Multiscale convergence of the inverse problem for chemotaxis in the Bayesian setting

TL;DR

This paper investigates the relationship between macro- and mesoscopic inverse problems in chemotaxis modeling within a Bayesian framework, proving the asymptotic equivalence of their posterior distributions.

Contribution

It establishes the asymptotic equivalence of the Bayesian posterior distributions for inverse problems at macro- and mesoscopic levels in chemotaxis models.

Findings

Proves asymptotic equivalence of posterior distributions

Links inverse problems across different modeling scales

Provides theoretical foundation for Bayesian chemotaxis analysis

Abstract

Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller-Segel equation and a chemotaxis kinetic equation. These two equations describe the organism movement at the macro- and mesoscopic level respectively, and are asymptotically equivalent in the parabolic regime. How the organism responds to a chemical stimulus is embedded in the diffusion/advection coefficients of the Keller-Segel equation or the turning kernel of the chemotaxis kinetic equation. Experiments are conducted to measure the time dynamics of the organisms' population level movement when reacting to certain stimulation. From this one infers the chemotaxis response, which constitutes an inverse problem. \\ In this paper we discuss the relation between both the macro- and mesoscopic inverse problems, each of which is associated to two different forward models. The discussion is presented in the Bayesian framework, where the posterior distribution of the turning kernel of the organism population is sought after. We prove the asymptotic equivalence of the two posterior distributions.