The approximation of the Feynman-Kac semigroups by systems of interacting particles is a very active research field, with applications in many different areas. In this paper, we study the parallelization of such approximations. The total population of particles is divided into sub-populations, referred to as islands. The particles within each island follow the usual selection / mutation dynamics. We Pierre Del Moral Centre INRIA Bordeaux Sud Ouest -351 Cours de la Libération, 33405 Talence Cedex, 2 C. Vergé and al.show that the evolution of each island is also driven by a Feynman-Kac semigroup, whose transition and potential can be explicitly related to ones of the original problem. Therefore, the same genetic type approximation of the Feynman-Kac semi-group may be used at the island level; each island might undergo selection / mutation algorithm. We investigate the impact of the population size within each island and the number of islands, and study different type of interactions. We find conditions under which introducing interactions between islands is beneficial. The theoretical results are supported by some Monte Carlo experiments. Keywords Particle approximation of Feynman-Kac flow, Island models, parallel implementation 1 Introduction Numerical approximation of Feynman-Kac semigroups by systems of interacting particles is a very active field of researchs. Interacting particle systems are increasingly used to sample complex high dimensional distributions in a wide range of applications including nonlinear filtering, data assimilation problems, rare event sampling, hidden Markov chain parameter estimation, stochastic control problems, financial mathematics; see for example [8], [2], [4], [1], [6] and the references therein. Let (En, En) n≥0 be a sequence of measurable spaces. Denote by B b (En) the Banach space of all bounded and measurable real valued functions f on En, equipped with the uniform norm. Let (gn) n∈N be a sequence of measurable potential functions, gn : En → R + . Let (Ω, F, P) be a probability space. In the sequel, all the processes are defined on this probability space. Let (Xn) n∈N be a non-homogenous i=1which are generated recursively. Typically, the update of the particles may be decomposed into a mutation and a selection step. For example, the bootstrap algorithm proceeds as follows. In the selection step the particles are first sampled with weights proportional to the potential functions. In the mutation step, a new generation of particles (X i n+1 ) N1 i=1 is generated from the selected particles using the kernel M n+1 . The asymptotic behavior of such particle approximation is now well understood (see [4] and [6]). Feynman-Kac measures appear naturally in the filtering problem for HiddenMarkov Model (HMM). Recall that a HMM is a pair of discrete time randomC. Vergé and al.processes (X, Y ) = (Xn, Yn) n∈N , where (Xn) n≥0 is the hidden state process (often called signal) and (Yn) n≥0 are the observations. To fix the ideas, Xn and Yn take values in X ⊂ R k and Y ⊂ R l . The state sequen...

The approximation of fixed-interval smoothing distributions is a key issue in inference for general state-space hidden Markov models (HMM). This contribution establishes non-asymptotic bounds for the Forward Filtering Backward Smoothing (FFBS) and the Forward Filtering Backward Simulation (FFBSi) estimators of fixed-interval smoothing functionals. We show that the rate of convergence of the Lq-mean errors of both methods depends on the number of observations T and the number of particles N only through the ratio T /N for additive functionals. In the case of the FFBS, this improves recent results providing bounds depending on T / √ N .

International audienceParticle smoothers are widely used algorithms allowing to approximate the smoothing distribution in hidden Markov models. Existing algorithms often suffer from slow computational time or degeneracy. We propose in this paper a way to improve any of them with a linear complexity in the number of particles. When iteratively applied to the degenerated Filter-Smoother, this method leads to an algorithm which turns out to outperform all other linear particle smoothers for a fixed computational tim