Neighborliness of Marginal Polytopes

TL;DR

This paper investigates the neighborliness of marginal polytopes in hierarchical models, providing new theoretical insights and explicit Markov bases, especially for binary variables and specific simplicial complexes.

Contribution

It establishes a neighborliness property based on the smallest non-face size and derives a Markov basis for binary models with complement-of-interval complexes.

Findings

Neighborliness depends on the smallest non-face cardinality.

Explicit Markov basis for binary models with complement-of-interval complexes.

Reduction of the general case to the binary case.

Abstract

A neighborliness property of marginal polytopes of hierarchical models, depending on the cardinality of the smallest non-face of the underlying simplicial complex, is shown. The case of binary variables is studied explicitly, then the general case is reduced to the binary case. A Markov basis for binary hierarchical models whose simplicial complexes is the complement of an interval is given.