How to implement the Bayes' formula in the age of ML?

TL;DR

This paper explores the implementation of Bayes' formula in nonlinear filtering, emphasizing optimal transportation methods and ML tools to address numerical challenges in control theory.

Contribution

It introduces a novel optimal transportation formulation for Bayes' formula and discusses its integration with machine learning for improved numerical implementation.

Findings

New optimal transportation approach for Bayes' formula

Connection to feedback particle filter methods

Enhanced numerical stability using ML techniques

Abstract

This chapter contains a self-contained introduction to the significance of Bayes' formula in the context of nonlinear filtering problems. Both discrete-time and continuous-time settings of the problem are considered in a unified manner. In control theory, the focus on optimization-based solution approaches is stressed together with a discussion of historical developments in this area (from 1960s onwards). The heart of this chapter contains a presentation of a novel optimal transportation formulation for the Bayes formula (developed recently by the first author) and its relationship to some of the prior joint work (feedback particle filter) from the authors. The presentation highlights how optimal transportation theory is leveraged to overcome some of the numerical challenges of implementing Bayes' law by enabling the use of machine learning (ML) tools.