Bayesian Nonparametric Inference in McKean-Vlasov models

TL;DR

This paper develops a Bayesian nonparametric framework for inferring interaction potentials in McKean-Vlasov models from noisy data, achieving fast convergence rates and near-parametric efficiency under certain regularity conditions.

Contribution

It introduces Gaussian process priors for nonparametric inference of interaction potentials in McKean-Vlasov equations, with proven convergence rates and conditions for consistent estimation.

Findings

Posterior mean estimators converge rapidly to true densities.

Consistent Sobolev-regular potential inference at rates approaching N^{-1/2}.

Method applies to noisy, discrete space-time measurements.

Abstract

We consider nonparametric statistical inference on a periodic interaction potential $W$ from noisy discrete space-time measurements of solutions $\rho=\rho_W$ of the nonlinear McKean-Vlasov equation, describing the probability density of the mean field limit of an interacting particle system. We show how Gaussian process priors assigned to $W$ give rise to posterior mean estimators that exhibit fast convergence rates for the implied estimated densities $\bar \rho$ towards $\rho_W$. We further show that if the initial condition $\phi$ is not too smooth and satisfies a standard deconvolvability condition, then one can consistently infer Sobolev-regular potentials $W$ at convergence rates $N^{-\theta}$ for appropriate $\theta>0$, where $N$ is the number of measurements. The exponent $\theta$ can be taken to approach $1/2$ as the regularity of $W$ increases corresponding to `near-parametric' models.