Data-informed posterior approximation for Bayesian linear inverse problems

TL;DR

This paper introduces a data-informed framework for Bayesian linear inverse problems that reduces computational complexity by focusing on a low-dimensional data space and employs a novel Krylov subspace method for efficient posterior approximation.

Contribution

It develops a rigorous data-informed approach that shifts the inference from parameter to data space and introduces a quotient-space Krylov method with integrated hyperparameter estimation.

Findings

The data-informed subspace is low-dimensional and captures the posterior update.

The proposed method achieves matrix-free posterior approximation and hyperparameter estimation.

Numerical experiments validate the theoretical framework and demonstrate effectiveness.

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.