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.

翻译：本文研究从非线性McKean-Vlasov方程解ρ=ρ_W的含噪离散时空测量数据中，对周期相互作用势W进行非参数统计推断的问题。该方程描述了相互作用粒子系统平均场极限的概率密度。我们证明，对W赋予高斯过程先验后，后验均值估计量能够使隐含估计密度ρ̄以快速收敛速度逼近ρ_W。进一步研究表明，若初始条件φ具有适当光滑性并满足标准反卷积条件，则可一致地以收敛速率N^{-θ}（其中θ>0，N为测量次数）推断出势函数W本身。随着W正则性的增加（对应"近参数"模型），指数θ可趋近于1/2。