Implicit inequality constraints in a binary tree model

TL;DR

This paper characterizes the geometric structure of binary tree Bayesian networks, highlighting the importance of polynomial equations and inequalities for accurate statistical analysis and diagnostics.

Contribution

It provides the complete set of polynomial equations and inequalities defining the model's geometry, emphasizing their role in statistical diagnostics.

Findings

Full geometric description of the model

Polynomial equations and inequalities characterize the model

Inequalities are crucial for correct statistical inference

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.