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

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，该问题依赖于最大概率点的精确计算，称为模态或最大后验估计量。这些自然出现在化学反应模型和具有深远意义的日常反问题中，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，以支持现代应用所要求的对无限维空间中这些点的严格处理。这类反应方程或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，由于这种分布通常太复杂而不实用，因此必须经常对其进行总结，将分布减少到单个点。除了均值和协方差等统计数据外，模态是这种类型的常用摘要，是分布下的“最可能”点。然而，在无限维空间的现代情况下，例如全球所有风场和温度场的空间，很难严格定义这些模态。此外，目前在如何使用最佳拟合最小化问题定义这些模式与完全贝叶斯分布观点之间存在根本的脱节：虽然模式显然是概率分布的粗略总结，但即使是目前已知的最强大的概率相似性概念也不足以确保这些所谓的简单总结的良好近似值。提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微积分的伽玛收敛工具结合起来，弥合这一差距，以便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，它将为逆问题提供稳健的离散不变解决方案，这些解决方案既具有统计上严格的意义，又能有效地满足外行决策者的需求。

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.