Data-informed posterior approximation for Bayesian linear inverse problems
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910239826116608 |
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| author | Li, Haibo |
| author_facet | Li, Haibo |
| contents | Computing posterior distributions in large-scale Bayesian linear inverse problems is challenging due to the high dimensionality of the parameter space. In this work, we develop a data-informed framework that shifts the computational focus from the parameter space to the data space. We rigorously characterize an intrinsically low-dimensional data space, establish its isometric embedding into the parameter space, and show that the prior-to-posterior update is confined to a data-informed subspace. This perspective allows posterior inference to be carried out in a reduced data-informed subspace. Based on this formulation, we propose a quotient-space Golub--Kahan bidiagonalization method to construct data-informed Krylov subspaces, and integrate empirical Bayesian inference into the iterative framework, enabling simultaneous hyperparameter estimation and posterior approximation in a matrix-free manner. Numerical experiments on representative problems support the theoretical framework and demonstrate the effectiveness of the resulting method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20770 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Data-informed posterior approximation for Bayesian linear inverse problems Li, Haibo Numerical Analysis Computing posterior distributions in large-scale Bayesian linear inverse problems is challenging due to the high dimensionality of the parameter space. In this work, we develop a data-informed framework that shifts the computational focus from the parameter space to the data space. We rigorously characterize an intrinsically low-dimensional data space, establish its isometric embedding into the parameter space, and show that the prior-to-posterior update is confined to a data-informed subspace. This perspective allows posterior inference to be carried out in a reduced data-informed subspace. Based on this formulation, we propose a quotient-space Golub--Kahan bidiagonalization method to construct data-informed Krylov subspaces, and integrate empirical Bayesian inference into the iterative framework, enabling simultaneous hyperparameter estimation and posterior approximation in a matrix-free manner. Numerical experiments on representative problems support the theoretical framework and demonstrate the effectiveness of the resulting method. |
| title | Data-informed posterior approximation for Bayesian linear inverse problems |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2605.20770 |