Electronic Journal of Statistics

Implicit inequality constraints in a binary tree model

Piotr Zwiernik and Jim Q. Smith

Full-text: Open access

Abstract

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.

Article information

Source
Electron. J. Statist., Volume 5 (2011), 1276-1312.

Dates
First available in Project Euclid: 19 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1319028569

Digital Object Identifier
doi:10.1214/11-EJS640

Mathematical Reviews number (MathSciNet)
MR2842906

Zentralblatt MATH identifier
1274.62355

Subjects
Primary: 62H05: Characterization and structure theory 62E15: Exact distribution theory
Secondary: 60K99: None of the above, but in this section 62F99: None of the above, but in this section

Keywords
Graphical models on trees binary data tree cumulants semialgebraic statistical models phylogenetic invariants inequality constraints

Citation

Zwiernik, Piotr; Smith, Jim Q. Implicit inequality constraints in a binary tree model. Electron. J. Statist. 5 (2011), 1276--1312. doi:10.1214/11-EJS640. https://projecteuclid.org/euclid.ejs/1319028569


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