正規化された分配凾数

正規化された分配凾数

$$\begin{array}{rcl}\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }& =& \exp (\log \left({\displaystyle \prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }}\right))\cdots A=\exp (\log \left(A\right))\\ & =& \exp (\sum _{i=1}^n\log (q(X_i;\theta {)}^{\beta }))\cdots \log \left(\prod _{i=1}^n{A}_i\right)=\sum _{i=1}^n\log \left(A_i\right)\\ & =& \exp (\beta \sum _{i=1}^n\log (q(X_i;\theta )))\\ & =& \exp (-n\beta (-\frac{1}{n}\sum _{i=1}^n\log \left(q(X_i;\theta )\right)))\cdots (-n)(-\frac{1}{n})=1\\ & =& \exp (-n\beta L_n(\theta ))\cdots L_n(\theta )=-\frac{1}{n}{\displaystyle \sum _{i=1}^n\log (q(X_i;\theta ))}\end{array}$$

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$$\begin{array}{rcl}{Z}_n(\beta )& =& {\int }_{\mathrm{\Theta }}\pi (\theta )\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }\mathrm{d}\theta \\ & =& {\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta L_n(\theta ))\mathrm{d}\theta \cdots \prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }=\exp (-n\beta L_n(\theta ))\\ & =& {\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta (L_n({\theta }_0)+K_n(\theta )))\mathrm{d}\theta \cdots L_n(\theta )=L_n({\theta }_0)+K_n(\theta )\\ & =& {\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta L_n({\theta }_0)-n\beta K_n(\theta ))\mathrm{d}\theta \\ & =& {\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta L_n({\theta }_0))\exp (-n\beta K_n(\theta ))\mathrm{d}\theta \\ & =& \exp (-n\beta L_n({\theta }_0)){\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta K_n(\theta ))\mathrm{d}\theta \\ & =& \exp (-n\beta L_n({\theta }_0))Z_n^{(0)}(\beta )\cdots Z_n^{(0)}(\beta )={\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta K_n(\theta ))\mathrm{d}\theta :\mathrm{正}\mathrm{規}\mathrm{化}\mathrm{さ}\mathrm{れ}\mathrm{た}\mathrm{分}\mathrm{配}\mathrm{凾}\mathrm{数}\\ & =& Z_n^{(0)}(\beta )\exp (-n\beta L_n({\theta }_0))\end{array}$$