{"id":2263,"date":"2020-07-31T11:00:00","date_gmt":"2020-07-31T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2263"},"modified":"2020-08-11T11:17:10","modified_gmt":"2020-08-11T02:17:10","slug":"cdirid-mv","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cdirid-mv","title":{"rendered":"\u30c7\u30a3\u30ea\u30af\u30ec\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011"},"content":{"rendered":"\n

\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2605\u2606\u2606<\/span><\/p>\n\n\n\n\n\n\n\n

\u3053\u306e\u8a18\u4e8b\u3067\u306f\u30c7\u30a3\u30ea\u30af\u30ec\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u3092\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u304c\u5206\u304b\u3089\u306a\u3044\u65b9\u306f\u662f\u975e\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002\u305d\u306e\u4ed6\u306e\u30c7\u30a3\u30ea\u30af\u30ec\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306f\uff1c\u30c7\u30a3\u30ea\u30af\u30ec\u5206\u5e03\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n\n

\n \n
\"\u30c7\u30a3\u30ea\u30af\u30ec\u5206\u5e03\"<\/div>\n
\n
\u30c7\u30a3\u30ea\u30af\u30ec\u5206\u5e03<\/div>\n
\u3000\u30c7\u30a3\u30ea\u30af\u30ec\u5206\u5e03\uff08Dirichlet distribution\uff09\u306f\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3001\u30d9\u30fc\u30bf\u5206\u5e03\u3092\u591a\u6b21\u5143\u5316\u3057\u305f\u3082\u306e\u306b\u306a\u308a\u307e\u3059\u3002\u3000\u30c7\u30a3\u30ea\u30af\u30ec\u5206\u5e03\u306f\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n

\u3000<\/p>\n\n\n\n

\u30c7\u30a3\u30ea\u30af\u30ec\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563<\/h2>\n\n\n\n
\u671f\u5f85\u5024\u3068\u5206\u6563<\/strong><\/div>\n
\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\alpha=(\\alpha_{1},\\cdots,\\alpha_{k})\\)\u306e\u30c7\u30a3\u30ea\u30af\u30ec\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim Dir(\\alpha)\\)\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\n
\\begin{align}\n\\mathrm{E}[X_{i}]=\\frac{\\alpha_{i}}{\\alpha_{0}},\\ \\ \\ \\mathrm{Var}[X_{i}]=\\frac{ \\alpha_{i}(\\alpha_{0}-\\alpha_{i}) }{ \\alpha_{0}^{2}(\\alpha_{0}+1) }\n\\end{align}<\/div>\n\u305f\u3060\u3057\u3001\\(\\alpha_{0}=\\sum_{i=1}^{n}\\alpha_{i}\\)\u3067\u3059\u3002\n<\/div>\n\n\n\n

\u671f\u5f85\u5024\u30fb\u5206\u6563\u3092\u6c42\u3081\u308b\u969b\u306b\u306f\uff1c\u671f\u5f85\u5024\u306e\u5b9a\u7fa9\uff1e<\/a>\u304a\u3088\u3073\uff1c\u5206\u6563\u306e\u5b9a\u7fa9\uff1e<\/a>\u3092\u4f7f\u7528\u3059\u308b\u306e\u3067\u3001\u899a\u3048\u3066\u3044\u306a\u3044\u65b9\u306f\u8a3c\u660e\u3092\u8aad\u3080\u524d\u306b\u4e00\u5ea6\u3001\u76ee\u3092\u901a\u3057\u3066\u304a\u3044\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\u30c7\u30a3\u30ea\u30af\u30ec\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u3092\u6c42\u3081\u308b\u306b\u3042\u305f\u3063\u3066\u6b21\u306e\u30ac\u30f3\u30de\u95a2\u6570\u306e\u7a4d\u5206\u516c\u5f0f\u3092\u7528\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u30ac\u30f3\u30de\u95a2\u6570\u306e\u7a4d\u5206\u516c\u5f0f<\/strong><\/div>\n
\u30ac\u30f3\u30de\u95a2\u6570\u306b\u306f\u6b21\u306e\u3088\u3046\u306a\u6027\u8cea\u304c\u3042\u308a\u307e\u3059\u3002\n
\\begin{align}\n\\int_{0}^{\\infty} x_{1}^{\\alpha_{1}-1}\\cdots x_{n}^{\\alpha_{n}-1}dx_{1}\\cdots dx_{n-1} = \\frac{\\Gamma(\\alpha_{1})\\cdots\\Gamma(\\alpha_{n})}{\\Gamma\\left( \\sum_{i=1}^{n}\\alpha_{i} \\right)}\n\\end{align}<\/div>\n\u305f\u3060\u3057\u3001\\(x_{i}\\geq0 (1\\leq i\\leq n-1),\\sum_{i=1}^{n-1}x_{i}\\leq 1\\)\u3067\u3059\u3002\n<\/div>\n\n\n\n

\u3000<\/p>\n\n\n\n

\u8a3c\u660e\uff08\u30ac\u30f3\u30de\u95a2\u6570\u306e\u7a4d\u5206\u516c\u5f0f\uff09<\/h4>\n\n\n\n

\u30ac\u30f3\u30de\u95a2\u6570\u306e\u5b9a\u7fa9\u304b\u3089

\\begin{align}
\\Gamma(\\alpha_{1})\\cdots\\Gamma(\\alpha_{n}) &= \\int_{0}^{\\infty}e^{-t_{1}}t_{1}^{\\alpha_{1}-1}dt_{1}\\cdots\\int_{0}^{\\infty}e^{-t_{n}}t_{n}^{\\alpha_{n}-1}dt_{n} \\\\
&= \\int_{0}^{\\infty}\\cdots\\int_{0}^{\\infty} e^{-t_{1}-\\cdots-t_{n}}t_{1}^{\\alpha_{1}-1}\\cdots t_{n}^{\\alpha_{n}-1}dt_{1}\\cdots dt_{n}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067
\\begin{align}
t_{1}=u_{1}y,\\ t_{2}=u_{2}y,\\ \\cdots\\ ,t_{n-1}=u_{n-1}y,\\ t_{n}=(1-u_{1}-\\cdots-u_{n-1})y
\\end{align}<\/div>
\u3067\u5909\u6570\u5909\u63db\u3092\u3057\u307e\u3059\u3002\u3053\u306e\u5909\u6570\u5909\u63db\u306b\u3088\u308b\u30e4\u30b3\u30d3\u30a2\u30f3\u306f\\(y^{n-1}\\)\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3001\u4e0a\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3068
\\begin{align}
\\int_{0}^{\\infty}e^{-y}y^{\\alpha_{1}+\\cdots+\\alpha_{n}-n}y^{n-1}dy\\times\\int u_{1}^{\\alpha_{1}-1}\\cdots u_{n-1}^{\\alpha_{n-1}-1}(1-u_{1}-\\cdots-u_{n-1})^{\\alpha_{n}-1}du_{1}\\cdots du_{n-1} \\\\
= \\int_{0}^{\\infty}e^{-y}y^{\\alpha_{1}+\\cdots+\\alpha_{n}-1}\\times\\int u_{1}^{\\alpha_{1}-1}\\cdots u_{n-1}^{\\alpha_{n-1}-1}u_{n}^{\\alpha_{n}-1}du_{1}\\cdots du_{n-1} \\\\
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u3053\u3053\u3067
\\begin{align}
u_{n} = 1-u_{1}-\\cdots-u_{n-1}
\\end{align}<\/div>
\u3067\u3059\u3002\u30ac\u30f3\u30de\u95a2\u6570\u306e\u5b9a\u7fa9\u304b\u3089
\\begin{align}
\\Gamma(\\alpha_{1})\\cdots\\Gamma(\\alpha_{n}) &= \\Gamma(\\alpha_{1}+\\cdots+\\alpha_{n})\\int u_{1}^{\\alpha_{1}-1}\\cdots u_{n-1}^{\\alpha_{n-1}-1}u_{n}^{\\alpha_{n}-1}du_{1}\\cdots du_{n-1}
\\end{align}<\/div>
\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3053\u3068\u304b\u3089\u4e21\u8fba\u3092\u6574\u7406\u3059\u308b\u3068\u3001\u30ac\u30f3\u30de\u95a2\u6570\u306e\u7a4d\u5206\u516c\u5f0f\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n

\u25a1<\/p>\n\n\n\n