An Introduction to Algebraic Statistics with Tensors

Zusatztext

This book provides an introduction to various aspects of Algebraic Statistics with the principal aim of supporting Masters and PhD students who wish to explore the algebraic point of view regarding recent developments in Statistics. The focus is on the background needed to explore the connections among discrete random variables. The main objects that encode these relations are multilinear matrices, i.e., tensors. The book aims to settle the basis of the correspondence between properties of tensors and their translation in Algebraic Geometry. It is divided into three parts, on Algebraic Statistics, Multilinear Algebra, and Algebraic Geometry. The primary purpose is to describe a bridge between the three theories, so that results and problems in one theory find a natural translation to the others. This task requires, from the statistical point of view, a rather unusual, but algebraically natural, presentation of random variables and their main classical features. The third part of the book can be considered as a short, almost self-contained, introduction to the basic concepts of algebraic varieties, which are part of the fundamental background for all who work in Algebraic Statistics.

Autorenportrait

Prof. Luca Chiantini is Full Professor of Geometry at the University of Siena (Italy). His research interests focus mainly on Algebraic Geometry and Multilinear Algebra, and include the theory of vector bundles on varieties and the study of secant spaces, which are the geometric counterpart of the theory of tensor ranks. In particular, he recently studied the relations between Multilinear Algebra and the theory of finite sets in projective spaces. Prof. Cristiano Bocci is Assistant Professor of Geometry at the University of Siena (Italy). His research concerns Algebraic Geometry, Commutative Algebra, and their applications. In particular, his current interests are focused on symbolic powers of ideals, Hadamard product of varieties, and the study of secant spaces. He also works in two interdisciplinary teams in the fields of Electronic Measurements and Sound Synthesis.

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