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

International audienceThis paper presents a new and systematic method of approximating exact nonlinear filters with finite dimensional filters, using the differential geometric approach to statistics. The projection filter is defined rigorously in the case of exponential families. A convenient exponential family is proposed which allows one to simplify the projection filter equation and to define an a posteriori measure of the local error of the projection filter approximation. Finally, simulation results are discussed for the cubic sensor problem

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