Reversible Markov chains: variational representations and ordering

TL;DR

This paper discusses variational methods for analyzing reversible Markov chains, focusing on spectral gaps, asymptotic variance, and conductance, providing tools for comparing chain efficiencies.

Contribution

It introduces and explains three variational representations that facilitate the comparison of reversible Markov chains' efficiencies.

Findings

Variational representations of spectral gaps using Dirichlet forms

A variational formula for asymptotic variance of ergodic averages

Equivalence between conductance and spectral gap

Abstract

This pedagogical document explains three variational representations that are useful when comparing the efficiencies of reversible Markov chains: (i) the Dirichlet form and the associated variational representations of the spectral gaps; (ii) a variational representation of the asymptotic variance of an ergodic average; and (iii) the conductance, and the equivalence of a non-zero conductance to a non-zero right spectral gap.