c2%\u30015%\u3067\u3042\u308b\u3068\u3059\u308b\u3002\u307e\u305f\u3001\u5de5\u5834A\u304b\u3089\u51fa\u8377\u3055\u308c\u305f\u88fd\u54c1\u304c\u5168\u4f53\u306e3\u5272\u3001\u5de5\u5834B\u304b\u3089\u51fa\u8377\u3055\u308c\u305f\u88fd\u54c1\u304c\u5168\u4f53\u306e7\u5272\u3092\u5360\u3081\u307e\u3059<br \/>\u30002\u3064\u306e\u5de5\u5834\u304b\u3089\u4f5c\u3089\u308c\u305f\u88fd\u54c1\u306e\u4e2d\u304b\u30891\u500b\u53d6\u308a\u51fa\u3057\u305f\u3068\u304d\u3001\u305d\u308c\u304c\u4e0d\u826f\u54c1\u3067\u3057\u305f\u3002\u305d\u308c\u304c\u5de5\u5834A\u3067\u51fa\u8377\u3055\u308c\u305f\u3082\u306e\u3067\u3042\u308b\u78ba\u7387\u3092\u6c42\u3081\u306a\u3055\u3044\u3002<\/div>\n<p>\u00a0<\/p>\n\n\n\n<p>\u53d6\u308a\u51fa\u3057\u305f\u88fd\u54c1\u304c\u5de5\u5834A\u3001B\u3067\u3064\u304f\u3089\u308c\u305f\u3068\u3044\u3046\u4e8b\u8c61\u3092\\(A\u3001B\\)\u3068\u3057\u307e\u3059\u3002\u3055\u3089\u306b\u3001\u53d6\u308a\u51fa\u3057\u305f\u88fd\u54c1\u304c\u4e0d\u826f\u54c1\u3067\u3042\u308b\u3068\u3044\u3046\u4e8b\u8c61\u3092\\(E\\)\u3068\u3057\u307e\u3059\u3002<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/conditional-probability\" class=\"aioseop-link\">\uff1c\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u306e\u8a18\u4e8b\uff1e<\/a>\u3067\\(\\mathrm{P}(E)=0.041\\)\u3067\u3042\u308b\u3053\u3068\u306f\u6c42\u3081\u307e\u3057\u305f\u3002\u3053\u306e\u3053\u3068\u3092\u7528\u3044\u308b\u3068\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u304b\u3089\u6c42\u3081\u305f\u3044\u78ba\u7387\\(\\mathrm{P}(A|E)\\)\u306f$$\\mathrm{P}(A|E)=\\frac{\\mathrm{P}(A)\\mathrm{P}(D|A)}{\\mathrm{P}(E)}=\\frac{0.30\\times 0.02}{0.041}\\approx 0.146$$\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"wp-block-liquid-speech-balloon liquid-speech-balloon-wrap liquid-speech-balloon-00  liquid-speech-balloon-short\"><div class=\"liquid-speech-balloon-avatar\"><\/div><div class=\"liquid-speech-balloon-text\" style=\"border-color:#d6ebf8;background-color:#d6ebf8;color:#000000\"><p>\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u306f\u591a\u5c11\u8907\u96d1\u3067\u3059\u304c\u3001\u30d9\u30a4\u30ba\u7d71\u8a08\u5b66\u3092\u52c9\u5f37\u3057\u305f\u3044\u65b9\u306f\u3001\u662f\u975e\u3068\u3082\u30de\u30b9\u30bf\u30fc\u3057\u3066\u304a\u304d\u307e\u3057\u3087\u3046\uff01<\/p><div class=\"liquid-speech-balloon-arrow\" style=\"border-color:transparent #d6ebf8 transparent transparent\"><\/div><\/div><\/div>\n\n\n\n<h3><span id=\"i-3\">\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u306e\u8a3c\u660e<\/span><\/h3>\n\n\n\n<div class=\"wp-block-liquid-speech-balloon liquid-speech-balloon-wrap liquid-speech-balloon-00  liquid-speech-balloon-short\"><div class=\"liquid-speech-balloon-avatar\"><\/div><div class=\"liquid-speech-balloon-text\" style=\"border-color:#d6ebf8;background-color:#d6ebf8;color:#000000\"><p>\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u306e\u8a3c\u660e\u306f\u3068\u3063\u3066\u3082\u7c21\u5358\u3067\u3059\u3002<br>\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u3092\u52c9\u5f37\u3057\u3066\u304a\u304f\u3068\u3059\u3050\u306b\u7406\u89e3\u3067\u304d\u307e\u3059\u3002<\/p><div class=\"liquid-speech-balloon-arrow\" style=\"border-color:transparent #d6ebf8 transparent transparent\"><\/div><\/div><\/div>\n\n\n\n<p>\u25a0\u8a3c\u660e<\/p>\n\n\n\n<p>\u307e\u305a\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u306e\u5b9a\u7fa9\u304b\u3089$$\\mathrm{P}(H_{i}|A)=\\frac{\\mathrm{P}(H_{i}\\cap A)}{\\mathrm{P}(A)} = \\frac{\\mathrm{P}(A|H_{i})\\cdot \\mathrm{P}(H_{i})}{\\mathrm{P}(A)}$$\u3068\u306a\u308a\u307e\u3059\u3002\u96c6\u5408\u8ad6\u306e\u6027\u8cea\u304b\u3089<br><div style=\"overflow-x: auto;\">\\begin{align}<br>A &amp;= A\\cap\\Omega\\\\<br>&amp;= A\\cap(H_{1}\\cup\\cdots\\cup H_{k})\\\\<br>&amp;= (A\\cap H_{1})\\cup\\cdots\\cup (A\\cap H_{k})<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002\u4e8b\u8c61\\((A\\cap H_{1}),\\cdots, (A\\cap H_{k})\\)\u306f\u304a\u4e92\u3044\u306b\u6392\u53cd\u3067\u3042\u308b\u3053\u3068\u3092\u7528\u3044\u308b\u3068\u3001\u78ba\u7387\u306e\u52a0\u6cd5\u5b9a\u7406\u304b\u3089$$\\mathrm{P}(A)=\\sum_{j=1}^{k}\\mathrm{P}(A\\cap H_{j})=\\sum_{j=1}^{k}\\mathrm{P}(H_{j})\\cdot\\mathrm{P}(A|H_{j})$$\u3068\u306a\u308b\u3053\u3068\u304b\u3089$$\\mathrm{P}(H_{i}|A)=\\frac{\\mathrm{P}(A|H_{i})\\cdot\\mathrm{P}(H_{i})}{\\sum_{j=1}^{k}\\mathrm{P}(H_{j})\\mathrm{P}(A|H_{j})}$$\u304c\u6210\u7acb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"has-text-align-right\">\u25a1<\/p>\n\n\n\n<p>\u203b\u3000\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u306b\u3064\u3044\u3066\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/conditional-probability\" class=\"aioseop-link\">\uff1c\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u306e\u8a18\u4e8b\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002<br>\u203b\u3000\u78ba\u7387\u306e\u6392\u53cd\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/set-calculation\" class=\"aioseop-link\">\uff1c\u4e8b\u8c61\u306e\u6f14\u7b97\u306e\u8a18\u4e8b\uff1e<\/a>\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002<br>\u203b\u3000\u78ba\u7387\u306e\u52a0\u6cd5\u5b9a\u7406\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/probability-def\" class=\"aioseop-link\">\uff1c\u78ba\u7387\u306e\u5b9a\u7fa9\u306e\u8a18\u4e8b\uff1e<\/a>\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<h3><span id=\"i-4\">\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u306e\u307e\u3068\u3081<\/span><\/h3>\n\n\n\n<p>\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u306f\u5f97\u3089\u308c\u305f\u7d50\u679c\u304b\u3089\u3001\u5143\u3005\u306e\u539f\u56e0\u304c\u8d77\u3053\u308b\u78ba\u7387\u3092\u4e88\u6e2c\u3059\u308b\u305f\u3081\u306e\u5b9a\u7406\u3067\u3059\u3002\u4e3b\u306b\u3001\u7e70\u308a\u8fd4\u3057\u6e2c\u5b9a\u304c\u96e3\u3057\u3044\u5834\u5408\u306b\u3088\u304f\u4f7f\u7528\u3055\u308c\u3001\u4e88\u6e2c\u30fb\u4eba\u5de5\u77e5\u80fd\u306a\u3069\u306b\u3088\u304f\u4f7f\u7528\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u306f\u3001\u30d9\u30a4\u30ba\u7d71\u8a08\u5b66\u306e\u57fa\u790e\u3068\u306a\u3063\u3066\u3044\u308b\u306e\u3067\u3001\u30d9\u30a4\u30ba\u7d71\u8a08\u5b66\u3092\u52c9\u5f37\u3057\u3066\u3044\u304f\u4e0a\u3067\u306f\u3068\u3066\u3082\u91cd\u8981\u306a\u6027\u8cea\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u9ad8\u6821\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u3000\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u3068\u95a2\u9023\u4ed8\u3051\u3066\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u3068\u3044\u3046\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u3002\u30a2\u30bf\u30ea\u3068\u30cf\u30ba\u30ec\u304c\u5165\u3063\u3066\u3044\u308b\u304f\u3058\u3092\u8003\u3048\u3066\u307f &#8230; 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