An Introduction to Algebraic Statistics with Tensors

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.

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.

"The book under review contributes to the literature in algebraic statistics by highlighting the role of tensors, which are a vital tool in algebraic statistical theory and methods." (Carlos Amendola, Mathematical Reviews, December, 2020)