Semiparametric Bernstein-von Mises theorems for reversible diffusions

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.