Geometry, moments and conditional independence trees with hidden variables

We study the geometry of the parameter space for Bayesian directed graphical models with hidden variables that have a tree structure and where all the nodes are binary. We show that the conditional independence statements implicit in such models can be expressed in terms of polynomial relationships among the central moments. This algebraic structure will enable us to identify the inequality constraints on the space of the manifest variables that are induced by the conditional independence assumptions as well as determine the degree of unidentifiability of the parameters associated with the hidden variables. By understanding the geometry of the sample space under this class of models we shall propose and discuss simple diagnostic methods