SIGEST

SIAM Review, Volume 65, Issue 3, Page 829-829, August 2023.
The SIGEST article in this issue, which comes from the SIAM/ASA Journal on Uncertainty Quantification, is “Bayesian Inverse Problems Are Usually Well-Posed,” by Jonas Latz. The author investigates the well-posedness of Bayesian approaches to inverse problems, generalizing the framework of well-posedness introduced by Andrew Stuart to a set of weaker assumptions. Well-posedness here is understood in the sense of Hadamard, that is, a solution exists, is unique, and continuously depends on the input data. Inverse problems are typically ill-posed due to properties of the model, a lack of data, and measurement noise. The Bayesian approach to inverse problems reformulates the quest for a solution to the inverse problem in terms of a quest for its posterior distribution, which is determined by the data likelihood and prior distribution of the solution, and which in contrast to the inverse problem itself should be well-posed. In the Bayesian context, well-posedness typically relates to existence, uniqueness, and Lipschitz continuity of the posterior distribution with respect to the data in the so-called Hellinger distance. In many practical applications such well-posedness is difficult, if not impossible, to verify. Moreover, the choice of the Hellinger distance as the right metric might not always be the best fitted depending on the problem at hand. This sets the starting point for the paper where the author introduces a new framework for well-posedness of Bayesian inverse problems in which he shows existence, uniqueness, and continuity with respect to various metrics for a large class of Bayesian inverse problems, with conditions that are either nonrestrictive or verifiable in practical settings. This paper gives a strong new mathematical foundation for Bayesian inverse problems. The underlying statistical and probabilistic concepts are explained comprehensively and comprehensibly and, thus, in a way that opens up the Bayesian approach for a large readership. For the SIGEST version of the paper the author introduced more background material to make it more accessible to a general audience and extended the conclusion and outlook section, summarizing developments in the field that happened since the publication of the original work and discussing future research directions.