指数分布族モデル / 共役な事前分布 / 予測分布

指数分布族モデル

$$\begin{array}{rcl}q(x;\theta )& =& v(x)\exp (f(\theta )\cdot g(x))\\ {\int }_{X}q(x;\theta )\mathrm{d}x& =& {\int }_{X}v(x)\exp (f(\theta )\cdot g(x))\mathrm{d}x\\ & =& 1\cdots \mathrm{確}\mathrm{率}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{こ}\mathrm{の}\mathrm{条}\mathrm{件}\mathrm{を}\mathrm{満}\mathrm{た}\mathrm{す}\mathrm{よ}\mathrm{う}\mathrm{に}v,f,g\mathrm{を}\mathrm{用}\mathrm{意}\mathrm{す}\mathrm{る}\end{array}$$

共役な事前分布

$$\begin{array}{rcl}\pi (\theta ;\varphi )& =& \frac{1}{z(\varphi )}\exp (f(\theta )\cdot \varphi )\\ {\int }_X\pi (\theta ;\varphi )\mathrm{d}x& =& {\int }_X\frac{1}{z(\varphi )}\exp (f(\theta )\cdot \varphi )\mathrm{d}x\\ & =& 1\cdots \mathrm{確}\mathrm{率}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{こ}\mathrm{の}\mathrm{条}\mathrm{件}\mathrm{を}\mathrm{満}\mathrm{た}\mathrm{す}\mathrm{よ}\mathrm{う}\mathrm{に}z,f,\varphi \mathrm{を}\mathrm{用}\mathrm{意}\mathrm{す}\mathrm{る}\\ z(\varphi )& =& {\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi )\mathrm{d}\theta \cdots \mathrm{む}\mathrm{し}\mathrm{ろ}z\mathrm{は}\mathrm{上}\mathrm{記}\mathrm{条}\mathrm{件}\mathrm{を}\mathrm{満}\mathrm{た}\mathrm{す}\mathrm{た}\mathrm{め}\mathrm{に}\mathrm{用}\mathrm{意}\mathrm{さ}\mathrm{れ}\mathrm{る}\end{array}$$

予測分布

$$\begin{array}{rcl}q(X_{n+1}|X^n)& =& \frac{1}{Z_n(\beta )}\int q(X_{n+1}|\theta {)}^{\beta }\prod _{i=1}^n\{q(X_i|\theta {)}^{\beta }\}\pi (\theta )\mathrm{d}\theta \\ & =& \frac{1}{Z_n(\beta )}{\int }_{\mathrm{\Theta }}\pi (\theta )q(X_{n+1}|\theta {)}^{\beta }{\displaystyle \prod _{i=1}^n{q(X_i|\theta )}^{\beta }\mathrm{d}\theta }\\ & =& \frac{1}{Z_n(\beta )}{\int }_{\mathrm{\Theta }}\pi (\theta ){\displaystyle \prod _{i=1}^{n+1}q(X_i|\theta {)}^{\beta }\mathrm{d}\theta }\dots {\displaystyle \prod _{i=1}^{n+1}q(X_i|\theta {)}^{\beta }=q(X_{n+1}|\theta {)}^{\beta }{\displaystyle \prod _{i=1}^{n}q(X_i|\theta {)}^{\beta }}}\\ & =& \frac{1}{Z_n(\beta )}{Z}_{n+1}(\beta )\dots Z_{n+1}(\beta )={\int }_{\mathrm{\Theta }}\pi (\theta ){\displaystyle \prod _{i=1}^{n+1}q(X_i|\theta {)}^{\beta }\mathrm{d}\theta }\\ & =& \frac{Z_{n+1}(\beta )}{Z_n(\beta )}\end{array}$$

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分配凾数

$$\begin{array}{rcl}{Z}_n(\beta )& =& {\int }_{\mathrm{\Theta }}\pi (\theta ;\varphi )\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }\mathrm{d}\theta \\ & =& {\int }_{\mathrm{\Theta }}\pi (\theta ;\varphi )\prod _{i=1}^n{[v(X_i)\exp (f(\theta )\cdot g(X_i))]}^{\beta }\mathrm{d}\theta \cdots q(X_i;\theta )=v(X_i)\exp (f(\theta )\cdot g(X_i))\\ & =& {\int }_{\mathrm{\Theta }}[\frac{1}{z(\varphi )}\exp (f(\theta )\cdot \varphi )]\prod _{i=1}^n{[v(X_i)\exp (f(\theta )\cdot g(X_i))]}^{\beta }\mathrm{d}\theta \cdots \pi (\theta ;\varphi )=\frac{1}{z(\varphi )}\exp (f(\theta )\cdot \varphi )\\ & =& \frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi )\prod _{i=1}^n{[v(X_i)\exp (f(\theta )\cdot g(X_i))]}^{\beta }\mathrm{d}\theta \cdots \pi (\theta ;\varphi )=\frac{1}{z(\varphi )}\exp (f(\theta )\cdot \varphi )\\ & =& \frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi )\prod _{i=1}^n[v(X_i{)}^{\beta }\exp {(f(\theta )\cdot g(X_i))}^{\beta }]\mathrm{d}\theta \cdots (ab{)}^c=a^c{b}^c\\ & =& \frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi )[\prod _{i=1}^{n}v(X_i{)}^{\beta }][\prod _{i=1}^n\exp {(f(\theta )\cdot g(X_i))}^{\beta }]\mathrm{d}\theta \cdots \prod _{i=1}^{n}AB=\prod _{i=1}^{n}A\prod _{i=1}^{n}B\\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi )[\prod _{i=1}^n\exp \left(\beta \{f\left(\theta \right)\cdot g\left(X_i\right)\}\right)]\mathrm{d}\theta \cdots \exp {\left(A\right)}^B=\exp \left(AB\right)\\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi )[\prod _{i=1}^n\exp (f\left(\theta \right)\cdot \beta g\left(X_i\right))]\mathrm{d}\theta \cdots c(A\cdot B)=A\cdot cB(c:\mathrm{定}\mathrm{数})\\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi +\sum _{i=1}^{n}f\left(\theta \right)\cdot \beta g\left(X_i\right))\mathrm{d}\theta \cdots \exp \left(A\right)\exp \left(B\right)=\exp (A+B)\\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi +f\left(\theta \right)\cdot \sum _{i=1}^n\beta g\left(X_i\right))\mathrm{d}\theta \cdots A\cdot B+A\cdot C=A\cdot (B+C)\\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \{\varphi +\sum _{i=1}^n\beta g\left(X_i\right)\})\mathrm{d}\theta \cdots A\cdot B+A\cdot C=A\cdot (B+C)\\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\mathrm{d}\theta \cdots {\hat{\varphi }}_{\beta }=\varphi +\sum _{i=1}^n\beta g\left(X_i\right)\\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}z({\hat{\varphi }}_{\beta })\cdots z({\hat{\varphi }}_{\beta })={\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\mathrm{d}\theta \\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\end{array}$$

自由エネルギー

$$\begin{array}{rcl}{F}_n(\beta )& =& -\frac{1}{\beta }\log Z_n(\beta )\\ & =& -\frac{1}{\beta }\log \left([\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\right)\\ & =& -\frac{1}{\beta }\log \left([\prod _{i=1}^{n}v(X_i{)}^{\beta }]\right)-\frac{1}{\beta }\log \left(\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\right)\\ & =& -\frac{1}{\beta }\log \left([v(X_1{)}^{\beta }v(X_2{)}^{\beta }\cdots v(X_n{)}^{\beta }]\right)-\frac{1}{\beta }\log \left(\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\right)\\ & =& -\frac{1}{\beta }\log \left({[v(X_1)v(X_2)\cdots v(X_n)]}^{\beta }\right)-\frac{1}{\beta }\log \left(\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\right)\\ & =& -\frac{1}{\beta }\beta \log \left([v(X_1)v(X_2)\cdots v(X_n)]\right)-\frac{1}{\beta }\log \left(\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\right)\\ & =& -\log \left([v(X_1)v(X_2)\cdots v(X_n)]\right)-\frac{1}{\beta }\log \left(\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\right)\\ & =& -\sum _{i=1}^n\log v(X_i)-\frac{1}{\beta }\log \left(\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\right)\end{array}$$

事後分布

$$\begin{array}{rcl}q(\theta ;X^n)& =& \frac{1}{Z_n(\beta )}\pi (\theta ;\varphi )\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }\\ & =& \frac{1}{Z_n(\beta )}[\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\\ & =& \frac{1}{[\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}}[\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\cdots Z_n(\beta )=[\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\\ & =& \frac{1}{z({\hat{\varphi }}_{\beta })}\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\\ & =& \pi (\theta ;{\hat{\varphi }}_{\beta })\end{array}$$

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上記事後分布で予測分布を書き直す

$$\begin{array}{rcl}q(x;X^n)& =& {\int }_{\mathrm{\Theta }}q(x;\theta {)}^{\beta }q(\theta ;X^n)\mathrm{d}\theta \cdots X^n\mathrm{か}\mathrm{ら}\theta \mathrm{を}，\theta \mathrm{か}\mathrm{ら}x\mathrm{の}\mathrm{推}\mathrm{測}\mathrm{確}\mathrm{率}(\mathrm{分}\mathrm{布})\mathrm{を}\mathrm{求}\mathrm{め}\mathrm{る}\\ & =& {\int }_{\mathrm{\Theta }}q(x;\theta {)}^{\beta }\pi (\theta ;{\hat{\varphi }}_{\beta })\mathrm{d}\theta \cdots {\hat{\varphi }}_{\beta }\mathrm{か}\mathrm{ら}\theta \mathrm{を}，\theta \mathrm{か}\mathrm{ら}x\mathrm{の}\mathrm{推}\mathrm{測}\mathrm{確}\mathrm{率}(\mathrm{分}\mathrm{布})\mathrm{を}\mathrm{求}\mathrm{め}\mathrm{る}\\ & =& {\int }_{\mathrm{\Theta }}q(x;\theta {)}^{\beta }\frac{1}{z({\hat{\varphi }}_{\beta })}\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\mathrm{d}\theta \\ & =& {\int }_{\mathrm{\Theta }}{(v(x)\exp (f(\theta )\cdot g(x)))}^{\beta }\frac{1}{z({\hat{\varphi }}_{\beta })}\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\mathrm{d}\theta \\ & =& v(x{)}^{\beta }\frac{1}{z({\hat{\varphi }}_{\beta })}\int \exp {(f(\theta )\cdot g(x))}^{\beta }\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\mathrm{d}\theta \\ & =& v(x{)}^{\beta }\frac{1}{z({\hat{\varphi }}_{\beta })}\int \exp (f(\theta )\cdot \beta g(x))\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\mathrm{d}\theta \\ & =& v(x{)}^{\beta }\frac{1}{z({\hat{\varphi }}_{\beta })}\int \exp (f(\theta )\cdot ({\hat{\varphi }}_{\beta }+\beta g(x)))\mathrm{d}\theta \\ & =& v(x{)}^{\beta }\frac{1}{z({\hat{\varphi }}_{\beta })}z({\hat{\varphi }}_{\beta }+\beta g(x))\\ & =& v(x{)}^{\beta }\frac{z({\hat{\varphi }}_{\beta }+\beta g(x))}{z({\hat{\varphi }}_{\beta })}\\ & =& v(x{)}^{\beta }\frac{z(\varphi +\beta g(X_1)+\cdots +\beta g(X_n)+\beta g(x))}{z(\varphi +\beta g(X_1)+\cdots +\beta g(X_n))}\end{array}$$