On Empirical Bayes Approach to Inverse Problems

Inverse problems arise naturally in many scientific disciplines, such as physics, imaging, tomography, medicine, material sciences and engineering, when one wants to extract information from indirect and noisy measurements. Observed are noisy results of certain (forward) operator evaluated at an element from certain Hilbert space. The general objective is to recover that element using the observed data. Some important inverse problems arise in the area of partial differential equations (PDEs) which describe some physical systems. We consider the inverse problem in a statistical setting and apply an empirical Bayes approach. We address the issue of (local) optimality in the framework of oracle inequalities which is stronger than the traditional minimax (global) optimality. The Bayesian modeling allows to solve a local version of the problem in the oracle formulation, leading to intrinsically adaptive results, i.e., we establish the optimality of the proposed procedure without knowledge of the smoothness/sparsity structure of the underlying function of interest.