Connections between sequential Bayesian inference and evolutionary dynamics

TL;DR

This paper rigorously establishes a connection between continuous-time Bayesian filtering equations and evolutionary dynamics, specifically linking the Kushner-Stratonovich equation with replicator-mutator models and gradient flows.

Contribution

It provides a rigorous mathematical link between sequential Bayesian inference and evolutionary processes, including new approximation methods and applications to filtering.

Findings

Discrete filtering equations converge to Stratonovich interpretation

Connections between nonlinear stochastic filtering and replicator-mutator dynamics are established

Gradient flow formulations are explored for filtering and sampling

Abstract

It has long been posited that there is a connection between the dynamical equations describing evolutionary processes in biology and sequential Bayesian learning methods. This manuscript describes new research in which this precise connection is rigorously established in the continuous time setting. Here we focus on a partial differential equation known as the Kushner-Stratonovich equation describing the evolution of the posterior density in time. Of particular importance is a piecewise smooth approximation of the observation path from which the discrete time filtering equations, which are shown to converge to a Stratonovich interpretation of the Kushner-Stratonovich equation. This smooth formulation will then be used to draw precise connections between nonlinear stochastic filtering and replicator-mutator dynamics. Additionally, gradient flow formulations will be investigated as well as a form of replicator-mutator dynamics which is shown to be beneficial for the misspecified model filtering problem. It is hoped this work will spur further research into exchanges between sequential learning and evolutionary biology and to inspire new algorithms in filtering and sampling.