TY - JOUR

T1 - A note on convergence of the equi-energy sampler

AU - Andrieu, Christophe

AU - Jasra, Ajay

AU - Doucet, Arnaud

AU - Del Moral, Pierre

PY - 2008/3

Y1 - 2008/3

N2 - In a recent article, 'The equi-energy sampler with applications statistical inference and statistical mechanics' [Ann. Stat., 34 (2006) 1581-1619], Kou, Zhou, and Wong presented a new stochastic simulation method called the equi-energy (EE) sampler. This technique is designed to simulate from a probability measure , 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 article, by Atchad and Liu [3] 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. [2] for 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.

AB - In a recent article, 'The equi-energy sampler with applications statistical inference and statistical mechanics' [Ann. Stat., 34 (2006) 1581-1619], Kou, Zhou, and Wong presented a new stochastic simulation method called the equi-energy (EE) sampler. This technique is designed to simulate from a probability measure , 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 article, by Atchad and Liu [3] 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. [2] for 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.

KW - Equi-energy sampler

KW - Non linear Markov chain Monte Carlo

KW - Poisson equation

KW - Uniform ergodicity

UR - http://www.scopus.com/inward/record.url?scp=40549112674&partnerID=8YFLogxK

U2 - 10.1080/07362990701857178

DO - 10.1080/07362990701857178

M3 - Article (Academic Journal)

AN - SCOPUS:40549112674

VL - 26

SP - 298

EP - 312

JO - Stochastic Analysis and Applications

JF - Stochastic Analysis and Applications

SN - 0736-2994

IS - 2

ER -