Efficient marginalization-based MCMC methods for hierarchical bayesian inverse problems

Hierarchical models in Bayesian inverse problems are characterized by an assumed prior probability distribution for the unknown state and measurement error precision, and hyper-priors for the prior parameters. Combining these probability models using Bayes' law often yields a posterior distribution that cannot be sampled from directly, even for a linear model with Gaussian measurement error and Gaussian prior, both of which we assume in this paper. In such cases, Gibbs sampling can be used to sample from the posterior [Bardsley, SIAM J. Sci. Comput., 34 (2012), pp. A1316-A1332], but problems arise when the dimension of the state is large. This is because the Gaussian sample required for each iteration can be prohibitively expensive to compute, and because the statistical efficiency of the Markov chain degrades as the dimension of the state increases. The latter problem can be mitigated using marginalization-based techniques, such as those found in [Fox and Norton, SIAM/ASA J. Uncertain. Quantif., 4 (2016), pp. 1191-1218; Joyce, Bardsley, and Luttman, SIAM J. Sci. Comput., 40 (2018), pp. B766-B787; Rue and Held, Monogr. Statist. Appl. Probab. 104, Chapman & Hall/CRC, Boca Raton, FL, 2005], but these can be computationally prohibitive as well. In this paper, we combine the low-rank techniques of [Brown, Saibaba, and Vallelian, SIAM/ASA J. Uncertain. Quantif., 6 (2018), pp. 1076-1100] with the marginalization approach of [Rue and L. Held, Monogr. Statist. Appl. Probab. 104, Chapman & Hall/CRC, Boca Raton, FL, 2005]. We consider two variants of this approach: delayed acceptance and pseudomarginalization. We provide a detailed analysis of the acceptance rates and computational costs associated with our proposed algorithms and compare their performances on two numerical test cases-image deblurring and inverse heat equation.