On Deducing Conditional Independence from d-Separation in Causal Graphs with Feedback (Research Note)

TL;DR

This paper challenges the extension of d-separation criteria to causal graphs with feedback, demonstrating that additional conditions beyond uniqueness are necessary for accurate conditional independence inference.

Contribution

It clarifies the limitations of applying d-separation in cyclic causal graphs and identifies the need for stronger conditions like causal dynamics for correctness.

Findings

D-separation does not always imply conditional independence in feedback networks.

Uniqueness of solutions alone is insufficient for d-separation validity.

Stronger conditions, such as guaranteed causal dynamics, are required.

Abstract

Pearl and Dechter (1996) claimed that the d-separation criterion for conditional independence in acyclic causal networks also applies to networks of discrete variables that have feedback cycles, provided that the variables of the system are uniquely determined by the random disturbances. I show by example that this is not true in general. Some condition stronger than uniqueness is needed, such as the existence of a causal dynamics guaranteed to lead to the unique solution.