Causal Analysis in Theory and Practice

From a previous correspondence with Eliezer S. Yudkowsky, Research Fellow, Singularity Institute for Artificial Intelligence, Santa Clara, CA

The following paragraph appears on p. 103, shortly after eq. 3.63 in my copy of Causality:

"To place this result in the context of our analysis in this chapter, we note that the class of semi-Markovian models satisfying assumption (3.62) corresponds to complete DAGs in which all arrowheads pointing to X_k originate from observed variables."

It looks to me like this is a sufficient, but not necessary, condition to satisfy 3.62. It appears to me that the necessary condition is that no confounder exist between any X_i and L_j with i < j and that no confounder exist between any X_i and the outcome variable Y. However, a confounding arc between any X_i and X_j, or a confounding arc between L_i and X_j with i <= j, should not render the causal effect non-identifiable. For example, even if a confounding arc exists between X₂ and X₃ (but no other confounding arcs exist in the model), the causal effect on Y of setting X₂=x₂ and X₃=x₃ should be the same as the distribution on Y if we observe x₂ and x₃.

It is also not necessary that the DAG be complete.