Log-Linear Models, Toric Varieties, and Their Markov Bases

My goal in this chapter of lecture notes is to introduce the class of loglinear models and study them from the algebraic perspective. There are at least three different ways to introduce log-linear models: in statistics they are also called discrete exponential families and in algebraic geometry they are known as toric varieties. Our goal is to introduce these models from all three perspective and show that they are equivalent. Then we take the approach suggested in the first chapter of lecture notes and study the ideals defining the log-linear model. They are known as toric ideals. Extensive further information about toric ideals can be found in [15]. In Section 3 of this chapter, we explain how the generators of the toric ideals are useful for performing conditional inference. In particular, these generating sets can be used to perform certain random walks to generate random samples from the hypergeometric distribution, and compute p-values of various statistical tests. In this context, generating sets for toric ideals are known as Markov bases. This connection was first made in [5]. In Section 4, we discuss the particular example of hierarchical models. In Section 5 we provide a description of some of the basic results about Markov bases of the hierarchical models. In Section 6 we describe some other interesting examples of log-linear models, in particular, models for analyzing ranked data and Hardy-Weinberg equilibrium.