\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\@writefile{toc}{\contentsline {section}{\numberline {1}Why EM Works}{1}{section.1}\protected@file@percent }
\newlabel{sec:whyworks}{{1}{1}{Why EM Works}{section.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Why There Is a Bound at All: Jensen}{1}{subsection.1.1}\protected@file@percent }
\newlabel{sec:jensen}{{1.1}{1}{Why There Is a Bound at All: Jensen}{subsection.1.1}{}}
\newlabel{eq:jensen}{{1}{1}{Why There Is a Bound at All: Jensen}{equation.1}{}}
\newlabel{eq:jensenlog}{{2}{1}{Why There Is a Bound at All: Jensen}{equation.2}{}}
\@writefile{toc}{\contentsline {paragraph}{Deriving the ELBO.}{2}{section*.1}\protected@file@percent }
\newlabel{eq:elbo}{{3}{2}{Deriving the ELBO}{equation.3}{}}
\newlabel{eq:elboA}{{4}{2}{Deriving the ELBO}{equation.4}{}}
\newlabel{eq:elboB}{{5}{2}{Deriving the ELBO}{equation.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}The KL Decomposition, and Why the E Step Makes the Bound Tight}{2}{subsection.1.2}\protected@file@percent }
\newlabel{sec:kldecomp}{{1.2}{2}{The KL Decomposition, and Why the E Step Makes the Bound Tight}{subsection.1.2}{}}
\newlabel{eq:decomp}{{7}{2}{The KL Decomposition, and Why the E Step Makes the Bound Tight}{equation.7}{}}
\newlabel{eq:estep}{{8}{2}{The KL Decomposition, and Why the E Step Makes the Bound Tight}{equation.8}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}The Monotonicity Theorem}{3}{subsection.1.3}\protected@file@percent }
\newlabel{sec:monotone}{{1.3}{3}{The Monotonicity Theorem}{subsection.1.3}{}}
\@writefile{toc}{\contentsline {paragraph}{Proof.}{3}{section*.2}\protected@file@percent }
\newlabel{eq:step1}{{9}{3}{Proof}{equation.9}{}}
\newlabel{eq:step2}{{10}{3}{Proof}{equation.10}{}}
\newlabel{eq:step3}{{11}{3}{Proof}{equation.11}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.4}EM Is Coordinate Ascent on the ELBO}{4}{subsection.1.4}\protected@file@percent }
\newlabel{sec:coordasc}{{1.4}{4}{EM Is Coordinate Ascent on the ELBO}{subsection.1.4}{}}
\@writefile{toc}{\contentsline {paragraph}{How this maps onto the board's route.}{4}{section*.3}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {2}Local Optima, MAP-EM, and Regularized Covariance}{4}{section.2}\protected@file@percent }
\newlabel{sec:optima}{{2}{4}{Local Optima, MAP-EM, and Regularized Covariance}{section.2}{}}
\@writefile{toc}{\contentsline {paragraph}{How bad are the local optima, really?}{4}{section*.4}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{The unbounded-likelihood degeneracy.}{5}{section*.5}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{MAP-EM is the fix.}{5}{section*.6}\protected@file@percent }
\newlabel{eq:mapsigma}{{12}{5}{MAP-EM is the fix}{equation.12}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}$k$-Means as a Limit, and Choosing $K$}{5}{section.3}\protected@file@percent }
\newlabel{sec:modelsel}{{3}{5}{$k$-Means as a Limit, and Choosing $K$}{section.3}{}}
\@writefile{toc}{\contentsline {paragraph}{$k$-means is zero-temperature EM.}{5}{section*.7}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Choosing $K$ --- the likelihood cannot.}{5}{section*.8}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4}EM's Original Problem: Missing Data}{6}{section.4}\protected@file@percent }
\newlabel{sec:missing}{{4}{6}{EM's Original Problem: Missing Data}{section.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5}Giving the Latent Variable Structure: Bayes Nets and HMMs}{7}{section.5}\protected@file@percent }
\newlabel{sec:hmm}{{5}{7}{Giving the Latent Variable Structure: Bayes Nets and HMMs}{section.5}{}}
\@writefile{toc}{\contentsline {paragraph}{``Not-so-naive Bayes'' $\to $ Bayes networks.}{7}{section*.9}\protected@file@percent }
\newlabel{eq:bayesnet}{{13}{7}{``Not-so-naive Bayes'' $\to $ Bayes networks}{equation.13}{}}
\@writefile{toc}{\contentsline {paragraph}{HMMs: one chain of latent variables.}{7}{section*.10}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Forward and backward probabilities.}{7}{section*.11}\protected@file@percent }
\newlabel{eq:alphabeta}{{14}{7}{Forward and backward probabilities}{equation.14}{}}
\newlabel{eq:fb}{{15}{7}{Forward and backward probabilities}{equation.15}{}}
\newlabel{eq:fbcombine}{{16}{8}{Forward and backward probabilities}{equation.16}{}}
\@writefile{toc}{\contentsline {paragraph}{Baum--Welch is EM.}{8}{section*.12}\protected@file@percent }
\newlabel{eq:bw}{{17}{8}{Baum--Welch is EM}{equation.17}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6}When the Posterior Is Intractable}{9}{section.6}\protected@file@percent }
\newlabel{sec:intractable}{{6}{9}{When the Posterior Is Intractable}{section.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.1}Topic Models: Mixtures With a Prior on the Weights}{9}{subsection.6.1}\protected@file@percent }
\newlabel{sec:topics}{{6.1}{9}{Topic Models: Mixtures With a Prior on the Weights}{subsection.6.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.2}Variational Inference, and the VAE}{9}{subsection.6.2}\protected@file@percent }
\newlabel{sec:vi}{{6.2}{9}{Variational Inference, and the VAE}{subsection.6.2}{}}
\@writefile{toc}{\contentsline {paragraph}{The one assumption we have been making.}{9}{section*.13}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Variational inference: give up on exactness, keep the bound.}{9}{section*.14}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{The VAE.}{9}{section*.15}\protected@file@percent }
\newlabel{eq:vae}{{18}{10}{The VAE}{equation.18}{}}
\@writefile{toc}{\contentsline {section}{\numberline {7}Further Reading}{10}{section.7}\protected@file@percent }
\newlabel{sec:reading}{{7}{10}{Further Reading}{section.7}{}}
\gdef \@abspage@last{11}