Bocci, Cristiano; Chiantini, Luca An introduction to algebraic statistics with tensors. (English) Zbl 1432.62006 Unitext 118. La Matematica per il 3+2. Cham: Springer (ISBN 978-3-030-24623-5/pbk; 978-3-030-24624-2/ebook). xix, 235 p. (2019). Here we have a well readable introduction to algebraic statistics. The word “tensor” in the title of the book does not refer to the usual apparatus called tensor calculus, as used e.g. in relativity theory and differential geometry, but here “tensor” is nothing but a synonym for “multidimensional matrices”, see page 81 of the book. The preface gives a short overview, and the authors warmly thank Fabio Rapallo for many improvements of the text.The 13 chapters of this book are headed as follows: 1. Systems of random variables and distributions, 2. Basic statistics, 3. Statistical models, 4. Complex projective algebraic statistics, 5. Conditional independence, 6. Tensors, 7. Symmetric tensors, 8. Marginalization and flattenings, 9. Elements of projective algebraic geometry, 10. Projective maps and the Chow’s theorem, 11. Dimension theory, 12. Secant varieties, 13. Groebner bases.This book is dedicated to Anthony Vito Geramita who passed away on June 22, 2016. Publisher’s description: “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.” Reviewer: Hans-Jürgen Schmidt (Potsdam) Cited in 8 Documents MSC: 62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics 13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra 14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry 15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra 62R01 Algebraic statistics 13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) 14N07 Secant varieties, tensor rank, varieties of sums of powers 15A69 Multilinear algebra, tensor calculus Keywords:algebraic statistics; tensors Biographic References: Geramita, Anthony Vito PDFBibTeX XMLCite \textit{C. Bocci} and \textit{L. Chiantini}, An introduction to algebraic statistics with tensors. Cham: Springer (2019; Zbl 1432.62006) Full Text: DOI