Abstract : This article analyses a new class of advanced particle Markov chain Monte Carlo
algorithms recently introduced by Andrieu, Doucet, and Holenstein (2010). We present
a natural interpretation of these methods in terms of well known unbiasedness properties
of Feynman-Kac particle measures, and a new duality with Feynman-Kac models.
This perspective sheds new light on the foundations and the mathematical analysis
of this class of methods. A key consequence is their equivalence with the Gibbs sampling
of a (many-body) Feynman-Kac target distribution. Our approach also presents a new
stochastic differential calculus based on geometric combinatorial techniques to derive
non-asymptotic Taylor type series for the semigroup of a class of particle Markov chain
Monte Carlo models around their invariant measures with respect to the population size
of the auxiliary particle sampler.
These results provide sharp quantitative estimates of the convergence rate of the
models with respect to the time horizon and the size of the systems. We illustrate the
direct implication of these results with sharp estimates of the contraction coefficient and
the Lyapunov exponent of the corresponding samplers, and explicit and non-asymptotic
L p -mean error decompositions of the law of the random states around the limiting
invariant measure. The abstract framework developed in the article also allows the
design of natural extensions to island (also called SMC 2 ) type particle methodologies.
We illustrate this general framework and results in the context of nonlinear filtering,
hidden Markov chain problems with fixed unknown parameters, and Feynman-Kac path-
integration models arising in computational physics and chemistry.