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This paper introduces in detail a new systematic method to construct approximate ®nite-dimensional solutions for the nonlinear ®ltering problem. Once a ®nite-dimensional family is selected, the nonlinear ®ltering equation is projected in Fisher metric on the corresponding manifold of densities, yielding the projection ®lter for the chosen family. The general de®nition of the projection ®lter is given, and its structure is explored in detail for exponential families. Particular exponential families which optimize the correction step in the case of discrete-time observations are given, and an a posteriori estimate of the local error resulting from the projection is de®ned. Simulation results comparing the projection ®lter and the optimal ®lter for the cubic sensor problem are presented. The classical concept of assumed density ®lter (ADF) is compared with the projection ®lter. It is shown that the concept of ADF is inconsistent in the sense that the resulting ®lters depend on the choice of a stochastic calculus, i.e. the Ito à or the Stratonovich calculus. It is shown that in the context of exponential families, the projection ®lter coincides with the Stratonovich-based ADF. An example is provided, which shows that this does not hold in general, for non-exponential families of densities.

International audienceWe present a new and systematic method of approximating exact nonlinear filters with finite dimensional filters, using the differential geometric approach in statistics. We define rigorously the projection filter in the case of exponential families. We propose a convenient exponential family, which allows one to simplify the projection filter equation, and to define an a posteriori measure of the performance of the projection filte

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We study the stability of the optimal filter w.r.t. its initial condition and w.r.t. the model for the hidden state and the observations in a general hidden Markov model, using the Hilbert projective metric. These stability results are then used to prove, under some mixing assumption, the uniform convergence to the optimal filter of several particle filters, such as the interacting particle filter and some other original particle filters. IntroductionThe stability of the optimal filter has become recently an active research area. Ocone and Pardoux have proved in [26] that the filter forgets its initial condition in the L p sense, without stating any rate of convergence. Recently, a new approach has been proposed using the Hilbert projective metric. This metric allows to get rid of the normalization constant in the Bayes formula, and reduces the problem to studying the linear equation satisfied by the unnormalized optimal filter. Using the Hilbert metric, stability results w.r.t. the initial condition have been proved by Atar and Zeitouni in [4], and some stability result w.r.t. the model have been proved by Le Gland and Mevel in [19,20] Independently, Del Moral and Guionnet have adopted in [9], for the same class of HMM, another approach based on semi-group techniques and on the Dobrushin ergodic coefficient, to derive stability results w.r.t. the initial condition, which are used to prove uniform convergence of the interacting particle system (IPS) approximation to the optimal predictor. New approaches have been proposed recently, to prove the stability of the optimal filter w.r.t. its initial condition, in the case of a noncompact state space, see e.g. Atar [1], Atar, Viens and Zeitouni [2], Budhiraja and Ocone [6,7].In this article, we use the approach based on the Hilbert metric to study the asymptotic behavior of the optimal filter, and to prove as in [9] the uniform convergence of several particle filters, such as the interacting particle filter (IPF) and some other original particle filters.A common assumption to prove stability results, see e.g. in [9, Theorem 2.4], is that the Markov transition kernels are mixing, which implies that the hidden state sequence is ergodic. Our results are obtained under the assumption that the nonnegative kernels describing the evolution of the unnormalized optimal filter, and incorporating simultaneously the Markov transition kernels and the likelihood functions, are mixing. This is a weaker assumption, see Proposition 3.9, which allows to consider some cases, similar to the case studied in [6], where the hidden state sequence is not ergodic, see Example 3.10. This point of view is further developped by Le Gland and Oudjane in [22] and by Oudjane and Rubenthaler in [28]. Our main contribution is to study also the stability of the optimal filter w.r.t. the model, when the local error is propagated by mixing kernels, and can be estimated in the Hilbert metric, in the total variation norm, or in a weaker distance suitable for random probability distributions.AMS 1991 subject...