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

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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.