  In estimation theory, the Kushner equation provides the evolution of the
probability density of the state of a dynamical system given continuous-time
observations. Building upon our recent work, we propose a new way to
approximate the solution of the Kushner equation through tractable variational
Gaussian approximations of two proximal losses associated with the propagation
and Bayesian update of the probability density. The first is a proximal loss
based on the Wasserstein metric and the second is a proximal loss based on the
Fisher metric. The solution to this last proximal loss is given by implicit
updates on the mean and covariance that we proposed earlier. These two
variational updates can be fused and shown to satisfy a set of stochastic
differential equations on the Gaussian's mean and covariance matrix. This
Gaussian flow is consistent with the Kalman-Bucy and Riccati flows in the
linear case and generalize them in the nonlinear one.
