Abstract A Bayesian network is a statistical model which can be presented graphically by a directed acyclic graph. The nodes in the graph are discrete random variables, and the edges encode dependencies between the variables. Bayesian nets can also be described algebraically as varieties of homogeneous prime ideals. In this talk we will discuss connections between algebraic properties of such ideals and combinatorial properties of the graphs. In particular, we would like to understand when the variety is toric and when the ideal is quadratic.