Hierarchical Bayesian Inverse Problems: A High-Dimensional Statistics Viewpoint

Abstract

This paper analyzes hierarchical Bayesian inverse problems using techniques from highdimensional statistics. Furthermore, our analysis leverages a property of hierarchical Bayesian regularizers that we call approximate decomposability to obtain non-asymptotic bounds on the reconstruction error attained by maximum a posteriori estimators. The new theory explains how hierarchical Bayesian models that exploit sparsity, group sparsity, and sparse representations of the unknown parameter can achieve accurate reconstructions in high-dimensional settings.