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.

该项目解决了在当代应用中的广泛影响的数学问题，该问题依赖于最大概率（称为模式或最大后验估计器）的准确计算。 这些自然出现在化学反应的模型和日常深度重要的重要性的逆问题中，例如医学成像，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，以基于现代应用所需的无限维空间中对这些点的严格处理。对这种反应方程或这种逆问题的完整解决方案是反应路径，图像，天气状态等的概率分布。但是，这种分布通常是实用的，因为它通常是实用的，因此它通常是为了分配，因此分配了一个分配，而不是单个分配。 除了统计数据（例如均值和协方差）外，一种模式是这种类型的常用摘要，是该分布下的“最有可能”点。 但是，在现代的无限维空间中，例如全球范围内所有风和温度场的空间，严格定义此类模式并不容易。 此外，目前在这些模式使用最佳最小化问题定义的方式与完全贝叶斯分布的观点之间存在基本脱节：虽然一种模式显然是概率分布的粗略总结，即使是当前最有力的概率概念，即使是当前最著名的概率概念，即使是在这些问题上都不足以确保这些问题的分析能够构成这些概述。借助各变化的伽马连接工具，为了为解决方案和最大后验估计问题的溶液和近似值提供稳固的数学基础。 因此，它将为反问题提供强大的离散不变解决方案，这些解决方案在统计上具有严格的含义，并且也与外行决策者的需求相对应。