  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.
