Issue ESAIM: PROC Volume 19, 2007 Conference Oxford sur les méthodes de Monte Carlo séquentielles Page(s) 22 - 31 DOI 10.1051/proc:071905 Published online 30 October 2007

ESAIM: Proc., 2007, Vol. 19, pp. 22-31
DOI: 10.1051/proc:071905

Stability of sequential Markov Chain Monte Carlo methods

Andreas Eberle and Carlo Marinelli

Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, 53115 Bonn, Germany

(Published online: 30 October 2007)

Abstract
Sequential Monte Carlo Samplers are a class of stochastic algorithms for Monte Carlo integral estimation w.r.t. probability distributions, which combine elements of Markov chain Monte Carlo methods and importance sampling/resampling schemes. We develop a stability analysis by funtional inequalities for a nonlinear flow of probability measures describing the limit behavior of the methods as the number of particles tends to infinity. Stability results are derived both under global and local assumptions on the generator of the underlying Metropolis dynamics. This allows us to prove that the combined methods sometimes have good asymptotic stability properties in multimodal setups where traditional MCMC methods mix extremely slowly. For example, this holds for the mean field Ising model at all temperatures.

Mathematics Subject Classification. 65C05, 60J25, 60B10, 47H20, 47D08

Key words: Markov Chain Monte Carlo, sequential Monte Carlo, importance sampling, spectral gap, Dirichlet forms, functional inequalities


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