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

Michael C. H. Choi; Lu-Jing Huang

doi:10.1007/s10959-019-00903-2

paper

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

arXiv:1810.11763 · doi:10.1007/s10959-019-00903-2

Abstract

Given a target distribution $μ$ and a proposal chain with generator $Q$ on a finite state space, in this paper we study two types of Metropolis-Hastings (MH) generator $M_{1} (Q, μ)$ and $M_{2} (Q, μ)$ 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 $ℓ^{1}$ minimizers between $Q$ and the set of $μ$ -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 $M_{2}$ .

17 pages. To appear in J. Theoret. Probab