3 - Algebra

Summary

The philosophy of algebraic statistics is that statistical models are algebraic varieties. We encountered many such models in Chapter 1. The purpose of this chapter is to give an elementary introduction to the relevant algebraic concepts, with examples drawn from statistics and computational biology.

Algebraic varieties are zero sets of systems of polynomial equations in several unknowns. These geometric objects appear in many contexts. For example, in genetics, the familiar Hardy–Weinberg curve is an algebraic variety (see Figure 3.1). In statistics, the distributions corresponding to independent random variables form algebraic varieties, called Segre varieties, that are well known to mathematicians. There are many questions one can ask about a system of polynomial equations; for example whether the solution set is empty, nonempty but finite, or infinite. Gröbner bases can be used to answer these questions.

Algebraic varieties can be described in two different ways, either by equations or parametrically. Each of these representations is useful. We encountered this duality in the Hammersley–Clifford Theorem which says that a graphical model can be described by conditional independence statements or by a polynomial parameterization. Clearly, methods for switching between these two representations are desirable. We discuss such methods in Section 3.2.

The study of systems of polynomial equations is the main focus of a central area in mathematics called algebraic geometry. This is a rich, beautiful, and well-developed subject, at whose heart lies a deep connection between algebra and geometry.