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.

中文翻译：

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西格斯特

《SIAM 评论》，第 65 卷，第 3 期，第 829-829 页，2023 年 8 月。
本期的 SIGEST 文章来自 SIAM/ASA 不确定性量化期刊，文章是乔纳斯·拉茨 (Jonas Latz) 撰写的“贝叶斯逆问题通常是适定的”。作者研究了贝叶斯逆问题方法的适定性，将 Andrew Stuart 引入的适定性框架推广到一组较弱的假设。这里的适定性是按照 Hadamard 的意义来理解的，即解存在、唯一，并且持续依赖于输入数据。由于模型的特性、数据的缺乏和测量噪声，逆问题通常是不适定的。反演问题的贝叶斯方法根据后验分布的探索重新表述了对反演问题解的探索，后验分布由解的数据似然性和先验分布决定，与逆问题本身相比，它应该是适定的。在贝叶斯背景下，适定性通常与后验分布相对于所谓的海灵格距离中的数据的存在性、唯一性和 Lipschitz 连续性相关。在许多实际应用中，验证这种适定性即使不是不可能，也是很困难的。此外，根据当前的问题，选择海灵格距离作为正确的度量可能并不总是最合适的。这为本文奠定了起点，作者介绍了贝叶斯反问题适定性的新框架，在该框架中他展示了一大类贝叶斯反问题的各种度量的存在性、唯一性和连续性，具有非限制性或在实际环境中可验证的条件。本文为贝叶斯逆问题提供了坚实的新数学基础。基本的统计和概率概念得到了全面、易懂的解释，从而为广大读者开放了贝叶斯方法。对于论文的 SIGEST 版本，作者介绍了更多的背景材料，以便更容易为普通读者所理解，并扩展了结论和展望部分，总结了自原始作品发表以来该领域发生的进展，并讨论了未来的研究方向。基本的统计和概率概念得到了全面、易懂的解释，从而为广大读者开放了贝叶斯方法。对于论文的 SIGEST 版本，作者介绍了更多的背景材料，以便更容易为普通读者所理解，并扩展了结论和展望部分，总结了自原始作品发表以来该领域发生的进展，并讨论了未来的研究方向。基本的统计和概率概念得到了全面、易懂的解释，从而为广大读者开放了贝叶斯方法。对于论文的 SIGEST 版本，作者介绍了更多的背景材料，以便更容易为普通读者所理解，并扩展了结论和展望部分，总结了自原始作品发表以来该领域发生的进展，并讨论了未来的研究方向。