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Uncertainty Quantification and Modern Statistical Inference

Periodic Reporting for period 4 - UQMSI (Uncertainty Quantification and Modern Statistical Inference)

Reporting period: 2020-03-01 to 2021-02-28

The objectives are the understanding the mathematical underpinnings of modern statistical inference techniques and uncertainty quantification procedures. As such methodology is widely used in sciences/medicine etc it has great long term impact on society. Research on the topics suggested in the grant proposal, and dissemination of research results at conferences, are of vital importance to foster the relation between society and modern data science, and its many cognate disciplines.
Research on the topics suggested in the grant proposal, resulting in fundamental progress that underpins the mathematical theory of statistical uncertainty quantification in modern data science. Several important research papers have been written, submitted and published by the end date of the grant in February 2021. The main focus has been on obtaining rigorous mathematical results that justify the use of Bayesian statistical methodology for inference with complex and high-dimensional data generating processes, such as nonlinear inverse problems arising with partial differential equations. A main line of progress was in terms of a sequence of results published in prestigious journals such as Annals of Statistics, Communications on Pure and Applied Mathematics, Journal of the European Mathematical Society, Inverse Problems, etc., that allow to justify Bayesian `credible sets' computed from posterior distributions in a statistical, frequentist way via Bernstein von Mises theorems. These ultimately rely on statistical `contraction rate theorems'. A focus has been on widely used Gaussian process priors.

Likewise, we have disseminated our research results at in person and remote conferences -- in particular the PI has given several invited lectures at major conferences, explaining to large audiences how the conducted research has changed our understanding of the discipline.
Substantially new results on the understanding of matrix completion procedures, and of Bayesian inference methods for diffusion models and non-linear inverse problems, were obtained. Some of the results obtained have paved the way for a rigorous mathematical theory of Bayesian inference in PDE models and in this regard have gone substantially beyond the state of the art.