Four lectures on probabilistic methods for data science

Methods of high-dimensional probability play a central role in applications for statistics, signal processing theoretical computer science and related fields. These lectures present a sample of particularly useful tools of high-dimensional probability, focusing on the classical and matrix Bernstein's inequality and the uniform matrix deviation inequality. We illustrate these tools with applications for dimension reduction, network analysis, covariance estimation, matrix completion and sparse signal recovery. The lectures are geared towards beginning graduate students who have taken a rigorous course in probability but may not have any experience in data science applications.Comment: Lectures given at 2016 PCMI Graduate Summer School in Mathematics of Data. Some typos, inaccuracies fixe

A Conversation with Eugenio Regazzini

Eugenio Regazzini was born on August 12, 1946 in Cremona (Italy), and took his degree in 1969 at the University "L. Bocconi" of Milano. He has held positions at the universities of Torino, Bologna and Milano, and at the University "L. Bocconi" as assistant professor and lecturer from 1974 to 1980, and then professor since 1980. He is currently professor in probability and mathematical statistics at the University of Pavia. In the periods 1989-2001 and 2006-2009 he was head of the Institute for Applications of Mathematics and Computer Science of the Italian National Research Council (C.N.R.) in Milano and head of the Department of Mathematics at the University of Pavia, respectively. For twelve years between 1989 and 2006, he served as a member of the Scientific Board of the Italian Mathematical Union (U.M.I.). In 2007, he was elected Fellow of the IMS and, in 2001, Fellow of the "Istituto Lombardo---Accademia di Scienze e Lettere." His research activity in probability and statistics has covered a wide spectrum of topics, including finitely additive probabilities, foundations of the Bayesian paradigm, exchangeability and partial exchangeability, distribution of functionals of random probability measures, stochastic integration, history of probability and statistics. Overall, he has been one of the most authoritative developers of de Finetti's legacy. In the last five years, he has extended his scientific interests to probabilistic methods in mathematical physics; in particular, he has studied the asymptotic behavior of the solutions of equations, which are of interest for the kinetic theory of gases. The present interview was taken in occasion of his 65th birthday.Comment: Published in at http://dx.doi.org/10.1214/11-STS362 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Professor Zdzisław Hellwig (1925–2013) a Giant in the Renaissance Style

On 8th November 2013, with great sadness we said goodbye to our dear Master and Teacher, Professor Zdzisław Hellwig. He walked away from us forever. Professor Zdzisław Hellwig (1925 – 2013) was a great man with impressive biography. Primarily professor Zdzisław Hellwig was prominent, widely recognized, eminent scholar of international standing in the field of statistics. His most important works are Elements of probability and mathematical statistics, Linear Regression and its applications in economics and Stochastic approximation. On 8th November 2013, with great sadness we said goodbye to our dear Master and Teacher, Professor Zdzisław Hellwig. He walked away from us forever. Professor Zdzisław Hellwig (1925 – 2013) was a great man with impressive biography. Primarily professor Zdzisław Hellwig was prominent, widely recognized, eminent scholar of international standing in the field of statistics. His most important works are Elements of probability and mathematical statistics, Linear Regression and its applications in economics and Stochastic approximation. His second field of achievements was econometrics. The rich scientific achievements in the field of econometrics of Professor Zdzisław Hellwig cover numerous studies dealing with the theory and application, including modeling of the socio – economic development, economic forecasting, and multidimensional comparative analysis and taxsonometrics. Professor Zdzisław Hellwig has a standing as economist. Professor Zdzisław Hellwig is a precursor of research in the field referred to as sustainable development, and early warning system for the national economy. He is considered pioneer of computer science in Poland. His international activities gave him the global scholar rank. Professor Zdzisław Hellwig was exceptionally gifted teacher and educator with long list of prominent followers. He has notable achievements as an organizer. Achievements of Professor Zdzisław Hellwig were widely acknowledged, both in the home university, countywide and abroad.

Changes from Classical Statistics to Modern Statistics and Data Science

A coordinate system is a foundation for every quantitative science, engineering, and medicine. Classical physics and statistics are based on the Cartesian coordinate system. The classical probability and hypothesis testing theory can only be applied to Euclidean data. However, modern data in the real world are from natural language processing, mathematical formulas, social networks, transportation and sensor networks, computer visions, automations, and biomedical measurements. The Euclidean assumption is not appropriate for non Euclidean data. This perspective addresses the urgent need to overcome those fundamental limitations and encourages extensions of classical probability theory and hypothesis testing , diffusion models and stochastic differential equations from Euclidean space to non Euclidean space. Artificial intelligence such as natural language processing, computer vision, graphical neural networks, manifold regression and inference theory, manifold learning, graph neural networks, compositional diffusion models for automatically compositional generations of concepts and demystifying machine learning systems, has been rapidly developed. Differential manifold theory is the mathematic foundations of deep learning and data science as well. We urgently need to shift the paradigm for data analysis from the classical Euclidean data analysis to both Euclidean and non Euclidean data analysis and develop more and more innovative methods for describing, estimating and inferring non Euclidean geometries of modern real datasets. A general framework for integrated analysis of both Euclidean and non Euclidean data, composite AI, decision intelligence and edge AI provide powerful innovative ideas and strategies for fundamentally advancing AI. We are expected to marry statistics with AI, develop a unified theory of modern statistics and drive next generation of AI and data science.Comment: 37 page