生成对抗网络推导

$\textbf{性质1：}$$For\ G\ fixed,\ the\ optimal\ discriminator\ D\ is$

$$D_G^{*}(\textbf{x})=\frac{q(\textbf{x})}{q(\textbf{x})+p(\textbf{x})}$$

$\textbf{证明：}$

$$\begin{align*} V(D,G) &=E_{q(\textbf{x})}[\log(D(\textbf{x}))]+E_{p(\textbf{z})}[\log(1-D(G(\textbf{z})))]\\ &=\int q(\textbf{x})\log(D(\textbf{x}))d\textbf{x}+\int\int p(\textbf{z})p(\textbf{x}|\textbf{z})\log(1-D(\textbf{x}))d\textbf{x}d\textbf{z} \end{align*}$$

固定$G$，我们的目标是最大化$V(G,D)$。因为$\ $
$p(\textbf{x})=\int p(\textbf{z})p(\textbf{x}|\textbf{z})d\textbf{z}$，所以

V(D,G)=∫q(x)log(D(x))dx+∫p(x)log(1−D(x))dx<