From 4b8c58903baa9ff8c508bda798492e10dde9cb7f Mon Sep 17 00:00:00 2001
From: =?utf8?q?Fran=C3=A7ois=20Fleuret?= <francois@fleuret.org>
Date: Wed, 28 Feb 2024 08:19:50 +0100
Subject: [PATCH] Update.

---
 elbo.tex | 16 +++++++++-------
 1 file changed, 9 insertions(+), 7 deletions(-)

diff --git a/elbo.tex b/elbo.tex
index fe91565..4c6cb24 100644
--- a/elbo.tex
+++ b/elbo.tex
@@ -76,24 +76,25 @@
 \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 i.i.d training samples $x_1, \dots, x_N$ 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).
@@ -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 +148,4 @@ $\theta$ and $\alpha$ to maximize
 it maximizes $\log \, p_\theta(x_n)$ and brings $q_\alpha(z \mid
 x_n)$ close to $p_\theta(z \mid x_n)$.
 
-
 \end{document}
-- 
2.20.1