Characterization of multi-way binary tables with uniform margins and fixed correlations

2026-01-12

AI Summary

This work addresses the non-uniqueness of the joint distribution of multivariate binary tables when only pairwise correlations are specified, which hinders the characterization of higher-order dependencies. The authors develop a geometric framework to characterize the feasible set of multidimensional binary distributions with uniform margins and fixed pairwise correlations. They establish for the first time that this feasible set forms a convex polytope, revealing its inherent symmetries and extremal ray structure, and provide a complete description of its boundary. By integrating tools from convex geometry, combinatorial optimization, and probabilistic modeling, the framework enables systematic exploration of higher-order dependence structures. Its flexibility and practical utility are demonstrated through an application to inter-rater agreement, where it facilitates realistic simulation and modeling scenarios.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In many applications involving binary variables, only pairwise dependence measures, such as correlations, are available. However, for multi-way tables involving more than two variables, these quantities do not uniquely determine the joint distribution, but instead define a family of admissible distributions that share the same pairwise dependence while potentially differing in higher-order interactions. In this paper, we introduce a geometric framework to describe the entire feasible set of such joint distributions with uniform margins. We show that this admissible set forms a convex polytope, analyze its symmetry properties, and characterize its extreme rays. These extremal distributions provide fundamental insights into how higher-order dependence structures may vary while preserving the prescribed pairwise information. Unlike traditional methods for table generation, which return a single table, our framework makes it possible to explore and understand the full admissible space of dependence structures, enabling more flexible choices for modeling and simulation. We illustrate the usefulness of our theoretical results through examples and a real case study on rater agreement.

Problem

• multi-way binary tables
• uniform margins
• fixed correlations
• joint distribution
• higher-order interactions

Innovation

• convex polytope
• binary tables
• higher-order interactions
• uniform margins
• pairwise correlations