指数分布族モデル
$$
\begin{eqnarray}
q(x;\theta)&=&v(x)\exp(f(\theta) \cdot g(x))\\
\int_X q(x;\theta) \mathrm{d}x
&=&\int_X v(x)\exp(f(\theta) \cdot g(x)) \mathrm{d}x\\
&=&1\;\cdots\;確率なのでこの条件を満たすようにv,f,gを用意する\\
\end{eqnarray}
$$
共役な事前分布
$$
\begin{eqnarray}
\pi(\theta;\phi)&=&\frac{1}{z(\phi)}\exp(f(\theta) \cdot \phi)\\
\int_X \pi(\theta;\phi) \mathrm{d}x&=&\int_X \frac{1}{z(\phi)}\exp(f(\theta) \cdot \phi)\mathrm{d}x\\
&=&1\;\cdots\;確率なのでこの条件を満たすようにz,f,\phiを用意する\\
z(\phi)&=&\int_\Theta \exp(f(\theta) \cdot \phi) \mathrm{d}\theta\;\cdots\;むしろzは上記条件を満たすために用意される\\
\end{eqnarray}
$$
予測分布
$$
\begin{eqnarray}
q(X_{n+1}|X^n)
&=&\frac{1}{Z_n(\beta)}
\int
q(X_{n+1}|\theta)^{\beta}
\;\prod_{i=1}^n \left\{q(X_i|\theta)^{\beta} \right\}
\;\pi(\theta)
\mathrm{d}\theta
\\
&=&\frac{1}{Z_n(\beta)}\int_{\Theta}{
\pi(\theta) q(X_{n+1}|\theta)^\beta \displaystyle\prod_{i=1}^n{q(X_{i}|\theta)}^\beta
\mathrm{d}\theta}\\
&=&\frac{1}{Z_n(\beta)}\int_{\Theta}{
\pi(\theta)\displaystyle\prod_{i=1}^{n+1}{q(X_{i}|\theta)^\beta
}\mathrm{d}\theta}
\;\dots\;\displaystyle\prod_{i=1}^{n+1}{q(X_{i}|\theta)^\beta}=q(X_{n+1}|\theta)^\beta\displaystyle\prod_{i=1}^n{q(X_{i}|\theta)^\beta}\\
&=&\frac{1}{Z_n(\beta)}Z_{n+1}(\beta)\;\dots\;Z_{n+1}(\beta)=\int_{\Theta}{\pi(\theta)\displaystyle\prod_{i=1}^{n+1}{q(X_{i}|\theta)^\beta}\mathrm{d}\theta}\\
&=&\frac{Z_{n+1}(\beta)}{Z_n(\beta)}\\
\end{eqnarray}
$$
分配凾数
$$
\begin{eqnarray}
Z_n(\beta)&=&\int_\Theta \pi(\theta;\phi) \prod_{i=1}^n q(X_i;\theta)^{\beta} \mathrm{d}\theta\\
&=&\int_\Theta
\pi(\theta;\phi)
\prod_{i=1}^n \left[
v(X_i)\exp(f(\theta) \cdot g(X_i))
\right]^\beta
\mathrm{d}\theta
\;\cdots\;q(X_i;\theta)=v(X_i)\exp(f(\theta) \cdot g(X_i))\\
&=&\int_\Theta
\left[
\frac{1}{z(\phi)}\exp(f(\theta) \cdot \phi)
\right]
\prod_{i=1}^n \left[
v(X_i)\exp(f(\theta) \cdot g(X_i))
\right]^\beta
\mathrm{d}\theta
\;\cdots\;\pi(\theta;\phi)=\frac{1}{z(\phi)}\exp(f(\theta) \cdot \phi)\\
&=&\frac{1}{z(\phi)}
\int_\Theta
\exp(f(\theta) \cdot \phi)
\prod_{i=1}^n \left[
v(X_i)\exp(f(\theta) \cdot g(X_i))
\right]^\beta
\mathrm{d}\theta
\;\cdots\;\pi(\theta;\phi)=\frac{1}{z(\phi)}\exp(f(\theta) \cdot \phi)\\
&=&\frac{1}{z(\phi)}
\int_\Theta
\exp(f(\theta) \cdot \phi)
\prod_{i=1}^n \left[
v(X_i)^\beta\exp(f(\theta) \cdot g(X_i))^\beta
\right]
\mathrm{d}\theta
\;\cdots\;(ab)^c=a^c b^c\\
&=&\frac{1}{z(\phi)}
\int_\Theta
\exp(f(\theta) \cdot \phi)
\left[
\prod_{i=1}^n v(X_i)^\beta
\right]
\left[
\prod_{i=1}^n
\exp(f(\theta) \cdot g(X_i))^\beta
\right]
\mathrm{d}\theta
\;\cdots\;\prod_{i=1}^nAB=\prod_{i=1}^nA\prod_{i=1}^nB\\
&=&\left[\prod_{i=1}^n v(X_i)^\beta\right]
\frac{1}{z(\phi)}
\int_\Theta
\exp(f(\theta) \cdot \phi)
\left[
\prod_{i=1}^n
\exp\left( \beta \left\{ f\left(\theta\right) \cdot g\left(X_i\right) \right\} \right)
\right]
\mathrm{d}\theta
\;\cdots\;\exp(A)^B=\exp(AB)\\
&=&\left[\prod_{i=1}^n v(X_i)^\beta\right]
\frac{1}{z(\phi)}
\int_\Theta
\exp(f(\theta) \cdot \phi)
\left[
\prod_{i=1}^n
\exp\left( f\left(\theta\right) \cdot \beta g\left(X_i\right) \right)
\right]
\mathrm{d}\theta
\;\cdots\;c(A\cdot B)= A\cdot cB \;(c:定数)\\
&=&\left[\prod_{i=1}^n v(X_i)^\beta\right]
\frac{1}{z(\phi)}
\int_\Theta
\exp\left(f(\theta) \cdot \phi
+ \sum_{i=1}^n f\left(\theta\right) \cdot \beta g\left(X_i\right) \right)
\mathrm{d}\theta
\;\cdots\;\exp(A)\exp(B)=\exp(A+B)\\
&=&\left[\prod_{i=1}^n v(X_i)^\beta\right]
\frac{1}{z(\phi)}
\int_\Theta
\exp\left(f(\theta) \cdot \phi
+ f\left(\theta\right) \cdot \sum_{i=1}^n \beta g\left(X_i\right) \right)
\mathrm{d}\theta
\;\cdots\;A\cdot B+A\cdot C=A\cdot (B+C)\\
&=&\left[\prod_{i=1}^n v(X_i)^\beta\right]
\frac{1}{z(\phi)}
\int_\Theta
\exp\left(f(\theta) \cdot \left\{\phi + \sum_{i=1}^n \beta g\left(X_i\right) \right\}\right)
\mathrm{d}\theta
\;\cdots\;A\cdot B+A\cdot C=A\cdot (B+C)\\
&=&\left[\prod_{i=1}^n v(X_i)^\beta\right]
\frac{1}{z(\phi)}
\int_\Theta
\exp\left(f(\theta) \cdot \hat{\phi}_{\beta}\right)
\mathrm{d}\theta
\;\cdots\;\hat{\phi}_{\beta} = \phi + \sum_{i=1}^n \beta g\left(X_i\right)\\
&=&\left[\prod_{i=1}^n v(X_i)^\beta\right]
\frac{1}{z(\phi)}
z(\hat{\phi}_{\beta})
\;\cdots\;z(\hat{\phi}_{\beta})=\int_\Theta \exp(f(\theta) \cdot \hat{\phi}_{\beta}) \mathrm{d}\theta\\
&=&
\left[\prod_{i=1}^n v(X_i)^\beta\right]
\frac{z(\hat{\phi}_{\beta})}{z(\phi)}
\\
\end{eqnarray}
$$
自由エネルギー
$$
\begin{eqnarray}
F_n(\beta)&=&-\frac{1}{\beta}\log{Z_n(\beta)}\\
&=&-\frac{1}{\beta}\log\left(\left[\prod_{i=1}^n v(X_i)^\beta\right]\frac{z(\hat{\phi}_{\beta})}{z(\phi)}\right)\\
&=&-\frac{1}{\beta}\log\left(\left[\prod_{i=1}^n v(X_i)^\beta\right]\right)
-\frac{1}{\beta}\log\left(\frac{z(\hat{\phi}_{\beta})}{z(\phi)}\right)\\
&=&-\frac{1}{\beta}\log\left(\left[v(X_1)^\beta v(X_2)^\beta \cdots v(X_n)^\beta\right]\right)
-\frac{1}{\beta}\log\left(\frac{z(\hat{\phi}_{\beta})}{z(\phi)}\right)\\
&=&-\frac{1}{\beta}\log\left(\left[v(X_1)v(X_2)\cdots v(X_n)\right]^\beta\right)
-\frac{1}{\beta}\log\left(\frac{z(\hat{\phi}_{\beta})}{z(\phi)}\right)\\
&=&-\frac{1}{\beta}\beta\log\left(\left[v(X_1)v(X_2)\cdots v(X_n)\right]\right)
-\frac{1}{\beta}\log\left(\frac{z(\hat{\phi}_{\beta})}{z(\phi)}\right)\\
&=&-\log\left(\left[v(X_1)v(X_2)\cdots v(X_n)\right]\right)
-\frac{1}{\beta}\log\left(\frac{z(\hat{\phi}_{\beta})}{z(\phi)}\right)\\
&=&-\sum_{i=1}^n \log{v(X_i)}-\frac{1}{\beta}\log\left(\frac{z(\hat{\phi}_{\beta})}{z(\phi)}\right)\\
\end{eqnarray}
$$
事後分布
$$
\begin{eqnarray}
q(\theta;X^n)&=&\frac{1}{Z_n(\beta)}\pi(\theta;\phi)\prod_{i=1}^n q(X_i;\theta)^\beta\\
&=&\frac{1}{Z_n(\beta)}\left[\prod_{i=1}^n v(X_i)^\beta\right]\frac{1}{z(\phi)}\exp(f(\theta)\cdot \hat{\phi}_{\beta})\\
&=&\frac{1}{\left[\prod_{i=1}^n v(X_i)^\beta\right]
\frac{z(\hat{\phi}_{\beta})}{z(\phi)}
}\left[\prod_{i=1}^n v(X_i)^\beta\right]\frac{1}{z(\phi)}\exp(f(\theta)\cdot \hat{\phi}_{\beta})
\;\cdots\;Z_n(\beta)=\left[\prod_{i=1}^n v(X_i)^\beta\right]\frac{z(\hat{\phi}_{\beta})}{z(\phi)}\\
&=&\frac{1}{z(\hat{\phi}_{\beta})}\exp(f(\theta)\cdot \hat{\phi}_{\beta})\\
&=&\pi(\theta;\hat{\phi}_{\beta})\\
\end{eqnarray}
$$
上記事後分布で予測分布を書き直す
$$
\begin{eqnarray}
q(x;X^n)
&=&\int_\Theta q(x;\theta)^{\beta}q(\theta;X^n)\mathrm{d}\theta\;\cdots\;X^nから\thetaを，\thetaからxの推測確率(分布)を求める\\
&=&\int_\Theta q(x;\theta)^{\beta}\pi(\theta;\hat{\phi}_{\beta})\mathrm{d}\theta\;\cdots\;\hat{\phi}_{\beta}から\thetaを，\thetaからxの推測確率(分布)を求める\\
&=&\int_\Theta q(x;\theta)^{\beta}\frac{1}{z(\hat{\phi}_{\beta})}\exp(f(\theta)\cdot \hat{\phi}_{\beta})\mathrm{d}\theta\\
&=&\int_\Theta \left(v(x)\exp(f(\theta) \cdot g(x))\right)^{\beta}\frac{1}{z(\hat{\phi}_{\beta})}\exp(f(\theta)\cdot \hat{\phi}_{\beta})\mathrm{d}\theta\\
&=&v(x)^{\beta}\frac{1}{z(\hat{\phi}_{\beta})}\int \exp(f(\theta) \cdot g(x))^{\beta}\exp(f(\theta)\cdot \hat{\phi}_{\beta})\mathrm{d}\theta\\
&=&v(x)^{\beta}\frac{1}{z(\hat{\phi}_{\beta})}\int \exp(f(\theta) \cdot {\beta}g(x))\exp(f(\theta)\cdot \hat{\phi}_{\beta})\mathrm{d}\theta\\
&=&v(x)^{\beta}\frac{1}{z(\hat{\phi}_{\beta})}\int \exp(f(\theta) \cdot (\hat{\phi}_{\beta}+{\beta}g(x)))\mathrm{d}\theta\\
&=&v(x)^{\beta}\frac{1}{z(\hat{\phi}_{\beta})} z(\hat{\phi}_{\beta}+{\beta}g(x))\\
&=&v(x)^{\beta}\frac{z(\hat{\phi}_{\beta}+{\beta}g(x))}{z(\hat{\phi}_{\beta})}\\
&=&v(x)^{\beta}\frac{z(\phi+{\beta}g(X_1)+\cdots+{\beta}g(X_n)+{\beta}g(x))}{z(\phi+{\beta}g(X_1)+\cdots+{\beta}g(X_n))}\\
\end{eqnarray}
$$
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