\newtheorem{theorem}{Theorem}
%\newtheorem{prop}{Proposition}
%\newtheorem{lemma}{Lemma}
%\newtheorem{fact}{Fact}
%\newtheorem{example}{Example}
\newtheorem{corollary}{Corollary}
\newtheorem{remark}{Remark}

\newcommand{\noise}{\text{noise}}
\newcommand{\sol}{\text{sol}}

\newcommand{\bp}{\mathbf{p}}
\newcommand{\bq}{\mathbf{q}}
\newcommand{\bM}{\mathbf{M}}
\newcommand{\bT}{\mathbf{T}}
\newcommand{\bC}{\mathbf{C}}
\newcommand{\bB}{\mathbf{B}}
\newcommand{\bA}{\mathbf{A}}
\newcommand{\bv}{\mathbf{v}}
\newcommand{\bD}{\mathbf{D}}
\newcommand{\bQ}{\mathbf{Q}}
% --------------------
\def\bx{{\bm x}}
\def\by{{\bf y}}
\def\bw{{\bf w}}
\def\bv{{\bf v}}
%\def\bm{{\bf m}}
\def\bn{{\bf n}}
\def\bu{{\bf u}}
\def\be{{\bf e}}
\def\bc{{\bf c}}
\def\bb{{\bf b}}
\def\bss{{\bf s}}

\newcommand{\mcS}{\mathcal{S}}

\newcommand{\nn}{\nonumber}
\newcommand{\1}{\mathbbm{1}^{n\times n}}


\newcommand{\la}{\lambda}
\newcommand{\eps}{\epsilon}
%\newcommand{\ed}{ {\bf{ED}}}
\newcommand{\ed}{ {\bf{D}}}
\newcommand{\ted}{ {\tilde{\bf{D}}}}
\newcommand{\ssucc}{ {\textrm{succ}}}
\newcommand{\fail}{ {\textrm{fail}}}
\newcommand{\thresh}{ {\textrm{thresh}}}

%\newcommand{\bern}{\text{Bern}_0}
\newcommand{\lafail}{\mathbf{\Lambda}_{\fail}}
\newcommand{\lasuc}{\mathbf{\Lambda}_{\ssucc}}
\newcommand{\tlasuc}{\tilde{\mathbf{\Lambda}}_{\ssucc}}
\newcommand{\order}[1]{\mathcal{O}({#1})}
\newcommand{\cdist}{{\bf{D}}_\Aa}
\newcommand{\tcdist}{{\tilde{\bf{D}}}_\Aa}


\newcommand{\su}{{\text{sum}}}
\newcommand{\ri}{\right>}
\newcommand{\li}{\left<}
\newcommand{\rt}{\right}
\newcommand{\lt}{\left}
\newcommand{\ttr}{\text{trace}}
\newcommand{\one}{\mathbbm{1}}

\newcommand{\U}{{\bf{U}}}
\newcommand{\PPP}{{\bf{P}}}


\newcommand{\G}{\mathcal{G}}

\newcommand{\Rr}{\mathcal{R}}
\newcommand{\Aa}{\mathcal{A}}
\newcommand{\MM}{\mathcal{M}}
\newcommand{\mE}{\mathcal{E}}

\newcommand{\E}{\mathbb{E}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\M}{\mathbf{M}}


\newcommand{\aaa}{{\bf a}}
\newcommand{\bbb}{{\bf b}}

\newcommand{\bX}{{\bar{X}}}

\newcommand{\X}{{\bf X}}
\newcommand{\Y}{{\bf Y}}


\newcommand{\Ao}{{\bf A_{obs}}}

\newcommand{\Lb}{{\bf L}}
\newcommand{\Lo}{{\bf L_{obs}}}

\newcommand{\Sb}{{\bf S}}
\newcommand{\So}{{\bf S_{obs}}}
\newcommand{\Sr}{{\bf S_{rest}}}

\newcommand{\Gout}{{\Gamma^{out}}}

\newcommand{\Ob}{\mathscr{O}}

\newcommand{\gsu}{\gamma_{\ssucc}}
\newcommand{\gfa}{\gamma_{\fail}}


\newcommand{\Eb}{{\bf E}}
\newcommand{\Eo}{{\bf E_{obs}}}
\newcommand{\Er}{{\bf E_{rest}}}
 \newcommand{\Fb}{{\bf F}}
\newcommand{\Ab}{{\bf A}}
\newcommand{\Wb}{{\bf W}}
\newcommand{\Qb}{{\bf Q}}
\newcommand{\Ib}{{\bf I}}
\newcommand{\Gb}{{\bf G}}
\newcommand{\Kb}{{\bf K}}
\newcommand{\Mb}{{\bf M}}
\newcommand{\mb}{{\bf m}}
\newcommand{\Nb}{{\bf N}}
\newcommand{\nb}{{\bf n}}
\newcommand{\pb}{{\bf p}}
\newcommand{\Pb}{{\bf P}}
\newcommand{\Rb}{{\bf R}}
\newcommand{\Bb}{{\bf B}}
\newcommand{\rb}{{\bf r}}
\newcommand{\ub}{{\bf u}}
\newcommand{\Ub}{{\bf U}}
\newcommand{\Tb}{{\bf T}}
\newcommand{\vb}{{\bf v}}
\newcommand{\Vb}{{\bf V}}
\newcommand{\Xb}{{\bf X}}
\newcommand{\xb}{{\bm x}}
\newcommand{\yb}{{\bf y}}
\newcommand{\Yb}{{\bf Y}}
\newcommand{\zb}{{\bf z}}
\newcommand{\Zb}{{\bf Z}}

\newcommand{\Pbr}{{\Pb\rb}}
