Semiparametric Bernstein-von Mises theorems for reversible diffusions

TL;DR

This paper proves a general semiparametric Bernstein-von Mises theorem for Bayesian inference in reversible diffusion models, enabling efficient uncertainty quantification for a wide class of functionals.

Contribution

It establishes a new theoretical framework for Bayesian nonparametric inference in multidimensional diffusion models, covering nonlinear functionals and specific priors.

Findings

Gaussian and Besov-Laplace priors achieve efficient inference

Theoretical results validated through numerical simulations

Broad applicability to functionals satisfying linearization conditions

Abstract

We establish a general semiparametric Bernstein-von Mises theorem for Bayesian nonparametric priors based on continuous observations in a periodic reversible multidimensional diffusion model. We consider a wide range of functionals satisfying an approximate linearization condition, including several nonlinear functionals of the invariant measure. Our result is applied to Gaussian and Besov-Laplace priors, showing these can perform efficient semiparametric inference and thus justifying the corresponding Bayesian approach to uncertainty quantification. Our theoretical results are illustrated via numerical simulations.