Efficient iterative methods for hyperparameter estimation in large-scale linear inverse problems
CoRR(2023)
Abstract
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.
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