Nonparametric Bayesian Inference for Stochastic Reaction-Diffusion Equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918087587004416 |
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| author | Altmeyer, Randolf Gaudlitz, Sascha |
| author_facet | Altmeyer, Randolf Gaudlitz, Sascha |
| contents | We consider the Bayesian nonparametric estimation of a nonlinear reaction function in a reaction-diffusion stochastic partial differential equation (SPDE). The likelihood is well-defined and tractable by the infinite-dimensional Girsanov theorem, and the posterior distribution is analysed in the growing domain asymptotic. Based on a Gaussian wavelet prior, the contraction of the posterior distribution around the truth at the minimax optimal rate is proved. The analysis of the posterior distribution is complemented by a semiparametric Bernstein--von Mises theorem. The proofs rely on the sub-Gaussian concentration of spatio-temporal averages of transformations of the SPDE, which is derived by combining the Clark-Ocone formula with bounds for the derivatives of the (marginal) densities of the SPDE. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06857 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonparametric Bayesian Inference for Stochastic Reaction-Diffusion Equations Altmeyer, Randolf Gaudlitz, Sascha Statistics Theory Probability Primary 62G20, 60H15, secondary 60H07, 62F15 We consider the Bayesian nonparametric estimation of a nonlinear reaction function in a reaction-diffusion stochastic partial differential equation (SPDE). The likelihood is well-defined and tractable by the infinite-dimensional Girsanov theorem, and the posterior distribution is analysed in the growing domain asymptotic. Based on a Gaussian wavelet prior, the contraction of the posterior distribution around the truth at the minimax optimal rate is proved. The analysis of the posterior distribution is complemented by a semiparametric Bernstein--von Mises theorem. The proofs rely on the sub-Gaussian concentration of spatio-temporal averages of transformations of the SPDE, which is derived by combining the Clark-Ocone formula with bounds for the derivatives of the (marginal) densities of the SPDE. |
| title | Nonparametric Bayesian Inference for Stochastic Reaction-Diffusion Equations |
| topic | Statistics Theory Probability Primary 62G20, 60H15, secondary 60H07, 62F15 |
| url | https://arxiv.org/abs/2507.06857 |