# PGM 读书笔记节选（十三）

$\displaystyle\min \mathrm{KL}(\tilde{p}(X), \Pr(X)) = \int \tilde{p}(X) \log \tilde{p}(X) \,\mathrm{d} X - \int \tilde{p}(X) \log \Pr(X) \,\mathrm{d} X \quad \Rightarrow \quad \max \sum_{i = 1}^N \log \Pr(X_i)$

$\displaystyle\Pr (X) = \prod_{x} \Pr(x \mid \pi (x))$

$\displaystyle \max \sum_{i = 1}^N \log\Pr(X_i) = \sum_x \sum_{i = 1}^N \Pr (x_i \mid \pi(x_i), \theta_x)$

$\displaystyle \max \sum_{i = 1}^N \log\Pr(X_i) = \sum_{i = 1}^N\sum_c \log\phi (x_c^{(i)}) - \log Z^{(i)}$

$\displaystyle \frac{\partial \ell}{\partial \theta_c} = \sum_{i = 1}^N f_c (x_c^{(i)}) - \langle f_c(x_c)\rangle \stackrel{\triangle}{=} \tilde{f}_c - \langle f_c (x_c) \rangle$

$\displaystyle \frac{\partial^2 \ell}{\partial \theta_c \partial \theta_{c'}^\top} = -\mathrm{cov} (f_c, f_{c'})$

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And the thing was very grievous in Abraham’s sight because of his son.

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