An approximate empirical Bayesian method for large-scale linear-Gaussian inverse problems

作     者：Zhou, Qingping Liu, Wenqing Li, Jinglai Marzouk, Youssef M. 

作者机构：Department of Mathematics Institute of Natural Sciences Shanghai Jiao Tong University Shanghai200240 China Department of Mathematics and Zhiyuan College Shanghai Jiao Tong University Shanghai200240 China Institute of Natural Sciences Department of Mathematics MOE Key Laboratory of Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai200240 China Department of Aeronautics and Astronautics Massachusetts Institute of Technology CambridgeMA02139 United States 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2017年

核心收录：

主　　题：Approximation theory 

摘      要：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 largescale 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. Copyright © 2017, The Authors. All rights reserved.