# Proximal Recursion for the Wonham Filter

arXiv: https://arxiv.org/abs/1909.05827 · HTML: https://mlsanity.com/paper/1909.05827/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Sep 2019.

## Built on by (most cited first, 4 of 4)

- [An Optimal Transport Formulation of Bayes' Law for Nonlinear Filtering Algorithms](https://mlsanity.com/paper/2203.11869/) (2022)
- [Proximal Mean Field Learning in Shallow Neural Networks](https://mlsanity.com/paper/2210.13879/) (2022)
- [A Distributed Algorithm for Measure-valued Optimization with Additive Objective](https://mlsanity.com/paper/2202.08930/) (2022)
- [Learning Concave Bid Shading Strategies in Online Auctions via Measure-valued Proximal Optimization](https://mlsanity.com/paper/2509.10693/) (2025)

## Abstract (arXiv)

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.