  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 the potential $W$ itself 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.
