  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.
