SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation

TL;DR

This paper develops SPDE-based methods to analyze nonparametric Bayesian posteriors, providing contraction rates, Bernstein von Mises results, and Laplace approximations in infinite-dimensional models, with applications to Gaussian inverse problems.

Contribution

It extends diffusion-based Bayesian analysis to infinite dimensions, deriving non-asymptotic contraction rates and Laplace approximations for complex models.

Findings

Established posterior contraction rates in Hilbert norms.

Proved finite-sample Bernstein von Mises theorems.

Demonstrated the approach on a Gaussian inverse problem.

Abstract

We derive posterior contraction rates (PCRs) and finite-sample Bernstein von Mises (BvM) results for non-parametric Bayesian models by extending the diffusion-based framework of Mou et al. (2024) to the infinite-dimensional setting. The posterior is represented as the invariant measure of a Langevin stochastic partial differential equation (SPDE) on a separable Hilbert space, which allows us to control posterior moments and obtain non-asymptotic concentration rates in Hilbert norms under various likelihood curvature and regularity conditions. We also establish a quantitative Laplace approximation for the posterior. The theory is illustrated in a nonparametric linear Gaussian inverse problem.