The geometry of independence tree models with hidden variables

In this paper we investigate the geometry of undirected discrete graphical models of trees when all the variables in the system are binary, where leaves represent the observable variables and where the inner nodes are unobserved. We obtain a full geometric description of these models which is given by polynomial equations and inequalities. We also give exact formulas for their parameters in terms of the marginal probability over the observed variables. Our analysis is based on combinatorial results generalizing the notion of cumulants and introduce a novel use of M\"{o}bius functions on partially ordered sets. The geometric structure we obtain links to the notion of a tree metric considered in phylogenetic analysis and to some interesting determinantal formulas involving hyperdeterminants of $2\times 2\times 2$ tables as defined in [19].