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

• URL: http://arxiv.org/abs/2203.11869v1
• Date: Tue, 22 Mar 2022 16:43:33 GMT
• Title: An Optimal Transport Formulation of Bayes' Law for Nonlinear Filtering Algorithms
• Authors: Amirhossein Taghvaei and Bamdad Hosseini
• Abstract summary: This paper presents a variational representation of the Bayes' law using optimal transportation theory. By imposing certain structure on the transport map, the solution to the variational problem is used to construct a Brenier-type map. The proposed methodology is used to derive the optimal transport form of the feedback particle filler (FPF) in the continuous-time limit.
• Score: 7.919213739992465
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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