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Generalized cut polytopes for binary hierarchical models

Jane Ivy Coons, Joseph Cummings, Benjamin Hollering and Aida Maraj

Vol. 14 (2023), No. 1, 17–36
Abstract

Marginal polytopes are important geometric objects that arise in statistics as the polytopes underlying hierarchical log-linear models. These polytopes can be used to answer geometric questions about these models, such as determining the existence of maximum likelihood estimates or the normality of the associated semigroup. Cut polytopes of graphs have been useful in analyzing binary marginal polytopes in the case where the simplicial complex underlying the hierarchical model is a graph. We introduce a generalized cut polytope that is isomorphic to the binary marginal polytope of an arbitrary simplicial. This polytope is full dimensional in its ambient space and has a natural switching operation among its facets that can be used to deduce symmetries between the facets of the correlation and binary marginal polytopes. We use this switching operation along with Bernstein and Sullivant’s characterization of unimodular simplicial complexes to find a complete -representation for many unimodular simplicial complexes.

Keywords
generalized cut polytope, marginal polytope, hierarchical model, half-space description
Mathematical Subject Classification
Primary: 52B05, 52B12, 52B35, 62R01
Milestones
Received: 26 September 2022
Revised: 14 November 2022
Accepted: 16 December 2022
Published: 28 November 2023
Authors
Jane Ivy Coons
St John’s College
University of Oxford
United Kingdom
Joseph Cummings
University of Notre Dame
IN
United States
Benjamin Hollering
Max Planck Institute for Mathematics in the Sciences
Leipzig
Germany
Aida Maraj
University of Michigan
Ann Arbor, MI
United States