SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911538921603072 |
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| author | Alberola-Boloix, Enric Casado-Telletxea, Ioar |
| author_facet | Alberola-Boloix, Enric Casado-Telletxea, Ioar |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_22468 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation Alberola-Boloix, Enric Casado-Telletxea, Ioar Machine Learning Statistics Theory 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. |
| title | SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2603.22468 |