We establish a bijection between marginal independence models on $n$ random variables and split closed order ideals in the poset of partial set partitions. We also establish that every discrete marginal independence model is toric in cdf coordinates. This generalizes results of Boege, Petrovic, and Sturmfels and Drton and Richardson, and provides a unified framework for discussing marginal independence models. Additionally, we provide an axiomatic characterization of marginal independence and we show that our set of axioms are sound and complete in the set of probability distributions. This follows the work of Geiger, Paz and Pearl who provided an analogous characterization of independence for statements involving 2 sets of random variables.

翻译：我们在$n$个随机变量的边际独立性模型与偏集合划分偏序集中的分裂闭序理想之间建立了一一对应关系。同时证明了在累积分布函数坐标下，所有离散边际独立性模型都是环面模型。该结果推广了Boege、Petrovic、Sturmfels以及Drton和Richardson的研究成果，为讨论边际独立性模型提供了统一框架。此外，我们给出了边际独立性的公理化刻画，并证明该公理系统在概率分布集合中具有可靠性和完备性。这项工作延续了Geiger、Paz和Pearl对涉及两组随机变量独立性陈述的类似刻画研究。