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 Sobolev-regular potentials $W$ 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：我们考虑从非线性McKean-Vlasov方程解$\rho=\rho_W$的含噪声离散时空测量中，对周期相互作用势$W$进行非参数统计推断，该方程描述了相互作用粒子系统平均场极限的概率密度。我们证明了赋予$W$的高斯过程先验如何产生后验均值估计量，使得隐含的估计密度$\bar \rho$能以较快的收敛速度逼近$\rho_W$。我们进一步证明，若初始条件$\phi$具有适度非光滑性且满足标准的解卷积条件，则能以$N^{-\theta}$的收敛速度（其中$N$为测量次数，$\theta>0$为适当参数）一致地推断Sobolev正则势$W$。随着$W$正则性的增强（对应"近参数"模型），指数$\theta$可趋近于$1/2$。