ABSTRACT
Conditional independence (CI) implications can also be interpreted as intersections of graphical models. As the example shows, the intersection of two graphical models need not be a graphical model. How can one compute this intersection? This chapter explores these questions and introduce tools from computational algebra for studying them. It provides an overview of basic ideas in algebraic geometry which are useful for the study of CI structures and graphical models. The chapter examines the ideals associated to families of CI statements, and explains how to apply the basic techniques to deduce CI implications. It illustrates the main ideas with some deeper examples coming from the literature. The chapter describes the vanishing ideal of a graphical model, which is a complete set of implicit restrictions for that model. It discusses the basic facts that hopefully make it possible for the reader to understand the phenomena and algorithms.
