  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 many-body Feynman-Kac models. This perspective sheds a new light on the
foundations and the mathematical analysis of this class of methods. A key
consequence is the equivalence between the backward and ancestral particle
Markov chain Monte Carlo methods, and 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 explicit
non-asymptotic Taylor type series of 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 quan- titative estimates of the convergence properties of conditional
particle Markov chain models with respect to the time horizon and the size of
the systems. We illustrate the implication of these results with sharp
estimates of the contraction coefficient and the Lyapunov exponent of
conditional particle samplers, and explicit and non-asymptotic Lp-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 SMC2) type particle methodologies.
