What is an inducing path in causal inference?

Question

In causal graphical models, an inducing path is defined as:

[Definition Inducing Path] An inducing path relative to L is a path on which every non-endpoint node X < L is a collider on the path (i.e., both edges incident to the node are into it) and every collider is an ancestor of an endpoint of the path.

My questions are:

1. What is the intuition of an inducing path? (e.g. why is it called that, or what is it doing?)
2. When is it useful?

I found another post that briefly mentions it, but I cannot find a laymen definition, or interpretation of it.

Answer 1

An inducing path is one where all non-endpoint nodes along the path are colliders and an ancestor of one of the endpoints. An example of an inducing paths is:

$X \rightarrow C1 \leftrightarrow C2 \leftrightarrow Y$, where also $C1 \rightarrow X; C2 \rightarrow X$.

An inducing path intuitively is a path between two non-adjacent nodes that cannot be d-separated. Therefore, the path is always "active" regardless of what variables you condition on.

It is useful in a MAG and PAG setting because it implies there are nodes that are non-adjacent in the true DAG, that will appear adjacent in the MAG/PAG. I.e. it "induces" an adjacency, even though the nodes are in fact non-adjacent.

This is useful for understanding because it means that adjacencies in a MAG/PAG setting are not the same as that of a DAG.