Open Access
April 2010 Interacting Markov chain Monte Carlo methods for solving nonlinear measure-valued equations
Pierre Del Moral, Arnaud Doucet
Ann. Appl. Probab. 20(2): 593-639 (April 2010). DOI: 10.1214/09-AAP628


We present a new class of interacting Markov chain Monte Carlo algorithms for solving numerically discrete-time measure-valued equations. The associated stochastic processes belong to the class of self-interacting Markov chains. In contrast to traditional Markov chains, their time evolutions depend on the occupation measure of their past values. This general methodology allows us to provide a natural way to sample from a sequence of target probability measures of increasing complexity. We develop an original theoretical analysis to analyze the behavior of these iterative algorithms which relies on measure-valued processes and semigroup techniques. We establish a variety of convergence results including exponential estimates and a uniform convergence theorem with respect to the number of target distributions. We also illustrate these algorithms in the context of Feynman–Kac distribution flows.


Download Citation

Pierre Del Moral. Arnaud Doucet. "Interacting Markov chain Monte Carlo methods for solving nonlinear measure-valued equations." Ann. Appl. Probab. 20 (2) 593 - 639, April 2010.


Published: April 2010
First available in Project Euclid: 9 March 2010

zbMATH: 1198.65024
MathSciNet: MR2650043
Digital Object Identifier: 10.1214/09-AAP628

Primary: 47H20 , 60G35 , 60J85 , 62G09
Secondary: 47D08 , 47G10 , 62L20

Keywords: Feynman–Kac formulae , Markov chain Monte Carlo methods , Metropolis–Hastings algorithm , self-interacting processes , Sequential Monte Carlo methods , time-inhomogeneous Markov chains

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.20 • No. 2 • April 2010
Back to Top