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On the eciency of adaptive MCMC algorithms Christophe Andrieu1 and Yves F. Atchad2

Summary: On the eciency of adaptive MCMC algorithms
Christophe Andrieu1 and Yves F. Atchadé2
(October 2005)
We study a class of adaptive Markov Chain Monte Carlo (MCMC) processes which aim at
behaving as an optimal target process via a learning procedure. We show, under appro-
priate conditions, that the adaptive process and the optimal (nonadaptive) MCMC pro-
cess share identical asymptotic properties. The special case of adaptive MCMC algorithms
governed by stochastic approximation is considered in details and we apply our results to
the adaptive Metropolis algorithm of Haario et al. (2001). We also propose a new class of
adaptive MCMC algorithms, called quasi-perfect adaptive MCMC which possesses appealing
theoretical and practical properties, as demonstrated through numerical simulations.
Key words: Adaptive Markov chains, Coupling, Markov Chain Monte Carlo, Metropolis
Algorithm, Stochastic Approximation, Rate of convergence.
MSC Numbers: 60J35, 60J22, 65C40
1 Introduction
Markov chain Monte Carlo (MCMC) is a popular computational method for generating samples
from virtually any distribution dened on a space X. These samples are often used to eciently
compute expectations with respect to by invoking some form of the law of large numbers. The
method consists of simulating an ergodic Markov chain {Xn, n 0} on X with transition probability


Source: Atchadé, Yves F. - Department of Statistics, University of Michigan


Collections: Mathematics