Tree-cumulants and the geometry of binary tree models

In this paper we investigate undirected discrete graphical tree models when all the variables in the system are binary, where leaves represent the observable variables and where all the inner nodes are unobserved. A novel approach based on the theory of partially ordered sets allows us to obtain a convenient parametrization of this model class. The construction of the proposed coordinate system mirrors the combinatorial definition of cumulants. A simple product-like form of the resulting parameterization gives insight into identifiability issues associated with this model class. In particular we provide necessary and sufficient conditions for such a model to be identified up to the switching of labels of the inner nodes. When these conditions hold we give explicit formulas for the parameters of the model. Whenever the model fails to be identified we use the new parameterization to describe the geometry of the unidentified parameter space. We illustrate these results using a simple example.