On nonlinear Markov chain Monte Carlo

Christophe Andrieu*, Ajay Jasra, Arnaud Doucet, Pierre Del Moral

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)

12 Citations (Scopus)

Abstract

Let P (E) be the space of probability measures on a measurable space (E, e). In this paper we introduce a class of nonlinear Markov chain Monte Carlo (MCMC) methods for simulating from a probability measure pi is an element of P(E). Nonlinear Markov kernels (see [Feyninan-Kac Formulae: Genealogical and Interacting Particle Systems with Applications (2004) Springer]) K : P(E) x E -> 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.

Original languageEnglish
Pages (from-to)987-1014
Number of pages28
JournalBernoulli
Volume17
Issue number3
DOIs
Publication statusPublished - Aug 2011

Keywords

  • Foster-Lyapunov condition
  • interacting Markov chains
  • nonlinear Markov kernels
  • Poisson equation
  • EQUI-ENERGY SAMPLER
  • METROPOLIS ALGORITHMS
  • U-STATISTICS
  • CONVERGENCE
  • ERGODICITY
  • INFERENCE

Projects

PERTURBED AND SELF INTERACTING

Andrieu, C.

1/10/051/10/10

Project: Research

Cite this

Andrieu, C., Jasra, A., Doucet, A., & Del Moral, P. (2011). On nonlinear Markov chain Monte Carlo. Bernoulli, 17(3), 987-1014. https://doi.org/10.3150/10-BEJ307