自由エネルギー

自由エネルギー

$$ \begin{eqnarray} F_n(\beta) &=&-\frac{1}{\beta}\log{Z_n(\beta)} \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/05/blog-post_92.html}{Z_n(\beta):分配凾数}\\ &=&-\frac{1}{\beta}\log{\left( Z_n^{(0)}(\beta) \prod^n_{i=1} q(X_i;\theta)^\beta \right)} \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/05/2.html}{Z_n(\beta)=Z_n^{(0)}(\beta)\prod^n_{i=1}q(X_i;\theta_0)^\beta}\\ &=&-\frac{1}{\beta}\left( \log{\left( \prod^n_{i=1} q(X_i;\theta)^\beta \right)} +\log{\left( Z_n^{(0)}(\beta) \right)} \right) \;\cdots\;\log(AB)=\log(A)+\log(B)\\ &=&-\frac{1}{\beta}\log{\left(\prod^n_{i=1} q(X_i;\theta)^\beta\right)} -\frac{1}{\beta}\log{\left(Z_n^{(0)}(\beta)\right)}\\ &=&-\frac{1}{\beta}\log{\left(\prod^n_{i=1} q(X_i;\theta)^\beta\right)} +F_n^{(0)}(\beta) \;\cdots\;F_n^{(0)}(\beta)=-\frac{1}{\beta}\log{\left(Z_n^{(0)}(\beta)\right)}:正規化された自由エネルギー\\ &=&-\frac{1}{\beta}\log{\left(\prod^n_{i=1} q(X_i;\theta)\right)^\beta} +F_n^{(0)}(\beta) \;\cdots\;\prod A^B=\left(\prod A\right)^B\\ &=&-\frac{1}{\beta}\beta\log{\left(\prod^n_{i=1} q(X_i;\theta)\right)} +F_n^{(0)}(\beta) \;\cdots\;\log{A^B}=B\log{A}\\ &=&-\log{\left(\prod^n_{i=1} q(X_i;\theta)\right)} +F_n^{(0)}(\beta)\\ &=&-\sum^n_{i=1}\log{q(X_i;\theta)} +F_n^{(0)}(\beta) \;\cdots\;\log{\left(\prod^n_{i=1} A_i\right)}=\sum^n_{i=1}{\left(\log{A_i}\right)}\\ &=&-n\frac{1}{n}\sum^n_{i=1}\log{q(X_i;\theta)} +F_n^{(0)}(\beta)\;\cdots\;n\frac{1}{n}=1\\ &=&n\left\{ -\frac{1}{n}\sum^n_{i=1}\log{q(X_i;\theta)}\right\} +F_n^{(0)}(\beta)\\ &=&nL_n(\theta_0)+F_n^{(0)}(\beta) \;\cdots\; \href{https://shikitenkai.blogspot.com/2020/05/blog-post_1.html}{-\frac{1}{n}\sum^n_{i=1}\log{q(X_i;\theta)}=L_n(\theta_0)}\\ \\ \end{eqnarray} $$