On the asymptotic variance of reversible Markov chain without cycles

TL;DR

This paper proves that for reversible Markov chains without cycles, the asymptotic variance cannot be improved, confirming a conjecture that acyclic structures minimize commute times and optimize performance.

Contribution

It provides a rigorous proof that acyclic reversible Markov chains have minimal asymptotic variance, confirming a conjecture about their optimality in MCMC performance.

Findings

Acyclic reversible Markov chains produce minimum commute times.

The proof confirms that no improvements are possible without cycles.

Acyclic structure leads to optimal asymptotic variance in MCMC.

Abstract

Markov chain Monte Carlo(MCMC) is a popular approach to sample from high dimensional distributions, and the asymptotic variance is a commonly used criterion to evaluate the performance. While most popular MCMC algorithms are reversible, there is a growing literature on the development and analyses of nonreversible MCMC. Chen and Hwang(2013) showed that a reversible MCMC can be improved by adding an antisymmetric perturbation. They also raised a conjecture that it can not be improved if there is no cycle in the corresponding graph. In this paper, we present a rigorous proof of this conjecture. The proof is based on the fact that the transition matrix with an acyclic structure will produce minimum commute time between vertices.