Bayesian Inverse Problems Are Usually Well-Posed

Jonas Latz*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)
65 Downloads (Pure)

Abstract

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.

Original languageEnglish
Pages (from-to)831-865
Number of pages35
JournalSIAM Review
Volume65
Issue number3
Early online date8 Aug 2023
DOIs
Publication statusPublished - Aug 2023

Keywords

  • Bayesian inference
  • inverse problems
  • Kullback-Leibler divergence
  • total variation
  • Wasserstein
  • well-posedness

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Mathematics
  • Applied Mathematics

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