Asymptotic shape and frequentist calibration of noisy-collocation posteriors

Establish the asymptotic shape and frequentist calibration of the full posterior distribution for Bayesian probabilistic numerical methods based on noisy collocation of nonlinear partial differential equations.

Background

The paper studies Bayesian probabilistic numerical methods for solving nonlinear partial differential equations from noisy evaluations at randomly sampled collocation points. It develops posterior contraction rates and Bernstein–von Mises theorems for truncated Gaussian-series priors, including Gaussian approximations of the full posterior in positive-order Sobolev spaces.

Before this work, related Gaussian-process methods primarily addressed deterministic or maximum-a-posteriori approximations, while posterior contraction had also been studied for Bayesian neural-network variants. The authors explicitly identify the asymptotic shape and frequentist calibration of the entire posterior distribution under noisy collocation as unresolved, motivating their function-space Bernstein–von Mises results.