Particle Markov chain Monte Carlo methods

Christophe Andrieu, Arnaud Doucet

Abstract:

Markov chain Monte Carlo and sequential Monte Carlo methods have emerged as
the two main tools to sample from high dimensional probability distributions.Although asymptotic
convergence of Markov chain Monte Carlo 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 indepen-
dently. We show here how it is possible to build efficient high dimensional proposal distributions
by using sequential Monte Carlo methods. This allows us not only to improve over standard
Markov chain Monte Carlo 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 Lévy-driven stochastic volatility model.