On the Convergence Rates of Some Adaptive Markov Chain Monte Carlo Algorithms

TL;DR

This paper analyzes the convergence rates of adaptive MCMC algorithms, showing IRMCMC converges at a rate of O(n^{-1}) under certain conditions, while the Equi-Energy sampler converges at O(n^{-1/2}).

Contribution

It provides theoretical convergence rate bounds for IRMCMC and the Equi-Energy sampler, highlighting their efficiency and limitations.

Findings

IRMCMC converges at O(n^{-1}) under regularity conditions.

The Equi-Energy sampler converges at O(n^{-1/2}).

IRMCMC does not generally converge faster than O(n^{-1}).

Abstract

This paper studies the mixing time of certain adaptive Markov Chain Monte Carlo algorithms. Under some regularity conditions, we show that the convergence rate of Importance Resampling MCMC (IRMCMC) algorithm, measured in terms of the total variation distance is $O(n^{-1})$, and by means of an example, we establish that in general, this algorithm does not converge at a faster rate. We also study the Equi-Energy sampler and establish that its mixing time is of order $O(n^{-1/2})$.