式展開: 自由エネルギー

自由エネルギー

自由エネルギー

\begin{array}{rcl} F_{n} (β) & = & - \frac{1}{β} \log Z_{n} (β) \dots Z_{n} (β) : 分 配 凾 数 \\ = & - \frac{1}{β} \log (Z_{n}^{(0)} (β) \prod_{i = 1}^{n} q (X_{i}; θ)^{β}) \dots Z_{n} (β) = Z_{n}^{(0)} (β) \prod_{i = 1}^{n} q (X_{i}; θ_{0})^{β} \\ = & - \frac{1}{β} (\log (\prod_{i = 1}^{n} q (X_{i}; θ)^{β}) + \log (Z_{n}^{(0)} (β))) \dots \log (A B) = \log (A) + \log (B) \\ = & - \frac{1}{β} \log (\prod_{i = 1}^{n} q (X_{i}; θ)^{β}) - \frac{1}{β} \log (Z_{n}^{(0)} (β)) \\ = & - \frac{1}{β} \log (\prod_{i = 1}^{n} q (X_{i}; θ)^{β}) + F_{n}^{(0)} (β) \dots F_{n}^{(0)} (β) = - \frac{1}{β} \log (Z_{n}^{(0)} (β)) : 正 規 化 さ れ た 自 由 エ ネ ル ギ ー \\ = & - \frac{1}{β} \log {(\prod_{i = 1}^{n} q (X_{i}; θ))}^{β} + F_{n}^{(0)} (β) \dots \prod A^{B} = {(\prod A)}^{B} \\ = & - \frac{1}{β} β \log (\prod_{i = 1}^{n} q (X_{i}; θ)) + F_{n}^{(0)} (β) \dots \log A^{B} = B \log A \\ = & - \log (\prod_{i = 1}^{n} q (X_{i}; θ)) + F_{n}^{(0)} (β) \\ = & - \sum_{i = 1}^{n} \log q (X_{i}; θ) + F_{n}^{(0)} (β) \dots \log (\prod_{i = 1}^{n} A_{i}) = \sum_{i = 1}^{n} (\log A_{i}) \\ = & - n \frac{1}{n} \sum_{i = 1}^{n} \log q (X_{i}; θ) + F_{n}^{(0)} (β) \dots n \frac{1}{n} = 1 \\ = & n {- \frac{1}{n} \sum_{i = 1}^{n} \log q (X_{i}; θ)} + F_{n}^{(0)} (β) \\ = & n L_{n} (θ_{0}) + F_{n}^{(0)} (β) \dots - \frac{1}{n} \sum_{i = 1}^{n} \log q (X_{i}; θ) = L_{n} (θ_{0}) \end{array}

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