Semi-parametric Bernstein-von Mises Theorem in a Parabolic PDE Problem
Adel Magra, Frank van der Meulen, Aad van der Vaart
TL;DR
The paper addresses Bayesian uncertainty quantification for a semiparametric inverse problem in a parabolic PDE: recovering the diffusivity theta and the absorption f from noisy observations of the solution. It develops a semi-parametric Bernstein-von Mises theorem for the marginal posterior of theta by combining posterior contraction for the full (theta,f) with a local asymptotic normality (LAN) expansion, and it characterizes the efficient information via a least favorable direction gamma. Central to the approach are PDE-based regularity results, a Gaussian prior on f, an independence assumption between theta and f, and a two-step contraction mechanism (direct problem then inverse problem). The results enable asymptotically valid Bayesian inference for theta at the parametric rate despite infinite-dimensional nuisance, with explicit conditions on priors, smoothness, and identifiability. Together with the detailed PDE analysis, the work advances uncertainty quantification for nonlinear inverse problems governed by PDEs.
Abstract
We consider the heat equation with absorption in a bounded domain of $\mathbb{R}^d$, where both the scalar diffusivity and the absorption function are unknown. We investigate a Bayesian approach for recovering the diffusivity from a noisy observation of the solution to the PDE over the domain. Given a Gaussian process prior on the absorption function, we derive a Bernstein-von Mises theorem for the marginal posterior distribution of the diffusivity under assumptions on the prior and on smoothness properties of the absorption.