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February 2012 Tree cumulants and the geometry of binary tree models
Piotr Zwiernik, Jim Q. Smith
Bernoulli 18(1): 290-321 (February 2012). DOI: 10.3150/10-BEJ338


In this paper we investigate undirected discrete graphical tree models when all the variables in the system are binary, where leaves represent the observable variables and where all the inner nodes are unobserved. A novel approach based on the theory of partially ordered sets allows us to obtain a convenient parametrization of this model class. The construction of the proposed coordinate system mirrors the combinatorial definition of cumulants. A simple product-like form of the resulting parametrization gives insight into identifiability issues associated with this model class. In particular, we provide necessary and sufficient conditions for such a model to be identified up to the switching of labels of the inner nodes. When these conditions hold, we give explicit formulas for the parameters of the model. Whenever the model fails to be identified, we use the new parametrization to describe the geometry of the unidentified parameter space. We illustrate these results using a simple example.


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Piotr Zwiernik. Jim Q. Smith. "Tree cumulants and the geometry of binary tree models." Bernoulli 18 (1) 290 - 321, February 2012.


Published: February 2012
First available in Project Euclid: 20 January 2012

zbMATH: 1235.62004
MathSciNet: MR2888708
Digital Object Identifier: 10.3150/10-BEJ338

Rights: Copyright © 2012 Bernoulli Society for Mathematical Statistics and Probability


Vol.18 • No. 1 • February 2012
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