Bayesian Nonparametric Inference in McKean-Vlasov models

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