Connections between sequential Bayesian inference and evolutionary dynamics

Area of Science:

• Mathematics
• Computational Biology
• Data Science

Background:

• A long-standing hypothesis suggests a link between biological evolutionary dynamics and sequential Bayesian learning.
• Establishing this connection rigorously in a continuous-time setting is crucial for advancing both fields.

Purpose of the Study:

• To rigorously establish the connection between dynamical equations in evolutionary biology and sequential Bayesian learning in continuous time.
• To explore novel algorithms for filtering and sampling by bridging these two domains.

Main Methods:

• Focus on the Kushner-Stratonovich equation for posterior density evolution.
• Utilize a piecewise smooth approximation of observation paths for discrete-time filtering equations.
• Investigate gradient flow formulations and specific replicator-mutator dynamics.

Main Results:

• Demonstrated convergence of discrete-time filtering equations to a Stratonovich interpretation of the Kushner-Stratonovich equation.
• Established precise connections between nonlinear stochastic filtering and replicator-mutator dynamics.
• Identified a beneficial form of replicator-mutator dynamics for misspecified model filtering.

Conclusions:

• The research provides a rigorous mathematical framework linking evolutionary biology and Bayesian learning.
• The findings are expected to stimulate further interdisciplinary research and inspire new computational algorithms.
• This work contributes to the theme issue on Partial Differential Equations in Data Science.