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Bayesian Nonparametric Inference in McKean-Vlasov models

Richard Nickl, Grigorios A. Pavliotis, Kolyan Ray

Abstract

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.

Bayesian Nonparametric Inference in McKean-Vlasov models

Abstract

We consider nonparametric statistical inference on a periodic interaction potential from noisy discrete space-time measurements of solutions 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 give rise to posterior mean estimators that exhibit fast convergence rates for the implied estimated densities towards . We further show that if the initial condition is not too smooth and satisfies a standard deconvolvability condition, then one can consistently infer Sobolev-regular potentials at convergence rates for appropriate , where is the number of measurements. The exponent can be taken to approach as the regularity of increases corresponding to `near-parametric' models.
Paper Structure (11 sections, 10 theorems, 92 equations)

This paper contains 11 sections, 10 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Notation
  4. Gaussian process regression in the McKean-Vlasov model
  5. Posterior contraction rates for the particle densities
  6. Examples of Gaussian priors
  7. Recovering the interaction potential
  8. Proofs
  9. Proof of Theorem \ref{['thm:GP_forward']}: forward contraction rate
  10. Proof of Theorems \ref{['stabest']} and \ref{['thm:deconvolution']}: stability estimate and deconvolution
  11. Regularity properties for the McKean-Vlasov PDE

Key Result

Theorem 1

Suppose $\phi \in H^\beta(\mathbb{T}^d)$ for some even integer $\beta \geq 3+d$ and let $\Pi_V$ be a Gaussian measure satisfying Condition cond:RKHS. Consider the rescaled Gaussian process prior from e:rescaled_GP projected onto a linear subspace $\mathcal{W}_N$ of $\mathcal{W}$ such that $\|\pi_{\m as $N\to \infty$, where

Theorems & Definitions (24)

  • Theorem 1
  • Example 1: Periodized Matérn process
  • Remark 1
  • Example 2: Truncated Fourier prior
  • Remark 2: Symmetric potentials
  • Remark 3: Almost finite-dimensional models
  • Theorem 2
  • Theorem 3
  • Remark 4: Deconvolution rates for almost finite-dimensional models
  • Example 3: Periodised symmetric multivariate Laplace distribution
  • ...and 14 more