Regularization of statistical inverse problems and the Bakushinskii veto

Details Regularization of statistical inverse problems and the Bakushinskii veto Regularization of statistical inverse problems and the Bakushinskii veto

2010-11-16

arxiv.org/abs/1011.3669v1

Inverse Problems 27 (2011) 115010

Mathematics

Statistics Theory

20 pages

Scientific paper

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.

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