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Bayesian Inverse Problems Are Usually Well-Posed
SIAM Review ( IF 10.2 ) Pub Date : 2023-08-08 , DOI: 10.1137/23m1556435 Jonas Latz
SIAM Review ( IF 10.2 ) Pub Date : 2023-08-08 , DOI: 10.1137/23m1556435 Jonas Latz
SIAM Review, Volume 65, Issue 3, Page 831-865, August 2023.
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.
中文翻译：
贝叶斯逆问题通常是适定的
《SIAM 评论》，第 65 卷，第 3 期，第 831-865 页，2023 年 8 月。
逆问题描述了将数学模型与观测数据相结合的任务——这是许多科学和工程学科的基本任务。此类任务的可解决性通常通过其适定性进行分类。如果一个问题有一个持续依赖于输入或数据的唯一解决方案，则该问题是适定的。逆问题通常是不适定的，但有时可以通过制定可能适定问题的方法来解决。常用的方法是解决反问题的变分法和贝叶斯法。对于贝叶斯方法，Stuart [Acta Numer., 19 (2010), pp. 451--559] 给出了假设，在该假设下，后验测度（贝叶斯逆问题的解）存在、唯一且是 Lipschitz相对于海灵格距离是连续的，因此是适定的。在这项工作中，我们简化并概括了这个概念：事实上，我们通过证明 Hellinger 距离、Wasserstein 距离和总变差距离的存在性、唯一性和连续性，以及相对于弱收敛性，分别在显着较弱的情况下证明了适定性。假设。大量实际相关的贝叶斯逆问题满足这些条件。通常可以在不分析底层数学模型的情况下验证条件——该模型可以被视为黑匣子。大量实际相关的贝叶斯逆问题满足这些条件。通常可以在不分析底层数学模型的情况下验证条件——该模型可以被视为黑匣子。大量实际相关的贝叶斯逆问题满足这些条件。通常可以在不分析底层数学模型的情况下验证条件——该模型可以被视为黑匣子。
更新日期：2023-08-08
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.
中文翻译：
贝叶斯逆问题通常是适定的
《SIAM 评论》，第 65 卷，第 3 期，第 831-865 页，2023 年 8 月。
逆问题描述了将数学模型与观测数据相结合的任务——这是许多科学和工程学科的基本任务。此类任务的可解决性通常通过其适定性进行分类。如果一个问题有一个持续依赖于输入或数据的唯一解决方案，则该问题是适定的。逆问题通常是不适定的，但有时可以通过制定可能适定问题的方法来解决。常用的方法是解决反问题的变分法和贝叶斯法。对于贝叶斯方法，Stuart [Acta Numer., 19 (2010), pp. 451--559] 给出了假设，在该假设下，后验测度（贝叶斯逆问题的解）存在、唯一且是 Lipschitz相对于海灵格距离是连续的，因此是适定的。在这项工作中，我们简化并概括了这个概念：事实上，我们通过证明 Hellinger 距离、Wasserstein 距离和总变差距离的存在性、唯一性和连续性，以及相对于弱收敛性，分别在显着较弱的情况下证明了适定性。假设。大量实际相关的贝叶斯逆问题满足这些条件。通常可以在不分析底层数学模型的情况下验证条件——该模型可以被视为黑匣子。大量实际相关的贝叶斯逆问题满足这些条件。通常可以在不分析底层数学模型的情况下验证条件——该模型可以被视为黑匣子。大量实际相关的贝叶斯逆问题满足这些条件。通常可以在不分析底层数学模型的情况下验证条件——该模型可以被视为黑匣子。
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