\relax \providecommand\hyper@newdestlabel[2]{} \providecommand\HyField@AuxAddToFields[1]{} \providecommand\HyField@AuxAddToCoFields[2]{} \@writefile{toc}{\contentsline {section}{\numberline {1}Mixture Models}{1}{section.1}\protected@file@percent } \newlabel{sec:model}{{1}{1}{Mixture Models}{section.1}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.1}One Bump Is Not Enough}{1}{subsection.1.1}\protected@file@percent } \newlabel{sec:why}{{1.1}{1}{One Bump Is Not Enough}{subsection.1.1}{}} \newlabel{eq:mixture}{{1}{1}{One Bump Is Not Enough}{equation.1}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.2}The Generative Story, and the Variable We Cannot See}{2}{subsection.1.2}\protected@file@percent } \newlabel{sec:latent}{{1.2}{2}{The Generative Story, and the Variable We Cannot See}{subsection.1.2}{}} \newlabel{eq:onehot}{{2}{2}{The Generative Story, and the Variable We Cannot See}{equation.2}{}} \newlabel{eq:joint}{{3}{2}{The Generative Story, and the Variable We Cannot See}{equation.3}{}} \newlabel{eq:obsll}{{4}{3}{The Generative Story, and the Variable We Cannot See}{equation.4}{}} \@writefile{toc}{\contentsline {section}{\numberline {2}Chicken and Egg: $k$-Means as EM}{3}{section.2}\protected@file@percent } \newlabel{sec:kmeans}{{2}{3}{Chicken and Egg: $k$-Means as EM}{section.2}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}The familiar case: hard $k$-means}{3}{subsection.2.1}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Softening the E step}{4}{subsection.2.2}\protected@file@percent } \newlabel{eq:softkm}{{5}{4}{Softening the E step}{equation.5}{}} \@writefile{toc}{\contentsline {section}{\numberline {3}The E Step: Bayes' Rule}{5}{section.3}\protected@file@percent } \newlabel{sec:estep}{{3}{5}{The E Step: Bayes' Rule}{section.3}{}} \newlabel{eq:estep}{{6}{5}{The E Step: Bayes' Rule}{equation.6}{}} \@writefile{toc}{\contentsline {section}{\numberline {4}The M Step}{6}{section.4}\protected@file@percent } \newlabel{sec:mstepsec}{{4}{6}{The M Step}{section.4}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.1}The Complete-Data Likelihood, and Why We Take Its Expectation}{6}{subsection.4.1}\protected@file@percent } \newlabel{sec:ell}{{4.1}{6}{The Complete-Data Likelihood, and Why We Take Its Expectation}{subsection.4.1}{}} \newlabel{eq:cdl1}{{7}{6}{The Complete-Data Likelihood, and Why We Take Its Expectation}{equation.7}{}} \newlabel{eq:cdl}{{8}{6}{The Complete-Data Likelihood, and Why We Take Its Expectation}{equation.8}{}} \newlabel{eq:Q}{{9}{6}{The Complete-Data Likelihood, and Why We Take Its Expectation}{equation.9}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Three Weighted Averages}{7}{subsection.4.2}\protected@file@percent } \newlabel{sec:mstep}{{4.2}{7}{Three Weighted Averages}{subsection.4.2}{}} \newlabel{eq:Nk}{{10}{7}{Three Weighted Averages}{equation.10}{}} \@writefile{toc}{\contentsline {paragraph}{\textcircled {\small 1} The means.}{7}{section*.1}\protected@file@percent } \newlabel{eq:quadderiv}{{11}{7}{\textcircled {\small 1} The means}{equation.11}{}} \newlabel{eq:mmu}{{12}{7}{\textcircled {\small 1} The means}{equation.12}{}} \@writefile{toc}{\contentsline {paragraph}{\textcircled {\small 2} The covariances.}{8}{section*.2}\protected@file@percent } \newlabel{eq:msigma}{{13}{8}{\textcircled {\small 2} The covariances}{equation.13}{}} \@writefile{toc}{\contentsline {paragraph}{\textcircled {\small 3} The mixing weights.}{8}{section*.3}\protected@file@percent } \newlabel{eq:mpi}{{14}{8}{\textcircled {\small 3} The mixing weights}{equation.14}{}} \@writefile{toc}{\contentsline {section}{\numberline {5}The Algorithm, Start to Finish}{9}{section.5}\protected@file@percent } \newlabel{sec:algorithm}{{5}{9}{The Algorithm, Start to Finish}{section.5}{}} \@writefile{toc}{\contentsline {paragraph}{Numerics --- the three that actually bite.}{9}{section*.4}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Initialization and stopping.}{10}{section*.5}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {6}Why It Works: the ELBO and Monotonicity}{10}{section.6}\protected@file@percent } \newlabel{sec:why-it-works}{{6}{10}{Why It Works: the ELBO and Monotonicity}{section.6}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {6.1}What EM Is Really Optimizing: the ELBO}{10}{subsection.6.1}\protected@file@percent } \newlabel{sec:elbo}{{6.1}{10}{What EM Is Really Optimizing: the ELBO}{subsection.6.1}{}} \newlabel{eq:elbo}{{15}{10}{What EM Is Really Optimizing: the ELBO}{equation.15}{}} \newlabel{eq:elbogmm}{{16}{11}{What EM Is Really Optimizing: the ELBO}{equation.16}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {6.2}Monotonicity: the Log-Likelihood Must Not Decrease}{11}{subsection.6.2}\protected@file@percent } \newlabel{sec:monotone}{{6.2}{11}{Monotonicity: the Log-Likelihood Must Not Decrease}{subsection.6.2}{}} \@writefile{toc}{\contentsline {section}{\numberline {7}Mixtures of Other Things}{13}{section.7}\protected@file@percent } \newlabel{sec:families}{{7}{13}{Mixtures of Other Things}{section.7}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.1}Same Machinery, Different Component: the Coins Problem}{13}{subsection.7.1}\protected@file@percent } \newlabel{sec:coins}{{7.1}{13}{Same Machinery, Different Component: the Coins Problem}{subsection.7.1}{}} \@writefile{toc}{\contentsline {paragraph}{The setup.}{13}{section*.6}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{The component likelihood.}{13}{section*.7}\protected@file@percent } \newlabel{eq:binom}{{17}{13}{The component likelihood}{equation.17}{}} \@writefile{toc}{\contentsline {paragraph}{E step.}{13}{section*.8}\protected@file@percent } \newlabel{eq:coinestep}{{18}{13}{E step}{equation.18}{}} \@writefile{toc}{\contentsline {paragraph}{M step.}{13}{section*.9}\protected@file@percent } \newlabel{eq:coinmstep}{{19}{13}{M step}{equation.19}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.2}Mixtures of Poissons}{14}{subsection.7.2}\protected@file@percent } \newlabel{sec:poisson}{{7.2}{14}{Mixtures of Poissons}{subsection.7.2}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.3}Choosing a Component Family}{14}{subsection.7.3}\protected@file@percent } \newlabel{sec:choosefamily}{{7.3}{14}{Choosing a Component Family}{subsection.7.3}{}} \@writefile{toc}{\contentsline {paragraph}{What changes, and what does not.}{15}{section*.10}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {8}What Goes Wrong: Local Optima, Label Switching, Identifiability}{15}{section.8}\protected@file@percent } \newlabel{sec:identify}{{8}{15}{What Goes Wrong: Local Optima, Label Switching, Identifiability}{section.8}{}} \@writefile{toc}{\contentsline {paragraph}{(1) Label switching (harmless).}{15}{section*.11}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{(2) Local optima (real, manageable).}{15}{section*.12}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{(3) Weak identifiability (not fixable by restarts).}{16}{section*.13}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {9}HW4: Worked Numbers, Map, and Common Bugs}{16}{section.9}\protected@file@percent } \newlabel{sec:hw4sec}{{9}{16}{HW4: Worked Numbers, Map, and Common Bugs}{section.9}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {9.1}Numbers You Can Check by Hand, and What to Expect on the HW Data}{16}{subsection.9.1}\protected@file@percent } \newlabel{sec:worked}{{9.1}{16}{Numbers You Can Check by Hand, and What to Expect on the HW Data}{subsection.9.1}{}} \@writefile{toc}{\contentsline {paragraph}{What convergence looks like.}{17}{section*.14}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {9.2}Problem-by-Problem Map, and Common Bugs}{18}{subsection.9.2}\protected@file@percent } \newlabel{sec:hw4}{{9.2}{18}{Problem-by-Problem Map, and Common Bugs}{subsection.9.2}{}} \@writefile{toc}{\contentsline {paragraph}{Problem 3 (A,B) --- EM for a Gaussian mixture (50 points).}{18}{section*.15}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Problem 3(C) --- the ELBO (10 points).}{18}{section*.16}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Problem 4 --- mixture of Binomials (50 points).}{19}{section*.17}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Problem 6 --- (theory) can the log-likelihood decrease? (25 points).}{19}{section*.18}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Problem 11 (optional) --- Naive Bayes with $K$ Gaussians per feature.}{19}{section*.19}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Common bugs.}{19}{section*.20}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {10}Notation, and Where This Goes Next}{19}{section.10}\protected@file@percent } \newlabel{sec:back}{{10}{19}{Notation, and Where This Goes Next}{section.10}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {10.1}Notation Bridge}{19}{subsection.10.1}\protected@file@percent } \newlabel{sec:bridge}{{10.1}{19}{Notation Bridge}{subsection.10.1}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {10.2}Where This Goes Next}{20}{subsection.10.2}\protected@file@percent } \newlabel{sec:next}{{10.2}{20}{Where This Goes Next}{subsection.10.2}{}} \gdef \@abspage@last{20}