  Feedback particle filter (FPF) is a Monte-Carlo (MC) algorithm to approximate
the solution of a stochastic filtering problem. In contrast to conventional
particle filters, the Bayesian update step in FPF is implemented via a
mean-field type feedback control law.
  The objective for this paper is to situate the development of FPF and related
controlled interacting particle system algorithms within the framework of
optimal transportation theory. Starting from the simplest setting of the Bayes'
update formula, a coupling viewpoint is introduced to construct particle
filters. It is shown that the conventional importance sampling resampling
particle filter implements an independent coupling. Design of optimal couplings
is introduced first for the simple Gaussian settings and subsequently extended
to derive the FPF algorithm. The final half of the paper provides a review of
some of the salient aspects of the FPF algorithm including the feedback
structure, algorithms for gain function design, and comparison with
conventional particle filters. The comparison serves to illustrate the benefit
of feedback in particle filtering.
