The Annals of Applied Probability

On the ergodicity properties of some adaptive MCMC algorithms

Christophe Andrieu and Éric Moulines

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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.

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Ann. Appl. Probab., Volume 16, Number 3 (2006), 1462-1505.

First available in Project Euclid: 2 October 2006

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Zentralblatt MATH identifier

Primary: 65C05: Monte Carlo methods 65C40: Computational Markov chains 60J27: Continuous-time Markov processes on discrete state spaces 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 93E35: Stochastic learning and adaptive control

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


Andrieu, Christophe; Moulines, Éric. On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Probab. 16 (2006), no. 3, 1462--1505. doi:10.1214/105051606000000286.

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