The correlation space of Gaussian latent tree models and model selection without fitting

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Abstract

We provide a complete description of possible distributions consistent with any Gaussian latent tree model. This description consists of polynomial equations and inequalities involving covariances between the observed variables. Testing inequality constraints can be done using the inverse Wishart distribution and this leads to simple preliminary assessment of tree-compatibility. To test equality constraints we employ general techniques of tetrad analyses. This approach is effective even for small sample sizes and can be easily adjusted to test either entire models or just particular macrostructures of a tree. Our methods are simple to implement and do not require fitting of the model. The versatility of the techniques is illustrated by performing exploratory and confirmatory tetrad analyses in linguistic and biological settings respectively.

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