Bayesian nonparametric inference in McKean–Vlasov models

Abstract

We consider nonparametric statistical inference on a periodic interaction potential W from noisy discrete space-time measurements of solutions ρ=ρW 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 ρ¯ towards ρW. We further show that if the initial condition ϕ is not too smooth and satisfies a standard deconvolvability condition, then one can consistently infer Sobolev-regular potentials W at convergence rates N−θ for appropriate θ>0, where N is the number of measurements. The exponent θ can be taken to approach 1/2 as the regularity of W increases corresponding to ‘near-parametric’ models.