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# Poset Representations for Sets of Elementary Triplets

*Proceedings of the 10th International Conference on Probabilistic Graphical Models*, PMLR 138:521-532, 2020.

#### Abstract

Semi-graphoid independence relations, composed of independence triplets, are typically exponentially large in the number of variables involved. For compact representation of such a relation, just a subset of its triplets, called a basis, are listed explicitly, while its other triplets remain implicit through a set of derivation rules. Two types of basis were defined for this purpose, which are the dominant-triplet basis and the elementary-triplet basis, of which the latter is commonly assumed to be significantly larger in size in general. In this paper we introduce the elementary po-triplet as a compact representation of multiple elementary triplets, by using separating posets. By exploiting this new representation, the size of an elementary-triplet basis can be reduced considerably. For computing the elementary closure of a starting set of po-triplets, we present an elegant algorithm that operates on the least and largest elements of the separating posets involved.