Exponential Hilbert series and hierarchical log-linear models

TL;DR

This paper introduces a new way to determine the rank of matrices associated with hierarchical log-linear models using exponential Hilbert series, simplifying calculations especially under certain symmetry conditions.

Contribution

It provides a novel characterization of the matrix rank via logarithmic transformations of the exponential Hilbert series, simplifying the analysis of hierarchical models.

Findings

Rank formula reduces to face vector when variables have equal outcomes.

Simplified rank computation for models satisfying Dehn-Sommerville relations.

Explicit formulas for model dimension and degrees of freedom.

Abstract

Consider a hierarchical log-linear model, given by a simplicial complex, $\Gamma$, and integer matrix $A_\Gamma$. We give a new characterization of the rank of $A_\Gamma$ given by a logarithmic transformation on the exponential Hilbert series of $\Gamma$. 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 $\Gamma$. If $\Gamma$ further satisfies the Dehn-Sommerville relations, then we give an exceptionally simple formula for computing the rank of $A_\Gamma$, and thus the dimension and the number of degrees of freedom of the model.