Convergence Rates of Attractive-Repulsive MCMC Algorithms

TL;DR

This paper analyzes the convergence rates of particle system MCMC algorithms with attractive and repulsive forces, providing explicit bounds for both bounded and unbounded state spaces.

Contribution

It establishes uniform and geometric ergodicity for these algorithms and derives explicit quantitative convergence rates using shift-coupling.

Findings

Uniform ergodicity with explicit rate on bounded state space

Geometric ergodicity with explicit rate on unbounded state space

Use of shift-coupling method for convergence bounds

Abstract

We consider MCMC algorithms for certain particle systems which include both attractive and repulsive forces, making their convergence analysis challenging. We prove that a version of these algorithms on a bounded state space is uniformly ergodic with an explicit quantitative convergence rate. We also prove that a version on an unbounded state-space is still geometrically ergodic, and then use the method of shift-coupling to obtain an explicit quantitative bound on its convergence rate.