  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. Gibbs sampling can be used to sample from
the posterior, 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, but these can
be computationally prohibitive as well. In this paper, we combine the low-rank
techniques of Brown, Saibaba, and Vallelian (2018) with the marginalization
approach of Rue and Held (2005). We consider two variants of this approach:
delayed acceptance and pseudo-marginalization. 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.
