M. Studeny: Semigraphoids and structures of probabilistic conditional independence. Annals of Mathematics and Artificial Intelligence 21 (1997), n. 1, pp. 71-98.

The concept of conditional independence has an important role in probabilistic reasoning, that is a branch of artificial intelligence where knowledge is modelled by means of a multidimensional finite-valued probability distribution. The probabilistic conditional independence structures are described by means of semigraphoids, that is the lists of conditional independence statements closed under four particular inference rules, which have at most two antecedents. It is known that every conditional independence model is a semigraphoid, but the converse is not true. In this paper, the semigraphoid closure of every couple of conditional independence statements is proved to be a conditional independence model. The substantial step to it is to show that every probabilistically sound inference rule for axiomatic characterization of conditional independence 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.

AMS classification 62H05, 60A05, 68T30, 03B30

probabilistic reasoning
(probabilistic) conditional independence
independency model
inference rule
axiomatic characterization of conditional independence

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