For the distributions of finitely many binary random variables, we study the interaction of restrictions of the supports with conditional independence constraints. We prove a generalization of the Hammersley-Clifford theorem for distributions whose support is a natural distributive lattice: that is, any distribution which has natural lattice support and satisfies the pairwise Markov statements of a graph must factor according to the graph. We also show a connection to the Hibi ideals of lattices.

翻译：针对有限多个二元随机变量的分布，我们研究了支撑集限制与条件独立性约束之间的相互作用。我们证明了Hammersley-Clifford定理在支撑集为自然分配格分布上的推广：即任何具有自然格支撑且满足图成对马尔可夫性质的分布，必然能按该图进行因子分解。我们还展示了该结果与格的Hibi理想之间的联系。