事後分布

最適パラメータにおける条件付き分布

$$ \begin{eqnarray} Z_n(\beta) &=& \href{https://shikitenkai.blogspot.com/2020/05/blog-post_36.html}{Z_n^{(0)}(\beta)\exp(-n\beta L_n(\theta_0))}\\ &=& \href{https://shikitenkai.blogspot.com/2020/05/2.html}{Z_n^{(0)}(\beta)\prod_{i=1}^n q(X_i;\theta_0)^\beta}\\ \prod_{i=1}^n q(X_i;\theta_0)^\beta &=&\exp(-n\beta L_n(\theta_0))\;\cdots\;最適パラメータにおける条件付き分布\\ \end{eqnarray} $$

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$$ \begin{eqnarray} q(\theta;X^n) &=&\frac{1}{Z_n(\beta)} \pi(\theta) \prod_{i=1}^n q(X_i;\theta)^\beta\\ &=&\frac{1}{Z_n^{(0)}(\beta)\exp(-n\beta L_n(\theta_0))} \pi(\theta) \prod_{i=1}^n q(X_i;\theta)^\beta \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/05/blog-post_36.html}{Z_n(\beta)=Z_n^{(0)}(\beta)\exp(-n\beta L_n(\theta_0))}\\ &=&\frac{1}{Z_n^{(0)}(\beta)} \frac{ \exp{\left(-n\beta L_n(\theta)\right)} } { \exp{\left(-n\beta L_n(\theta_0)\right)} } \pi(\theta) \;\cdots\;\prod_{i=1}^nq(X_i;\theta)^\beta=\exp(-n\beta L_n(\theta))\\ &=&\frac{1}{Z_n^{(0)}(\beta)} \frac{ \exp{\left(-n\beta (L_n(\theta_0)+K_n(\theta))\right)} } { \exp{\left(-n\beta L_n(\theta_0)\right)} } \pi(\theta) \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/05/blog-post_1.html}{L_n(\theta)=L_n(\theta_0)+K_n(\theta)}\\ &=&\frac{1}{Z_n^{(0)}(\beta)} \frac{ \exp{\left(-n\beta L_n(\theta_0)\right)}\exp{\left(-n\beta K_n(\theta)\right)} } { \exp{\left(-n\beta L_n(\theta_0)\right)} } \pi(\theta) \;\cdots\;\exp{(A+B)}=\exp{(A)}\exp{(B)}\\ &=&\frac{1}{Z_n^{(0)}(\beta)}\exp{\left(-n\beta K_n(\theta)\right)}\pi\left(\theta\right)\\ \end{eqnarray} $$