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Particle Markov chain Monte Carlo methods

by Christophe Andrieu, Arnaud Doucet and Roman Holenstein

Markov chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC) methods have emerged as the two main tools to sample from high dimensional probability distributions. Although asymptotic convergence of MCMC algorithms is ensured under weak assumptions, the performance of these algorithms is unreliable when the proposal distributions that are used to explore the space are poorly chosen and/or if highly correlated variables are updated independently. We show how it is possible to build efficient high dimensional proposal distributions by using SMC methods. This allows us not only to improve over standard MCMC schemes but also to make Bayesian inference feasible for a large class of statistical models where this was not previously so. We demonstrate these algorithms on a non-linear state space model and a LÚvy-driven stochastic volatility model.

Key words: Markov chain Monte Carlo; Sequential Monte Carlo methods; State-space modes; Bayesian inference; Dirichlet process mixtures

Full text of the paper (pdf), to appear as a Read Paper in the Journal of the Royal Statistical Society, B.