Poisson's equation in nonlinear filtering

TL;DR

This paper offers a variational perspective on continuous-time nonlinear filtering, interpreting the filter as a gradient flow of an energy functional, and derives an exact feedback particle filter algorithm.

Contribution

It introduces a variational framework for nonlinear filtering and derives an exact feedback particle filter based on first variation analysis.

Findings

Nonlinear filter dynamics can be viewed as a gradient flow.

The feedback particle filter is shown to be exact when initialized correctly.

The approach provides a new variational interpretation of nonlinear filtering.

Abstract

The aim of this paper is to provide a variational interpretation of the nonlinear filter in continuous time. A time-stepping procedure is introduced, consisting of successive minimization problems in the space of probability densities. The weak form of the nonlinear filter is derived via analysis of the first-order optimality conditions for these problems. The derivation shows the nonlinear filter dynamics may be regarded as a gradient flow, or a steepest descent, for a certain energy functional with respect to the Kullback-Leibler divergence. The second part of the paper is concerned with derivation of the feedback particle filter algorithm, based again on the analysis of the first variation. The algorithm is shown to be exact. That is, the posterior distribution of the particle matches exactly the true posterior, provided the filter is initialized with the true prior.