P - D - Separation in Causal Networks

Spirtes, Glymour and Scheines 19] formulated a Conjecture that a direct dependence test and a head-to-head meeting test would suuce to construe directed acyclic graph decompositions of a joint probability distribution (bayesian network) for which Pearl's d-separation 2] applies. This Conjecture was later shown to be a direct conse-2 MIECZYSLAW A. KLOPOTEK quence of a result of Pearl and Verma 21], cited as Theorem 1 in 13], see also Theorem 3.4. in 20]). This paper is intended to prove this Conjecture in a new way, by introducing the concept of p-d-separation (partial dependency separation). While Pearl's d-separation works with bayesian networks, p-d-separation is intended to apply to causal networks: that is partially oriented networks in which orientations are given to only to those edges, that express statistically connrmed causal innuence, whereas undirected edges express existence of direct innuence without possibility of determination of direction of causation. As a consequence of the particular way of proving the validity of this Conjecture, an algorithm for construction of all the directed acyclic graphs (dags) carrying the available independence information is also presented. The notion of a partially oriented graph (pog) is introduced and within this graph the notion of p-d-separation is deened. It is demonstrated that the p-d-separation within the pog is equivalent to d-separation in all derived dags.