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：我们研究大规模线性逆问题的贝叶斯方法，重点关注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遵循马尔可夫链蒙特卡罗方法，适用于超参数极少的小规模问题，但无法在未知参数数量巨大的问题中实现计算可行性。本文描述了一种经验贝叶斯方法，通过最大化边缘后验（即超参数在数据条件下的概率密度）来估计超参数，并利用估计值计算逆参数的后验。针对先验协方差矩阵的平方根与逆矩阵无法计算的问题，我们提出了一种基于广义Golub-Kahan双对角化方法，用以近似边缘后验，并寻找使近似边缘后验最小化的超参数。地震层析成像和大气层析成像的数值结果表明了该方法的准确性、鲁棒性和潜在优势。