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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

Abstract

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.

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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. https://doi.org/10.1214/09-AAP628

Information

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

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

Subjects:
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

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Vol.20 • No. 2 • April 2010
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