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August 2011 On nonlinear Markov chain Monte Carlo
Christophe Andrieu, Ajay Jasra, Arnaud Doucet, Pierre Del Moral
Bernoulli 17(3): 987-1014 (August 2011). DOI: 10.3150/10-BEJ307

Abstract

Let $\mathscr{P}(E)$ be the space of probability measures on a measurable space $(E,\mathcal{E})$. In this paper we introduce a class of nonlinear Markov chain Monte Carlo (MCMC) methods for simulating from a probability measure $\pi\in\mathscr{P}(E)$. Nonlinear Markov kernels (see [Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications (2004) Springer]) $K:\mathscr{P}(E)\times E\rightarrow\mathscr{P}(E)$ can be constructed to, in some sense, improve over MCMC methods. However, such nonlinear kernels cannot be simulated exactly, so approximations of the nonlinear kernels are constructed using auxiliary or potentially self-interacting chains. Several nonlinear kernels are presented and it is demonstrated that, under some conditions, the associated approximations exhibit a strong law of large numbers; our proof technique is via the Poisson equation and Foster–Lyapunov conditions. We investigate the performance of our approximations with some simulations.

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Christophe Andrieu. Ajay Jasra. Arnaud Doucet. Pierre Del Moral. "On nonlinear Markov chain Monte Carlo." Bernoulli 17 (3) 987 - 1014, August 2011. https://doi.org/10.3150/10-BEJ307

Information

Published: August 2011
First available in Project Euclid: 7 July 2011

zbMATH: 1241.60037
MathSciNet: MR2817614
Digital Object Identifier: 10.3150/10-BEJ307

Keywords: Foster–Lyapunov condition , interacting Markov chains , nonlinear Markov kernels , Poisson equation

Rights: Copyright © 2011 Bernoulli Society for Mathematical Statistics and Probability

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Vol.17 • No. 3 • August 2011
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