  A new formulation of the particle filter for nonlinear filtering is
presented, based on concepts from optimal control, and from the mean-field game
theory. The optimal control is chosen so that the posterior distribution of a
particle matches as closely as possible the posterior distribution of the true
state given the observations. This is achieved by introducing a cost function,
defined by the Kullback-Leibler (K-L) divergence between the actual posterior,
and the posterior of any particle.
  The optimal control input is characterized by a certain Euler-Lagrange (E-L)
equation, and is shown to admit an innovation error-based feedback structure.
For diffusions with continuous observations, the value of the optimal control
solution is ideal. The two posteriors match exactly, provided they are
initialized with identical priors. The feedback particle filter is defined by a
family of stochastic systems, each evolving under this optimal control law.
  A numerical algorithm is introduced and implemented in two general examples,
and a neuroscience application involving coupled oscillators. Some preliminary
numerical comparisons between the feed- back particle filter and the bootstrap
particle filter are described.
