Data Analysis and Related Applications, Volume 1

24Approximate Bayesian Inference Using the Mean-Field Distribution

Dynamical systems representing populations of interacting heterogeneous individuals are rarely studied and validated within a Bayesian framework, with the notable exception of Schneider et al. (2006), dealing with a model of plants in competition for a light resource. The reasons for this lack of coverage of a subject with such significant stakes (agriculture, crowd dynamics) are to be found in the computational difficulties posed by the problem of inference when the size of the population is large. In this chapter, we will focus on dynamical systems admitting a mean-field limit distribution when the population’s size tends to infinity, such as the flocking models presented in Carrillo et al. (2010). We introduce a numerical scheme to simulate the mean-field distribution, which is a partial differential transport equation solution, and we use these simulations to simplify the likelihood distributions associated with Bayesian inference problems arising when the population is only partially observed.

24.1. Introduction

Population models may be used to assess, from data, the interaction laws governing the individual dynamics (Bongini et al. 2017; Lu et al. 2019). In most of these models, the interaction of an individual with the rest of the population is represented by means of some statistics, potentially depending on the state variables of the whole population. These statistics can take the form of the average ...