We celebrate Sankt Hans Aften with a series of lectures by internationally outstanding researchers, focusing on current developments and challenges in the field of mathematical statistics. There will also be food, drink, and plenty of time and space for discussion.
Marc Hoffmann: Some statistical inference results for interacting particle models in a mean-field limit
We propose a theoretical statistical analysis for systems of interacting diffusions, possibly with common noise and/or degenerate diffusion components, in a mean-field regime. These models are more or less widely used in finance, MFG, systemic risk analysis, behaviourial sociology or ecology. We consider several inference issues such as: i) nonparametric estimation of the solution of the underlying Fokker–Planck type equation or the drift of the system, ii) testing for the interaction between components, iii) estimation of the interaction range between particles. This talk is based on joint results with C. Fonte and L. Della Maestra.
Richard J. Samworth: Optimal nonparametric testing of Missing Completely At Random, and its connections to compatibility
Given a set of incomplete observations, we study the nonparametric problem of testing whether data are Missing Completely At Random (MCAR). Our first contribution is to characterise precisely the set of alternatives that can be distinguished from the MCAR null hypothesis. This reveals interesting and novel links to the theory of Fréchet classes (in particular, compatible distributions) and linear programming, that allow us to propose MCAR tests that are consistent against all detectable alternatives. We define an incompatibility index as a natural measure of ease of detectability, establish its key properties, and show how it can be computed exactly in some cases and bounded in others. Moreover, we prove that our tests can attain the minimax separation rate according to this measure, up to logarithmic factors. Our methodology does not require any complete cases to be effective, and is available in the R package MCARtest. The talk is based on https://arxiv.org/abs/2205.08627.
Johannes Schmidt-Hieber: Statistical learning in biological neural networks
Compared to artificial neural networks (ANNs), the brain learns faster, generalizes better to new situations and consumes much less energy. ANNs are motivated by the functioning of the brain, but differ in several crucial aspects. In particular, it is biologically implausible that the learning of the brain is based on gradient descent. In this talk we look at the brain as a statistical method for supervised learning. The main contribution is to relate the local updating rule of the connection parameters in biological neural networks (BNNs) to a zero-order optimization method. The talk is based on https://arxiv.org/abs/2301.11777.