# SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation

arXiv: https://arxiv.org/abs/2603.22468 · HTML: https://mlsanity.com/paper/2603.22468/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 70 references that are in the map.
- First posted on arXiv 23 Mar 2026.

## Builds on (most cited first, 12 of 12)

- [Infinite-Dimensional Diffusion Models](https://mlsanity.com/paper/2302.10130/) (2023)
- [Continuous-Time Functional Diffusion Processes](https://mlsanity.com/paper/2303.00800/) (2023)
- [Conditional score-based diffusion models for Bayesian inference in infinite dimensions](https://mlsanity.com/paper/2305.19147/) (2023)
- [Dimension-independent likelihood-informed MCMC](https://mlsanity.com/paper/1411.3688/) (2014)
- [Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation](https://mlsanity.com/paper/2303.04772/) (2023)
- [Optimal low-rank approximations of Bayesian linear inverse problems](https://mlsanity.com/paper/1407.3463/) (2014)
- [A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems](https://mlsanity.com/paper/1308.6221/) (2013)
- [Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problems](https://mlsanity.com/paper/2405.15676/) (2024)
- [Malliavin Calculus for Score-based Diffusion Models](https://mlsanity.com/paper/2503.16917/) (2025)
- [Are minimizers of the Onsager-Machlup functional strong posterior modes?](https://mlsanity.com/paper/2212.04275/) (2022)
- [A Malliavin-Gamma calculus approach to Score Based Diffusion Generative models for random fields](https://mlsanity.com/paper/2505.13189/) (2025)
- [Strong posterior contraction rates via Wasserstein dynamics](https://mlsanity.com/paper/2203.10754/) (2022)

## Abstract (arXiv)

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.