Exponential Hilbert series and hierarchical log-linear models

11/07/2022

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by   Wayne A. Johnson, et al.

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Consider a hierarchical log-linear model, given by a simplicial complex, Γ, and integer matrix A_Γ. We give a new characterization of the rank of A_Γ given by a logarithmic transformation on the exponential Hilbert series of Γ. We show that, if each random variable in X has the same number of possible outcomes, then this formula reduces to a simple description in terms of the face vector of Γ. If Γ further satisfies the Dehn-Sommerville relations, then we give an exceptionally simple formula for computing the rank of A_Γ, and thus the dimension and the number of degrees of freedom of the model.