事後分布

最適パラメータにおける条件付き分布

$$\begin{array}{rcl}{Z}_n(\beta )& =& Z_n^{(0)}(\beta )\exp (-n\beta L_n({\theta }_0))\\ & =& Z_n^{(0)}(\beta )\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }\\ \prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }& =& \exp (-n\beta L_n({\theta }_0))\cdots \mathrm{最}\mathrm{適}\mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}\mathrm{に}\mathrm{お}\mathrm{け}\mathrm{る}\mathrm{条}\mathrm{件}\mathrm{付}\mathrm{き}\mathrm{分}\mathrm{布}\end{array}$$

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$$\begin{array}{rcl}q(\theta ;X^n)& =& \frac{1}{Z_n(\beta )}\pi (\theta )\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }\\ & =& \frac{1}{Z_n^{(0)}(\beta )\exp (-n\beta L_n({\theta }_0))}\pi (\theta )\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }\cdots Z_n(\beta )=Z_n^{(0)}(\beta )\exp (-n\beta L_n({\theta }_0))\\ & =& \frac{1}{Z_n^{(0)}(\beta )}\frac{\exp (-n\beta L_n(\theta ))}{\exp (-n\beta L_n({\theta }_0))}\pi (\theta )\cdots \prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }=\exp (-n\beta L_n(\theta ))\\ & =& \frac{1}{Z_n^{(0)}(\beta )}\frac{\exp (-n\beta (L_n({\theta }_0)+K_n(\theta )))}{\exp (-n\beta L_n({\theta }_0))}\pi (\theta )\cdots L_n(\theta )=L_n({\theta }_0)+K_n(\theta )\\ & =& \frac{1}{Z_n^{(0)}(\beta )}\frac{\exp (-n\beta L_n({\theta }_0))\exp (-n\beta K_n(\theta ))}{\exp (-n\beta L_n({\theta }_0))}\pi (\theta )\cdots \exp (A+B)=\exp (A)\exp (B)\\ & =& \frac{1}{Z_n^{(0)}(\beta )}\exp (-n\beta K_n(\theta ))\pi \left(\theta \right)\end{array}$$