Multivariate Feedback Particle Filter Rederived from the Splitting-Up Scheme

The multivariate feedback particle filter (FPF) is formulated from the viewpoint of splitting-up methods. The essential difference between this formulation and the formal derivation is that instead of one-time control at a discrete time instant, we consider the updating stage as a stochastic flow of particles in each time interval. This allows to easily obtain a consistent stochastic flow by comparing the Kolmogorov forward equation of particles and the updating part of the Kushner’s equation in the splitting-up method. Moreover, if an optimal stochastic flow exists, the convergence of the splitting-up method can be studied by passing to an FPF with a continuous time. To guarantee the existence of a stochastic flow, we validate the Poincaré inequality for the alternating distributions, given the time discretization and the observation path, under mild conditions on the nonlinear filtering system and the initial state. Besides, re-examining the original derivation of the FPF, we show that the optimal transport map between the prior and the posterior is an $f$-divergence invariant in the abstract Bayesian inference framework.