Regularization of statistical inverse problems and the Bakushinskii veto

TL;DR

This paper demonstrates that Bakushinskii's veto, which restricts data-driven regularization in deterministic inverse problems, does not extend to the statistical setting, allowing for new approaches in statistical inverse problem regularization.

Contribution

The work shows that Bakushinskii's theorem does not hold in the statistical context and introduces new concepts for analyzing statistical inverse problems.

Findings

Bakushinskii's veto does not apply to statistical inverse problems

New concepts for statistical regularization are developed

Counterexamples or reformulations depend on probability distribution classes

Abstract

In the deterministic context Bakushinskii's theorem excludes the existence of purely data driven convergent regularization for ill-posed problems. We will prove in the present work that in the statistical setting we can either construct a counter example or develop an equivalent formulation depending on the considered class of probability distributions. Hence, Bakushinskii's theorem does not generalize to the statistical context, although this has often been assumed in the past. To arrive at this conclusion, we will deduce from the classic theory new concepts for a general study of statistical inverse problems and perform a systematic clarification of the key ideas of statistical regularization.