Information Geometric Nonlinear Filtering

TL;DR

This paper introduces an information geometric framework for continuous-time nonlinear filtering, representing posterior distributions on a Hilbert manifold and connecting information flow with the process's quadratic variation.

Contribution

It develops a novel geometric representation of nonlinear filters using a Hilbert information manifold and derives intrinsic evolution equations for exponential filters.

Findings

Posterior distributions satisfy a stochastic differential equation on a Hilbert manifold.

Flows of Shannon information relate to the quadratic variation of posterior processes.

Finite-dimensional exponential filters are encompassed within this geometric framework.

Abstract

This paper develops information geometric representations for nonlinear filters in continuous time. The posterior distribution associated with an abstract nonlinear filtering problem is shown to satisfy a stochastic differential equation on a Hilbert information manifold. This supports the Fisher metric as a pseudo-Riemannian metric. Flows of Shannon information are shown to be connected with the quadratic variation of the process of posterior distributions in this metric. Apart from providing a suitable setting in which to study such information-theoretic properties, the Hilbert manifold has an appropriate topology from the point of view of multi-objective filter approximations. A general class of finite-dimensional exponential filters is shown to fit within this framework, and an intrinsic evolution equation, involving Amari's $-1$-covariant derivative, is developed for such filters. Three example systems, one of infinite dimension, are developed in detail.