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Zwiernik, Piotr; Smith, Jim Q.

Tree cumulants and the geometry of binary tree models. (English) Zbl 1235.62004

Bernoulli 18, No. 1, 290-321 (2012).
Summary: 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.

Cited in 7 Documents

MSC:

62A99 Foundational topics in statistics
62H99 Multivariate analysis
05C05 Trees
06A06 Partial orders, general
60J99 Markov processes
05C90 Applications of graph theory

Keywords:

binary data; central moments; conditional independence; cumulants; general Markov models; graphical models on trees; hidden data; identifiability; Möbius function
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References:

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