\[ \newcommand{\figleft}{{\em (Left)}} \newcommand{\figcenter}{{\em (Center)}} \newcommand{\figright}{{\em (Right)}} \newcommand{\figtop}{{\em (Top)}} \newcommand{\figbottom}{{\em (Bottom)}} \newcommand{\captiona}{{\em (a)}} \newcommand{\captionb}{{\em (b)}} \newcommand{\captionc}{{\em (c)}} \newcommand{\captiond}{{\em (d)}} \newcommand{\newterm}[1]{{\bf #1}} \def\figref#1{figure~\ref{#1}} \def\Figref#1{Figure~\ref{#1}} \def\twofigref#1#2{figures \ref{#1} and \ref{#2}} \def\quadfigref#1#2#3#4{figures \ref{#1}, \ref{#2}, \ref{#3} and \ref{#4}} \def\secref#1{section~\ref{#1}} \def\Secref#1{Section~\ref{#1}} \def\twosecrefs#1#2{sections \ref{#1} and \ref{#2}} \def\secrefs#1#2#3{sections \ref{#1}, \ref{#2} and \ref{#3}} \def\eqref#1{equation~\ref{#1}} \def\Eqref#1{Equation~\ref{#1}} \def\plaineqref#1{\ref{#1}} \def\chapref#1{chapter~\ref{#1}} \def\Chapref#1{Chapter~\ref{#1}} \def\rangechapref#1#2{chapters\ref{#1}--\ref{#2}} \def\algref#1{algorithm~\ref{#1}} \def\Algref#1{Algorithm~\ref{#1}} \def\twoalgref#1#2{algorithms \ref{#1} and \ref{#2}} \def\Twoalgref#1#2{Algorithms \ref{#1} and \ref{#2}} \def\partref#1{part~\ref{#1}} \def\Partref#1{Part~\ref{#1}} \def\twopartref#1#2{parts \ref{#1} and \ref{#2}} \def\ceil#1{\lceil #1 \rceil} \def\floor#1{\lfloor #1 \rfloor} \def\1{\bm{1}} \newcommand{\train}{\mathcal{D}} \newcommand{\valid}{\mathcal{D_{\mathrm{valid}}}} \newcommand{\test}{\mathcal{D_{\mathrm{test}}}} \def\eps{{\epsilon}} \def\reta{{\textnormal{$\eta$}}} \def\ra{{\textnormal{a}}} \def\rb{{\textnormal{b}}} \def\rc{{\textnormal{c}}} \def\rd{{\textnormal{d}}} \def\re{{\textnormal{e}}} \def\rf{{\textnormal{f}}} \def\rg{{\textnormal{g}}} \def\rh{{\textnormal{h}}} \def\ri{{\textnormal{i}}} \def\rj{{\textnormal{j}}} \def\rk{{\textnormal{k}}} \def\rl{{\textnormal{l}}} \def\rn{{\textnormal{n}}} \def\ro{{\textnormal{o}}} \def\rp{{\textnormal{p}}} \def\rq{{\textnormal{q}}} \def\rr{{\textnormal{r}}} \def\rs{{\textnormal{s}}} \def\rt{{\textnormal{t}}} \def\ru{{\textnormal{u}}} \def\rv{{\textnormal{v}}} \def\rw{{\textnormal{w}}} \def\rx{{\textnormal{x}}} \def\ry{{\textnormal{y}}} \def\rz{{\textnormal{z}}} \def\rvepsilon{{\mathbf{\epsilon}}} \def\rvtheta{{\mathbf{\theta}}} \def\rva{{\mathbf{a}}} \def\rvb{{\mathbf{b}}} \def\rvc{{\mathbf{c}}} \def\rvd{{\mathbf{d}}} \def\rve{{\mathbf{e}}} \def\rvf{{\mathbf{f}}} \def\rvg{{\mathbf{g}}} \def\rvh{{\mathbf{h}}} \def\rvu{{\mathbf{i}}} \def\rvj{{\mathbf{j}}} \def\rvk{{\mathbf{k}}} \def\rvl{{\mathbf{l}}} \def\rvm{{\mathbf{m}}} \def\rvn{{\mathbf{n}}} \def\rvo{{\mathbf{o}}} \def\rvp{{\mathbf{p}}} \def\rvq{{\mathbf{q}}} \def\rvr{{\mathbf{r}}} \def\rvs{{\mathbf{s}}} \def\rvt{{\mathbf{t}}} \def\rvu{{\mathbf{u}}} \def\rvv{{\mathbf{v}}} \def\rvw{{\mathbf{w}}} \def\rvx{{\mathbf{x}}} \def\rvy{{\mathbf{y}}} \def\rvz{{\mathbf{z}}} \def\erva{{\textnormal{a}}} \def\ervb{{\textnormal{b}}} \def\ervc{{\textnormal{c}}} \def\ervd{{\textnormal{d}}} \def\erve{{\textnormal{e}}} \def\ervf{{\textnormal{f}}} \def\ervg{{\textnormal{g}}} \def\ervh{{\textnormal{h}}} \def\ervi{{\textnormal{i}}} \def\ervj{{\textnormal{j}}} \def\ervk{{\textnormal{k}}} \def\ervl{{\textnormal{l}}} \def\ervm{{\textnormal{m}}} \def\ervn{{\textnormal{n}}} \def\ervo{{\textnormal{o}}} \def\ervp{{\textnormal{p}}} \def\ervq{{\textnormal{q}}} \def\ervr{{\textnormal{r}}} \def\ervs{{\textnormal{s}}} \def\ervt{{\textnormal{t}}} \def\ervu{{\textnormal{u}}} \def\ervv{{\textnormal{v}}} \def\ervw{{\textnormal{w}}} \def\ervx{{\textnormal{x}}} \def\ervy{{\textnormal{y}}} \def\ervz{{\textnormal{z}}} \def\rmA{{\mathbf{A}}} \def\rmB{{\mathbf{B}}} \def\rmC{{\mathbf{C}}} \def\rmD{{\mathbf{D}}} \def\rmE{{\mathbf{E}}} \def\rmF{{\mathbf{F}}} \def\rmG{{\mathbf{G}}} \def\rmH{{\mathbf{H}}} \def\rmI{{\mathbf{I}}} \def\rmJ{{\mathbf{J}}} \def\rmK{{\mathbf{K}}} \def\rmL{{\mathbf{L}}} \def\rmM{{\mathbf{M}}} \def\rmN{{\mathbf{N}}} \def\rmO{{\mathbf{O}}} \def\rmP{{\mathbf{P}}} \def\rmQ{{\mathbf{Q}}} \def\rmR{{\mathbf{R}}} \def\rmS{{\mathbf{S}}} \def\rmT{{\mathbf{T}}} \def\rmU{{\mathbf{U}}} \def\rmV{{\mathbf{V}}} \def\rmW{{\mathbf{W}}} \def\rmX{{\mathbf{X}}} \def\rmY{{\mathbf{Y}}} \def\rmZ{{\mathbf{Z}}} \def\ermA{{\textnormal{A}}} \def\ermB{{\textnormal{B}}} \def\ermC{{\textnormal{C}}} \def\ermD{{\textnormal{D}}} \def\ermE{{\textnormal{E}}} \def\ermF{{\textnormal{F}}} \def\ermG{{\textnormal{G}}} \def\ermH{{\textnormal{H}}} \def\ermI{{\textnormal{I}}} \def\ermJ{{\textnormal{J}}} \def\ermK{{\textnormal{K}}} \def\ermL{{\textnormal{L}}} \def\ermM{{\textnormal{M}}} \def\ermN{{\textnormal{N}}} \def\ermO{{\textnormal{O}}} \def\ermP{{\textnormal{P}}} \def\ermQ{{\textnormal{Q}}} \def\ermR{{\textnormal{R}}} \def\ermS{{\textnormal{S}}} \def\ermT{{\textnormal{T}}} \def\ermU{{\textnormal{U}}} \def\ermV{{\textnormal{V}}} \def\ermW{{\textnormal{W}}} \def\ermX{{\textnormal{X}}} \def\ermY{{\textnormal{Y}}} \def\ermZ{{\textnormal{Z}}} \def\vzero{{\bm{0}}} \def\vone{{\bm{1}}} \def\vmu{{\bm{\mu}}} \def\vtheta{{\bm{\theta}}} \def\va{{\bm{a}}} \def\vb{{\bm{b}}} \def\vc{{\bm{c}}} \def\vd{{\bm{d}}} \def\ve{{\bm{e}}} \def\vf{{\bm{f}}} \def\vg{{\bm{g}}} \def\vh{{\bm{h}}} \def\vi{{\bm{i}}} \def\vj{{\bm{j}}} \def\vk{{\bm{k}}} \def\vl{{\bm{l}}} \def\vm{{\bm{m}}} \def\vn{{\bm{n}}} \def\vo{{\bm{o}}} \def\vp{{\bm{p}}} \def\vq{{\bm{q}}} \def\vr{{\bm{r}}} \def\vs{{\bm{s}}} \def\vt{{\bm{t}}} \def\vu{{\bm{u}}} \def\vv{{\bm{v}}} \def\vw{{\bm{w}}} \def\vx{{\bm{x}}} \def\vy{{\bm{y}}} \def\vz{{\bm{z}}} \def\evalpha{{\alpha}} \def\evbeta{{\beta}} \def\evepsilon{{\epsilon}} \def\evlambda{{\lambda}} \def\evomega{{\omega}} \def\evmu{{\mu}} \def\evpsi{{\psi}} \def\evsigma{{\sigma}} \def\evtheta{{\theta}} \def\eva{{a}} \def\evb{{b}} \def\evc{{c}} \def\evd{{d}} \def\eve{{e}} \def\evf{{f}} \def\evg{{g}} \def\evh{{h}} \def\evi{{i}} \def\evj{{j}} \def\evk{{k}} \def\evl{{l}} \def\evm{{m}} \def\evn{{n}} \def\evo{{o}} \def\evp{{p}} \def\evq{{q}} \def\evr{{r}} \def\evs{{s}} \def\evt{{t}} \def\evu{{u}} \def\evv{{v}} \def\evw{{w}} \def\evx{{x}} \def\evy{{y}} \def\evz{{z}} \def\mA{{\bm{A}}} \def\mB{{\bm{B}}} \def\mC{{\bm{C}}} \def\mD{{\bm{D}}} \def\mE{{\bm{E}}} \def\mF{{\bm{F}}} \def\mG{{\bm{G}}} \def\mH{{\bm{H}}} \def\mI{{\bm{I}}} \def\mJ{{\bm{J}}} \def\mK{{\bm{K}}} \def\mL{{\bm{L}}} \def\mM{{\bm{M}}} \def\mN{{\bm{N}}} \def\mO{{\bm{O}}} \def\mP{{\bm{P}}} \def\mQ{{\bm{Q}}} \def\mR{{\bm{R}}} \def\mS{{\bm{S}}} \def\mT{{\bm{T}}} \def\mU{{\bm{U}}} \def\mV{{\bm{V}}} \def\mW{{\bm{W}}} \def\mX{{\bm{X}}} \def\mY{{\bm{Y}}} \def\mZ{{\bm{Z}}} \def\mBeta{{\bm{\beta}}} \def\mPhi{{\bm{\Phi}}} \def\mLambda{{\bm{\Lambda}}} \def\mSigma{{\bm{\Sigma}}} \newcommand{\tens}[1]{\bm{\mathsfit{#1}}} \def\tA{{\tens{A}}} \def\tB{{\tens{B}}} \def\tC{{\tens{C}}} \def\tD{{\tens{D}}} \def\tE{{\tens{E}}} \def\tF{{\tens{F}}} \def\tG{{\tens{G}}} \def\tH{{\tens{H}}} \def\tI{{\tens{I}}} \def\tJ{{\tens{J}}} \def\tK{{\tens{K}}} \def\tL{{\tens{L}}} \def\tM{{\tens{M}}} \def\tN{{\tens{N}}} \def\tO{{\tens{O}}} \def\tP{{\tens{P}}} \def\tQ{{\tens{Q}}} \def\tR{{\tens{R}}} \def\tS{{\tens{S}}} \def\tT{{\tens{T}}} \def\tU{{\tens{U}}} \def\tV{{\tens{V}}} \def\tW{{\tens{W}}} \def\tX{{\tens{X}}} \def\tY{{\tens{Y}}} \def\tZ{{\tens{Z}}} \def\gA{{\mathcal{A}}} \def\gB{{\mathcal{B}}} \def\gC{{\mathcal{C}}} \def\gD{{\mathcal{D}}} \def\gE{{\mathcal{E}}} \def\gF{{\mathcal{F}}} \def\gG{{\mathcal{G}}} \def\gH{{\mathcal{H}}} \def\gI{{\mathcal{I}}} \def\gJ{{\mathcal{J}}} \def\gK{{\mathcal{K}}} \def\gL{{\mathcal{L}}} \def\gM{{\mathcal{M}}} \def\gN{{\mathcal{N}}} \def\gO{{\mathcal{O}}} \def\gP{{\mathcal{P}}} \def\gQ{{\mathcal{Q}}} \def\gR{{\mathcal{R}}} \def\gS{{\mathcal{S}}} \def\gT{{\mathcal{T}}} \def\gU{{\mathcal{U}}} \def\gV{{\mathcal{V}}} \def\gW{{\mathcal{W}}} \def\gX{{\mathcal{X}}} \def\gY{{\mathcal{Y}}} \def\gZ{{\mathcal{Z}}} \def\sA{{\mathbb{A}}} \def\sB{{\mathbb{B}}} \def\sC{{\mathbb{C}}} \def\sD{{\mathbb{D}}} \def\sF{{\mathbb{F}}} \def\sG{{\mathbb{G}}} \def\sH{{\mathbb{H}}} \def\sI{{\mathbb{I}}} \def\sJ{{\mathbb{J}}} \def\sK{{\mathbb{K}}} \def\sL{{\mathbb{L}}} \def\sM{{\mathbb{M}}} \def\sN{{\mathbb{N}}} \def\sO{{\mathbb{O}}} \def\sP{{\mathbb{P}}} \def\sQ{{\mathbb{Q}}} \def\sR{{\mathbb{R}}} \def\sS{{\mathbb{S}}} \def\sT{{\mathbb{T}}} \def\sU{{\mathbb{U}}} \def\sV{{\mathbb{V}}} \def\sW{{\mathbb{W}}} \def\sX{{\mathbb{X}}} \def\sY{{\mathbb{Y}}} \def\sZ{{\mathbb{Z}}} \def\emLambda{{\Lambda}} \def\emA{{A}} \def\emB{{B}} \def\emC{{C}} \def\emD{{D}} \def\emE{{E}} \def\emF{{F}} \def\emG{{G}} \def\emH{{H}} \def\emI{{I}} \def\emJ{{J}} \def\emK{{K}} \def\emL{{L}} \def\emM{{M}} \def\emN{{N}} \def\emO{{O}} \def\emP{{P}} \def\emQ{{Q}} \def\emR{{R}} \def\emS{{S}} \def\emT{{T}} \def\emU{{U}} \def\emV{{V}} \def\emW{{W}} \def\emX{{X}} \def\emY{{Y}} \def\emZ{{Z}} \def\emSigma{{\Sigma}} \newcommand{\etens}[1]{\mathsfit{#1}} \def\etLambda{{\etens{\Lambda}}} \def\etA{{\etens{A}}} \def\etB{{\etens{B}}} \def\etC{{\etens{C}}} \def\etD{{\etens{D}}} \def\etE{{\etens{E}}} \def\etF{{\etens{F}}} \def\etG{{\etens{G}}} \def\etH{{\etens{H}}} \def\etI{{\etens{I}}} \def\etJ{{\etens{J}}} \def\etK{{\etens{K}}} \def\etL{{\etens{L}}} \def\etM{{\etens{M}}} \def\etN{{\etens{N}}} \def\etO{{\etens{O}}} \def\etP{{\etens{P}}} \def\etQ{{\etens{Q}}} \def\etR{{\etens{R}}} \def\etS{{\etens{S}}} \def\etT{{\etens{T}}} \def\etU{{\etens{U}}} \def\etV{{\etens{V}}} \def\etW{{\etens{W}}} \def\etX{{\etens{X}}} \def\etY{{\etens{Y}}} \def\etZ{{\etens{Z}}} \newcommand{\pdata}{p_{\rm{data}}} \newcommand{\ptrain}{\hat{p}_{\rm{data}}} \newcommand{\Ptrain}{\hat{P}_{\rm{data}}} \newcommand{\pmodel}{p_{\rm{model}}} \newcommand{\Pmodel}{P_{\rm{model}}} \newcommand{\ptildemodel}{\tilde{p}_{\rm{model}}} \newcommand{\pencode}{p_{\rm{encoder}}} \newcommand{\pdecode}{p_{\rm{decoder}}} \newcommand{\precons}{p_{\rm{reconstruct}}} \newcommand{\E}{\mathbb{E}} \newcommand{\Ls}{\mathcal{L}} \newcommand{\R}{\mathbb{R}} \newcommand{\emp}{\tilde{p}} \newcommand{\lr}{\alpha} \newcommand{\reg}{\lambda} \newcommand{\rect}{\mathrm{rectifier}} \newcommand{\softmax}{\mathrm{softmax}} \newcommand{\sigmoid}{\sigma} \newcommand{\softplus}{\zeta} \newcommand{\KL}{D_{\mathrm{KL}}} \newcommand{\Var}{\mathrm{Var}} \newcommand{\standarderror}{\mathrm{SE}} \newcommand{\Cov}{\mathrm{Cov}} \newcommand{\normlzero}{L^0} \newcommand{\normlone}{L^1} \newcommand{\normltwo}{L^2} \newcommand{\normlp}{L^p} \newcommand{\normmax}{L^\infty} \newcommand{\parents}{Pa} % See usage in notation.tex. Chosen to match Daphne's book. \DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\Tr}{Tr} \let\ab\allowbreak \]