Graphical Representation of Asymmetric Graphoid Structures

Independence models induced by some uncertainty measures (e.g. conditional probability, possibility) do not obey the usual graphoid properties, since they do not satisfy the symmetry property. They are efficiently representable through directed acyclic l-graphs by using L-separation criterion. In this paper, we show that in general there is not a l-graph which describes completely all the independence statements of a given model; hence we introduce in this context the notion of minimal I-map and we show how to build it, given an ordering on the variables. In addition, we prove that, for any ordering, there exists an I-map for any asymmetric graphoid structure.