Fast Mixing of Metropolis-Hastings with Unimodal Targets

TL;DR

This paper provides a quantitative bound demonstrating that the Metropolis-Hastings algorithm mixes rapidly for unimodal targets with not-too-heavy tails, filling a gap in the theoretical literature.

Contribution

It offers a new, sharper bound on the mixing time of Metropolis-Hastings for unimodal distributions using the drift-and-minorization framework.

Findings

Provides a generic bound based on Lyapunov functions.

Uses path arguments for sharper bounds.

Fills a gap in the theoretical understanding of MCMC mixing times.

Abstract

A well-known folklore result in the MCMC community is that the Metropolis-Hastings algorithm mixes quickly for any unimodal target, as long as the tails are not too heavy. Although we've heard this fact stated many times in conversation, we are not aware of any quantitative statement of this result in the literature, and we are not aware of any quick derivation from well-known results. The present paper patches this small gap in the literature, providing a generic bound based on the popular "drift-and-minorization" framework of Rosenthal (1995). Our main contribution is to study two sublevel sets of the Lyapunov function and use path arguments in order to obtain a sharper general bound than what can typically be obtained from multistep minorization arguments.