Toric and Non-toric Bayesian Networks

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 provably 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 [J. Symbolic Comput., 39 (2005), pp. 331-355] for this class. In addition, we give a necessary condition on the underlying directed acyclic graph for when all relations are quadratic.