Hierarchical Bayesian Inverse Problems: A High-Dimensional Statistics Viewpoint

Abstract

SIAM Review, Volume 67, Issue 3, Page 543-575, August 2025.
Abstract.This paper analyzes hierarchical Bayesian inverse problems using techniques from high-dimensional statistics. Our analysis leverages a property of hierarchical Bayesian regularizers that we call approximate decomposability to obtain nonasymptotic 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.