  In this paper we investigate the geometry of a discrete Bayesian network
whose graph is a tree all of whose variables are binary and the only observed
variables are those labeling its leaves. We provide the full geometric
description of these models which is given by a set of polynomial equations
together with a set of complementary implied inequalities induced by the
positivity of probabilities on hidden variables. The phylogenetic invariants
given by the equations can be useful in the construction of simple diagnostic
tests. However, in this paper we point out the importance of also incorporating
the associated inequalities into any statistical analysis. The full
characterization of these inequality constraints derived in this paper helps us
determine how and why routine statistical methods can break down for this model
class.
