SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation

2026-03-23

AI Summary

This work investigates the non-asymptotic characterization of posterior contraction rates and finite-sample Bernstein–von Mises (BvM) behavior in nonparametric Bayesian models. By interpreting the posterior distribution as the invariant measure of a Langevin stochastic partial differential equation (SPDE) on a separable Hilbert space, the study extends diffusion-based analytical frameworks to infinite-dimensional settings for the first time. This approach enables precise control over posterior moments and yields sharp non-asymptotic concentration rates in Hilbert norm, along with a quantitative Laplace approximation of the posterior. In the context of nonparametric linear Gaussian inverse problems, the theory elucidates how likelihood curvature and prior regularity jointly govern posterior contraction rates and finite-sample BvM phenomena.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience
• Security and Privacy: Large-scale security measurements

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.

Problem

• posterior contraction
• nonparametric Bayesian
• Laplace approximation
• stochastic partial differential equation
• Bernstein von Mises

Innovation

• stochastic partial differential equation
• posterior contraction rate
• nonparametric Bayesian inference
• Laplace approximation
• Bernstein–von Mises theorem