  This paper presents a variational representation of the Bayes' law using
optimal transportation theory. The variational representation is in terms of
the optimal transportation between the joint distribution of the (state,
observation) and their independent coupling. By imposing certain structure on
the transport map, the solution to the variational problem is used to construct
a Brenier-type map that transports the prior distribution to the posterior
distribution for any value of the observation signal. The new formulation is
used to derive the optimal transport form of the Ensemble Kalman filter (EnKF)
for the discrete-time filtering problem and propose a novel extension of EnKF
to the non-Gaussian setting utilizing input convex neural networks. Finally,
the proposed methodology is used to derive the optimal transport form of the
feedback particle filler (FPF) in the continuous-time limit, which constitutes
its first variational construction without explicitly using the nonlinear
filtering equation or Bayes' law.
