p-d-Separation -- A Concept for Expressing Dependence/Independence
  Relations in Causal Networks
  Spirtes, Glymour and Scheines formulated a Conjecture that a direct
dependence test and a head-to-head meeting test would suffice to construe
directed acyclic graph decompositions of a joint probability distribution
(Bayesian network) for which Pearl's d-separation applies. This Conjecture was
later shown to be a direct consequence of a result of Pearl and Verma. This
paper is intended to prove this Conjecture in a new way, by exploiting 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 confirmed causal
influence, whereas undirected edges express existence of direct influence
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 defined. It is demonstrated that the p-d-separation
within the pog is equivalent to d-separation in all derived dags.
