Grid Based Nonlinear Filtering Revisited: Recursive Estimation & Asymptotic Optimality

TL;DR

This paper revisits grid-based recursive filtering for Markov processes, introducing relaxed conditions for convergence and asymptotic optimality, with a focus on linear-algebraic techniques for simplicity.

Contribution

It proposes new relaxed conditions for the convergence and optimality of grid-based filters, including the notion of conditional regularity for stochastic kernels.

Findings

Established strong pathwise convergence of filters to MMSE estimators.

Introduced the concept of conditional regularity of stochastic kernels.

Extended convergence results to filtering of functionals and state prediction.

Abstract

We revisit the development of grid based recursive approximate filtering of general Markov processes in discrete time, partially observed in conditionally Gaussian noise. The grid based filters considered rely on two types of state quantization: The \textit{Markovian} type and the \textit{marginal} type. We propose a set of novel, relaxed sufficient conditions, ensuring strong and fully characterized pathwise convergence of these filters to the respective MMSE state estimator. In particular, for marginal state quantizations, we introduce the notion of \textit{conditional regularity of stochastic kernels}, which, to the best of our knowledge, constitutes the most relaxed condition proposed, under which asymptotic optimality of the respective grid based filters is guaranteed. Further, we extend our convergence results, including filtering of bounded and continuous functionals of the state, as well as recursive approximate state prediction. For both Markovian and marginal quantizations, the whole development of the respective grid based filters relies more on linear-algebraic techniques and less on measure theoretic arguments, making the presentation considerably shorter and technically simpler.