Update.

[tex.git] / elbo.tex

diff --git a/elbo.tex b/elbo.tex
index 239a657..563ec3c 100644 (file)
--- a/elbo.tex
+++ b/elbo.tex
@@ -76,30 +76,31 @@
 \setlength{\abovedisplayshortskip}{2ex}
 \setlength{\belowdisplayshortskip}{2ex}
 
-\vspace*{-4ex}
+\vspace*{-3ex}
 
 \begin{center}
 {\Large The Evidence Lower Bound}
 
-\vspace*{1ex}
+\vspace*{2ex}
 
 Fran\c cois Fleuret
 
+%% \vspace*{2ex}
+
 \today
 
-\vspace*{-1ex}
+%% \vspace*{-1ex}
 
 \end{center}
 
-Given a training set $x_1, \dots, x_N$ that follows an unknown
-distribution $\mu_X$, we want to fit a model $p_\theta(x,z)$ to it,
-maximizing
+Given i.i.d training samples $x_1, \dots, x_N$ we want to fit a model
+$p_\theta(x,z)$ to it, maximizing
 %
 \[
 \sum_n \log \, p_\theta(x_n).
 \]
 %
-If we do not have a analytical form of the marginal $p_\theta(x_n)$
+If we do not have an analytical form of the marginal $p_\theta(x_n)$
 but only the expression of $p_\theta(x_n,z)$, we can get an estimate
 of the marginal by sampling $z$ with any distribution $q$
 %
@@ -134,6 +135,8 @@ since this maximization pushes that KL term down, it also aligns
 $p_\theta(z \mid x_n)$ and $q(z)$, and we may get a worse
 $p_\theta(x_n)$ to bring $p_\theta(z \mid x_n)$ closer to $q(z)$.
 
+\medskip
+
 However, all this analysis is still valid if $q$ is a parameterized
 function $q_\alpha(z \mid x_n)$ of $x_n$. In that case, if we optimize
 $\theta$ and $\alpha$ to maximize
@@ -145,5 +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}