Exact Convergence Rate Analysis of the Independent Metropolis-Hastings Algorithms

TL;DR

This paper derives the exact convergence rates for Independent Metropolis-Hastings algorithms on both general and discrete state spaces, revealing that different initializations share the same convergence speed.

Contribution

It provides the first precise convergence rate analysis for IMH algorithms and shows initializations do not affect the convergence speed.

Findings

Exact convergence rates for IMH on general state spaces

Initializations do not impact convergence speed

Sharp bounds on convergence rates for IMH algorithms

Abstract

A well-known difficult problem regarding Metropolis-Hastings algorithms is to get sharp bounds on their convergence rates. Moreover, a fundamental but often overlooked problem in Markov chain theory is to study the convergence rates for different initializations. In this paper, we study the two issues mentioned above of the Independent Metropolis-Hastings (IMH) algorithms on both general and discrete state spaces. We derive the exact convergence rate and prove that the IMH algorithm's different deterministic initializations have the same convergence rate. We get the exact convergence speed for IMH algorithms on general state spaces.