Analysis of maximum a posteriori estimators: Common convergence theories for Bayesian and variational inverse problems

最大后验估计量分析：贝叶斯和变分逆问题的常见收敛理论

• 批准号: 415980428
• 负责人: Professor Dr. Timothy Sullivan, Ph.D.
• 金额: --
• 依托单位: Mathematics Institute
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/415980428?language=en
• 关键词: Analysis; maximum; posteriori; estimators; Common

This project addresses a mathematical problem of broad impact in contemporary applications that rely upon the accurate calculation of points of maximum probability, known as modes or maximum a posteriori estimators. These arise naturally in models of chemical reactions and in inverse problems of everyday far-reaching importance such as medical imaging, weather and climate prediction, and machine learning. The proposed research project will provide the necessary mathematical analysis to underpin the rigorous treatment of these points in infinite-dimensional spaces, as demanded by modern applications.The full solution to a reaction equation or inverse problem of this kind is a probability distribution over reaction paths, images, weather states etc. However, because this distribution is in general too complicated to be practical, it must often be summarised, reducing the distribution to a single point. In addition to statistics such as means and covariances, a mode is a commonly-used summary of this type, being a "most likely" point under the distribution. However, in the modern case of infinite-dimensional spaces, e.g. the space of all wind and temperature fields over the whole globe, it is not easy to rigorously define such modes. Moreover, there is currently a fundamental disconnect between how these modes are defined using best-fit minimisation problems and the fully Bayesian distributional viewpoint: while a mode is clearly a crude summary of a probability distribution, even the strongest currently-known notions of similarity between probabilities are not sufficient to ensure good approximation of these supposedly simple summaries.The proposed project will provide the missing analysis to bridge this gap by bringing recent advances in inverse problems theory together with the tools of Gamma-convergence from variational calculus, in order to provide a solid mathematical basis for the solution and approximation of maximum a posteriori estimation problems. It will thereby offer robust discretisation-invariant solutions to inverse problems that both have statistically rigorous meaning and also usefully correspond to the needs of lay decision-makers.

该项目解决了一个在当代应用中具有广泛影响的数学问题，该应用依赖于最大概率点的准确计算，称为模式或最大后验估计。 这些自然出现在化学反应模型和日常意义深远的逆问题中，如医学成像、天气和气候预测以及机器学习。 拟议的研究项目将提供必要的数学分析，以支持在无限维空间中严格处理这些点，正如现代应用所要求的那样。反应方程或这类反问题的完整解是反应路径，图像，天气状态等的概率分布。然而，由于这种分布通常过于复杂而不实用，它必须经常被概括，将分布减少到一个点。 除了均值和协方差等统计量之外，众数是这种类型的常用汇总，是分布下的“最可能”点。 然而，在无限维空间的现代情况下，例如，在整个地球仪上的所有风场和温度场的空间中，严格地定义这样的模式是不容易的。 此外，目前在如何使用最佳拟合最小化问题和完全贝叶斯分布观点定义这些模式之间存在根本性的脱节：虽然众数显然是概率分布的粗略概括，即使是目前最强大的-已知的概率之间相似性的概念不足以确保这些假定简单的摘要的良好近似。拟议的项目将提供缺失的分析，通过将反问题理论的最新进展与变分法的Gamma收敛工具结合起来，弥合这一差距，为最大后验估计问题的解决和近似提供坚实的数学基础。 因此，它将提供强大的离散不变的解决方案，既有统计上严格的意义，也有益地对应于外行决策者的需求的反问题。