The Codegree, Weak Maximum Likelihood Threshold, and the Gorenstein Property of Hierarchical Models
Joseph Johnson, Seth Sullivant
TL;DR
This work connects polyhedral geometry with algebraic statistics by relating the codegree of marginal polytopes to the weak maximum likelihood threshold in hierarchical log-linear models, establishing a general lower bound $\mathrm{codeg}(\mathrm{Marg}(A)) \le \mathrm{wmlt}(\mathcal{M}_A)$ and identifying equality under normality. It proposes a conjectural exact formula for codegree in a broad setting and proves it in several cases, while applying these insights to the Gorenstein property via fiber-product techniques. A complete classification of decomposable Gorenstein hierarchical models is given, along with conjectural classifications for binary cases and supporting computational evidence. Overall, the paper advances a cohesive framework linking lattice polytopes, MLE existence thresholds, and Gorenstein geometry to understand hierarchical models and their data-size requirements.
Abstract
The codegree of a lattice polytope is the smallest integer dilate that contains a lattice point in the relative interior. The weak maximum likelihood threshold of a statistical model is the smallest number of data points for which there is a non-zero probability that the maximum likelihood estimate exists. The codegree of a marginal polytope is a lower bound on the maximum likelihood threshold of the associated log-linear model, and they are equal when the marginal polytope is normal. We prove a lower bound on the codegree in the case of hierarchical log-linear models and provide a conjectural formula for the codegree in general. As an application, we study when the marginal polytopes of hierarchical models are Gorenstein, including a classification of Gorenstein decomposable models, and a conjectural classification of Gorenstein binary hierarchical models.