Abstract: Nonlinear filtering is a branch of Bayesian estimation in which a “signal” process is progressively estimated from the history of a related “observations” process. Nonlinear filters are typically expressed in terms of stochastic differential equations for the posterior distribution of the signal, which is nearly always infinite-dimensional (in the sense that it cannot be represented by a finite number of statistics). For example, nonlinear filters for diffusion processes can often be expressed in terms of stochastic pdes for the conditional densities. More generally, the natural “state space” for a nonlinear filter is a suitably rich family of probability measures having an appropriate topology, and the so-called “statistical manifolds” of Information Geometry are obvious candidates.
Starting with a brief introduction to Information Geometry, the talk will concentrate on an infinite-dimensional Hilbert manifold, M, as developed in [2], and outline its application to nonlinear filtering. The Hilbert manifold is less technically demanding than the (highly inclusive) exponential Orlicz manifold of [3], and lends itself to the L2 theory of stochastic integration. Although developed with nonlinear filtering in mind, it is potentially of broad application in Information Geometry.
The M-valued nonlinear filter can be used as a basis for projective approximations of the type proposed in [1].
[1] Brigo, D., Hanzon, B. and Le Gland, F. (1999) Approximate nonlinear filtering on exponential manifolds of densities, Bernoulli, 5, pp. 495-534.
[2] Newton, N. J. (2012), An infinite-dimensional statistical manifold modelled on Hilbert space, J. Functional Anal., 263, pp. 1661-1681.
[3] Pistone, G. and Sempi, C. (1995) An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one, Annals of Statistics, 23, pp. 1543-1561.