Given a finite set of variables N, conditional independence (CI) structures establish a link between A, B, and C, which are subsets of N, in the form “A is conditionally independent of B, given C.” These types of structures are used in computational data modeling and analysis for a variety of applications, some of which are mentioned in chapter 1 of this book. This book looks at the probabilistic version of conditional independence structures, in which a probability measure P over N is introduced into the statement.
The book has a very good introductory chapter (chapter 2), in which the basic elements of conditional independence structures are explained. Chapter 2 establishes the basic formal notation to describe these structures and their statements. The chapter also covers the different classes of probability measures that are used in probabilistic CI structures.
CI structures are often represented in graphical formats. These may be undirected graphs, acyclic directed graphs, or classical chain graphs, to mention a few. Chapter 3 covers the graphical representation of CI in detail, with an extended section on advanced graphical models.
Chapter 4 provides the basics of the particular notion of imsets, introduced briefly at the end of chapter 2. Imsets are integer-valued functions over N. Chapter 4 focuses on structural imsets, which are effectively a combination of elementary imsets.
Chapters 5 and 6 may be of particular interest to language researchers, since they expand on the elements of the CI structures. Chapter 5 covers probabilistic models and how they can be described. Chapter 6 covers equivalence and implication.
Chapters 7 and 8 may be of particular interest to artificial intelligence researchers. Chapter 7 covers the problem with representative choice. Chapter 8 covers learning, classifying approaches to learning CI from sets of data into two groups: significance tests and quality criterion.
The book concludes with open problems (chapter 9), those that are still outstanding in the theory and implementation of probabilistic CI structures and models, and their operations.
Overall, the book is very well organized and presented. It reads very well, and introduces an interesting area of discrete mathematics that has potential in knowledge representation, modeling, and probabilistic reasoning, especially in the field of artificial intelligence.