Book Review: Algebraic statistics

Algebraic statistics studies the interplay of algebra and statistics.Both of these fields are usually broadly interpreted.Algebra can include algebraic geometry, combinatorics, discrete mathematics, optimization, and so on.Statistics is usually meant to include also aspects of probability theory and data science in general.Algebraic statistics has the MSC number 62R01, a dedicated journal Algebraic statistics, and a vibrant community.There are several spezialized textbooks on algebraic statistics themes, e.g., [1-3, 6, 8], but Seth Sullivant's text is the first to aim to capture the state of algebraic statistics in its entirety.Algebraic statistics is not centered around one definition like topology, which studies topological spaces, or group theory, which studies groups.Rather, algebraic statistics is a realm of ideas and a community of researchers who constantly discover new areas of activity connected to algebra and data science.Importantly, the transfer of ideas goes in both directions.Mathematical structures in the applications motivate new fundamental theorems, and algebraic theorems solve statistical problems.Seth Sullivant's new book is a general introduction and exposition, aiming to take account of the state of the art.After a few introductory chapters with prerequisites, each new chapter goes to a different topic and lays out the foundations of another success story of algebraic statistics.To give a flavor of this, we showcase conditional independence here, which, as Sullivant writes, "is a natural place to begin to study the connection between probabilistic concepts and methods from algebraic geometry".Conditional independence constraints are used to describe a statistical model, that is, a set of probability distributions in which a true, data-generating distribution is sought.Conditional independence constraints are natural because humans, when doing science, always think about cause and effect, dependence and independence.While discovering actual causality using statistics is another rapidly developing and debated topic [4,5], here we content ourselves with statistical (in)dependence.To quote again: "conditional independence constraints are simple and intuitive restrictions on probability densities that allow one to express the notion that two sets of random variables are unrelated, typically given knowledge of the values of a third set of random variables".Chapter 4 of Algebraic statistics contains an introduction to conditional independence.Here we illustrate the connection to algebra on distributions of random variables with finitely many discrete values.Another important class with many algebraic structures are normally distributed random variables.In general, (conditional) independence constraints concern factorization properties of joint densities of a collection of random variables, but for these two cases, conditional independence constraints take the form of polynomial conditions on the densities.For discrete random variables, this means polynomial conditions on the elementary probabilities of events.In the case of normally distributed random variables, the conditions are rank conditions on submatrices of covariances.