VAE推导过程

$$\begin{gathered} \log P_{\theta }(x)=\log \int P_{\theta }(z,x)dz \\ =\log \int q_{\varphi }(z|x)\frac{P_{\theta }(z,x)}{q_{\varphi }(z|x)}dz \\ \ge {\mathbb{E}}_q\varphi (z|x)\log \frac{P_{\theta }(z,x)}{q_{\varphi }(z|x)}=L \end{gathered}$$
$$\begin{gathered} L=\mathbb{E}\log \frac{P\theta (x|z)P_{\theta }(z)}{q\varphi (z|x)} \\ ={\mathbb{E}}_q\varphi (z|x)\log P_{\theta }(x|z)+{\mathbb{E}}_{q_{\varphi }(z|x)}\log \frac{P_{\theta }(z)}{q_{\varphi }(z|x)} \\ ={\mathbb{E}}_q\varphi (z|x)\log P_{\theta }(x|z)-KL(q_{\varphi }(z|x)||P_{\theta }(z)) \end{gathered}$$