Toric and non-toric Bayesian networks

TL;DR

This paper explores the algebraic structure of Bayesian networks, introducing toric Bayesian nets, proving a conjecture about their relations, and identifying conditions for quadratic relations based on the graph structure.

Contribution

It characterizes toric Bayesian networks, provides the first example of a non-toric Bayesian net, and proves a conjecture on quadratic relations within this class.

Findings

Characterization of toric Bayesian networks

First example of a non-toric Bayesian net

Proof of Garcia, Stillman, and Sturmfels' conjecture

Abstract

In this paper we study Bayesian networks from a commutative algebra perspective. We characterize a class of toric Bayesian nets, and provide the first example of a Bayesian net which is proved non-toric under any linear change of variables. Concerning the class of toric Bayesian nets, we study their quadratic relations and prove a conjecture by Garcia, Stillman, and Sturmfels for this class. In addition, we give a necessary condition on the underlying directed acyclic graph for when all relations are quadratic.