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We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. Starting from ZFC, the exposition in this first part includes relation and order theory as well as a construction of…
In applied mathematics and related disciplines, the modeling-simulation-optimization workflow is a prominent scheme, with mathematical models and numerical algorithms playing a crucial role. For these types of mathematical research data,…
This paper revisits the principle of uniform convergence in statistical learning, discusses how it acts as the foundation behind machine learning, and attempts to gain a better understanding of the essential problem that current deep…
Statistical learning theory is the foundation of machine learning, providing theoretical bounds for the risk of models learned from a (single) training set, assumed to issue from an unknown probability distribution. In actual deployment,…
Deeper insight leads to better practice. We show how the study of the foundations of quantum mechanics has led to new pictures of open systems and to a method of computation which is practical and can be used where others cannot. We…
It is argued from several points of view that quantum probabilities might play a role in statistical settings. New approaches toward quantum foundations have postulates that appear to be equally valid in macroscopic settings. One such…
This book introduces the mathematical foundations and techniques that lead to the development and analysis of many of the algorithms that are used in machine learning. It starts with an introductory chapter that describes notation used…
Machine learning algorithms learn a desired input-output relation from examples in order to interpret new inputs. This is important for tasks such as image and speech recognition or strategy optimisation, with growing applications in the IT…
This book covers a wide range of problems involving the applications of stochastic processes, stochastic calculus, large deviation theory, group representation theory and quantum statistics to diverse fields in dynamical systems,…
This introduction to Bayesian statistics presents the main concepts as well as the principal reasons advocated in favour of a Bayesian modelling. We cover the various approaches to prior determination as well as the basis asymptotic…
This is a writeup of lectures on "statistics" that have evolved from the initial version for the 2009 Hadron Collider Physics Summer School at CERN to versions for other venues and, most recently, for the African School of Fundamental…
Statistics has moved beyond the frequentist-Bayesian controversies of the past. Where does this leave our ability to interpret results? I suggest that a philosophy compatible with statistical practice, labeled here statistical pragmatism,…
This paper attempts to situate statistics in relation to qualitative research methods and other means of "finding out". It compares and contrasts aspects of qualitative research methods and statistical inquiry and attempts to answer the…
Although not as wide, and popular, as that of quantum mechanics, the investigation of fundamental aspects of statistical mechanics constitutes an important research field in the building of modern physics. Besides the interest for itself,…
In this paper, we propose standard statistical tools as a solution to commonly highlighted problems in the explainability literature. Indeed, leveraging statistical estimators allows for a proper definition of explanations, enabling…
Recent developments in the mathematical foundations of quantum mechanics have brought the theory closer to that of classical probability and statistics. On the other hand, the unique character of quantum physics sets many of the questions…
Differential privacy is effective in sharing information and preserving privacy with a strong guarantee. As social network analysis has been extensively adopted in many applications, it opens a new arena for the application of differential…
Statistical mechanics is one of the most powerful and elegant tools in the quantitative sciences. One key virtue of statistical mechanics is that it is designed to examine large systems with many interacting degrees of freedom, providing a…
This is an annotated bibliography on estimation and inference results for queues and related stochastic models. The purpose of this document is to collect and categorise works in the field, allowing for researchers and practitioners to…
Causal inference is a critical research topic across many domains, such as statistics, computer science, education, public policy and economics, for decades. Nowadays, estimating causal effect from observational data has become an appealing…