An approximate empirical Bayesian method for large-scale linear-Gaussian inverse problems
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1705.07646.pdf
Description
Submitted version
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999.17 KB
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Adobe PDF
Checksum (MD5)
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Author(s)
Zhou, Qingping
Liu, Wenqing
Li, Jinglai
Marzouk, Youssef M
Date Issued
June 2018
Journal
Inverse Problems
Publisher
IOP Publishing
Citation
Zhou, Qingping, et al. "An approximate empirical Bayesian method for large-scale linear-Gaussian inverse problems." Inverse Problems 34, 9 (June 2018) © 2018 IOP Publishing Ltd.
Version
Original manuscript
Abstract
We study Bayesian inference methods for solving linear inverse problems, focusing on hierarchical formulations where the prior or the likelihood function depend on unspecified hyperparameters. In practice, these hyperparameters are often determined via an empirical Bayesian method that maximizes the marginal likelihood function, i.e. the probability density of the data conditional on the hyperparameters. Evaluating the marginal likelihood, however, is computationally challenging for large-scale problems. In this work, we present a method to approximately evaluate marginal likelihood functions, based on a low-rank approximation of the update from the prior covariance to the posterior covariance. We show that this approximation is optimal in a minimax sense. Moreover, we provide an efficient algorithm to implement the proposed method, based on a combination of the randomized SVD and a spectral approximation method to compute square roots of the prior covariance matrix. Several numerical examples demonstrate good performance of the proposed method.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1103/10.1088/1361-6420/aac287
