Data-informed posterior approximation for Bayesian linear inverse problems

Haibo Li

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arXiv · 2605.20770

Data-informed posterior approximation for Bayesian linear inverse problems

Haibo Li

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Abstract

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.

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BibTeX RIS

Haibo Li. 2026-05-20. Data-informed posterior approximation for Bayesian linear inverse problems. https://arxiv.org/abs/2605.20770

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