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Implicit inequality constraints in a binary tree model. (English) Zbl 1274.62355

Summary: 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.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62E15 Exact distribution theory in statistics
62F15 Bayesian inference
68T35 Theory of languages and software systems (knowledge-based systems, expert systems, etc.) for artificial intelligence
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References:

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