A note on convergence of the equi-energy sampler

TL;DR

This paper provides a new convergence proof for the equi-energy sampler, a stochastic simulation method for sampling from complex probability measures, addressing previous incomplete proofs and focusing on the single feeding chain case.

Contribution

It introduces a novel convergence proof for the equi-energy sampler based on the Poisson equation, clarifying its correctness for the single feeding chain scenario.

Findings

New proof of convergence using Poisson equation

Highlights difficulties in analyzing equi-energy algorithms

Focuses on the single feeding chain case

Abstract

In a recent paper `The equi-energy sampler with applications statistical inference and statistical mechanics' [Ann. Stat. 34 (2006) 1581--1619], Kou, Zhou & Wong have presented a new stochastic simulation method called the equi-energy (EE) sampler. This technique is designed to simulate from a probability measure $\pi$, perhaps only known up to a normalizing constant. The authors demonstrate that the sampler performs well in quite challenging problems but their convergence results (Theorem 2) appear incomplete. This was pointed out, in the discussion of the paper, by Atchad\'e & Liu (2006) who proposed an alternative convergence proof. However, this alternative proof, whilst theoretically correct, does not correspond to the algorithm that is implemented. In this note we provide a new proof of convergence of the equi-energy sampler based on the Poisson equation and on the theory developed in Andrieu et al. (2007) for \emph{Non-Linear} Markov chain Monte Carlo (MCMC). The objective of this note is to provide a proof of correctness of the EE sampler when there is only one feeding chain; the general case requires a much more technical approach than is suitable for a short note. In addition, we also seek to highlight the difficulties associated with the analysis of this type of algorithm and present the main techniques that may be adopted to prove the convergence of it.