Modern Probability theory and its application
4EU+ Summer School
The Summer School is taking place in Warsaw, at the Faculty of Mathematics, Informatics and Mechanics.
University of Warsaw will book reasonably priced rooms (single and double ones) for all willing participants at Hera Guest House. The rooms are equipped with a WC and a wash basin. There is a common bathroom.
25 MA students + 5 PhD students of universities associated within the 4EU + Alliance, from the fields of mathematics and computer science as well as students with an interdisciplinary background in computational science applications.
In order to apply please contact your local 4EU+ Flagship 3 Coordinator. The application deadline is 10 August 2021 - this is the date when the local Coordinators of FL3 should send the list of participants to Warsaw.
There is a 4EU+ mobility program available to support participants to attend the event. The program provides support to mobility and may differ between Universities allied. Ask your local 4EU+ Flagship 3 Coordinator for details.
University of Copenhagen
Causality: Models, Learning, and Invariance
University of Warsaw
Sharp convex geometric inequalities via probabilistic approach
Mixing times of random walks on random graphs
Mellin transform: theory and application to multiplicative measurement error models
Statistical depth: Nonparametric analysis of multivariate data
After a PhD at Paris Diderot University under the supervision of Stéphane Boucheron and Justin Salez, and a postdoc at IMPA in Rio de Janeiro under the supervision of Roberto Oliveira, I am now Maître de Conférences (Assistant Professor) at Sorbonne University in Paris. My research interest are in Probability and Statistics. I am particularly interested in mixing times of Markov chains and in concentration inequalities.
Quantifying the speed of convergence of random walks on graphs is a question of both theoretical and practical interest: fast-mixing random walks allow to sample quickly from potentially huge networks. In some situations, convergence occurs at a very precise time: the distance to equilibrium remains close to 1 for a long time and then abruptly drops to 0. This is the cutoff phenomenon. Clearly, if there is cutoff, one would want to know it. Unfortunately, proving its occurrence on a given graph is often a challenging task. But one might ask: what is the mixing behavior of a RW on a “typical” graph? Our course will be devoted to this question. We will first give a general introduction to Markov chains mixing times and to the cutoff phenomenon, and then focus on the case f random walks on random graphs, mainly random regular graphs and random graphs with given degrees.
Sergio Brenner Miguel received his Bachelors and Masters degrees in Mathematics from the Heidelberg University. Since 2018, he is a PhD student at Heidelberg University, Germany, under the supervision of Jan Johannes. His research interests include non-parametric Statistics, Survival analysis and in particular the multiplicative measurements error model with application to Stochastic Volatility models.
In modern statistics the inference for additive noise models is an intensely researched field where the underlying inverse problem is well-understood. This field borrows several results from functional analysis and approximation theory, for instance the rich theory of Fourier transform and Sobolev spaces. For multiplicative measurements errors the existing strategies are based on a logarithmic transformation of the data which may allow for constructing estimators. However, they impede a deep understanding of the underlying inverse problem. In recent years, the rich theory of Mellin transforms for multiplicative measurement error models has become popular among researchers. This lecture starts with an introduction to the Mellin transform and the corresponding Sobolev space theory. More precisely, we will use the Mellin transform to construct approximations of square- integrable functions and analyze the behavior of the approximation error, the so-called bias under several smoothness assumptions.
The main part of the lecture focuses on the estimation of several quantities of interest based on data with multiplicative measurement error, for example, the density, the survival function and the cumulative distribution function. Here, we will consider both point wise risks and global risk estimation. We will develop the minimax theory and show the adaptivity of the presented estimators. We then proceed to consider dependency structures in the given sample and study their impacts on the resulting risk. For the practical part of this courses, we will use the technique of Monte Carlo simulation to visualize the behaviors of the approximations from the first part of the course and the estimators for the quantities of interest from the second part of the course.
By numerous examples, we will contrast the developed theory with the empirical behavior of the estimators in our simulation study. To do so, we will use the programming language R. More precisely, we will work with R Studio. The participants of this course will receive pre-written code to be able to participate at this practical course without any deeper practical knowledge with the programming language R.
The learning goals of this lecture include a basic understanding of the theory of Mellin transform, with a focus on its usage for non-parametric Statistics. We aim at establishing intuition on the applicability, interpretation, and limitations of such techniques and their role in modern statistical research.
Stanislav Nagy obtained his PhD in 2016 from KU Leuven (Belgium) in Mathematics and Charles University (Czech Republic) in Probability and Mathematical Statistics. His research focuses on nonparametric and robust analysis of multivariate and functional data. He tends to combine statistical theory with approaches from probability, geometry, functional analysis, or machine learning. He has co-authored more than 30 research articles and co-edited two books of conference proceedings. Currently, he works as an Assistant Professor at the Department of Probability and Mathematical Statistics of Charles University in Prague.
The nonparametric statistics of univariate datasets is based on the quantiles, ranks, and order statistics, and by extension, on the natural ordering of the real line. In multivariate and non-Euclidean spaces, however, no canonical ordering of the sample points exists. Thus, the nonparametrics of more complex datasets is much less developed, and already notions such as the median are not possible to be defined directly. Statistical depth is a tool devised to address those problems. For a given point x and a (probability) measure P on the same space, the depth D is an index that evaluates how much "centrally positioned", or "deep", the point is with respect to the geometry of the mass of P. The higher the depth D(x;P) is, the more representative of P the point x is. That way, it is possible to order points in general spaces in a centre-outwards sense, from the most typical points to the peripheral ones. A depth allows us to generalise quantiles. For example, for given P, the point attaining the maximum depth value with respect to P can be seen as a multivariate analogue of the median of P. The depth is not the density - while the density at x involves probabilities of small balls around x only and is, therefore, a local concept, the depth as a generalisation of quantiles is defined globally, reflecting the shape of the whole distribution P. There is no unique depth function - many different depths have been proposed in the past 30 years in the literature, with entirely different characteristics and uses. In the series of talks, we introduce the statistical depth in Euclidean spaces, explore its properties, and consider applications to data analysis. We draw connections of depth functions to the research outside multivariate statistics, particularly in geometry and machine learning. Depths for non-Euclidean data, such as the depths in functional data analysis or the depth for non-linear datasets will also be considered. We will see that many interesting open problems still stimulate active research in this area of statistics.
Piotr Nayar obtained his Ph.D. from the University of Warsaw in 2014, working under the supervision of Krzysztof Oleszkiewicz. In 2014-2015 he was a postdoctoral research assistant in the Institute for Mathematics and its Applications at University of Minnesota. He then spent one year working as a postdoctoral researcher at University of Pennsylvania. In 2016-2017 he held a postdoctoral position at the Technion, before joining the Institute of Mathematics at University of Warsaw as an assistant professor. His research interests include probability, convex geometry, and inequalities in information theory.
Understanding geometry of convex sets is a fundamental problem in mathematics, both from theoretical and applied point of view. If the dimension is large, one usually needs nontrivial tools to be able to attack even the most basic problems. Several such tools are based on probability theory. We shall focus on the techniques related to Gaussian random vectors and use them to analyze extremal sections of balls in the p-th norm in Rn. This is done by providing a link between these geometric problems and the so-called Khinchine type probabilistic inequalities. We also provide a relation between the cube slicing problems and the logarithmic Brunn-Minkowski conjecture. As a motivation the classical Brunn-Minkowski inequality will be discussed together with its most famous applications, including the solution to the isoperimetric problem. In the second part of the lecture series we describe a link between convex geometry and information theory. We shall focus on the Shannon differential entropy and introduce basic properties of this functional. We will discuss the Entropy Power Inequality together with its consequences and mention several related open problems. It is assumed that the audience is familiar with linear algebra, basic probability theory (including Gaussian random vectors in Rn) and basic measure theory (integration in Rn).
Jonas is professor in statistics at the Department of Mathematical Sciences at the University of Copenhagen. Previously, he has been at the Max-Planck-Institute for Intelligent Systems in Tuebingen and at the Seminar for Statistics, ETH Zurich. He studied Mathematics at the University of Heidelberg and the University of Cambridge and obtained his PhD jointly from MPI and ETH. He is interested in inferring causal relationships from different types of data and in building statistical methods that are robust with respect to distributional shifts. In his research, Jonas seeks to combine theory, methodology, and applications. His work relates to areas such as computational statistics, causal inference, graphical models, independence testing or high-dimensional statistics.
In science, we often want to understand how a system reacts under interventions (e.g., under gene knock-out experiments or a change of policy). These questions go beyond statistical dependences and can therefore not be answered by standard regression or classification techniques. In this tutorial we will learn about the powerful language of causality and recent developments in the field. No prior knowledge about causality is required.
More precisely, we introduce structural causal models and formalize interventional distributions. We define causal effects and show how to compute them if the causal structure is known. We discuss assumptions under which causal structure becomes identifiable from observational (and interventional) data and describe corresponding methodology. If time allows, we present connections between causality and distributional robustness.