  Uncertainty propagation and filtering can be interpreted as gradient flows
with respect to suitable metrics in the infinite dimensional manifold of
probability density functions. Such a viewpoint has been put forth in recent
literature, and a systematic way to formulate and solve the same for linear
Gaussian systems has appeared in our previous work where the gradient flows
were realized via proximal operators with respect to Wasserstein metric arising
in optimal mass transport. In this paper, we derive the evolution equations as
proximal operators with respect to Fisher-Rao metric arising in information
geometry. We develop the linear Gaussian case in detail and show that a
template two step optimization procedure proposed earlier by the authors still
applies. Our objective is to provide new geometric interpretations of known
equations in filtering, and to clarify the implication of different choices of
metric.
