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.