Open Access
August 2006 On the ergodicity properties of some adaptive MCMC algorithms
Christophe Andrieu, Éric Moulines
Ann. Appl. Probab. 16(3): 1462-1505 (August 2006). DOI: 10.1214/105051606000000286


In this paper we study the ergodicity properties of some adaptive Markov chain Monte Carlo algorithms (MCMC) that have been recently proposed in the literature. We prove that under a set of verifiable conditions, ergodic averages calculated from the output of a so-called adaptive MCMC sampler converge to the required value and can even, under more stringent assumptions, satisfy a central limit theorem. We prove that the conditions required are satisfied for the independent Metropolis–Hastings algorithm and the random walk Metropolis algorithm with symmetric increments. Finally, we propose an application of these results to the case where the proposal distribution of the Metropolis–Hastings update is a mixture of distributions from a curved exponential family.


Download Citation

Christophe Andrieu. Éric Moulines. "On the ergodicity properties of some adaptive MCMC algorithms." Ann. Appl. Probab. 16 (3) 1462 - 1505, August 2006.


Published: August 2006
First available in Project Euclid: 2 October 2006

zbMATH: 1114.65001
MathSciNet: MR2260070
Digital Object Identifier: 10.1214/105051606000000286

Primary: 60J27 , 60J35 , 65C05 , 65C40
Secondary: 93E35

Keywords: Adaptive Markov chain Monte Carlo , martingale , Metropolis–Hastings algorithm , Poisson method , randomly varying truncation , self-tuning algorithm , state-dependent noise , stochastic approximation

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.16 • No. 3 • August 2006
Back to Top