\usepackage{amsfonts}       % blackboard math symbols
\usepackage{nicefrac}       % compact symbols for 1/2, etc.
\usepackage{microtype}      % microtypography
\usepackage{xcolor}         % colors
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{physics}
\usepackage{graphicx}
\usepackage{bbm}
\usepackage{amsthm}

\DeclareMathOperator{\f}{\mathbf{f}}

\DeclareMathOperator{\bb}{\boldsymbol{a}}
\DeclareMathOperator{\ba}{\boldsymbol{b}}
\DeclareMathOperator{\bbq}{\boldsymbol{q}}
\DeclareMathOperator{\bbp}{\boldsymbol{p}}
\DeclareMathOperator{\bm}{\boldsymbol{m}}
\DeclareMathOperator{\bbv}{\boldsymbol{v}}
\DeclareMathOperator{\bx}{\boldsymbol{x}}
\DeclareMathOperator{\by}{\boldsymbol{y}}
\DeclareMathOperator{\bz}{\boldsymbol{z}}

\DeclareMathOperator{\dx}{\textrm{d}\boldsymbol{x}}
\DeclareMathOperator{\bepsilon}{\boldsymbol{\epsilon}}
\DeclareMathOperator{\brho}{\boldsymbol{\rho}}
\DeclareMathOperator{\btheta}{\boldsymbol{\theta}}
\DeclareMathOperator{\bmu}{\boldsymbol{\mu}}

\DeclareMathOperator{\dnu}{\textrm{d}\boldsymbol{\nu}}
\DeclareMathOperator{\dbx}{\textrm{d}\boldsymbol{x}}


\DeclareMathOperator{\bpsi}{\boldsymbol{\psi}}



\DeclareMathOperator{\bG}{\boldsymbol{G}}
\DeclareMathOperator{\bI}{\boldsymbol{I}}
\DeclareMathOperator{\bJ}{\boldsymbol{J}}
\DeclareMathOperator{\bL}{\boldsymbol{L}}
\DeclareMathOperator{\bM}{\boldsymbol{M}}
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\DeclareMathOperator{\bQ}{\boldsymbol{Q}}
\DeclareMathOperator{\bW}{\boldsymbol{W}}
\DeclareMathOperator{\bX}{\boldsymbol{X}}
\DeclareMathOperator{\bY}{\boldsymbol{Y}}

\DeclareMathOperator{\bzero}{\boldsymbol{0}}
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\DeclareMathOperator{\cN}{\mathcal{N}}
\DeclareMathOperator{\cM}{\mathcal{M}}

\DeclareMathOperator{\Ex}{\mathbb{E}}
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\DeclareMathOperator{\bMd}{\boldsymbol{M}_{\partial}}

\newcommand{\inp}[2]{\left\langle #1, #2 \right\rangle}

% Proofs
% \theoremstyle{remark}
\newtheorem{thm}{Theorem}
\newtheorem{proposition}{Proposition}
\newtheorem{observation}{Observation}
\newtheorem{assumption}{Assumption}


% Create macros for abbreviations. 
\newcommand{\ournamefull}{Metric-agnostic Geodesic Slice Sampler}
\newcommand{\ourmethod}{MAGSS}
\newcommand{\metaourmethod}{Meta-MAGSS}
\newcommand{\propfirst}{
\begin{proposition}
\emph{
     An MCMC sampler targeting the Hausdorff density on a Riemannian manifold $\mathcal{M}$ with metric tensor $\bG(\bx)$ also targets the correct distribution on the Euclidean space.
     }
\end{proposition}
}

% \newcommand{\propsecond}{
% \begin{observation}
% \emph{
%     Let $p(\bx)$ be a smooth density function. Let $(\bx_t, \bbv_t)$ be the geodesic flow with initial conditions $(\bx_0,\bbv_0)$ such that $p(x_0)>0$ with respect to the Inverse Generative metric. Then, for $t$ such that $p(\bx_t) \to 0$ we have $\norm{\bbv_t}_{2}>\norm{\bbv_t}_{0}$.
%     } \label{obs:invgen}
% \end{observation}
% }

% \newcommand{\propthird}{
% \begin{observation}
% \emph{
%     Let $p(\bx)$ be a smooth density function. Let $(\bx_t, \bbv_t)$ be the geodesic flow with initial conditions $(\bx_0,\bbv_0)$  with respect to the Inverse Monge metric, such that $\bx_0$ is a local maximum.
%     Then $\norm{\bbv_t}_{2}\geq \norm{\bbv_0}_{2}$ 
%     for all $t\neq 0$.
%     } \label{prop:obsmonge}
% \end{observation}
% }


