Tree-cumulants and the identifiability of Bayesian 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 the inner nodes are unobserved. A novel approach based on the theory of partially ordered sets allows us to obtain a convenient parameterization of the 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 the condition for such a model to be identified at least up to switching labels of the inner nodes. In this case explicit formulas are given 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 subspace. We illustrate these results through a simple example.