Nonparametric Bayesian Inference for Stochastic Reaction-Diffusion Equations

2025/07/09 by Altmeyer, Randolf, Gaudlitz, Sascha
#60H15 #62F15 #FOS: Mathematics #Primary 62G20 #Probability (math.PR) #Statistics Theory (math.ST) #secondary 60H07

paper · doi:10.48550/arxiv.2507.06857

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.

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