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The semialgebraic description of tree models for binary data

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 obtain a full semialgebraic geometric description of these models which is given by polynomial equations and inequalities. Our analysis is based on combinatorial results generalizing the notion of cumulants so that they apply to the models under analysis. The geometric structure we obtain links to the notion of a tree metric considered in phylogenetic analysis and to some interesting determinantal formulas involving the hyperdeterminant of 2x2x2 tables.

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