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We study Bayesian methods for large-scale linear inverse problems, focusingon the challenging task of hyperparameter estimation. Typical hierarchicalBayesian formulations that follow a Markov Chain Monte Carlo approach arepossible for small problems with very few hyperparameters but are notcomputationally feasible for problems with a very large number of unknownparameters. In this work, we describe an empirical Bayesian (EB) method toestimate hyperparameters that maximize the marginal posterior, i.e., theprobability density of the hyperparameters conditioned on the data, and then weuse the estimated values to compute the posterior of the inverse parameters.For problems where the computation of the square root and inverse of priorcovariance matrices are not feasible, we describe an approach based on thegeneralized Golub-Kahan bidiagonalization to approximate the marginal posteriorand seek hyperparameters that minimize the approximate marginal posterior.Numerical results from seismic and atmospheric tomography demonstrate theaccuracy, robustness, and potential benefits of the proposed approach.

Efficient iterative methods for hyperparameter estimation in large-scale linear inverse problems