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The codegree of a lattice polytope is the smallest integer dilate thatcontains a lattice point in the relative interior. The weak maximum likelihoodthreshold of a statistical model is the smallest number of data points forwhich there is a non-zero probability that the maximum likelihood estimateexists. The codegree of a marginal polytope is a lower bound on the maximumlikelihood threshold of the associated log-linear model, and they are equalwhen the marginal polytope is normal. We prove a lower bound on the codegree inthe case of hierarchical log-linear models and provide a conjectural formulafor the codegree in general. As an application, we study when the marginalpolytopes of hierarchical models are Gorenstein, including a classification ofGorenstein decomposable models, and a conjectural classification of Gorensteinbinary hierarchical models.

The Codegree, Weak Maximum Likelihood Threshold, and the Gorenstein Property of Hierarchical Models