% % A useful set of commands \usepackage{mathtools} \usepackage{dsfont} \usepackage[colorinlistoftodos]{todonotes} \usepackage{booktabs} \usepackage{xfrac} \usepackage{bbm} \usepackage{algpseudocode} \usepackage{algorithm} \usepackage{algorithmicx} \usepackage[most]{tcolorbox} \usepackage{xparse} \usepackage{lipsum} \usepackage{changepage} \usepackage{enumitem} \newcommand{\qedwhite}{\hfill \ensuremath{\Box}} %http://www-db.stanford.edu/~manku/latex.html %The itemize environment can be replaced by: \newcommand{\squishlist}{ \begin{list}{$\bullet$} { \setlength{\itemsep}{0pt} \setlength{\parsep}{3pt} \setlength{\topsep}{3pt} \setlength{\partopsep}{0pt} \setlength{\leftmargin}{1.5em} \setlength{\labelwidth}{1em} \setlength{\labelsep}{0.5em} } } \newcommand{\squishlisttwo}{ \begin{list}{$\bullet$} { \setlength{\itemsep}{0pt} \setlength{\parsep}{0pt} \setlength{\topsep}{0pt} \setlength{\partopsep}{0pt} \setlength{\leftmargin}{2em} \setlength{\labelwidth}{1.5em} \setlength{\labelsep}{0.5em} } } \newcommand{\squishend}{ \end{list} } %Example usage: \squishlist %% \begin{itemize} %\item First item %\item Second item %\squishend %% \end{itemize} \newcommand{\denselist}{\itemsep 0pt\topsep-6pt\partopsep-6pt} \newtheorem{thm}{Theorem}[section] \newtheorem{cor}{Corollary}[section] \newtheorem{defn}{Definition}[section] \newenvironment{mythm}{{\bf Theorem}}{} \newcommand{\tm}{\tilde{m}} \newcommand{\tv}{\tilde{v}} % For bold vector symbols \newcommand{\myvec}[1]{\boldsymbol{#1}} \newcommand{\myvecsym}[1]{\boldsymbol{#1}} \newcommand{\ind}[1]{\mathbb{I}(#1)} \newcommand{\vzero}{\myvecsym{0}} \newcommand{\vone}{\myvecsym{1}} \newcommand{\valpha}{\myvecsym{\alpha}} \newcommand{\vbeta}{\myvecsym{\beta}} \newcommand{\vchi}{\myvecsym{\chi}} \newcommand{\vdelta}{\myvecsym{\delta}} \newcommand{\vDelta}{\myvecsym{\Delta}} \newcommand{\vepsilon}{\myvecsym{\epsilon}} \newcommand{\vell}{\myvecsym{\ell}} \newcommand{\veta}{\myvecsym{\eta}} \newcommand{\vgamma}{\myvecsym{\gamma}} \newcommand{\vGamma}{\myvecsym{\Gamma}} \newcommand{\vmu}{\myvecsym{\mu}} \newcommand{\vnu}{\myvecsym{\nu}} \newcommand{\vkappa}{\myvecsym{\kappa}} \newcommand{\vlambda}{\myvecsym{\lambda}} \newcommand{\vLambda}{\myvecsym{\Lambda}} \newcommand{\vLambdaBar}{\overline{\vLambda}} \newcommand{\vomega}{\myvecsym{\omega}} \newcommand{\vOmega}{\myvecsym{\Omega}} \newcommand{\vphi}{\myvecsym{\phi}} \newcommand{\vPhi}{\myvecsym{\Phi}} \newcommand{\vpi}{\myvecsym{\pi}} \newcommand{\vpsi}{\myvecsym{\psi}} \newcommand{\vPsi}{\myvecsym{\Psi}} \newcommand{\vtheta}{\myvecsym{\theta}} \newcommand{\vTheta}{\myvecsym{\Theta}} \newcommand{\vsigma}{\myvecsym{\sigma}} \newcommand{\vSigma}{\myvecsym{\Sigma}} \newcommand{\vtau}{\myvecsym{\tau}} \newcommand{\vupsilon}{\myvecsym{\upsilon}} \newcommand{\vxi}{\myvecsym{\xi}} \newcommand{\vxn}{\vx^{(n)}} \newcommand{\vmuY}{\vb} \newcommand{\vmuMu}{\vmu_{x}} \newcommand{\vmuMuGivenY}{\vmu_{x|y}} \newcommand{\vSigmaMu}{\vSigma_{x}} \newcommand{\vSigmaMuInv}{\vSigma_{x}^{-1}} \newcommand{\vSigmaMuGivenY}{\vSigma_{x|y}} \newcommand{\vSigmaMuGivenYinv}{\vSigma_{x|y}^{-1}} \newcommand{\vSigmaY}{\vSigma_{y}} \newcommand{\vSigmaYinv}{\vSigma_{y}^{-1}} %\newcommand{\vmuY}{\vmu_{y}} %\newcommand{\vmuMu}{\vmu_{\mu}} %\newcommand{\vmuMuGivenY}{\vmu_{\mu|y}} %\newcommand{\vSigmaMu}{\vSigma_{\mu}} %\newcommand{\vSigmaMuInv}{\vSigma_{\mu}^{-1}} %\newcommand{\vSigmaMuGivenY}{\vSigma_{\mu|y}} %\newcommand{\vSigmaMuGivenYinv}{\vSigma_{\mu|y}^{-1}} %\newcommand{\vSigmaY}{\vSigma_{y}} %\newcommand{\vSigmaYinv}{\vSigma_{y}^{-1}} \newcommand{\muY}{\mu_{y}} \newcommand{\muMu}{\mu_{\mu}} \newcommand{\muMuGivenY}{\mu_{\mu|y}} \newcommand{\SigmaMu}{\Sigma_{\mu}} \newcommand{\SigmaMuInv}{\Sigma_{\mu}^{-1}} \newcommand{\SigmaMuGivenY}{\Sigma_{\mu|y}} \newcommand{\SigmaMuGivenYinv}{\Sigma_{\mu|y}^{-1}} \newcommand{\SigmaY}{\Sigma_{y}} \newcommand{\SigmaYinv}{\Sigma_{y}^{-1}} \newcommand{\hatf}{\hat{f}} \newcommand{\haty}{\hat{y}} \newcommand{\const}{\mbox{const}} \newcommand{\sigmoid}{\mbox{sigm}} \newcommand{\one}{(1)} \newcommand{\two}{(2)} \newcommand{\va}{\myvec{a}} \newcommand{\vb}{\myvec{b}} \newcommand{\vc}{\myvec{c}} \newcommand{\vd}{\myvec{d}} \newcommand{\ve}{\myvec{e}} \newcommand{\vf}{\myvec{f}} \newcommand{\vg}{\myvec{g}} \newcommand{\vh}{\myvec{h}} \newcommand{\vj}{\myvec{j}} \newcommand{\vk}{\myvec{k}} \newcommand{\vl}{\myvec{l}} \newcommand{\vm}{\myvec{m}} \newcommand{\vn}{\myvec{n}} \newcommand{\vo}{\myvec{o}} \newcommand{\vp}{\myvec{p}} \newcommand{\vq}{\myvec{q}} \newcommand{\vr}{\myvec{r}} \newcommand{\vs}{\myvec{s}} \newcommand{\vt}{\myvec{t}} \newcommand{\vu}{\myvec{u}} \newcommand{\vv}{\myvec{v}} \newcommand{\vw}{\myvec{w}} \newcommand{\vws}{\vw_s} \newcommand{\vwh}{\hat{\vw}} \newcommand{\vx}{\myvec{x}} %\newcommand{\vx}{\myvec{x}} \newcommand{\vxt}{\myvec{\tilde{x}}} \newcommand{\vy}{\myvec{y}} \newcommand{\vyt}{\myvec{\tilde{y}}} \newcommand{\vz}{\myvec{z}} \newcommand{\vA}{\myvec{A}} \newcommand{\vB}{\myvec{B}} \newcommand{\vC}{\myvec{C}} \newcommand{\vD}{\myvec{D}} \newcommand{\vE}{\myvec{E}} \newcommand{\vF}{\myvec{F}} \newcommand{\vG}{\myvec{G}} \newcommand{\vH}{\myvec{H}} \newcommand{\vI}{\myvec{I}} \newcommand{\vJ}{\myvec{J}} \newcommand{\vK}{\myvec{K}} \newcommand{\vL}{\myvec{L}} \newcommand{\vM}{\myvec{M}} \newcommand{\vN}{\myvec{N}} \newcommand{\vO}{\myvec{O}} \newcommand{\vP}{\myvec{P}} \newcommand{\vQ}{\myvec{Q}} \newcommand{\vR}{\myvec{R}} \newcommand{\vS}{\myvec{S}} \newcommand{\vT}{\myvec{T}} \newcommand{\vU}{\myvec{U}} \newcommand{\vV}{\myvec{V}} \newcommand{\vW}{\myvec{W}} \newcommand{\vX}{\myvec{X}} %\newcommand{\vXs}{\vX_{\vs}} \newcommand{\vXs}{\vX_{s}} \newcommand{\vXt}{\myvec{\tilde{X}}} \newcommand{\vY}{\myvec{Y}} \newcommand{\vZ}{\myvec{Z}} \newcommand{\vxtest}{\myvec{x}_*} \newcommand{\vytest}{\myvec{y}_*} \newcommand{\ftrue}{f_{true}} \newcommand{\myprec}{\mbox{prec}} \newcommand{\precw}{\lambda_{w}} % precision of weights (alpha) \newcommand{\precy}{\lambda_{y}} % precision of y (beta) \newcommand{\fbar}{\overline{f}} \newcommand{\xmybar}{\overline{x}} \newcommand{\ybar}{\overline{y}} \newcommand{\zbar}{\overline{z}} \newcommand{\vxbar}{\overline{\vx}} \newcommand{\vXbar}{\overline{\vX}} \newcommand{\vybar}{\overline{\vy}} \newcommand{\vYbar}{\overline{\vY}} \newcommand{\vzbar}{\overline{\vz}} \newcommand{\vZbar}{\overline{\vZ}} \newcommand{\xbar}{\overline{x}} \newcommand{\Xbar}{\overline{X}} \newcommand{\Ybar}{\overline{Y}} \newcommand{\Gbar}{\overline{G}} \newcommand{\Jbar}{\overline{J}} \newcommand{\Lbar}{\overline{L}} \newcommand{\Nbar}{\overline{N}} %\newcommand{\Qbar}{\overline{Q}} \newcommand{\Qbar}{\overline{Q}} \newcommand{\Tbar}{\overline{T}} \newcommand{\Sbar}{\overline{S}} \newcommand{\vSbar}{\overline{\vS}} \newcommand{\Rbar}{\overline{R}} \newcommand{\vtaubar}{\overline{\vtau}} \newcommand{\vtbar}{\overline{\vt}} \newcommand{\vsbar}{\overline{\vs}} \newcommand{\htilde}{\tilde{h}} \newcommand{\vhtilde}{\tilde{\vh}} \newcommand{\Dtilde}{\tilde{D}} \newcommand{\Ftilde}{\tilde{F}} \newcommand{\wtilde}{\tilde{w}} \newcommand{\ptilde}{\tilde{p}} \newcommand{\pemp}{p_{emp}} \newcommand{\pstar}{p^*} \newcommand{\xtilde}{\tilde{x}} \newcommand{\Xtilde}{\tilde{X}} \newcommand{\ytilde}{\tilde{y}} \newcommand{\Ytilde}{\tilde{Y}} \newcommand{\vxtilde}{\tilde{\vx}} \newcommand{\vytilde}{\tilde{\vy}} \newcommand{\ztilde}{\tilde{\z}} \newcommand{\vztilde}{\tilde{\vz}} \newcommand{\vthetaMAP}{\hat{\vtheta}_{MAP}} \newcommand{\vthetaS}{\vtheta^{(s)}} \newcommand{\vthetahat}{\hat{\vtheta}} \newcommand{\thetahat}{\hat{\theta}} \newcommand{\thetabar}{\overline{\theta}} \newcommand{\vthetabar}{\overline{\vtheta}} \newcommand{\pibar}{\overline{\pi}} \newcommand{\vpibar}{\overline{\vpi}} %\newcommand{\subsubsubsection}[1]{\paragraph{#1}} \newcommand{\choice}[2]{\left(\!\!\! \begin{array}{c} #1 \\ #2\end{array} \!\!\!\right)} \newcommand{\half}{\frac{1}{2}} \newcommand{\defeq}{\stackrel{\rm def}{=}} \newcommand{\real}{\mathbb{R}} \newcommand{\given}{\|} \newcommand{\indep}[2]{{#1} \perp {#2}} \newcommand{\condindep}[3]{{#1} \perp {#2} | {#3}} \newcommand{\condindepG}[3]{{#1} \perp_G {#2} | {#3}} \newcommand{\condindepP}[3]{{#1} \perp_p {#2} | {#3}} \newcommand{\depend}[2]{{#1} \not \perp {#2}} \newcommand{\conddepend}[3]{{#1} \not \perp {#2} | {#3}} \newcommand{\trans}[1]{{#1}^{\mathtt{T}}} \newcommand{\inv}[1]{{#1}^{-1}} \newcommand{\ra}{\rightarrow} \newcommand{\lra}{\leftrightarrow} \newcommand{\Ra}{\Rightarrow} %\newcommand{\rv}{r.v.} \newcommand{\la}{\leftarrow} \newcommand{\tr}{\mbox{tr}} \newcommand{\st}{\mbox{ s.t. }} \newcommand{\dom}{\mbox{dom}} \newcommand{\bel}{\mbox{bel}} \newcommand{\dsep}{\mbox{dsep}} \newcommand{\sep}{\mbox{sep}} \newcommand{\entails}{\models} \newcommand{\range}{\mbox{range}} \newcommand{\myspan}{\mbox{span}} \newcommand{\nullspace}{\mbox{nullspace}} \newcommand{\adj}{\mbox{adj}} \newcommand{\nbd}{\mbox{nbd}} \newcommand{\nbr}{\mbox{nbr}} \newcommand{\anc}{\mbox{anc}} \newcommand{\desc}{\mbox{desc}} \newcommand{\pred}{\mbox{pred}} \newcommand{\nondesc}{\mbox{nondesc}} \newcommand{\pa}{\pi} \newcommand{\ch}{\mbox{ch}} \newcommand{\mb}{\mbox{mb}} \newcommand{\connects}{\sim} \newcommand{\betadist}{\mbox{Beta}} \newcommand{\Betadist}{\mbox{Beta}} \newcommand{\bernoulli}{\mbox{Ber}} \newcommand{\Ber}{\mbox{Ber}} \newcommand{\Binom}{\mbox{Bin}} \newcommand{\NegBinom}{\mbox{NegBinom}} \newcommand{\binomdist}{\mbox{Bin}} \newcommand{\cauchy}{\mbox{Cauchy}} \newcommand{\DE}{\mbox{DE}} \newcommand{\Dir}{\mbox{Dir}} \newcommand{\discrete}{\calM} \newcommand{\Discrete}{\calM} \newcommand{\expdist}{\mbox{Exp}} \newcommand{\expon}{\mbox{Expon}} \newcommand{\gammadist}{\mbox{Ga}} \newcommand{\Ga}{\mbox{Ga}} \newcommand{\gauss}{{\cal N}} \newcommand{\IG}{\mbox{IG}} \newcommand{\IGauss}{\mbox{InvGauss}} \newcommand{\IW}{\mbox{IW}} \newcommand{\Laplace}{\mbox{Lap}} \newcommand{\Mu}{\mbox{Mu}} \newcommand{\Multi}{\mbox{Mu}} \newcommand{\NIX}{NI\chi^2} \newcommand{\GIX}{NI\chi^2} \newcommand{\NIG}{\mbox{NIG}} \newcommand{\GIG}{\mbox{NIG}} \newcommand{\NIW}{\mbox{NIW}} \newcommand{\GIW}{\mbox{NIW}} \newcommand{\MVNIW}{\mbox{NIW}} \newcommand{\NW}{\mbox{NWi}} \newcommand{\MVNIG}{\mbox{NIG}} \newcommand{\NGdist}{\mbox{NG}} \newcommand{\prob}{p} \newcommand{\Poi}{\mbox{Poi}} \newcommand{\Student}{{\cal T}} \newcommand{\student}{{\cal T}} \newcommand{\Wishart}{\mbox{Wi}} \newcommand{\Wi}{\mbox{Wi}} \newcommand{\unif}{\mbox{U}} \newcommand{\etr}{\mbox{etr}} \newcommand{\softmax}{\calS} \newcommand{\soft}{\mbox{soft}} \newcommand{\cond}{\mbox{cond}} \newcommand{\sign}{\mbox{sign}} \newcommand{\sgn}{\mbox{sgn}} \newcommand{\iid}{\mbox{iid}} \newcommand{\mle}{\mbox{mle}} \newcommand{\myiff}{\mbox{iff}} \newcommand{\pd}{\mbox{pd}} \newcommand{\pdf}{\mbox{pdf }} \newcommand{\cdf}{\mbox{cdf}} \newcommand{\pmf}{\mbox{pmf}} \newcommand{\wrt}{\mbox{wrt}} \newcommand{\matlab}{{\sc MATLAB}} \newcommand{\NETLAB}{{\sc NETLAB}} \newcommand{\MLABA}{\mbox{PMTK}} \newcommand{\BLT}{\mbox{PMTK}} \newcommand{\PMTK}{\mbox{PMTK}} \newcommand{\mywp}{\mbox{wp}} \newcommand{\KLpq}[2]{\mathrm{KL}\left[{#1}\|{#2}\right]} \newcommand{\KL}{\mbox{KL}} \newcommand{\MI}{\mathbb{I}} \newcommand{\MIxy}[2]{\mathbb{I}\left({#1};{#2}\right)} \newcommand{\MIxyz}[3]{\mathbb{I}\left({#1};{#2}|{#3}\right)} \newcommand{\entropy}[1]{\mathbb{H}\left({#1}\right)} \newcommand{\entropypq}[2]{\mathbb{H}\left({#1}, {#2}\right)} \newcommand{\vvec}{\mbox{vec}} \newcommand{\kron}{\otimes} \newcommand{\dof}{\mbox{dof}} %\newcommand{\E}{E} \newcommand{\E}{\mathbb{E}} \newcommand{\energy}{E} \newcommand{\expectAngle}[1]{\langle #1 \rangle} %\newcommand{\expect}[1]{\mathbb{E}\left[ {#1} \right]} \newcommand{\expect}[2]{\mathds{E}_{{#1}} \left[ {#2} \right]} \newcommand{\expectGiven}[3]{\mathds{E}_{{#1}} \left[ {#2} \mid {#3} \right]} \newcommand{\Var}{\mbox{Var}} \newcommand{\VarGiven}[3]{\mbox{Var}_{{#1}}\left[ {#2} \mid {#3}\right]} %\newcommand{\Var}{\mathbb{V}} \newcommand{\var}[1]{\mbox{var}\left[{#1}\right]} \newcommand{\std}[1]{\mbox{std}\left[{#1}\right]} \newcommand{\varQ}[2]{\mbox{var}_{{#2}}\left[{#1}\right]} \newcommand{\cov}[1]{\mbox{cov}\left[{#1}\right]} %\newcommand{\mode}[1]{\mbox{mode}\left[{#1}\right]} \newcommand{\median}[1]{\mbox{median}\left[{#1}\right]} \newcommand{\diag}{\mbox{diag}} \newcommand{\blkdiag}{\mbox{blkdiag}} \newcommand{\bias}{\mbox{bias}} \newcommand{\union}{\cup} \newcommand{\intersect}{\cap} \newcommand{\size}{\mbox{size}} \newcommand{\trace}{\mbox{trace}} \newcommand{\myc}{c} \newcommand{\myi}{i} \newcommand{\myj}{j} \newcommand{\myk}{k} \newcommand{\myn}{n} \newcommand{\myq}{q} \newcommand{\mys}{s} \newcommand{\myt}{t} \newcommand{\supp}{\mbox{supp}} \newcommand{\calA}{{\cal A}} \newcommand{\calB}{{\cal B}} \newcommand{\calC}{{\cal C}} \newcommand{\calD}{{\cal D}} \newcommand{\calDx}{{\cal D}_x} \newcommand{\calE}{{\cal E}} \newcommand{\cale}{{\cal e}} \newcommand{\calF}{{\cal F}} \newcommand{\calG}{{\cal G}} \newcommand{\calH}{{\cal H}} \newcommand{\calHX}{{\cal H}_X} \newcommand{\calHy}{{\cal H}_y} \newcommand{\calI}{{\cal I}} \newcommand{\calK}{{\cal K}} \newcommand{\calM}{{\cal M}} \newcommand{\calN}{{\cal N}} \newcommand{\caln}{{\cal n}} \newcommand{\calNP}{{\cal NP}} \newcommand{\calMp}{\calM^+} \newcommand{\calMm}{\calM^-} \newcommand{\calMo}{\calM^o} \newcommand{\Ctest}{C_*} \newcommand{\calL}{{\cal L}} \newcommand{\calP}{{\cal P}} \newcommand{\calq}{{\cal q}} \newcommand{\calQ}{{\cal Q}} \newcommand{\calR}{{\cal R}} \newcommand{\calS}{{\cal S}} \newcommand{\calSstar}{\calS_*} \newcommand{\calT}{{\cal T}} \newcommand{\calV}{{\cal V}} \newcommand{\calv}{{\cal v}} \newcommand{\calX}{{\cal X}} \newcommand{\calY}{{\cal Y}} \newcommand{\Lone}{$\ell_1$} \newcommand{\Ltwo}{$\ell_2$} \newcommand{\mya}{\mbox{a}} \newcommand{\myat}{\alpha_{t|t-1}} \newcommand{\score}{\mbox{score}} \newcommand{\AIC}{\mbox{AIC}} \newcommand{\BIC}{\mbox{BIC}} \newcommand{\BICcost}{\mbox{BIC-cost}} \newcommand{\scoreBIC}{\mbox{score-BIC}} \newcommand{\scoreBICL}{\mbox{score-BIC-L1}} \newcommand{\scoreL}{\mbox{score-L1}} \newcommand{\ecoli}{\mbox{{\it E. coli}}} \newcommand{\doPearl}{\mbox{do}} \newcommand{\data}{\calD} \newcommand{\model}{\calM} \newcommand{\dataTrain}{\calD_{\mbox{train}}} \newcommand{\dataTest}{\calD_{\mbox{test}}} \newcommand{\dataValid}{\calD_{\mbox{valid}}} \newcommand{\futuredata}{\tilde{\calD}} \newcommand{\algo}{\calA} \newcommand{\fitAlgo}{\calF} \newcommand{\predictAlgo}{\calP} \newcommand{\err}{\mbox{err}} \newcommand{\logit}{\mbox{logit}} \newcommand{\parent}{\mbox{pa}} %%%%%%%%%%% Hoyt \newcommand{\conv}[1]{\,\,\,\displaystyle{\operatorname*{\longrightarrow}^{\,_{#1}\,}}\,\,\,} \newcommand{\dconv}{\conv{D}} \newcommand{\pconv}{\conv{P}} \newcommand{\asconv}{\conv{AS}} \newcommand{\lpconv}[1]{\conv{L^{#1}}} \DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{n} \newcommand{\condSet}{\mathcal{S}} \newcommand{\condSetC}{\mathcal{\lnot S}}