Nonparametric Bayesian inference for reversible multi-dimensional diffusions

TL;DR

This paper develops nonparametric Bayesian methods for estimating the drift in reversible multi-dimensional diffusions, achieving optimal convergence rates by leveraging the process's reversibility and applying Gaussian and p-exponential priors.

Contribution

It introduces a general posterior contraction rate theorem for the drift gradient in reversible diffusions and demonstrates minimax optimal convergence with specific priors.

Findings

Proves a general posterior contraction rate theorem for the drift gradient.

Shows Gaussian and p-exponential priors achieve minimax optimal rates.

Validates the theoretical results with convergence in multi-dimensional settings.

Abstract

We study nonparametric Bayesian models for reversible multi-dimensional diffusions with periodic drift. For continuous observation paths, reversibility is exploited to prove a general posterior contraction rate theorem for the drift gradient vector field under approximation-theoretic conditions on the induced prior for the invariant measure. The general theorem is applied to Gaussian priors and $p$-exponential priors, which are shown to converge to the truth at the minimax optimal rate over Sobolev smoothness classes in any dimension.