  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.
