The direct L geometric structure on a manifold of probability densities with applications to Filtering

In this paper we introduce a projection method for the space of probability distributions based on the differential geometric approach to statistics. This method is based on a direct L2 metric as opposed to the usual Hellinger distance and the related Fisher Information metric. We explain how this apparatus can be used for the nonlinear filtering problem, in relationship also to earlier projection methods based on the Fisher metric. Past projection filters focused on the Fisher metric and the exponential families that made the filter correction step exact. In this work we introduce the mixture projection filter, namely the projection filter based on the direct L2 metric and based on a manifold given by a mixture of pre-assigned densities. The resulting prediction step in the filtering problem is described by a linear differential equation, while the correction step can be made exact. We analyze the relationship of a specific class of L2 filters with the Galerkin based nonlinear filters, and highlight the differences with our approach, concerning particularly the continuous–time observations filtering problems.