Semi-parametric Bernstein-von Mises in Linear Inverse Problems
Adel Magra, Aad van der Vaart, Harry van Zanten
TL;DR
The paper develops a semi-parametric Bernstein-von Mises theorem for the marginal posterior of a scalar θ in linear inverse problems with a parameter-dependent operator K_θ, under a two-observation white-noise model. The core approach leverages a local asymptotic normality expansion and a least favorable direction γ_{θ,f} to characterize bias and prior-insensitivity, yielding asymptotic normality of the θ-posterior with efficient information, despite an infinite-dimensional nuisance f. The authors instantiate the theory in two canonical applications—the heat equation for thermal diffusivity and a semi-blind deconvolution problem—and analyze the theorem under Gaussian process and p-exponential priors, providing contraction-rate conditions and explicit forms for the least favorable direction. They further complement the theory with posterior simulations that illustrate the zone of valid BvM behavior and the impact of prior regularity on uncertainty quantification. Overall, the work bridges Bayesian and frequentist perspectives in semi-parametric inverse problems with operator uncertainty, enabling valid uncertainty quantification for θ in challenging ill-posed settings.
Abstract
We consider a Bayesian approach for the recovery of scalar parameters arising in inverse problems. We consider a general signal-in white noise model where we have access to two independent noisy observations of a function, and of a linear transformation of the function. The linear operator is unknown up to a scalar parameter. We present a Bernstein-von Mises theorem for the marginal posterior of the scalar under regularity assumptions of the operator. We further derive Bernstein-von Mises results for different priors and apply them to two concrete examples: the recovery of the thermal diffusivity in a heat equation problem, and the recovery of a location parameter in a semi-blind deconvolution problem.