Extra Seminar Mathematics- Dr. E.Belitser

Title: Uncertainty Quantification in high-dimensional models

Abstract:
The topic Uncertainty Quantification (UQ) has actually been emerging simultaneously in many areas: besides mathematical statistics, also in numerical analysis, computational applied mathematics, numerical solutions of stochastic PDE (and PDE’s), physics, astronomy, finance, geophysics, life sciences, epidemiology/public health, and many other fields. The notion of UQ has different meanings in different fields, statistics has an advantage that this notion has already been developed and studied in the context of confidence sets. Nowadays, the interest to this topic within statistical community has been revived and even intensified. The problem is recognized to be as very challenging even for standard high-dimensional settings, as one has to deal with the rather subtle "deceptiveness phenomenon", which basically means impossibility of simultaneous fulfillment of the uniform coverage and optimal size properties for any confidence ball.

We propose a general framework of projection structures and study the UQ problem within this framework by using empirical Bayes and penalization methods. The approach is local, robust and provides refined non-asymptotic exponential probabilistic bounds. To address the deceptiveness phenomenon, we introduce the novel "excessive bias restriction" under which we establish the local (oracle) confidence optimality of the constructed confidence ball. The obtained general results deliver a whole avenue of results (many new ones and some known in the literature) for many important particular models and structures as consequences, with examples of structures such as smoothness, sparsity and biclustering.