An Optimal Transport Formulation of Bayes' Law for Nonlinear Filtering Algorithms

TL;DR

This paper introduces a variational approach to Bayesian filtering using optimal transportation theory, leading to new formulations of EnKF and FPF that handle non-Gaussian distributions without relying on traditional filtering equations.

Contribution

It develops a novel variational framework for Bayesian filtering based on optimal transport, enabling extensions to non-Gaussian settings with neural network parameterizations.

Findings

Derived an optimal transport form of EnKF for discrete-time filtering.

Extended EnKF to non-Gaussian distributions using input convex neural networks.

Formulated the FPF in continuous time as a variational problem without explicit Bayes' law.

Abstract

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.