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.

翻译：在许多涉及二元变量的应用中，仅可获得成对依赖性度量（如相关性）。然而，对于涉及两个以上变量的多维列联表，这些量并不能唯一确定联合分布，而是定义了一个容许分布族，这些分布共享相同的成对依赖性，但可能在高阶交互作用上存在差异。本文引入了一个几何框架来描述具有均匀边际的此类联合分布的整个可行集。我们证明该容许集构成一个凸多面体，分析了其对称性质，并刻画了其极方向。这些极值分布为理解在保持给定成对信息的同时高阶依赖结构可能如何变化提供了基本见解。与传统的表生成方法（仅返回单个表）不同，我们的框架使得探索和理解依赖结构的整个容许空间成为可能，从而为建模和模拟提供了更灵活的选择。我们通过示例和一个关于评分者一致性的实际案例研究阐明了理论结果的有用性。