%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Beginning of Wei's macro command %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\renewcommand{\thefootnote}{\fnsymbol{footnote}}
% \setlist{leftmargin=5mm}
\usepackage[utf8]{inputenc} % allow utf-8 input
\usepackage[T1]{fontenc}    % use 8-bit T1 fonts
\usepackage{comment}
\usepackage{bm}
%\usepackage{hyperref}       % hyperlinks
\usepackage{url}            % simple URL typesetting
\usepackage{booktabs}       % professional-quality tables
\usepackage{amsfonts}       % blackboard math symbols
\usepackage{amsmath,amssymb}

\usepackage{empheq}
\usepackage{amsthm}
\usepackage{nicefrac}       % compact symbols for 1/2, etc.
\usepackage{microtype}      % microtypography
\usepackage{enumerate,mathrsfs}
\usepackage{enumerate,natbib,mathrsfs}
\usepackage{tablefootnote}
\usepackage{wrapfig}
\usepackage{graphicx}
\usepackage{subfigure}
\usepackage{float}
\usepackage{enumitem}
\usepackage{multirow}
\usepackage{extarrows}
\usepackage[OT1]{fontenc}
\usepackage{mathtools}

% \usepackage[bookmarks=false]{hyperref}

\usepackage{xcolor}
\definecolor{darkblue}{rgb}{0.0, 0.0, 0.5}

% \usepackage{hyperref}
% \hypersetup{colorlinks=true,citecolor=red,linkcolor=red}
% \usepackage[margin=1in]{geometry}

% \definecolor{red}{rgb}{1.0, 0.0, 0.0}
\hypersetup{
  pdffitwindow=true,
  pdfstartview={FitH},
  pdfnewwindow=true,
  colorlinks,
  linktocpage=true,
  linkcolor=red,
  urlcolor=red,
  citecolor=darkblue
}
\newcommand{\Wei}[1]{{\color{red}[Wei: #1]}} 

\usepackage{hyperref}
\hypersetup{colorlinks=true,citecolor=darkblue,linkcolor=red}


\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}

\newtheorem{exercise}{Exercise}

\newcommand{\xeta}{x}
\newcommand{\mpi}{\cal \pi}

\def\ceil#1{\lceil #1 \rceil}
\def\floor#1{\lfloor #1 \rfloor}

\newcommand{\bbeta}{{\boldsymbol \beta}}
\newcommand{\bxi}{{\boldsymbol \xi}}
\newcommand{\bphi}{{\boldsymbol \phi}}
\newcommand{\bpsi}{{\boldsymbol \psi}}
\newcommand{\bvarphi}{{\boldsymbol \varphi}}
\newcommand{\bu}{{\boldsymbol u}}
\newcommand{\bx}{{\boldsymbol x}}
\newcommand{\by}{{\boldsymbol y}}
\newcommand{\bd}{{\boldsymbol d}}
\newcommand{\bZ}{{\boldsymbol Z}}
\newcommand{\bX}{{\mathrm{X}}}
\newcommand{\bW}{{\boldsymbol W}}
\newcommand{\bB}{{\mathrm{B}}}
\newcommand{\bY}{{\mathrm{Y}}}
\newcommand{\bt}{{\boldsymbol t}}
\newcommand{\bn}{\mathbf{n}}
\newcommand{\bw}{{\boldsymbol w}}
\newcommand{\bbf}{{\boldsymbol f}}
\newcommand{\bA}{{\boldsymbol A}}
\newcommand{\bE}{{\boldsymbol E}}
\newcommand{\bi}{{\boldsymbol i}}
\newcommand{\bC}{{\boldsymbol C}}
\newcommand{\bR}{{\boldsymbol R}}
\newcommand{\bK}{{\boldsymbol K}}
\newcommand{\bL}{{\boldsymbol L}}
\newcommand{\bV}{{\boldsymbol V}}
\newcommand{\bU}{{\boldsymbol U}}
\newcommand{\bH}{{\boldsymbol H}}
\newcommand{\bI}{{\boldsymbol I}}
\newcommand{\bJ}{{\boldsymbol J}}
\newcommand{\bp}{{\boldsymbol p}}
\newcommand{\bS}{{\boldsymbol S}}
\newcommand{\br}{{\boldsymbol r}}
\newcommand{\mPi}{{\mathrm{\Pi}}}
\newcommand{\MM}{{\mathcal{M}}}
\newcommand{\MK}{{\mathcal{K}}}
\newcommand{\var}{{\mbox{Var}}}
\newcommand{\cov}{{\mbox{Cov}}}
\newcommand{\MX}{{\mathcal{X}}}
\newcommand{\bchi}{{\boldsymbol{\chi}}}
\newcommand{\btheta}{{\boldsymbol{\theta}}}
\newcommand{\bUpsilon}{{\boldsymbol{\Upsilon}}}
\newcommand{\bDmis}{{\boldsymbol D}^{{\rm mis}}}
\newcommand{\bDobs}{{\boldsymbol D}^{{\rm obs}}}
\newcommand{\tbXmis}{{\tilde{\boldsymbol X}}^{{\rm  mis}}}
\newcommand{\tbxmis}{{\tilde{\boldsymbol x}}^{{\rm  mis}}}
\newcommand{\tXmis}{{\tilde{X}}^{{\rm  mis}}}
\newcommand{\txmis}{{\tilde{x}}^{{\rm  mis}}}
\newcommand{\Dmis}{D^{{\rm  mis}}}
\newcommand{\Dobs}{D^{{\rm  obs}}}
\newcommand{\bdmis}{{\boldsymbol d}^{{\rm  mis}}}
\newcommand{\bdobs}{{\boldsymbol d}^{{\rm  obs}}}
\newcommand{\dmis}{d^{{\rm  mis}}}
\newcommand{\dobs}{d^{{\rm  obs}}}
\newcommand{\bTheta}{{\boldsymbol \Theta}}
\newcommand{\bthetaL}{{\btheta_{\Lambda}}}
\newcommand{\bvartheta}{{\boldsymbol \vartheta}}
\newcommand{\bepsilon}{{\boldsymbol \epsilon}}
\newcommand{\bmu}{{\boldsymbol \mu}}
\newcommand{\bsigma}{{\boldsymbol \sigma}}
\newcommand{\bSigma}{{\boldsymbol \Sigma}}
\newcommand{\bzeta}{{\boldsymbol \zeta}}
\newcommand{\bGamma}{{\boldsymbol \varphi}}
\newcommand{\bvarepsilon}{{\boldsymbol \varepsilon}}
\newcommand{\seta}{\sigma_{\eta}}
\newcommand{\corr}{\mbox{Corr}}
\newcommand{\bdelta}{{\boldsymbol \delta}}
\newcommand{\bPi}{{\boldsymbol{\mathrm{\Pi}}}}


\newcommand{\la}{\langle}
\newcommand{\hP}{\hat\dbP}
\newcommand{\wt}{\widetilde}
\newcommand{\dd}{\mathcal{\dagger}}
\newcommand{\ts}{\mathsf{T}}
\newcommand{\tone}{t_{1}}
\newcommand{\ttwo}{t_{2}}
\newcommand{\tthree}{t_{3}}
\newcommand{\tfour}{t_{4}}
%%%%%%%%%%%
\newcommand{\pn}{P^{\tau}\nabla}

%%%%%%%%%%%new comment

\newcommand{\oo}{\overline}
\newcommand{\LDP}{{{LDP }}}
\newcommand{\MDP}{{{MDP }}}
\newcommand{\LDPx}{{{LDP}}}
\newcommand{\Las}{{\La^*}}
\newcommand{\ea}{\end{array}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\beaa}{\begin{eqnarray*}}
\newcommand{\eeaa}{\end{eqnarray*}}
\newcommand{\Remark}{\noindent{\bf Remark:}\  }
\newcommand{\Remarks}{\noindent{\bf Remarks:}\  }
\newcommand{\iL}{(L(\rho))^{-1}}

\def\neg{\negthinspace}

\def\dbB{\mathbb{B}}
\def\dbC{\mathbb{C}}
\def\dbD{\mathbb{D}}
\def\dbE{\mathbb{E}}
\def\dbF{\mathbb{F}}
\def\dbG{\mathbb{G}}
\def\dbH{\mathbb{H}}
\def\dbI{\mathbb{I}}
\def\dbJ{\mathbb{J}}
\def\dbK{\mathbb{K}}
\def\dbL{\mathbb{L}}
\def\dbM{\mathbb{M}}
\def\dbN{\mathbb{N}}
\def\dbP{\mathbb{P}}
\def\dbR{\mathbb{R}}
\def\dbS{\mathbb{S}}
\def\dbQ{\mathbb{Q}}
\def\hE{\mathbb{E}}
\def\E{\mathbb{E}}
\def\Dom{{\rm dom}}

%
%Lower case Greek letters
%
\def\a{\alpha}
\def\b{\beta}
\def\g{\varphi}
\def\d{\delta}
\def\e{\varepsilon}
\def\z{\zeta}
\def\k{\kappa}
\def\l{\lambda}
\def\m{\mu}
\def\n{\nu}
\def\si{\sigma}
\def\t{\tau}
\def\f{\psi}
\def\th{\theta}
\def\h{\widehat}
%
%Upper case Greek letters
%
%Upper case Greek letters
%
\def\G{\varphi}
\def\D{\Delta}
\def\Th{\Theta}
\def\L{\Lambda}
\def\Si{\Sigma}
\def\F{\Phi}
\def\infinity{\rotatebox{90}{8}}

%
%Calligraphic Capitals
%
\def\cA{{\cal A}}
\def\cB{{\cal B}}
\def\cC{{\cal C}}
\def\cD{{\cal D}}
\def\cE{{\cal E}}
\def\cF{{\cal F}}
\def\cG{{\cal G}}
\def\cH{{\cal H}}
\def\cI{{\cal I}}
\def\cJ{{\cal J}}
\def\cK{{\cal K}}
\def\cL{{\cal L}}
\def\cM{{\cal M}}
\def\cN{{\cal N}}
\def\cO{{\cal O}}
\def\cP{{\cal P}}
\def\cQ{{\cal Q}}
\def\cR{{\cal R}}
\def\cS{{\cal S}}
\def\cT{{\cal T}}
\def\cU{{\cal U}}
\def\cV{{\cal V}}
\def\cW{{\cal W}}
\def\cX{{\cal X}}
\def\cY{{\cal Y}}
\def\cZ{{\cal Z}}

%
%Hollow
\def\hA{\mathbb{A}}
\def\hB{\mathbb{B}}
\def\hC{\mathbb{C}}
\def\hD{\mathbb{D}}
\def\hE{\mathbb{E}}
\def\hF{\mathbb{F}}
\def\hG{\mathbb{G}}
\def\hH{\mathbb{H}}
\def\hI{\mathbb{I}}
\def\hJ{\mathbb{J}}
\def\hK{\mathbb{K}}
\def\hL{\mathbb{L}}
\def\hM{\mathbb{M}}
\def\hN{\mathbb{N}}
\def\hO{\mathbb{O}}
\def\hP{\mathbb{P}}
\def\hQ{\mathbb{Q}}
\def\hR{\mathbb{R}}
\def\hS{\mathbb{S}}
\def\hT{\mathbb{T}}
\def\hU{\mathbb{U}}
\def\hV{\mathbb{V}}
\def\hW{\mathbb{W}}
\def\hX{\mathbb{X}}
\def\hY{\mathbb{Y}}
\def\hZ{\mathbb{Z}}

%\mathscr
\def\sA{\mathscr{A}}
\def\sB{\mathscr{B}}
\def\sC{\mathscr{C}}
\def\sD{\mathscr{D}}
\def\sE{\mathscr{E}}
\def\scF{\mathscr{F}}
\def\sG{\mathscr{G}}
\def\sH{\mathscr{H}}
\def\sI{\mathscr{I}}
\def\sJ{\mathscr{J}}
\def\sK{\mathscr{K}}
\def\scL{\mathscr{L}}
\def\sM{\mathscr{M}}
\def\sN{\mathscr{N}}
\def\sO{\mathscr{O}}
\def\sP{\mathscr{P}}
\def\sQ{\mathscr{Q}}
\def\sR{\mathscr{R}}
\def\sS{\mathscr{S}}
\def\sT{\mathscr{T}}
\def\sU{\mathscr{U}}
\def\sV{\mathscr{V}}
\def\sW{\mathscr{W}}
\def\sX{\mathscr{X}}
\def\sY{\mathscr{Y}}
\def\sZ{\mathscr{Z}}
%
\def\no{\noindent}
\def\eq{\eqalign}
\def\ss{\smallskip}
\def\ms{\medskip}
\def\bs{\bigskip}
\def\q{\quad}
\def\qq{\qquad}
\def\hb{\hbox}
\def\pa{\partial}
\def\cd{\cdot}
\def\cds{\cdots}
\def\lan{{\langle}}
\def\ran{{\rangle}}
\def\td{\nabla}
\def\bD{{\bf D}}
\def\bF{{\bf F}}
\def\bG{{\bf G}}
\def\bx{{\bf x}}
\def\by{{\bf y}}
\def\bz{{\bf z}}

\def\tr{\hbox{\rm tr}}



\def\KL{\text{KL}}

% \def\qed{ \hfill \vrule width.25cm height.25cm depth0cm\smallskip}
% \def\qed{ \hfill \vrule width.25cm height.25cm depth0cm\smallskip}
% \renewcommand{\qed}{\hfill\square}
\def\qed{ \hfill\qedsymbol}

\newcommand{\dfnn}{\stackrel{\triangle}{=}}
\newcommand{\basa}{\begin{assumption}}
\newcommand{\easa}{\end{assumption}}
\newcommand{\tbar}{\overline{t}}
\newcommand{\xbar}{\overline{x}}
\newcommand{\bas}{\begin{assum}}
\newcommand{\eas}{\end{assum}}
\newcommand{\lime}{\lim_{\epsilon \rightarrow 0}}
\newcommand{\zep}{z^\epsilon}
\newcommand{\bep}{b^\epsilon}
\newcommand{\hbep}{\hat{b}^\epsilon}
\newcommand{\half}{\frac{1}{2}}


\def\dw{\mathbf{w}}
\def\dd{\mathrm{d}}
\def\limsup{\mathop{\overline{\rm lim}}}
\def\liminf{\mathop{\underline{\rm lim}}}
\def\ua{\mathop{\uparrow}}
\def\da{\mathop{\downarrow}}
\def\Ra{\mathop{\Rightarrow}}
\def\La{\mathop{\Leftarrow}}
\def\lan{\mathop{\langle}}
\def\ran{\mathop{\rangle}}
\def\embed{\mathop{\hookrightarrow}}
\def\esssup{\mathop{\rm esssup}}
\def\essinf{\mathop{\rm essinf}}
\def\limw{\mathop{\buildrel w\over\rightharpoonup}}
\def\limws{\mathop{\buildrel *\over\rightharpoonup}}
\def\lims{\mathop{\buildrel s\over\rightarrow}} 
\def\limP2{\,\mathop{\buildrel \Pi_2\over\longrightarrow\,}}
\def\lq{\leqno}
\def\rq{\eqno}
\def\pa{\partial}
\def\h{\widehat}
\def\wt{\widetilde}
\def\vr{\vrule width 1.7 pt height 6.8 pt depth 2.5pt} \def\cd{\cdot}
\def\cds{\cdots}
\def\ae{\hbox{\rm -a.e.{ }}}
\def\as{\hbox{\rm -a.s.{ }}}
\def\sgn{\hbox{\rm sgn$\,$}}
\def\meas{\hbox{\rm meas$\,$}}
\def\supp{\hbox{\rm supp$\,$}}
%\def\span{\hbox{\rm span$\,$}}
\def\co{\mathop{{\rm co}}}
\def\coh{\mathop{\overline{\rm co}}}
\def\cl{\overline}
\def\codim{\hbox{\rm codim$\,$}}
\def\Int{\hbox{\rm Int$\,$}}
\def\diam{\hbox{\rm diam$\,$}}
\def\deq{\mathop{\buildrel\D\over=}}
\def\tr{\hbox{\rm tr$\,$}}
\def\deq{\mathop{\buildrel\D\over=}}
\def\Re{\hbox{\rm Re$\,$}}
\def\bnm{{\,|\neg\neg|\neg\neg|\neg\neg|\,}}

\def\ind{{\perp\neg\neg\neg\perp}}

\def\dis{\displaystyle}
\def\wt{\widetilde}
\def\dh{\dot{h}}
\def\dF{\dot{F}}
\def\bF{{\bf F}}
\def\cad{c\`{a}dl\`{a}g}
\def\cag{c\`{a}gl\`{a}d~}
\def\bP{{\bf P}}
\def\1{{\bf 1}}
\def\by{{\bf y}}

\def\hSM{\widehat {\cS\!\cM}^2}

\def\:{\!:\!}
\def\reff#1{{\rm(\ref{#1})}}
\def \proof{{\noindent \bf Proof\quad}}
\def \rawproof{{\noindent \bf Sketch of proof\quad}}

\def\hE{\mathbb{E}}
\def\cE{{\cal E}}


\def\hE{\mathbb{E}}
\def\cE{{\cal E}}


\def\pop{\bigotimes}
\def\hE{\mathbb{E}}
\def\cE{{\cal E}}


\newtheorem{assump}{Assumption}
\newtheorem{assumpB}{Assumption}
\renewcommand\theassump{A\arabic{assump}}
\renewcommand\theassumpB{B\arabic{assumpB}}

\def\hbt#1{\textcolor{blue}{[#1]}}

\usepackage{algorithm}
\usepackage{algorithmic}

\newtheorem{remark}{Remark}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{assumption}{Assumption}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% End of Wei's macro command %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%