Asymptotically Optimal Discrete Time Nonlinear Filters From Stochastically Convergent State Process Approximations

TL;DR

This paper develops a theoretical framework for approximating optimal nonlinear filters in discrete time using stochastically convergent state process approximations, ensuring strong convergence properties.

Contribution

It introduces a method to construct approximate filters that converge to the true optimal filter under general conditions, providing a rigorous basis for heuristic filtering approaches.

Findings

Approximate filters converge strongly to the true nonlinear filter.

Convergence is uniform in probability over measurable sets.

Provides a quantitative basis for heuristic filtering methods.

Abstract

We consider the problem of approximating optimal in the Minimum Mean Squared Error (MMSE) sense nonlinear filters in a discrete time setting, exploiting properties of stochastically convergent state process approximations. More specifically, we consider a class of nonlinear, partially observable stochastic systems, comprised by a (possibly nonstationary) hidden stochastic process (the state), observed through another conditionally Gaussian stochastic process (the observations). Under general assumptions, we show that, given an approximating process which, for each time step, is stochastically convergent to the state process, an approximate filtering operator can be defined, which converges to the true optimal nonlinear filter of the state in a strong and well defined sense. In particular, the convergence is compact in time and uniform in a completely characterized measurable set of probability measure almost unity, also providing a purely quantitative justification of Egoroff's Theorem for the problem at hand. The results presented in this paper can form a common basis for the analysis and characterization of a number of heuristic approaches for approximating optimal nonlinear filters, such as approximate grid based techniques, known to perform well in a variety of applications.