SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 1, Page 451-482, January 2020.
The subject of this article is the introduction of a new concept of the well-posedness of Bayesian inverse problems. The conventional concept of (Lipschitz, Hellinger) well-posedness in [A. M. Stuart, Acta Numer., 19 (2010), pp. 451--559] is difficult to verify in practice and may be inappropriate in some contexts. Our concept simply replaces the Lipschitz continuity of the posterior measure in the Hellinger distance by continuity in an appropriate distance between probability measures. Aside from the Hellinger distance, we investigate well-posedness with respect to weak convergence, the total variation distance, the Wasserstein distance, and also the Kullback--Leibler divergence. We demonstrate that the weakening to continuity is tolerable and that the generalization to other distances is important. The main results of this article are proofs of well-posedness with respect to some of the aforementioned distances for large classes of Bayesian inverse problems. Here, little or no information about the underlying model is necessary, making these results particularly interesting for practitioners using black-box models. We illustrate our findings with numerical examples motivated from machine learning and image processing.

中文翻译：

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贝叶斯逆问题的适定性

SIAM / ASA不确定性量化期刊，第8卷，第1期，第451-482页，2020年1月。
本文的主题是介绍贝叶斯逆问题的适定性的新概念。[AM Stuart，Acta Numer。，19（2010），pp.451--559]中的（Lipschitz，Hellinger）适定性的传统概念在实践中难以验证，在某些情况下可能不合适。我们的概念只是通过概率测度之间适当距离的连续性来代替Hellinger距离中后验测度的Lipschitz连续性。除了Hellinger距离外，我们还研究了弱收敛，总变化距离，Wasserstein距离以及Kullback-Leibler散度方面的适定性。我们证明了减弱到连续性是可以容忍的，并且推广到其他距离很重要。本文的主要结果是针对大类贝叶斯逆问题的上述某些距离的适定性证明。在这里，关于基础模型的信息很少或没有必要，这对于使用黑匣子模型的从业者而言尤其有趣。我们以机器学习和图像处理为动机的数值示例说明了我们的发现。