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Update.
author
François Fleuret
<francois@fleuret.org>
Sat, 24 Feb 2024 11:11:36 +0000
(12:11 +0100)
committer
François Fleuret
<francois@fleuret.org>
Sat, 24 Feb 2024 11:11:36 +0000
(12:11 +0100)
elbo.tex
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diff --git
a/elbo.tex
b/elbo.tex
index
6875ddf
..
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(file)
--- a/
elbo.tex
+++ b/
elbo.tex
@@
-71,16
+71,23
@@
\begin{document}
\begin{document}
-\vspace*{0ex}
+\setlength{\abovedisplayskip}{2ex}
+\setlength{\belowdisplayskip}{2ex}
+\setlength{\abovedisplayshortskip}{2ex}
+\setlength{\belowdisplayshortskip}{2ex}
+
+\vspace*{-4ex}
\begin{center}
{\Large The Evidence Lower Bound}
\begin{center}
{\Large The Evidence Lower Bound}
+\vspace*{1ex}
+
Fran\c cois Fleuret
\today
Fran\c cois Fleuret
\today
-\vspace*{1ex}
+\vspace*{
-
1ex}
\end{center}
\end{center}
@@
-102,12
+109,14
@@
p_\theta(x_n) & = \int_z p_\theta(x_n,z) dz \\
& = \expect_{Z \sim q(z)} \left[\frac{p_\theta(x_n,Z)}{q(Z)}\right].
\end{align*}
%
& = \expect_{Z \sim q(z)} \left[\frac{p_\theta(x_n,Z)}{q(Z)}\right].
\end{align*}
%
-So if we
wanted to maximize $p_\theta(x_n)$ alone, we could
sample a
+So if we sample a
$Z$ with $q$ and maximize
%
\begin{equation*}
$Z$ with $q$ and maximize
%
\begin{equation*}
-\frac{p_\theta(x_n,Z)}{q(Z)}
.\label{eq:estimator}
+\frac{p_\theta(x_n,Z)}{q(Z)}
,
\end{equation*}
\end{equation*}
+%
+we do maximize $p_\theta(x_n)$ on average.
But we want to maximize $\sum_n \log \, p_\theta(x_n)$. If we use the
$\log$ of the previous expression, we can decompose its average value
But we want to maximize $\sum_n \log \, p_\theta(x_n)$. If we use the
$\log$ of the previous expression, we can decompose its average value