From: François Fleuret Date: Sat, 2 Mar 2024 00:04:42 +0000 (+0100) Subject: Update. X-Git-Url: https://fleuret.org/cgi-bin/gitweb/gitweb.cgi?a=commitdiff_plain;h=5c3ff032a4d2fc50d96f8f94672086ddde45ca75;p=tex.git Update. --- diff --git a/elbo.tex b/elbo.tex index 4c6cb24..563ec3c 100644 --- a/elbo.tex +++ b/elbo.tex @@ -148,4 +148,20 @@ $\theta$ and $\alpha$ to maximize it maximizes $\log \, p_\theta(x_n)$ and brings $q_\alpha(z \mid x_n)$ close to $p_\theta(z \mid x_n)$. +\medskip + +A point that may be important in practice is +% +\begin{align*} + & \expect_{Z \sim q_\alpha(z \mid x_n)} \left[ \log \frac{p_\theta(x_n,Z)}{q_\alpha(Z \mid x_n)} \right] \\ + & = \expect_{Z \sim q_\alpha(z \mid x_n)} \left[ \log \frac{p_\theta(x_n \mid Z) p_\theta(Z)}{q_\alpha(Z \mid x_n)} \right] \\ + & = \expect_{Z \sim q_\alpha(z \mid x_n)} \left[ \log \, p_\theta(x_n \mid Z) \right] \\ + & \hspace*{7em} - \dkl(q_\alpha(z \mid x_n) \, \| \, p_\theta(z)). +\end{align*} +% +This form is useful because for certain $p_\theta$ and $q_\alpha$, for +instance if they are Gaussian, the KL term can be computed exactly +instead of through sampling, which removes one source of noise in the +optimization process. + \end{document}