Bayesian Nonparametric Inference in McKean-Vlasov models
<|reference_start|>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 $\rho=\rho_W$ 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.<|reference_end|>
@article{nickl2024bayesian,
    title={Bayesian Nonparametric Inference in McKean-Vlasov models},
    author={Richard Nickl, Grigorios A. Pavliotis, Kolyan Ray},
    journal={arXiv preprint arXiv:2404.16742},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.16742},
    primaryClass={math.ST cs.NA math.AP math.NA stat.TH}
}