Interacting Markov Chain Monte Carlo Methods For Solving Nonlinear Measure-Valued Equations

Pierre Del Moral 1, 2 Arnaud Doucet 3, 4
2 ALEA - Advanced Learning Evolutionary Algorithms
Inria Bordeaux - Sud-Ouest, UB - Université de Bordeaux, CNRS - Centre National de la Recherche Scientifique : UMR5251
Abstract : We present a new interacting Markov chain Monte Carlo methodology 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 evolution may depend on the occupation measure of the 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 behaviour of these algorithms as the time parameter tends to infinity. This analysis relies on measure-valued processes and semigroup techniques. We present a variety of convergence results including exponential estimates and a uniform convergence theorem with respect to the number of target distributions, yielding what seems to be the first results of this kind for this class of self-interacting models. We also illustrate these models in the context of Feynman-Kac distribution flows.
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The Annals of Applied Probability : an official journal of the institute of mathematical statistics, The Institute of Mathematical Statistics, 2010, 20 (2), pp.593-639
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Pierre Del Moral, Arnaud Doucet. Interacting Markov Chain Monte Carlo Methods For Solving Nonlinear Measure-Valued Equations. The Annals of Applied Probability : an official journal of the institute of mathematical statistics, The Institute of Mathematical Statistics, 2010, 20 (2), pp.593-639. 〈inria-00227508v4〉

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