Geometry, moments and Bayesian networks with hidden variables

The purpose of this paper is to present a systematic way of analysing the geometry of the probability spaces for a particular class of Bayesian networks with hidden variables. It will be shown that the conditional independence statements implicit in such graphical models can be neatly expressed as simple polynomial relationships among central moments. This algebraic framework will enable us to explore and identify the structural constraints on the sample space induced by models with tree strcutures and therefore characterise the families of distributions consistent with such conditional independence assumptions.