Nonparametric Bayesian Inference for Stochastic Reaction-Diffusion Equations

TL;DR

This paper develops a Bayesian nonparametric approach to estimate reaction functions in stochastic reaction-diffusion equations, proving optimal convergence rates and a Bernstein--von Mises theorem for the posterior distribution.

Contribution

It introduces a Gaussian wavelet prior for reaction functions in SPDEs and establishes minimax optimal posterior contraction and asymptotic normality results.

Findings

Posterior contracts at minimax optimal rate

Gaussian wavelet prior is effective for SPDEs

Bernstein--von Mises theorem holds for the posterior

Abstract

We consider the Bayesian nonparametric estimation of a nonlinear reaction function in a reaction-diffusion stochastic partial differential equation (SPDE). The likelihood is well-defined and tractable by the infinite-dimensional Girsanov theorem, and the posterior distribution is analysed in the growing domain asymptotic. Based on a Gaussian wavelet prior, the contraction of the posterior distribution around the truth at the minimax optimal rate is proved. The analysis of the posterior distribution is complemented by a semiparametric Bernstein--von Mises theorem. The proofs rely on the sub-Gaussian concentration of spatio-temporal averages of transformations of the SPDE, which is derived by combining the Clark-Ocone formula with bounds for the derivatives of the (marginal) densities of the SPDE.