Tools for computing primary decompositions and applications to ideals associated to Bayesian networks

Abstract

These notes, written for the CIMPA School on “Systems of polynomial equations” (Argentina, 2003), have two goals: to present the underlying ideas and tools for computing primary decompositions of ideals, and to apply these techniques to a recent interesting class of ideals related to statistics. Primary decompositions are an important notion both in algebraic geometry and for applications. There are several algorithms available (the two closest to what we present are [GTZ88] and [SY96]). A good overview of the state of the art is the paper [DGP99]. Primary decompositions, and related computations, such as finding minimal and associated primes, the radical of an ideal, and the equidimensional decomposition of an ideal, are all implemented in most specialized computer algebra systems, such as CoCoa ([CNR00]), Macaulay 2 ([GS]), and Singular ([GPS01]). Several years ago, these algorithms and their implementations could handle only very small examples. Now, with improved implementations, and more efficient computers, larger ideals can be handled. However, if the number of indeterminates is large, the implemented algorithms often are unable to find a primary decomposition, or even to find the minimal primes. This is the case for many of the ideals associated to Bayesian networks that we consider here. Our first goal in these lectures is to describe some basic methods for manipulating components of an ideal. We put these together into an algorithm for primary decomposition, but, we hope that some of the students will have ideas about novel ways to combine these techniques to a more efficient algorithm! Our second goal is to define some interesting ideals, called Markov ideals, associated to a Bayesian network. In applications, Bayesian networks have been used in many ways, e.g. in machine learning, in vision and speech recognition, in attempting to reconstruct gene regulatory networks, and in the analysis of DNA micro-array data. These Markov ideals provide a striking link between multivariate statistics and algebra and geometry. In these lectures, we do little more than provide a glimpse into this very interesting and potentially powerful relationship. Here is one short glimpse: hidden variables in some Markov models correspond to secant loci of Segre embeddings of products of projective spaces (see [GSS] for details). Although this relationship is not yet in these notes, if there is time, we will discuss it at CIMPA, as it is related to some of the computational tools we will cover.