On the Ergodicity Properties of Some Adaptive MCMC Algorithms

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 conver...

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Bibliographic Details
Published in:The Annals of applied probability 2006-08, Vol.16 (3), p.1462-1505
Main Authors: Andrieu, Christophe, Moulines, Éric
Format: Article
Language:English
Subjects:
60J27
60J35
65C05
65C40
93E35
Adaptive Markov chain Monte Carlo
Approximation
Ergodic theory
Integers
Logical proofs
Markov chains
martingale
Martingales
Metropolis–Hastings algorithm
Metropolitan areas
Perceptron convergence procedure
Poisson equation
Poisson method
randomly varying truncation
self-tuning algorithm
state-dependent noise
stochastic approximation
Transition probabilities
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Summary: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.
ISSN:1050-5164
2168-8737
DOI:10.1214/105051606000000286