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This article serves as an introduction to the study of networks of social systems. First, we introduce the reader to key mathematical tools to study social networks, including mathematical representations of networks and essential…
The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of…
Algorithms for continuous optimization problems have a rich history of design and innovation over the past several decades, in which mathematical analysis of their convergence and complexity properties plays a central role. Besides their…
This tutorial investigates the convergence of statistical mechanics and learning theory, elucidating the potential enhancements in machine learning methodologies through the integration of foundational principles from physics. The tutorial…
In many data analysis applications, there is a need to explain why a surprising or interesting result was produced by a query. Previous approaches to explaining results have directly or indirectly used data provenance (input tuples…
We discuss various universality aspects of numerical computations using standard algorithms. These aspects include empirical observations and rigorous results. We also make various speculations about computation in a broader sense.
Motivated by a recent series of diametrically opposed articles on the relative value of statistical methods for the analysis of path diagrams in the social sciences, we discuss from a primarily theoretical perspective selected fundamental…
Here practical aspects of conducting research via computer simulations are discussed. The following issues are addressed: software engineering, object-oriented software development, programming style, macros, make files, scripts, libraries,…
One can argue that one of the main roles of the subject of statistics is to characterize what the evidence in collected data says about questions of scientific interest. There are two broad questions that we will refer to as the estimation…
We begin our journey by recalling the fundamentals of Probability Theory that underlie one of its most significant applications to real-world problems: Parametric Estimation. Throughout the text, we systematically develop this theme by…
This lecture addresses some general ideas behind numerical computations ranging from representation of numbers in computers to stability and accuracy of standard algorithms for some simple mathematical problems.
Simulation studies are computer experiments that involve creating data by pseudorandom sampling. The key strength of simulation studies is the ability to understand the behaviour of statistical methods because some 'truth' (usually some…
The aim of this chapter is twofold. In the first part we will provide a brief overview of the mathematical and statistical foundations of graphical models, along with their fundamental properties, estimation and basic inference procedures.…
Forecasting and optimisation are two major fields of operations research that are widely used in practice. These methods have contributed to each other growth in several ways. However, the nature of the relationship between these two fields…
Foundation models can be disruptive for future AI development by scaling up deep learning in terms of model size and training data's breadth and size. These models achieve state-of-the-art performance (often through further adaptation) on a…
We consider the relationship between learnability of a "base class" of functions on a set $X$, and learnability of a class of statistical functions derived from the base class. For example, we refine results showing that learnability of a…
Regression methods dominate the practice of biostatistical analysis, but biostatistical training emphasises the details of regression models and methods ahead of the purposes for which such modelling might be useful. More broadly,…
There is a mismatch between the standard theoretical analyses of statistical machine learning and how learning is used in practice. The foundational assumption supporting the theory is that we can represent features and models using…
This is an introduction to measure theory, integration and function spaces, with all the needed preliminaries included, and with some applications included as well. We first discuss some basic motivations, coming from discrete probability,…
This paper generalizes the notion of stochastic order to a relation between probability measures over arbitrary measurable spaces. This generalization is motivated by the observation that for the stochastic ordering of two stationary Markov…