On hitting time, mixing time and geometric interpretations of Metropolis-Hastings reversiblizations

TL;DR

This paper introduces a new Metropolis-Hastings generator $M_2$ that, along with the classical $M_1$, offers improved mixing properties and a geometric interpretation as $\, ext{l}^1$-minimizers between the proposal generator and reversible generators.

Contribution

The paper defines a novel MH generator $M_2$, compares it with $M_1$ across multiple metrics, and provides a geometric framework extending previous results.

Findings

$M_2$ has superior mixing properties than $M_1$.

Explicit spectral analysis for Metropolised independent sampling.

Laplace transform order of the fastest strong stationary time between $M_1$ and $M_2$.

Abstract

Given a target distribution $\mu$ and a proposal chain with generator $Q$ on a finite state space, in this paper we study two types of Metropolis-Hastings (MH) generator $M_1(Q,\mu)$ and $M_2(Q,\mu)$ in a continuous-time setting. While $M_1$ is the classical MH generator, we define a new generator $M_2$ that captures the opposite movement of $M_1$ and provide a comprehensive suite of comparison results ranging from hitting time and mixing time to asymptotic variance, large deviations and capacity, which demonstrate that $M_2$ enjoys superior mixing properties than $M_1$. To see that $M_1$ and $M_2$ are natural transformations, we offer an interesting geometric interpretation of $M_1$, $M_2$ and their convex combinations as $\ell^1$ minimizers between $Q$ and the set of $\mu$-reversible generators, extending the results by Billera and Diaconis (2001). We provide two examples as illustrations. In the first one we give explicit spectral analysis of $M_1$ and $M_2$ for Metropolised independent sampling, while in the second example we prove a Laplace transform order of the fastest strong stationary time between birth-death $M_1$ and $M_2$.