Great resource! Thank you.

+1: Can you please expand on #2? What does it mean for MAGa to be the projection of the causal DAG (including latent P) into the observed variables?

First of all, thank you. An eye-opening answer. Secondly, may I ask if you've seen this material covered any other place than in the referenced paper by Jiji Zhang? Any other resources you'd recommend? In particular I'd love to see applications of this theory; a few examples would be nice; the paper lacks any examples/applications of the theory.

+1: on #1) great; thank you. on #2) I thought visibility of a directed edge implies the absence of a confounder of the endpoints #3) can you please expand on what an inducing path is?

#1 was the heart of the question; perfect! But #2 is just as important as we want to draw the right conclusion from our finding that the path is visible

I asked about parent vs. ancestor because I suspect that we can only ever establish that a variable is an ancestor of another as opposed to a parent. I'd love to hear what others have to say about this.

I learned DAGs from Pearl's "Causal Inference in Statistics: A Primer". In it, every DAG is assumed to be complete (so we are assuming away the possibility of unmeasured confounders). Like you I was interpreting a directed edge from A to B as saying unambiguously that A causes B and that there is no unmeasured confounder between A and B (by assumption that the DAG is complete). However, with FCI we allow the possibility of unmeasured confounders, the outputs are PAGs as opposed to DAGs, and here, somewhat surprisingly, a directed edge has a slightly different interpretation.

A directed edge from A to B really only says that B does not cause A. That is the biggest thing we learn from it.

+1: Thank you for pointing me to pcalg's implementation of visibleEdge. While PC assumes that the DAG is known, which is in contrast to FCI which allows unmeasured confounders, this would still be insightful. I also initially thought it was a contradiction. It turns out there isn't a contradiction here. As the FCI authors themselves explain in Tetrad a directed edge means "a directed edge from A to B: A is a cause of B; it may be a direct or indirect cause that may include other measured variables. Also, there may be an unmeasured confounder of A and B".

I have not. I am not familiar with it. Can you help understand how falsifying the graph would help with determining visibility of a directed edge?