Open Access
2011 Implicit inequality constraints in a binary tree model
Piotr Zwiernik, Jim Q. Smith
Electron. J. Statist. 5: 1276-1312 (2011). DOI: 10.1214/11-EJS640


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.


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Piotr Zwiernik. Jim Q. Smith. "Implicit inequality constraints in a binary tree model." Electron. J. Statist. 5 1276 - 1312, 2011.


Published: 2011
First available in Project Euclid: 19 October 2011

zbMATH: 1274.62355
MathSciNet: MR2842906
Digital Object Identifier: 10.1214/11-EJS640

Primary: 62E15 , 62H05
Secondary: 60K99 , 62F99

Keywords: Binary data , Graphical models on trees , inequality constraints , phylogenetic invariants , semialgebraic statistical models , tree cumulants

Rights: Copyright © 2011 The Institute of Mathematical Statistics and the Bernoulli Society

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