Marginal Independence and Partial Set Partitions

Marginal independence models are often studied by associating a combinatorial structure to describe the independence relations between random variables. For instance, Drton and Richardson [3] used bidirected graphs to develop a family of marginal independence models for discrete random variables, and Boege, Petrovíc, and Sturmfels [1] used simplicial complexes. The marginal independence models of [3] fit more broadly into the class of graphical models [4, 5]. Each of these structures describes a very different family of models, with very little overlap between them. In both cases it had been shown that when we restrict to discrete random variables, we obtain toric models after a linear change of coordinates. However, there are collections of marginal independence statements that cannot be expressed either as graphs or simplicial complexes. This motivates the proposal of a more general combinatorial structure which we use in our treatment of marginal independence.