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

Proximal Recursion for the Wonham Filter