An Introduction to Algebraic Statistics with Tensors

This book provides an introduction to various aspects of Algebraic Statistics with the principal aim of supporting Master's 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.

Introduces a setting in which problems of Algebraic Statistics have a natural translation in terms of tensor analysis Includes a self-contained manual of Algebraic Geometry for the study of spaces of tensors Contains a description of relations between Algebraic Geometry and Multilinear Algebra

Autorentext

Cristiano Bocci is a full Professor in geometry at the Department of Information Engineering and Mathematics of the Università di Siena, Italy. His research interests focus on Algebraic Geometry, Combinatorial Commutative Algebra, Algebraic Statistics and Biomathematics. Enrico Carlini is a full Professor in geometry at the Department of Mathematical Sciences of the Politecnico di Torino, in Turin, Italy. His research interests focus on the interactions between Algebraic Geometry and Commutative Algebra and on their applications.

Klappentext

PART I: Algebraic Statistics.- 1 Systems of Random Variables and Distributions.- 2 Basic Statistics.- 3 Statistical models.- 4 Complex projective algebraic statistics.- 5 Conditional independence.- PART II: Multilinear Algebra.- 6 Tensors.- 7 Symmetric tensors.- 8 Marginalisation and attenings.- PART III: Commutative Algebra and Algebraic Geometry.- 9 Elements of Projective Algebraic Geometry.- 10 Projective maps and the Chow's Theorem.- 11 Dimension Theory.- 12 Secant varieties.- 13 Groebner bases.

Inhalt
PART I: Algebraic Statistics.- 1 Systems of Random Variables and Distributions.- 2 Basic Statistics.- 3 Statistical models.- 4 Complex projective algebraic statistics.- 5 Conditional independence.- PART II: Multilinear Algebra.- 6 Tensors.- 7 Symmetric tensors.- 8 Marginalisation and attenings.- PART III: Commutative Algebra and Algebraic Geometry.- 9 Elements of Projective Algebraic Geometry.- 10 Projective maps and the Chow's Theorem.- 11 Dimension Theory.- 12 Secant varieties.- 13 Groebner bases.