main-content

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.

### Chapter 1. Systems of Random Variables and Distributions

Abstract
This section contains the basic definitions with which we will construct our statistical theory. It is important to point out right away that in the field of Algebraic Statistics, a still rapidly developing area of study, the basic definitions are not yet standardized. Therefore, the definitions which we shall use in this text can differ significantly (more in form than in substance) from those of other texts.
Cristiano Bocci, Luca Chiantini

### Chapter 2. Basic Statistics

Abstract
In this chapter, we focus on some basic examples in Probability and Statistics. We phrase these concepts using the language and definitions we have given in the previous chapter.
Cristiano Bocci, Luca Chiantini

### Chapter 3. Statistical Models

Abstract
In this chapter, we introduce the concept of model, essential point of statistical inference. The concept is reviewed here by our algebraic interpretation.
Cristiano Bocci, Luca Chiantini

### Chapter 4. Complex Projective Algebraic Statistics

Abstract
No, we are not exaggerating. We are instead simplifying. In fact, many of the phenomena associated with the main statistic models are better understood if studied, at least in the first measure, from the projective point of view and on an algebraically closed numerical field. The main link between Algebraic Statistics and Projective Algebraic Geometry is based on the constructions of this chapter.
Cristiano Bocci, Luca Chiantini

### Chapter 5. Conditional Independence

Abstract
An intermediate case between total independence and generic situations of the dependence of random variables concerns the so-called conditional independence.
Cristiano Bocci, Luca Chiantini

### Chapter 6. Tensors

Abstract
The main objects of multi-linear algebra that we will use in the study of Algebraic Statistics are multidimensional matrices, that we will call tensors.
Cristiano Bocci, Luca Chiantini

### Chapter 7. Symmetric Tensors

Abstract
In this chapter, we make a specific analysis of the behavior of symmetric tensors, with respect to the rank and the decomposition. We will see, indeed, that besides their utility to understand some models of random systems, symmetric tensors have a relevant role in the study of the algebra and the computational complexity of polynomials.
Cristiano Bocci, Luca Chiantini

### Chapter 8. Marginalization and Flattenings

Abstract
We collect in this chapter some of the most useful operations on tensors, in view of the applications to Algebraic Statistics.
Cristiano Bocci, Luca Chiantini

### Chapter 9. Elements of Projective Algebraic Geometry

Abstract
The scope of this part of the book is to provide a quick introduction to the main tools of the algebraic geometry of projective spaces that are necessary to understand some aspects of algebraic models in Statistics.
Cristiano Bocci, Luca Chiantini

### Chapter 10. Projective Maps and the Chow’s Theorem

Abstract
The chapter contains the proof of the Chow’s Theorem, a fundamental result for algebraic varieties with an important consequence for the study of statistical models. It states that, over an algebraically closed field, like $$\mathbb C$$, the image of a projective (or multiprojective) variety X under a projective map is a Zariski closed subset of the target space, i.e., it is itself a projective variety.
Cristiano Bocci, Luca Chiantini

### Chapter 11. Dimension Theory

Abstract
The concept of dimension of a projective variety is a fairly intuitive but surprisingly delicate invariant, from an algebraic point of view.
Cristiano Bocci, Luca Chiantini

### Chapter 12. Secant Varieties

Abstract
The study of the rank of tensors has a natural geometric counterpart in the study of secant varieties. Secant varieties or, more generally, joins are a relevant object for several researches on the properties of projective varieties.
Cristiano Bocci, Luca Chiantini

### Chapter 13. Groebner Bases

Abstract
Groebner bases represent the most powerful tool for computational algebra, in particular for the study of polynomial ideals. In this chapter, based on [1, Chap. 2], we give a brief survey on the subject. For a deeper study of it, we suggest [1, 2].
Cristiano Bocci, Luca Chiantini