\newcommand{\mathbold}[1]{\bm{#1}}
\newcommand{\mbf}[1]{\mathbf{#1}}
\newcommand{\vect}[1]{\mbf{#1}}
\newcommand{\vectb}[1]{\mathbold{#1}}

\newcommand{\T}{\top}    %
\newcommand{\dd}{\,\mathrm{d}} %
\newcommand{\E}{\mathbb{E}}    %
\newcommand{\R}{\mathbb{R}}    %
\newcommand{\real}{\mathbb{R}}    %
\newcommand{\N}{\mathcal{N}}   %
\newcommand{\G}{\mathcal{G}}   %
\newcommand{\U}{\mathcal{U}}   %
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\chol}{chol}
\DeclareMathOperator{\dchol}{dchol}
\DeclareMathOperator{\Cov}{Cov}
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\gammad}{Gamma}
\DeclareMathOperator{\expd}{Exp}
\DeclareMathOperator{\sech}{sech}
\DeclareMathOperator{\erf}{erf}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\argmax}{arg\,max}
\newcommand{\KL}[2]{\mathrm{D}_\mathrm{KL}\left[#1\,\|\,#2\right]}
\newcommand{\hinge}[2]{\max \left(0, 1 #1 #2 \right)}
\newcommand{\pkg}[1]{\texttt{#1}} %
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}

\newcommand{\valpha}[0]{\mathbold{\alpha}}
\newcommand{\vbeta}[0]{\mathbold{\beta}}
\newcommand{\vgamma}[0]{\mathbold{\gamma}}
\newcommand{\vchi}[0]{\mathbold{\chi}}
\newcommand{\vepsilon}[0]{\mathbold{\varepsilon}}
\newcommand{\vrho}[0]{\mathbold{\rho}}
\newcommand{\veta}[0]{\mathbold{\eta}}
\newcommand{\vmu}[0]{\mathbold{\mu}}
\newcommand{\vomega}[0]{\mathbold{\omega}}
\newcommand{\vsigma}[0]{\mathbold{\sigma}}
\newcommand{\vxi}[0]{\mathbold{\xi}}
\newcommand{\vtheta}[0]{\mathbold{\theta}}
\newcommand{\vvartheta}[0]{\mathbold{\vartheta}}
\newcommand{\vTheta}[0]{\mathbold{\Theta}}
\newcommand{\vzeta}[0]{\mathbold{\zeta}}
\newcommand{\vlambda}[0]{\mathbold{\lambda}}
\newcommand{\vpsi}[0]{\mathbold{\psi}}
\newcommand{\MPsi}[0]{\mathbold{\Psi}}
\newcommand{\MPhi}[0]{\mathbold{\Phi}}
\newcommand{\MSigma}[0]{\mathbold{\Sigma}}
\newcommand{\MTheta}[0]{\mathbold{\Theta}}
\newcommand{\MLambda}[0]{\mathbold{\Lambda}}
\newcommand{\invchisq}[0]{\mathrm{Inv\text{-}}\chi^2}
\renewcommand{\mid}[0]{\,|\,}
\newcommand{\diff}[0]{\,\mathrm{d}}
\newcommand{\vzero}[0]{\mathbold{0}}
\newcommand{\vone}[0]{\mathbold{1}}

\newcommand{\RL}{\mathrm{L}}
\newcommand{\RM}{\mathrm{M}}
\newcommand{\RQ}{\mathrm{Q}}
\newcommand{\RS}{\mathrm{S}}
\newcommand{\RU}{\mathrm{U}}

\newcommand{\rb}{\mathrm{b}}
\newcommand{\rc}{\mathrm{c}}
\newcommand{\rrm}{\mathrm{m}}
\newcommand{\ro}{\mathrm{o}}
\newcommand{\rs}{\mathrm{s}}
\newcommand{\rrq}{\mathrm{q}}

\newcommand{\va}{\mbf{a}}
\newcommand{\vb}{\mbf{b}}
\newcommand{\vc}{\mbf{c}}
\newcommand{\vd}{\mbf{d}}
\newcommand{\ve}{\mbf{e}}
\newcommand{\vf}{\mbf{f}}
\newcommand{\vg}{\mbf{g}}
\newcommand{\vh}{\mbf{h}}
\newcommand{\vi}{\mbf{i}}
\newcommand{\vj}{\mbf{j}}
\newcommand{\vk}{\mbf{k}}
\newcommand{\vl}{\mbf{l}}
\newcommand{\vm}{\mbf{m}}
\newcommand{\vn}{\mbf{n}}
\newcommand{\vo}{\mbf{o}}
\newcommand{\vp}{\mbf{p}}
\newcommand{\vq}{\mbf{q}}
\newcommand{\vr}{\mbf{r}}
\newcommand{\vt}{\mbf{t}}
\newcommand{\vu}{\mbf{u}}
\newcommand{\vv}{\mbf{v}}
\newcommand{\vw}{\mbf{w}}
\newcommand{\vx}{\mbf{x}}
\newcommand{\vy}{\mbf{y}}
\newcommand{\vz}{\mbf{z}}
\newcommand{\MA}{\mbf{A}}
\newcommand{\MB}{\mbf{B}}
\newcommand{\MC}{\mbf{C}}
\newcommand{\MD}{\mbf{D}}
\newcommand{\MF}{\mbf{F}}
\newcommand{\MG}{\mbf{G}}
\newcommand{\MH}{\mbf{H}}
\newcommand{\MI}{\mbf{I}}
\newcommand{\MJ}{\mbf{J}}
\newcommand{\MK}{\mbf{K}}
\newcommand{\ML}{\mbf{L}}
\newcommand{\MM}{\mbf{M}}
\newcommand{\MP}{\mbf{P}}
\newcommand{\MQ}{\mbf{Q}}
\newcommand{\MR}{\mbf{R}}
\newcommand{\MS}{\mbf{S}}
\newcommand{\MT}{\mbf{T}}
\newcommand{\MU}{\mbf{U}}
\newcommand{\MV}{\mbf{V}}
\newcommand{\MW}{\mbf{W}}
\newcommand{\MX}{\mbf{X}}
\newcommand{\MY}{\mbf{Y}}
\newcommand{\MZ}{\mbf{Z}}

\newcommand{\GP}{\mathcal{GP}}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\Mt}{\mathcal{M}_t}
\newcommand{\ELBO}{\underbar{$\mathcal{L}$}}
\newcommand{\ELBOU}{$\mathcal{L}_*$}
\newcommand{\bigo}{\mathcal{O}}

\newcommand{\inducing}{\mathcal{I}}
\newcommand{\minibatch}{\mathcal{M}}
\newcommand{\ind}{\mathcal{I}}
\newcommand{\mbatch}{\mathcal{M}}
\newcommand{\ty}{\tilde{y}}
\newcommand{\tvy}{\tilde{\vy}}
\newcommand{\tm}{\tilde{m}}
\newcommand{\tvm}{\tilde{\vm}}
\newcommand{\hatm}{\widehat{m}}
\newcommand{\hvm}{\widehat{\vm}}
\newcommand{\tv}{\tilde{v}}
\newcommand{\hv}{\widehat{v}}
\newcommand{\tvv}{\tilde{\vv}}
\newcommand{\hvv}{\widehat{\vv}}
\newcommand{\tsigma}{\tilde{\sigma}}
\newcommand{\gauss}{\mathcal{N}}
\newcommand{\myexpect}{\mathbb{E}}
\newcommand{\rnd}[1]{\left(#1\right)}
\newcommand{\sqr}[1]{\left[#1\right]}
\newcommand{\crl}[1]{\left\{#1\right\}}
\newcommand{\notind}{\bar{\mathcal{I}}}
\newcommand{\dkls}[3]{\mathbb{D}_\mathrm{KL}^{#1}[#2 \, \|\, #3]}
\newcommand{\tlambda}{\widehat{\lambda}}
\newcommand{\talpha}{\widehat{\alpha}}
\newcommand{\tLambda}{\widehat{\Lambda}}
\newcommand{\tvLambda}{\widehat{\boldsymbol{\Lambda}}}
\newcommand{\tvlambda}{\widehat{\boldsymbol{\lambda}}}
\newcommand{\tvalpha}{\widehat{\boldsymbol{\alpha}}}
\newtheorem{thm}{Theorem}{}
\newtheorem{remark}{Remark}{}
\newcommand{\data}{\mathcal{D}}
\newcommand{\bdata}{\bar{\mathcal{D}}}
\newcommand{\MFeat}{\MU}
\newcommand{\vfeat}{\vu}
\newcommand{\feat}{u}
\newcommand{\indset}{\mathcal{Z}}
\newcommand{\sigmoid}{\mathcal{S}}
\newcommand{\vphi}{\boldsymbol{\phi}}
\newcommand{\loss}[2]{\ell \left(#1, #2\right)}

\newcommand{\entropy}{\mathcal{H}}
\newcommand{\ang}[1]{\left\langle#1\right\rangle}
\newcommand{\pms}[1]{\ensuremath{{\scriptstyle\pm #1}}}

\definecolor{darkgreen}{rgb}{0.0, 0.5, 0.0}

\usepackage{xspace}
\newcommand{\eg}{\textit{e.g.}\@\xspace}
\newcommand{\ie}{\textit{i.e.}\@\xspace}
\newcommand{\cf}{\textit{cf.}\@\xspace}
\newcommand{\etc}{\textit{etc.}\@\xspace}
\newcommand{\etal}{\textit{et~al.}\@\xspace}