p-d-Separation -- A Concept for Expressing Dependence/Independence Relations in Causal Networks

TL;DR

This paper introduces p-d-separation, a new concept for expressing dependence and independence in causal networks, and provides a proof of a related conjecture along with an algorithm for constructing relevant directed acyclic graphs.

Contribution

It presents p-d-separation as a novel method for analyzing causal networks with partially oriented edges and proves its equivalence to d-separation in derived DAGs.

Findings

p-d-separation is equivalent to d-separation in derived DAGs.

An algorithm for constructing all DAGs consistent with given independence information.

The concept applies to partially oriented causal networks with both directed and undirected edges.

Abstract

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.