\u5b8c\u5168\u306b\u7de8\u96c6\u4e2d\u3067\u3059…<\/p>\n\n\n\n
\u3053\u306e\u4e16\u754c\u3067\u8d77\u3053\u308b\u69d8\u3005\u306a\u73fe\u8c61\u306f\u306f\u3069\u306e\u3088\u3046\u306a\u30e1\u30ab\u30cb\u30ba\u30e0\u3067\u5f15\u304d\u8d77\u3053\u3055\u308c\u308b\u306e\u304b\u3002\u4f8b\u3048\u3070\u3001\u3069\u3046\u3044\u3063\u305f\u30e1\u30ab\u30cb\u30ba\u30e0\u3067\u5730\u7403\u306f\u6e29\u6696\u5316\u3059\u308b\u306e\u304b\u3002\u795e\u3067\u306a\u3044\u79c1\u305f\u3061\u306f\u5f53\u7136\u305d\u306e\u30e1\u30ab\u30cb\u30ba\u30e0\u3001\u3064\u307e\u308a\u771f\u5b9f\u3092\u77e5\u3089\u306a\u3044\u3002\u3086\u3048\u306b\u305d\u306e\u30e1\u30ab\u30cb\u30ba\u30e0\u3092\u306a\u3093\u3068\u304b\u3057\u3066\u30e2\u30c7\u30eb\u5316\u3057\u3066\u4e88\u6e2c\u3057\u3088\u3046\u3068\u3059\u308b\u3002\u3053\u308c\u306f\u57fa\u672c\u7684\u306b\u79d1\u5b66\u304c\u884c\u304a\u3046\u3068\u3057\u3066\u3044\u308b\u3053\u3068\u3060\u3002\u8a70\u307e\u308b\u3068\u3053\u308d\u3001\u79d1\u5b66\u306f\u4e16\u754c\u306e\u771f\u5b9f\u3092\u77e5\u3089\u306a\u3044\u79c1\u305f\u3061\u304c\u306a\u3093\u3068\u304b\u3057\u3066\u771f\u5b9f\u306b\u8fd1\u3065\u3053\u3046\u3068\u3059\u308b\u4f5c\u696d\uff08\u30e2\u30c7\u30ea\u30f3\u30b0\uff09\u3067\u3042\u308b\u3002\u3057\u304b\u3057\u3053\u3053\u3067\u7591\u554f\u304c\u6d6e\u304b\u3073\u4e0a\u304c\u308b\u3002\u79c1\u305f\u3061\u306f\u771f\u5b9f\u3092\u77e5\u3089\u306a\u3044\u306e\u306b\u3001\u306a\u305c\u3069\u306e\u30e2\u30c7\u30eb\u304c\u771f\u5b9f\u306b\u8fd1\u3044\u3068\u8a00\u3046\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u304b\u3002\u3082\u3061\u308d\u3093\u5e38\u8b58\u7684\u306b\u8003\u3048\u308b\u3068\u3001\u7269\u7406\u5b66\u306e\u3088\u3046\u306b\u57fa\u672c\u7684\u306b\u4e88\u6e2c\u304c\u5b8c\u74a7\u306b\u3067\u304d\u308b\u5834\u5408\u3001\u305d\u306e\u3088\u3046\u306a\u30e2\u30c7\u30eb\u306f\u6b63\u3057\u3044\u3001\u771f\u5b9f\uff08\u3082\u3057\u304f\u306f\u771f\u5b9f\u3078\u306e\u826f\u3044\u8fd1\u4f3c\uff09\u3067\u3042\u308b\u3068\u8a00\u3046\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u4e88\u6e2c\u304c\u307b\u307c\u5b8c\u74a7\u306a\u5834\u5408\u3001\u3082\u3057\u305d\u306e\u30e2\u30c7\u30eb\uff08\u7406\u8ad6\uff09\u304c\u771f\u5b9f\u3068\u7570\u306a\u3063\u3066\u3044\u308b\u306e\u3067\u3042\u308c\u3070\u3001\u3042\u307e\u308a\u306b\u3082\u30df\u30e9\u30af\u30eb\u3068\u3044\u3046\u3082\u306e\u3060\u308d\u3046\u3002\u3057\u304b\u3057\u79d1\u5b66\u306f\u7269\u7406\u5b66\u306e\u3088\u3046\u306b\u7cbe\u7dfb\u306b\u69cb\u7bc9\u3055\u308c\u305f\u3082\u306e\u3060\u3051\u3067\u306f\u306a\u3044\uff08\u4f8b\u3048\u3070\u3001\u91cf\u5b50\u529b\u5b66\u306e\u4e88\u6e2c\u7cbe\u5ea6\u306f\u5927\u5909\u9ad8\u3044\uff09\u3002\u305d\u306e\u307b\u304b\u306e\u9818\u57df\u3067\u306f\u8907\u6570\u306e\u7af6\u5408\u3059\u308b\u30e2\u30c7\u30eb\u304c\u3042\u308b\u5834\u5408\u3082\u3042\u308b\u3002\u305d\u3046\u3044\u3063\u305f\u5834\u5408\u306b\u3001\u79c1\u305f\u3061\u306f\u771f\u5b9f\u3092\u77e5\u3089\u306a\u3044\u306e\u306b\u3001\u306a\u305c\u3069\u306e\u30e2\u30c7\u30eb\u304c\u771f\u5b9f\u306b\u8fd1\u3044\u3068\u8a00\u3046\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u304b\u3001\u3068\u3044\u3046\u3053\u3068\u3092\u5c11\u3057\u7d71\u8a08\u7684\u306a\u8996\u70b9\u304b\u3089\u8003\u3048\u3066\u307f\u305f\u3044\u3002\u3082\u3061\u308d\u3093\u3053\u308c\u306f\u6a5f\u80fd\u4e3b\u7fa9\u3068\u845b\u85e4\u7406\u8ad6\u306e\u5bfe\u7acb\u306a\u3069\u3068\u3044\u3063\u305f\u793e\u4f1a\u79d1\u5b66\u3084\u54f2\u5b66\u306e\u30e2\u30c7\u30eb\u9078\u629e\u306b\u7c21\u5358\u306b\u5f53\u3066\u306f\u307e\u308b\u3082\u306e\u3067\u306f\u306a\u3044\u304c\u3001\u305d\u308c\u3067\u3082\u5c11\u3057\u8003\u3048\u3066\u307f\u3088\u3046\u3002\u4eca\u56de\u8003\u3048\u308b\u306e\u306fELBO\u3001AIC\u3001MCMC\u3001CFTP\u3068\u3044\u3046\u7d71\u8a08\u5b66\u306e\u624b\u6cd5\u3067\u3001\u3053\u308c\u3089\u306f\u5b9f\u306f\u304b\u306a\u308a\u5de7\u5999\u306a\u6570\u5b66\u306e\u4f7f\u7528\u65b9\u6cd5\u3067\u771f\u5b9f\u3092\u77e5\u3089\u306a\u3044\u306e\u306b\u3001\u3069\u306e\u30e2\u30c7\u30eb\u304c\u771f\u5b9f\u306b\u8fd1\u3044\u304b\u3092\u8a55\u4fa1\u3057\u3066\u304f\u308c\u308b\u3082\u306e\u3060\u3002\u7279\u306bELBO\u3001AIC\u306f\u305d\u306e\u5178\u578b\u3060\u3002<\/p>\n\n\n\n
\u3053\u3053\u3067\u4f55\u304c\u884c\u308f\u308c\u3066\u3044\u308b\u306e\u304b\u3002\u4ee5\u4e0b\u5c11\u3057\u30c6\u30af\u30cb\u30ab\u30eb\u306b\u306a\u308b\u306e\u3067\u3001\u7279\u306bELBO\u3068AIC\u304c\u4f55\u3092\u884c\u3063\u3066\u3044\u308b\u306e\u304b\u3092\u6bd4\u55a9\u3092\u4f7f\u3063\u3066\u8aac\u660e\u3057\u3088\u3046\u3002\u5730\u7403\u306b\u306f\u5357\u5317\u306e\u65b9\u5411\u306e\u4f4d\u7f6e\u3092\u793a\u3059\u7def\u5ea6\u3068\u6771\u897f\u65b9\u5411\u306e\u4f4d\u7f6e\u3092\u793a\u3059\u7d4c\u5ea6\u304c\u5b58\u5728\u3059\u308b\u3002\u5b9f\u306f\u7def\u5ea6\u306e\u8a08\u6e2c\u306f\u661f\u3092\u57fa\u6e96\u3068\u3057\u3066\u884c\u3046\u3053\u3068\u304c\u3067\u304d\u3001\u6614\u304b\u3089\u305d\u308c\u306f\u884c\u308f\u308c\u3066\u3044\u305f\u3002\u3060\u304b\u3089\u5927\u822a\u6d77\u6642\u4ee3\u306b\u3082\u305d\u308c\u306f\u884c\u308f\u308c\u3066\u3044\u305f\u3002\u3057\u304b\u3057\u6771\u897f\u65b9\u5411\u306e\u4f4d\u7f6e\u3092\u793a\u3059\u7d4c\u5ea6\u306e\u8a08\u7b97\u306f\u96e3\u3057\u304f\u3053\u308c\u306b\u3088\u308a\u591a\u304f\u306e\u4e8b\u6545\u304c\u8d77\u3053\u3063\u3066\u3044\u305f\u3002\u3061\u306a\u307f\u306b\u6614\u306f\u7d50\u3073\u76ee\uff08\u30ce\u30c3\u30c8\uff09\u306e\u3042\u308b\u30ed\u30fc\u30d7\u3092\u3064\u3051\u305f\u4e38\u592a\uff08\u30ed\u30b0\uff09\u3092\u6d77\u306b\u6295\u3052\u5165\u308c\u3066\u8ddd\u96e2\u3092\u6e2c\u3063\u3066\u3044\u305f\u3002\u3060\u304b\u3089\u30ce\u30c3\u30c8\u304c\u8239\u306e\u901f\u5ea6\u306e\u5358\u4f4d\u3067\u3001\u30ed\u30b0\u3092\u5165\u308c\u308b\u3053\u3068\u3001\u4f55\u304b\u306b\u5165\u308b\u3053\u3068\u306f\u30ed\u30b0\u30a4\u30f3\u3067\u3042\u308b\u3002\u3055\u3066\u3001\u3053\u3053\u3067\u30e8\u30fc\u30ed\u30c3\u30d1\u304b\u3089\u898b\u305f\u3068\u304d\u306b\u30a2\u30e1\u30ea\u30ab\u5927\u9678\u306f\u897f\u306b\u3042\u308b\u306e\u3060\u304c\u3001\u6771\u897f\u65b9\u5411\u306e\u4f4d\u7f6e\u3092\u793a\u3059\u7d4c\u5ea6\u306e\u8a08\u7b97\u304c\u3067\u304d\u306a\u3044\u304b\u3089\u6b63\u78ba\u306b\u30a2\u30e1\u30ea\u30ab\u5927\u9678\u306f\u3069\u3053\u306b\u3042\u308b\u304b\u308f\u304b\u3089\u306a\u3044\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002\u3064\u307e\u308a\u30a2\u30e1\u30ea\u30ab\u5927\u9678\u306e\u771f\u5b9f\u306e\u4f4d\u7f6e\u306f\u308f\u304b\u3089\u306a\u3044\u3002\u3067\u306f\u30a2\u30e1\u30ea\u30ab\u5927\u9678\u306e\u6b63\u78ba\u306a\u4f4d\u7f6e\u3092\u77e5\u3089\u306a\u3044\u3068\u30a2\u30e1\u30ea\u30ab\u5927\u9678\u306b\u8fd1\u3065\u3051\u306a\u3044\u306e\u304b\u3068\u3044\u3046\u3068\u305d\u3046\u3067\u306f\u306a\u304f\u3066\u3001\u897f\u306b\u884c\u3051\u3070\u884c\u304f\u307b\u3069\u30a2\u30e1\u30ea\u30ab\u5927\u9678\u306b\u306f\u8fd1\u3065\u304f\u3002\u30a2\u30e1\u30ea\u30ab\u5927\u9678\u306e\u6b63\u78ba\u306a\u4f4d\u7f6e\u306f\u308f\u304b\u3089\u306a\u3044\u304c\u3001\u305d\u308c\u306f\u5b9a\u6570\u3068\u3057\u3066\u6271\u3044\u3001\u3068\u308a\u3042\u3048\u305a\u897f\u3078\u897f\u3078\u3068\u5411\u3044\u897f\u3078\u306e\u8ddd\u96e2\u3092\u7e2e\u3081\u308c\u3070\u3001\u30a2\u30e1\u30ea\u30ab\u5927\u9678\u306b\u306f\u8fd1\u3065\u304f\u3002<\/p>\n\n\n\n
ELBO\u3068AIC<\/strong><\/p>\n\n\n\n \u30d9\u30a4\u30ba\u7d71\u8a08<\/a>\\[P(z \\mid x) = \\frac{P(x \\mid z) P(z)}{P(x)}\\]\u3067\u79c1\u305f\u3061\u304c\u6c42\u3081\u305f\u3044\u3082\u306e\u306f\u4e8b\u5f8c\u78ba\u7387\\[P(z \\mid x)\\]\u306a\u306e\u3060\u304c\u3001\u3053\u308c\u3092\u77e5\u308b\u305f\u3081\u306b\u306f\\[P(z \\mid x) = \\frac{P(x \\mid z) \\, P(z)}{P(x)}\\]\u304c\u5fc5\u8981\u3068\u306a\u308b\u3002\u3057\u304b\u3057\u30c7\u30a3\u30e1\u30f3\u30b7\u30e7\u30f3\u304c\u9ad8\u3044\u5834\u5408\uff08\u30d1\u30e9\u30e1\u30fc\u30bf\u304c\u591a\u3044\u5834\u5408\uff09\u6b63\u898f\u5316\u5b9a\u6570\u3067\u3042\u308b\\[P(x) = \\int P(x \\mid z) \\, P(z) \\, dz\\]\u3092\u6c42\u3081\u308b\u3053\u3068\u306f\u96e3\u3057\u3044\u5834\u5408\u304c\u591a\u3044\uff08\u6b63\u898f\u5316\u5b9a\u6570\u304c\u306a\u3044\u3068\u78ba\u7387\u3068\u3057\u3066\u6210\u7acb\u3057\u306a\u3044\uff09\u3002\u3086\u3048\u306b\\[P(z \\mid x)\\]\u3092\u5225\u306e\u7c21\u5358\u306a\u5206\u5e03\\[Q(z)\\]\u3067\u8fd1\u4f3c\u3059\u308b\u3068\u3044\u3046\u65b9\u6cd5\u304c\u4f7f\u308f\u308c\u308b\u3002\u3053\u306e\u6642\u306b\u3001\\[D_{\\text{KL}}(Q(z) || P(z \\mid x))\\]\u306e\u30ab\u30eb\u30d0\u30c3\u30af\u30fb\u30e9\u30a4\u30d6\u30e9\u30fc\u60c5\u5831\u91cf\u3068\u547c\u3070\u308c\u308b\u4e8c\u3064\u306e\u5206\u5e03\u306e\u8ddd\u96e2\uff08\u306e\u3088\u3046\u306a\u3082\u306e\uff09\u3092\u6e2c\u308b\u3053\u3068\u3067\\[Q(z)\\]\u3092\\[P(z \\mid x)\\]\u306b\u8fd1\u4f3c\u3055\u305b\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3057\u304b\u3057\u771f\u5b9f\u306e\u5206\u5e03\u3067\u3042\u308b\\[P(z \\mid x)\\]\u81ea\u4f53\u306e\u5206\u5e03\u3092\u77e5\u3089\u306a\u3044\u306e\u306b\u3069\u3046\u3057\u3066\\[D_{\\text{KL}}(Q(z) || P(z \\mid x))\\]\u306e\u30ab\u30eb\u30d0\u30c3\u30af\u30fb\u30e9\u30a4\u30d6\u30e9\u30fc\u60c5\u5831\u91cf\u3092\u6e2c\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u304b\u3002\u3053\u3053\u3067\u5909\u5206\u4e0b\u9650\uff08Evidence Lower Bound; ELBO\uff09\u304c\u767b\u5834\u3059\u308b\uff08\u30c7\u30a3\u30e1\u30f3\u30b7\u30e7\u30f3\u304c\u9ad8\u3044\u5834\u5408\u3001\u78ba\u7387\u306f\u639b\u3051\u7b97\u3067\u3042\u308a\u3001\u3068\u3066\u3082\u5c0f\u3055\u304f\u306a\u3063\u3066\u3057\u307e\u3046\u3068\u3044\u3046\u7406\u7531\u3068log\u306f\u639b\u3051\u7b97\u3092\u8db3\u3057\u7b97\u306b\u5909\u63db\u3059\u308b\u3068\u3044\u3046\u7406\u7531\u3067log\u304c\u3088\u304f\u4f7f\u308f\u308c\u308b\uff09\u3002\\[Q(z)\\]\u3068\\[P(z \\mid x)\\]\u306e\u8ddd\u96e2\u306f<\/p>\n\n\n\n \\[D_{\\text{KL}}(Q(z) \\| P(z \\mid x)) = \\int Q(z) \\log \\frac{Q(z)}{P(z \\mid x)} \\, dz\\]<\/p>\n\n\n\n \u30d9\u30a4\u30ba\u306e\u30eb\u30fc\u30eb\u306b\u3088\u308a<\/p>\n\n\n\n \\[P(z \\mid x) = \\frac{P(x, z)}{P(x)} \\quad \\Rightarrow \\quad \\log P(z \\mid x) = \\log P(x, z) – \\log P(x)\\]<\/p>\n\n\n\n \u3053\u308c\u3092\u30ab\u30eb\u30d0\u30c3\u30af\u30e9\u30a4\u30d6\u30e9\u30fc\u60c5\u5831\u91cf\u306b\u5165\u308c\u3066<\/p>\n\n\n\n \\[D_{\\text{KL}}(Q(z) \\| P(z \\mid x)) = \\int Q(z) \\left[ \\log Q(z) – \\log P(x, z) + \\log P(x) \\right] dz\\]<\/p>\n\n\n\n \u5b9a\u6570 \\[\\log P(x)\\]\u306f\\[z\\]\u306b\u4f9d\u5b58\u3057\u306a\u3044\u306e\u3067\u5916\u306b\u51fa\u3057\u3001<\/p>\n\n\n\n \\[D_{\\text{KL}}(Q \\| P) = \\int Q(z) \\log Q(z) \\, dz – \\int Q(z) \\log P(x, z) \\, dz + \\log P(x)\\]<\/p>\n\n\n\n \u66f8\u304d\u63db\u3048\u308b\u3068<\/p>\n\n\n\n \\[D_{\\text{KL}}(Q \\| P) = \\mathbb{E}_{Q}[\\log Q(z)] – \\mathbb{E}_{Q}[\\log P(x, z)] + \\log P(x)\\]<\/p>\n\n\n\n \u3068\u306a\u308b\u3002\u3053\u3053\u3067<\/p>\n\n\n\n \\[\\log P(x)\\]\u3092\u5206\u3051\u308b\u3068\u3001<\/p>\n\n\n\n \\[\\log P(x) = D_{\\text{KL}}(Q(z) \\| P(z \\mid x)) + \\left( \\mathbb{E}_{Q}[\\log P(x, z)] – \\mathbb{E}_{Q}[\\log Q(z)] \\right)\\]<\/p>\n\n\n\n \u3053\u3053\u3067Evidence Lower Bound (ELBO)\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n\n\n\n \\[\\text{ELBO}(Q) = \\mathbb{E}_{Q}[\\log P(x, z)] – \\mathbb{E}_{Q}[\\log Q(z)]\\]<\/p>\n\n\n\n \u305d\u3046\u3059\u308b\u3068\u5f0f\u5168\u4f53\u306f<\/p>\n\n\n\n \\[\\log P(x) = \\text{ELBO}(Q) + D_{\\text{KL}}(Q(z) \\| P(z \\mid x))\\]<\/p>\n\n\n\n \u3068\u306a\u308a\u3001ELBO\u3092\u6700\u5927\u5316\u3059\u308b\u3053\u3068\u304c\\[D_{\\text{KL}}(Q(z) | P(z \\mid x))\\]\u3092\u6700\u5c0f\u5316\u3055\u308c\u308b\u3053\u3068\u3068\u306a\u308b\u3002\u79c1\u305f\u3061\u306f\u771f\u5b9f\u306e\u5206\u5e03\u3067\u3042\u308b\\[P(z \\mid x)\\]\u81ea\u4f53\u306e\u5206\u5e03\u3092\u77e5\u3089\u306a\u3044\u304b\u3089\\[D_{\\text{KL}}(Q(z) | P(z \\mid x))\\]\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u306a\u3044\u306b\u3082\u95a2\u308f\u3089\u305a\u3001ELBO\u3068\u3044\u3046\u308f\u304b\u308b\u90e8\u5206\u3092\u6700\u5927\u5316\u3055\u305b\u308b\u3053\u3068\u3067\u771f\u5b9f\u306b\u8fd1\u3065\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n\n\n\n \u3053\u308c\u306fAIC\u306e\u884c\u3063\u305f\u6570\u5b66\u7684\u51e6\u7406\u3068\u306b\u3066\u3044\u308b\u3002AIC\u306f\u5b9f\u969b\u306e\u78ba\u7387\u5206\u5e03\u3092f\u3001\u81ea\u5206\u306e\u30e2\u30c7\u30eb\u3092g\u3068\u3057\u305f\u6642\u306b\u3001\u305d\u308c\u3089\u306e\u9593\u306e\u8ddd\u96e2\u3092\u6e2c\u308b\u30e1\u30ab\u30cb\u30ba\u30e0\u3060\u3002\u3057\u304b\u3057\u5f53\u7136\u306a\u304c\u3089\u5b9f\u969b\u306e\u78ba\u7387\u5206\u5e03f\u3092\u79c1\u305f\u3061\u306f\u77e5\u3089\u306a\u3044\u3002\u77e5\u3063\u3066\u3044\u305f\u3089\u305d\u308c\u3092\u30e2\u30c7\u30eb\u5316\u3059\u308b\u5fc5\u8981\u3059\u3089\u306a\u3044\u3002\u3067\u306f\u3069\u3046\u3057\u3066\u305d\u308c\u3089\u306e\u9593\u306e\u8ddd\u96e2\u3092\u6e2c\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u304b\u3002\u3053\u3053\u3067\u3082\u30ab\u30eb\u30d0\u30c3\u30af\u30fb\u30e9\u30a4\u30d6\u30e9\u30fc\u60c5\u5831\u91cf\u3092\u4f7f\u3044\u4e8c\u3064\u306e\u5206\u5e03\u306e\u8ddd\u96e2\uff08\u306e\u3088\u3046\u306a\u3082\u306e\uff09\u3092\u306f\u304b\u308b\u3002<\/p>\n\n\n\n \\[D_{\\text{KL}}(f | g) = \\int f(x) \\log \\frac{f(x)}{g(x \\mid \\theta)} \\, dx\\]<\/p>\n\n\n\n \\[D_{\\mathrm{KL}}(f | g) = \\mathbb{E}{f} \\left[ -\\log g(x \\mid \\hat{\\theta}) \\right] \u2013 \\mathbb{E}{f} \\left[ -\\log f(x) \\right]\\]<\/p>\n\n\n\n \u3053\u3053\u3067<\/p>\n\n\n\n \\[\\mathbb{E}_{f}\\left[-\\log f(x)\\right]\\]\u306f\u771f\u306e\u5206\u5e03\u306e\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc\u3067\u3042\u308b\u3002\u771f\u306e\u5206\u5e03\u306e\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc\u306f\u30e2\u30c7\u30eb\u306b\u4f9d\u5b58\u3057\u306a\u3044\u5b9a\u6570\uff08C\uff09\u3068\u3057\u3066\u6271\u3046\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n\n\n\n \u3086\u3048\u306b\\[\\mathbb{E}_{f}\\left[-\\log g(x \\mid \\hat{\\theta})\\right]\\]\u3092\u6700\u5c0f\u5316\u3059\u308b\u3068\u3001\u30ab\u30eb\u30d0\u30c3\u30af\u30e9\u30a4\u30d6\u30e9\u30fc\u60c5\u5831\u91cf\u306e\u8ddd\u96e2\u306f\u6700\u5c0f\u5316\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u3053\u3067\u3053\u308c\u306f<\/p>\n\n\n\n \\[\\mathbb{E}_f[-2 \\log g(x \\mid \\hat{\\theta})] \\approx -2 \\log L(\\hat{\\theta}) + 2k = \\mathrm{AIC}\\]<\/p>\n\n\n\n \u306b\u8fd1\u4f3c\u3055\u308c\u308b\uff082\u3068\u3044\u3046\u6570\u5b57\u306f\u5c24\u5ea6\u6bd4\u691c\u5b9a\u3084\u03c7\u4e8c\u4e57\u306a\u3069\u3068\u306e\u6574\u5408\u6027\u3092\u5408\u308f\u305b\u308b\u305f\u3081\u306e\u6570\u5b57\u3060\uff09\u3002\u3064\u307e\u308aELBO\u3082AIC\u3082\u771f\u5b9f\u3092\u77e5\u3089\u306a\u3044\u306e\u306b\u30af\u30ec\u30d0\u30fc\u306a\u6570\u5b66\u7684\u306a\u64cd\u4f5c\u3092\u884c\u3046\u3053\u3068\u306b\u3088\u3063\u3066\u3001\u771f\u5b9f\u306b\u8fd1\u3065\u304f\u3053\u3068\u3092\u53ef\u80fd\u306b\u3059\u308b\u3002<\/p>\n\n\n\n MCMC\u3068CTPF<\/strong><\/p>\n\n\n\n MCMC\u3068CTPF\u3082\u307e\u305f\u771f\u5b9f\u3092\u77e5\u3089\u306a\u3044\u306b\u3082\u95a2\u308f\u3089\u305a\u3001\u771f\u5b9f\u304b\u3089\u306e\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3092\u53ef\u80fd\u306b\u3059\u308b\u3002<\/p>\n\n\n\n \u30e2\u30f3\u30c6\u30ab\u30eb\u30ed\u6cd5\u306f\u4e71\u6570\u306b\u3088\u308a\u3042\u308b\u78ba\u7387\u5206\u5e03P(x)\u304b\u3089\u306e\u30b5\u30f3\u30d7\u30eb\u3092\u751f\u6210\u3059\u308b\u65b9\u6cd5\u306a\u306e\u3060\u304c\u3001\u5b9f\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u6570\u304c\u591a\u3044\u5834\u5408\uff08\u30c7\u30a3\u30e1\u30f3\u30b7\u30e7\u30f3\u304c\u9ad8\u3044\u5834\u5408\uff09\u5f80\u5f80\u306b\u3057\u3066P(x)\u304b\u3089\u76f4\u63a5\u72ec\u7acb\u30b5\u30f3\u30d7\u30eb\u3092\u5f97\u308b\u3053\u3068\u306f\u96e3\u3057\u3044\u3002\u3053\u306e\u3088\u3046\u306a\u5834\u5408\u306b\u30de\u30eb\u30b3\u30d5\u9023\u9396\u30e2\u30f3\u30c6\u30ab\u30eb\u30ed\u6cd5\uff08MCMC\uff09\u3092\u4f7f\u3044P(x)\u306e\u72ec\u7acb\u30b5\u30f3\u30d7\u30eb\u3092\u4f5c\u308a\u51fa\u3059\u3002MCMC\u306f\u30e9\u30f3\u30c0\u30e0\u30a6\u30a9\u30fc\u30af\u3068\u547c\u3070\u308c\u308b\u72ec\u7acb\u3067\u306a\u3044\uff08\u524d\u306e\u72b6\u614b\u306b\u4f9d\u5b58\u3057\u305f\uff09\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u624b\u6cd5\u3092\u7528\u3044\u308b\uff08\u30de\u30eb\u30b3\u30d5\u9023\u9396\uff09\u3002\u72ec\u7acb\u3067\u306a\u3044\u30b5\u30f3\u30d7\u30eb\u306f\u7d71\u8a08\u7684\u306b\u554f\u984c\u306a\u306e\u3060\u304c\u3001MCMC\u306f\u975e\u5e38\u306b\u5de7\u5999\u306b\u72ec\u7acb\u3067\u306a\u3044\u30b5\u30f3\u30d7\u30eb\u304b\u3089\u72ec\u7acb\u30b5\u30f3\u30d7\u30eb\u3092\u4f5c\u308b\u3053\u3068\u3092\u53ef\u80fd\u306b\u3059\u308b\u3002\u5177\u4f53\u7684\u306b\u306f\u30e9\u30f3\u30c0\u30e0\u30a6\u30a9\u30fc\u30af\u304c\u72b6\u614b\u7a7a\u9593\u306e\u4e2d\u3067\u3069\u3053\u3078\u3067\u3082\u884c\u3051\u308b\uff08irreducible\uff09\u3001\u30b5\u30a4\u30af\u30eb\u306b\u9665\u3089\u306a\u3044\u3001\u5b9a\u5e38\u7684\u3067\u3042\u308b\uff08\u6642\u9593\u3092\u7d4c\u3066\u3082\u5e73\u5747\u3068\u5206\u6563\u304c\u5909\u5316\u3057\u306a\u3044\uff09\u3001\u3068\u3044\u3046\u4e09\u3064\u306e\u6761\u4ef6\uff08\u30a8\u30eb\u30b4\u30fc\u30c9\u5b9a\u7406\uff09\u3092\u6e80\u305f\u305b\u3070\u3001P(x)\u3092\u77e5\u3089\u306a\u3044\u306b\u3082\u95a2\u308f\u3089\u305a\uff08\uff01\uff09\u3001\u305d\u3057\u3066\u30e9\u30f3\u30c0\u30e0\u30a6\u30a9\u30fc\u30af\u3068\u547c\u3070\u308c\u308b\u72ec\u7acb\u3067\u306a\u3044\uff08\u524d\u306e\u72b6\u614b\u306b\u4f9d\u5b58\u3057\u305f\uff09\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u624b\u6cd5\u3092\u7528\u3044\u308b\u306b\u3082\u95a2\u308f\u3089\u305a\uff08\uff01\uff09\u3001P(x)\u304b\u3089\u5f97\u3089\u308c\u308b\u6b63\u3057\u3044\u72ec\u7acb\u30b5\u30f3\u30d7\u30eb\u3092\u4f5c\u308a\u51fa\u3059\u3053\u3068\u3092\u53ef\u80fd\u306b\u3059\u308b\uff08\uff01\uff09\u3002\u3057\u304b\u3057MCMC\u3092\u305d\u306e\u307e\u307e\u4f7f\u3063\u305f\u306e\u3067\u306f\u4e8c\u3064\u306e\u554f\u984c\u304c\u3042\u308b\u3002\u4e00\u3064\u306f\u30e9\u30f3\u30c0\u30e0\u30a6\u30a9\u30fc\u30af\u304c\u3042\u307e\u308a\u306b\u9045\u3044\u3053\u3068\u3067\u3042\u308a\u3001\u3053\u308c\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u6570\u304c\u591a\u3044\u5834\u5408\uff08\u3064\u307e\u308a\u30c7\u30a3\u30e1\u30f3\u30b7\u30e7\u30f3\u304c\u9ad8\u3044\u5834\u5408\uff09\u81f4\u547d\u7684\u3060\u3002\u3082\u3046\u4e00\u3064\u306f\u3069\u3053\u307e\u3067\u30e9\u30f3\u30c0\u30e0\u30a6\u30a9\u30fc\u30af\u3092\u3059\u308c\u3070\u6b63\u3057\u3044\u30b5\u30f3\u30d7\u30eb\u304c\u5f97\u3089\u308c\u308b\u306e\u304b\u308f\u304b\u3089\u306a\u3044\u3068\u3044\u3046\u70b9\u306b\u3042\u308b\u3002\u3053\u308c\u306f\u3064\u307e\u308a\u4f55\u304c\u6b63\u3057\u3044\u30b5\u30f3\u30d7\u30eb\u306a\u306e\u304b\u898b\u6975\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u306a\u3044\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u304b\u3089\u3084\u306f\u308a\u554f\u984c\u3060\u3002<\/p>\n\n\n\n \u4e00\u3064\u76ee\u306e\u30e9\u30f3\u30c0\u30e0\u30a6\u30a9\u30fc\u30af\u304c\u9045\u3044\u3068\u3044\u3046\u554f\u984c\u306f\u3001\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306a\u3069\u306e\u624b\u6cd5\u3092\u7528\u3044\u308b\u3053\u3068\u3067\u56de\u907f\u3067\u304d\u308b\uff08\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u30fb\u30e2\u30f3\u30c6\u30ab\u30eb\u30ed\u3002NUTS\u306f\u3055\u3089\u306b\u4e0a\u3092\u3044\u304f\uff09\u3002\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306f\u7269\u7406\u5b66\u3067\u3088\u304f\u51fa\u3066\u304f\u308b\u30cf\u30df\u30eb\u30c8\u30f3\u5f62\u5f0f\u306e\u3053\u3068\u3067\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306f\u5f53\u521d\u3001F\uff1dma\u3001\u3064\u307e\u308a\u529b\u306f\u8cea\u91cf\u3068\u52a0\u901f\u5ea6\u3092\u639b\u3051\u5408\u308f\u305b\u305f\u3082\u306e\u3067\u3042\u308b\u3001\u3068\u3044\u3046\u5f62\u3067\u63d0\u793a\u3055\u308c\u305f\u304c\u3001\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u5f62\u5f0f\u306f\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u3092\u529b\u3067\u306f\u306a\u304f\u30a8\u30cd\u30eb\u30ae\u30fc\u3068\u6349\u3048\u3001\u4f4d\u7f6e\u30a8\u30cd\u30eb\u30ae\u30fc\uff08x\uff09\u3068\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\uff08p\uff09\u3067\u8868\u3055\u308c\u308b\u3002\u7269\u7406\u5b66\u306e\u4e16\u754c\u3067\u306f\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u5f62\u5f0f\u306f\u53e4\u5178\u529b\u5b66\u3068\u91cf\u5b50\u529b\u5b66\u306e\u6a4b\u6e21\u3057\u3092\u3059\u308b\u5f62\u5f0f\u306b\u306a\u308b\u3002\u30e9\u30f3\u30c0\u30e0\u30a6\u30a9\u30fc\u30af\u306e\u72b6\u614b\u7a7a\u9593\u3092\u53e4\u5178\u529b\u5b66\u306e\u4f4d\u76f8\u7a7a\u9593\u3068\u8003\u3048\u3001\u30d1\u30e9\u30e1\u30fc\u30bfx\u3092\u305d\u306e\u4f4d\u76f8\u7a7a\u9593\u306e\u30e9\u30f3\u30c9\u30b9\u30b1\u30fc\u30d7\u306e\u4e2d\u3067\u52d5\u304d\u56de\u308b\u7c92\u5b50\u3068\u8003\u3048\u308b\u3002\u305d\u3057\u3066\u30e9\u30f3\u30c0\u30e0\u30a6\u30a9\u30fc\u30af\u306f\u7c92\u5b50\u304c\u53e4\u5178\u529b\u5b66\u7684\u306a\u6c7a\u5b9a\u8ad6\u7684\u6cd5\u5247\u306b\u3088\u3063\u3066\u72b6\u614b\u7a7a\u9593\u306e\u4e2d\u3092\u52d5\u304f\u3053\u3068\u306b\u306a\u308b\u3002\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u30fb\u30e2\u30f3\u30c6\u30ab\u30eb\u30ed\u3067\u306f\u307e\u305a\u6700\u521d\u306e\u63d0\u6848\u3068\u3057\u3066\u30e9\u30f3\u30c0\u30e0\u306b\u521d\u671f\u306e\u4f4d\u7f6e\uff08x\uff09\u30e2\u30e1\u30f3\u30bf\u30e0\uff08p\uff09\u3068\u4f4d\u7f6e\u3092\u9078\u3076\u3053\u3068\u306b\u3088\u3063\u3066\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u306e\u4e2d\u306b\u30e9\u30f3\u30c0\u30e0\u3055\u3092\u5c0e\u5165\u3059\u308b\u3002\u3053\u306e\u63d0\u6848\u306f\u5fc5\u305a\u53d7\u3051\u5165\u308c\u3089\u308c\u308b\u3002\u305d\u3057\u3066\u5f8c\u306f\u7c92\u5b50\u304c\u30e9\u30f3\u30c0\u30e0\u30a6\u30a9\u30fc\u30af\u306b\u3088\u3063\u3066\u6c7a\u5b9a\u8ad6\u7684\u6cd5\u5247\u306b\u5f93\u3063\u3066\u52d5\u304f\u3053\u3068\u306b\u306a\u308b\u304b\u3089\u3001\u5f8c\u306f\u30e9\u30f3\u30c0\u30e0\u30a6\u30a9\u30fc\u30af\u3092\u6c7a\u5b9a\u8ad6\u7684\u6cd5\u5247\u306b\u5f93\u3063\u3066\u8a08\u7b97\u3059\u308b\u3060\u3051\u3060\u3002<\/p>\n\n\n\n \u7269\u7406\u7684\u4e16\u754c\u89b3\u306e\u4e2d\u3067\u306f\u30a8\u30cd\u30eb\u30ae\u30fc\u4fdd\u5b58\u5247\u304c\u5b58\u5728\u3057\u3001\u30e9\u30f3\u30c0\u30e0\u30a6\u30a9\u30fc\u30af\u306e\u72b6\u614b\u7a7a\u9593\u3082\u53e4\u5178\u529b\u5b66\u7684\u306a\u4e16\u754c\u89b3\u3067\u3042\u308b\u306e\u3067\u3001\u5f53\u7136\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u4fdd\u5b58\u3055\u308c\u308b\u3002\u3057\u304b\u3057\u5b9f\u969b\u306e\u7269\u7406\u7a7a\u9593\u306f\u7a4d\u5206\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u9023\u7d9a\u7684\u4e16\u754c\u3067\u3042\u308b\u304c\u3001\u30e2\u30f3\u30c6\u30ab\u30eb\u30ed\u306f\u305d\u308c\u3092\u96e2\u6563\u7684\u6570\u5b57\u3067\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u3057\u3066\u3044\u308b\u306e\u3067\u3001\u6642\u3068\u3057\u3066\u30ce\u30a4\u30ba\uff08\u6469\u64e6\uff09\u7684\u306a\u3053\u3068\u304c\u751f\u3058\u3066\u3057\u307e\u3044\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u304c\u4e0a\u6607\u3059\u308b\u3053\u3068\u304c\u3042\u308b\u3002\u30e9\u30f3\u30c0\u30e0\u30a6\u30a9\u30fc\u30af\u306e\u524d\u5f8c\u3092\u8a08\u7b97\u3057\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u304c\u4fdd\u5b58\u3055\u308c\u3066\u3044\u308c\u3070\u3001\u65b0\u3057\u3044\u4f4d\u7f6e\u3092\u53d7\u3051\u5165\u308c\u308b\u3002\u30a8\u30cd\u30eb\u30ae\u30fc\u304c\u4e0a\u6607\u3057\u305f\u5834\u5408\u306f\u4e8b\u524d\u306b\u6c7a\u5b9a\u3055\u308c\u305f\u3042\u308b\u78ba\u7387\u3067\u53d7\u3051\u5165\u308c\u306a\u3044\u3068\u3044\u3046\u5224\u65ad\u3092\u4e0b\u3059\u3002<\/p>\n\n\n\n \u30e9\u30f3\u30c9\u30b9\u30b1\u30fc\u30d7\u306e\u5f62\u306f\u4f4d\u7f6e\u30a8\u30cd\u30eb\u30ae\u30fc\\[U(x)=-log\u03c0(x)\\]\u3067\u8868\u3055\u308c\u3001\u7c92\u5b50\u306f\u30e2\u30e1\u30f3\u30bf\u30e0\uff08\u6163\u6027\uff09\u3092\u6301\u3063\u3066\u3044\u308b\u3002\u305d\u3046\u8003\u3048\u308b\u3068\u3001\u5168\u4f53\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u306f<\/p>\n\n\n\n \\[ H(x, p) = U(x) + K(p) \\]<\/p>\n\n\n\n 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