TY - JOUR

T1 - Bayesian Inverse Problems Are Usually Well-Posed

AU - Latz, Jonas

N1 - Funding Information:
\ast Published electronically August 8, 2023. This paper originally appeared in SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Number 1, 2020, pages 451--482, under the title ``On the Well-Posedness of Bayesian Inverse Problems."" https://doi.org/10.1137/23M1556435 Funding: The original work [54] was supported by DFG and Technische Universita\"t Mu\"nchen through the International Graduate School of Science and Engineering within project 10.02 BAYES. \dagger Maxwell Institute for Mathematical Sciences and School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK (j.latz@hw.ac.uk).
Funding Information:
The original work [54] was supported by DFG and Technische Universität München through the International Graduate School of Science and Engineering within project 10.02 BAYES.
Publisher Copyright:
© 2023 Society for Industrial and Applied Mathematics.

PY - 2023/8

Y1 - 2023/8

N2 - Inverse problems describe the task of blending a mathematical model with observational data-a fundamental task in many scientific and engineering disciplines. The solvability of such a task is usually classified through its well-posedness. A problem is well-posed if it has a unique solution that depends continuously on input or data. Inverse problems are usually ill-posed, but can sometimes be approached through a methodology that formulates a possibly well-posed problem. Usual methodologies are the variational and the Bayesian approach to inverse problems. For the Bayesian approach, Stuart [Acta Numer., 19 (2010), pp. 451-559] has given assumptions under which the posterior measure-the Bayesian inverse problem's solution-exists, is unique, and is Lipschitz continuous with respect to the Hellinger distance and, thus, well-posed. In this work, we simplify and generalize this concept: Indeed, we show well-posedness by proving existence, uniqueness, and continuity in Hellinger distance, Wasserstein distance, and total variation distance, and with respect to weak convergence, respectively, under significantly weaker assumptions. An immense class of practically relevant Bayesian inverse problems satisfies those conditions. The conditions can often be verified without analyzing the underlying mathematical model-the model can be treated as a black box.

AB - Inverse problems describe the task of blending a mathematical model with observational data-a fundamental task in many scientific and engineering disciplines. The solvability of such a task is usually classified through its well-posedness. A problem is well-posed if it has a unique solution that depends continuously on input or data. Inverse problems are usually ill-posed, but can sometimes be approached through a methodology that formulates a possibly well-posed problem. Usual methodologies are the variational and the Bayesian approach to inverse problems. For the Bayesian approach, Stuart [Acta Numer., 19 (2010), pp. 451-559] has given assumptions under which the posterior measure-the Bayesian inverse problem's solution-exists, is unique, and is Lipschitz continuous with respect to the Hellinger distance and, thus, well-posed. In this work, we simplify and generalize this concept: Indeed, we show well-posedness by proving existence, uniqueness, and continuity in Hellinger distance, Wasserstein distance, and total variation distance, and with respect to weak convergence, respectively, under significantly weaker assumptions. An immense class of practically relevant Bayesian inverse problems satisfies those conditions. The conditions can often be verified without analyzing the underlying mathematical model-the model can be treated as a black box.

KW - Bayesian inference

KW - inverse problems

KW - Kullback-Leibler divergence

KW - total variation

KW - Wasserstein

KW - well-posedness

UR - http://www.scopus.com/inward/record.url?scp=85169893977&partnerID=8YFLogxK

U2 - 10.1137/23M1556435

DO - 10.1137/23M1556435

M3 - Article

AN - SCOPUS:85169893977

SN - 0036-1445

VL - 65

SP - 831

EP - 865

JO - SIAM Review

JF - SIAM Review

IS - 3

ER -