  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 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.
