Proximal Recursion for the Wonham Filter

TL;DR

This paper presents a novel perspective on the Wonham filter by framing its posterior flow as a limit of proximal recursions on the probability simplex, extending previous work on Kalman-Bucy filters.

Contribution

It introduces a new gradient flow interpretation of the Wonham filter using proximal recursions, connecting stochastic filtering with geometric optimization methods.

Findings

Posterior flow can be viewed as a small time-step limit of proximal recursions.

Extends previous proximal recursion frameworks from Kalman-Bucy to Wonham filter.

Provides a geometric perspective on the evolution of the posterior distribution.

Abstract

This paper contributes to the emerging viewpoint that governing equations for dynamic state estimation, conditioned on the history of noisy measurements, can be viewed as gradient flow on the manifold of joint probability density functions with respect to suitable metrics. Herein, we focus on the Wonham filter where the prior dynamics is given by a continuous time Markov chain on a finite state space; the measurement model includes noisy observation of the (possibly nonlinear function of) state. We establish that the posterior flow given by the Wonham filter can be viewed as the small time-step limit of proximal recursions of certain functionals on the probability simplex. The results of this paper extend our earlier work where similar proximal recursions were derived for the Kalman-Bucy filter.