Semigraphoids and structures of probabilistic conditional independence

The concept of conditional independence (CI) has an important role in probabilistic reasoning, that is a branch of artificial intelligence where knowledge is modeled by means of a multidimensional finite-valued probability distribution. The structures of probabilistic CI are described by means of semigraphoids, that is lists of CI-statements closed under four concrete inference rules, which have at most two antecedents. It is known that every CI-model is a semigraphoid, but the converse is not true. In this paper, the semigraphoid closure of every couple of CI-statements is proved to be a CI-model. The substantial step to it is to show that every probabilistically sound inference rule for axiomatic characterization of CI properties (= axiom), having at most two antecedents, is a consequence of the semigraphoid inference rules. Moreover, all potential dominant triplets of the mentioned semigraphoid closure are found.