SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation

Enric Alberola-Boloix; Ioar Casado-Telletxea

stat.MLMar 23, 2026

SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation

Authors: Enric Alberola-Boloix, Ioar Casado-Telletxea

Organizations: Basque Center for Applied Mathematics (BCAM), Bilbao (Spain)

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.

Posterior Contraction

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May 19, 2026J/K move · Enter open · S save

Corrected Integrated Laplace Approximation for Bayesian Inference in Latent Gaussian Models

Jinlin Lai, Charles C. Margossian, Daniel R. Sheldon

Manning College of Information and Computer Sciences University of Massachusetts Amherst · Department of Statistics University of British Columbia

Latent Variable Models Importance Sampling Bayesian Inference Hierarchical Bayesian Modeling

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Latent Gaussian models (LGMs) are a popular class of Bayesian hierarchical models that include Gaussian processes, as well as certain spatial models and mixed-effect models. Efficient Bayesian inference of LGMs often requires marginalizing out the latent variables. For LGMs with a non-Gaussian likelihood, exact marginalization is not possible and a popular approach is to do approximate marginalization with an integrated Laplace approximation (ILA). Using ILA produces an approximate posterior which, in some settings, can differ significantly from the correct posterior, which impacts downstream applications. We propose an importance sampling scheme to correct the error introduced by ILA. By increasing the number of samples in importance sampling, the posterior with ILA converges to the correct posterior. This idea is realized with various techniques, including pseudo-marginalization, quasi-Monte Carlo and randomized quasi-Monte Carlo. We implement our methods in an automatic differentiation framework to support gradient-based algorithms when doing inference on the hyperparameters. For the latter, we specifically consider the use of Hamiltonian Monte Carlo. We demonstrate the benefits of reduced error in various applied models.

Abstract

Posterior Contraction

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