Bayesian Nonparametric Inference in McKean-Vlasov models
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910783305154560 |
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| author | Nickl, Richard Pavliotis, Grigorios A. Ray, Kolyan |
| author_facet | Nickl, Richard Pavliotis, Grigorios A. Ray, Kolyan |
| contents | We consider nonparametric statistical inference on a periodic interaction potential $W$ from noisy discrete space-time measurements of solutions $ρ=ρ_W$ 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 $\bar ρ$ towards $ρ_W$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_16742 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bayesian Nonparametric Inference in McKean-Vlasov models Nickl, Richard Pavliotis, Grigorios A. Ray, Kolyan Statistics Theory Numerical Analysis Analysis of PDEs We consider nonparametric statistical inference on a periodic interaction potential $W$ from noisy discrete space-time measurements of solutions $ρ=ρ_W$ 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 $\bar ρ$ towards $ρ_W$. 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. |
| title | Bayesian Nonparametric Inference in McKean-Vlasov models |
| topic | Statistics Theory Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2404.16742 |