  We propose a dimensionality reduction method for infinite-dimensional
measure-valued evolution equations such as the Fokker-Planck partial
differential equation or the Kushner-Stratonovich resp. Duncan-Mortensen-Zakai
stochastic partial differential equations of nonlinear filtering, with
potential applications to signal processing, quantitative finance, heat flows
and quantum theory among many other areas. Our method is based on the
projection coming from a duality argument built in the exponential statistical
manifold structure developed by G. Pistone and co-authors. The choice of the
finite dimensional manifold on which one should project the infinite
dimensional equation is crucial, and we propose finite dimensional exponential
and mixture families. This same problem had been studied, especially in the
context of nonlinear filtering, by D. Brigo and co-authors but the $L^2$
structure on the space of square roots of densities or of densities themselves
was used, without taking an infinite dimensional manifold environment space for
the equation to be projected. Here we re-examine such works from the
exponential statistical manifold point of view, which allows for a deeper
geometric understanding of the manifold structures at play. We also show that
the projection in the exponential manifold structure is consistent with the
Fisher Rao metric and, in case of finite dimensional exponential families, with
the assumed density approximation. Further, we show that if the sufficient
statistics of the finite dimensional exponential family are chosen among the
eigenfunctions of the backward diffusion operator then the statistical-manifold
or Fisher-Rao projection provides the maximum likelihood estimator for the
Fokker Planck equation solution. We finally try to clarify how the finite
dimensional and infinite dimensional terminology for exponential and mixture
spaces are related.
