A Mean field theory of nonlinear filtering

Pierre del Moral (1, 2) , Frédéric Patras (3) , Sylvain Rubenthaler (3)

1 ALEA - Advanced Learning Evolutionary Algorithms
2 IMB - Institut de Mathématiques de Bordeaux
3 JAD - Laboratoire Jean Alexandre Dieudonné

We present a mean field particle theory for the numerical approximation of Feynman-Kac path integrals in the context of nonlinear filtering. We show that the conditional distribution of the signal paths given a series of noisy and partial observation data is approximated by the occupation measure of a genealogical tree model associated with mean field interacting particle model. The complete historical model converges to the McKean distribution of the paths of a nonlinear Markov chain dictated by the mean field interpretation model. We review the stability properties and the asymptotic analysis of these interacting processes, including fluctuation theorems and large deviation principles. We also present an original Laurent type and algebraic tree-based integral representations of particle block distributions. These sharp and non asymptotic propagations of chaos properties seem to be the first result of this type for mean field and interacting particle systems.

Mots clés: Feynman-Kac measures; nonlinear filtering; interacting particle systems; historical and genealogical tree models; central limit theorems; Gaussian fields; propagations of chaos; trees and forests; combinatorial enumeration

Domaines: Probabilités [math.PR]

Pierre del Moral, Frédéric Patras, Sylvain Rubenthaler. A Mean field theory of nonlinear filtering. Dan Crisan, Boris L. Rozovskii. The Oxford Handbook of Nonlinear Filtering, Oxford University Press, pp.705-740, 2011, Oxford Handbooks in Mathematics, 978-0-19-953290-2. ⟨inria-00537331⟩