We study Bayesian methods for large-scale linear inverse problems, focusing on the challenging task of hyperparameter estimation. Typical hierarchical Bayesian formulations that follow a Markov Chain Monte Carlo approach are possible for small problems with very few hyperparameters but are not computationally feasible for problems with a very large number of unknown parameters. In this work, we describe an empirical Bayesian (EB) method to estimate hyperparameters that maximize the marginal posterior, i.e., the probability density of the hyperparameters conditioned on the data, and then we use the estimated values to compute the posterior of the inverse parameters. For problems where the computation of the square root and inverse of prior covariance matrices are not feasible, we describe an approach based on the generalized Golub-Kahan bidiagonalization to approximate the marginal posterior and seek hyperparameters that minimize the approximate marginal posterior. Numerical results from seismic and atmospheric tomography demonstrate the accuracy, robustness, and potential benefits of the proposed approach.

翻译：我们研究针对大规模线性逆问题的贝叶斯方法，重点聚焦于超参数估计这一具有挑战性的任务。典型的层次贝叶斯公式采用马尔可夫链蒙特卡洛方法，仅适用于超参数极少的简单问题，但在未知参数规模庞大的复杂问题中不具备计算可行性。本文提出一种经验贝叶斯方法，通过最大化边缘后验（即给定数据条件下超参数的概率密度）来估计超参数，继而利用这些估计值计算逆问题的后验参数。针对先验协方差矩阵的平方根及逆矩阵无法直接计算的场景，我们提出基于广义Golub-Kahan双对角化技术的近似方法，通过最小化近似边缘后验来估计超参数。地震层析成像与大气层析成像的数值实验表明，该方法在精度、鲁棒性及潜在应用价值方面均具有显著优势。