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A Mean Field Theory of Nonlinear Filtering Pierre Del Moral, Frédéric Patras, Sylvain Rubenthaler

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A Mean Field Theory of Nonlinear Filtering Pierre Del Moral, Frédéric Patras, Sylvain Rubenthaler A Mean Field Theory of Nonlinear Filtering Pierre del Moral, Frédéric Patras, Sylvain Rubenthaler To cite this version: Pierre del Moral, Frédéric Patras, Sylvain Rubenthaler. A Mean Field Theory of Nonlinear Filtering. [Research Report] RR-6437, INRIA. 2008. inria-00238398v3 HAL Id: inria-00238398 https://hal.inria.fr/inria-00238398v3 Submitted on 5 Feb 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE A Mean Field Theory of Nonlinear Filtering Pierre Del Moral — Frédéric Patras — Sylvain Rubenthaler N° 6437 February 2008 Thème NUM apport de recherche ISSN 0249-6399 ISRN INRIA/RR--6437--FR+ENG A Mean Field Theory of Nonlinear Filtering Pierre Del Moral∗, Fr´ed´eric Patrasy , Sylvain Rubenthaler z Th`emeNUM | Syst`emesnum´eriques Equipes-Projets´ CQFD Rapport de recherche n° 6437 | February 2008 | 36 pages Abstract: We present a mean field particle theory for the numerical approx- imation 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 mea- sure 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 princi- ples. 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. Key-words: Feynman-Kac measures, nonlinear filtering, interacting particle systems, historical and genealogical tree models, central limit theorems, Gaus- sian fields, propagations of chaos, trees and forests, combinatorial enumeration. ∗ Centre INRIA Bordeaux et Sud-Ouest & Institut de Math´ematiquesde Bordeaux , Uni- versit´ede Bordeaux I, 351 cours de la Lib´eration33405 Talence cedex, France, Pierre.Del- [email protected]. y CNRS UMR 6621, Universit´ede Nice, Laboratoire de Math´ematiquesJ. Dieudonn´e,Parc Valrose, 06108 Nice Cedex 2, France. z CNRS UMR 6621, Universit´ede Nice, Laboratoire de Math´ematiquesJ. Dieudonn´e,Parc Valrose, 06108 Nice Cedex 2, France. Centre de recherche INRIA Bordeaux – Sud Ouest Domaine Universitaire - 351, cours de la Libération 33405 Talence Cedex Téléphone : +33 5 40 00 69 00 Une th´eoriechamp moyen du filtrage non lin´eaire R´esum´e: Nous exposons ici une th´eorieparticulaire de type champ moyen pour la r´esolutionnum´eriquedes int´egralesde chemins de Feynman utilis´eesen fil- trage non lin´eaire.Nous d´emontrons que les lois conditionnelles des trajectoires d'un signal bruit´eet partiellement observ´epeuvent ^etrecalcul´ees `apartir des mesures d'occupation d'arbres g´en´ealogiquesassoci´es`ades syst`emes de partic- ules en interaction. Le processus historique caract´erisant l'´evolution ancestrale compl`eteconverge vers la mesure de McKean des trajectoires d'une cha^ınede Markov non lin´eairedict´eepar l'interpr´etationchamp moyen du mod`elede fil- trage. Nous passons en revue les propri´et´esde stabilit´eet les r´esultatsd'analyse asymptotique de ces processus en interaction, avec notamment des th´eor`emes de fluctuations et des principes de grandes d´eviations. Nous exposons aussi des d´eveloppements faibles et non asymptotiques des distributions de blocs de particules en termes combinatoire de for^etset d'arbres de coalescences. Ces propri´et´esfines de propagations du chaos semblent ^etreles premiers r´esultats de ce type pour des syst`emesde particules en interaction de type champ moyen. Mots-cl´es: Mesures de Feynman-Kac, filtrage non lin´eaire,syst`emesde particules en interaction, processus historique et mod`elesd'arbres g´en´ealogiques, th´eor`emesde la limite centrale, champs gaussiens, propri´et´esde propagations du chaos, combinatoire d'arbres et de for^et. A Mean Field Theory of Nonlinear Filtering 3 1 Introduction 1.1 A mean field theory of nonlinear filtering The filtering problem consists in computing the conditional distributions of a state signal given a series of observations. The signal/observation pair sequence (Xn;Yn)n≥0 is defined as a Markov chain which takes values in some product of measurable spaces (En ×Fn)n≥0. We further assume that the initial distribution ν0 and the Markov transitions Pn of the pair process (Xn;Yn) have the form ν0(d(x0; y0)) = g0(x0; y0) η0(dx0) q0(dy0) Pn((xn−1; yn−1); d(xn; yn)) = Mn(xn−1; dxn) gn(xn; yn) qn(dyn) (1.1) where gn are strictly positive functions on (En × Fn) and qn is a sequence of measures on Fn. The initial distribution η0 of the signal Xn, the Markov transitions Mn and the likelihood functions gn are assumed to be known. The main advantage of this general and abstract set-up comes from the fact that it applies directly without further work to traditional real valued or multidimensional problems, as well as to smoothing and path estimation filtering models. For instance the signal 0 0 Xn = (X0;:::;Xn) may represents the path from the origin up to time n of an auxiliary Markov 0 0 0 chain Xn taking values in some measurable state space (En; En). A version of the conditional distributions ηn of the signal states Xn given their noisy observations Yp = yp up to time p < n is expressed in terms of a flow of Feynman-Kac measures associated with the distribution of the paths of the signal and weighted by the collection of likelihood potential functions. To better connect the mean field particle approach to existing alternatives methods, it is convenient at this point to make a couple of remarks. Firstly, using this functional representation it is tempting to approximate this measures using a crude Monte Carlo method based on independent simula- tions of the paths of the signal weighted by products of the likelihood functions from the origin up to time n. This strategy gives satisfactory results when the signal paths are sufficiently stable using a simple weight regularization to avoid their degeneracy with respect to the time parameter. We refer to [12, 14, 33] for further details. Precise comparisons of the fluctuation variances associated with this Monte Carlo method and the mean feld particle models presented in the present article have been studied in [23]. Another commonly used strategy in Bayesian statistics and in stochastic engineering is the well-known Monte Carlo Markov chain method (abbreviate MCMC methods). The idea is to interpret the conditional distributions ηn as the invariant measure of a suitably chosen Markov chain. This technique as to main drawbacks. The first one is that we need to run the underlying chain for very long times. This burning period is difficult to estimate, and it is often too long to tackle filtering problems with high frequency observation sequences such as those arising in radar processing. The second difficulty is that the conditional distributions ηn vary with the time parameter and we need to chose at each time an appropriate MCMC algorithm. In contrast to the two ideas discussed above, the mean field particle strategy presented in this article can be interpreted as a stochastic and adaptive grid RR n° 0123456789 4 Del Moral, Patras & Rubenthaler approximation model. Loosely speaking, this particle technique consists in in- terpreting the weights likelihood as branching selection rates. No calibration of the convergence to equilibrium is needed, the conditional measures variations are automatically updated by the stochastic particle model. The first rigorous study in this field seems to be the article [15] published in 1996 on the appli- cations of interacting particle methods to nonlinear estimation problems. This article provides the first convergence results for a series of heuristic genetic type schemes presented in the beginning of the 1990's in three independent chains of articles on nonlinear filtering [36, 35], [39], and [2, 40, 30, 32]. In the end of the 1990's, three other independent works [6, 7, 8] proposed another class of particle branching variants for solving continuous-time filtering problems. For a more thorough discussion on the origins and the analysis these mod- els, we refer the reader to the research monograph of the first author [11]. The latter is dedicated to Feynman-Kac and Boltzmann-Gibbs measures with their genealogical and interacting particle system interpretations, as well as their applications in physics, in biology and in engineering sciences. Besides their im- portant application in filtering, Feynman-Kac type particle models also arise in the spectral analysis of Schrdinger type operators, rare events estimation, as well as in macromolecular and directed polymers simulations. In this connection, we also mention that the mean field theory presented here applies without further work to study the analysis of a variety of heuristic like models introduced in stochastic engineering since the beginning of the 1950's, including genetic type algorithms, quantum and sequential Monte Carlo methods, pruned enrichment molecular simulations, bootstrap and particle filters, and many others. For a rather thorough discussion on these rather well known application areas, the interested reader is also recommended to consult the book [34], and the ref- erences therein.
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