Projection Geometric Methods for Linear and Nonlinear Filtering Problems

Abstract

In this thesis, we review the infinite-dimensional space containing the solution of a broad range of stochastic filtering problems, and outline the substantive differences between the foundations of finite dimensional information geometry and Pistone’s extension to infinite dimensions characterizing the substantive differences between the two geometries with respect to the geometric structures needed for projection theorems such as a dually flat affine manifold preserving the affine and convex geometry of the set of all probability measures with the same support, the notion of orthogonal complement between the different tangent representation which are key for the generalized Pythagorean theorem, and the key notion of exponential and mixture parallel transport needed for projecting a point on a submanifold. We also explore the projection method proposed by Brigo and Pistone for reducing the dimensionality of infinite-dimensional measure valued evolution equations from the infinite-dimensional space in which they are written, that is the infinitedimensional statistical manifold of Pistone, onto a finite-dimensional exponential subfamily using a local generalized projection theorem that is a non-parameteric analog of the generalized projection theorem proposed by Amari. Also, we explore using standard arguments the projection idea in the discrete state space with focus on building intuition and using computational examples to understand properties of the projection method. We establish two novel results regarding the impact of the boundary and choosing a subfamily that does not contain the initial condition of the problem. We demostrate, when the evolution process approaches the boundary of the space, the projection method fails completely due to the classical boundary relating to the vanishing of the tangent spaces at the boundary. We also show the impact of choosing a subfamily to project onto that does not contain the initial condition of the problem showing that, in certain directions, the approximation by projection changes from the true value due to solving a different differential equation than if we are to start from within the low-dimensional manifold. We also study the importance of having a sufficient statistics of the exponential subfamily to lie in the span of the left eigenfunctions of the infinitesimal generator of the process we wish to project using computational experiments.