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On the ergodicity properties of some adaptive MCMC algorithms

Christophe Andrieu·Éric Moulines
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Abstract

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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Christophe Andrieu, Éric Moulines. 2006-10-10. On the ergodicity properties of some adaptive MCMC algorithms. https://doi.org/10.1214/105051606000000286

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