The geometry of conditional independence tree models with hidden variables

In this paper we investigate the geometry of undirected graphical models of trees when all the variables in the system are binary and some of them are hidden. We obtain a full description of those models which is given by polynomial equations and inequalities and give exact formulas for their parameters in terms of the marginal probability over the observed variables. We also show how correlations link to tree metrics considered in phylogenetics. Finally, a new system of coordinates is given that is intrinsically related to the phylogenetic tree models and which allows us to classify phylogenetic invariants.