The content of this answer is mainly taken from the paper:
Zhang, Jiji. "On the completeness of orientation rules for causal
discovery in the presence of latent confounders and selection bias."
Artificial Intelligence 172.16-17 (2008): 1873-1896.
For DAGs we have the two nice facts that
- The node $Z$ is a collider between $X$ and $Y$, i.e.:
$$
X \to Z \leftarrow Y
$$
iff any set $d$-separating $X$ and $Y$ does not contain $Z$ and that
- $Z$ is not a collider between $X$ and $Y$, i.e.:
$$
X \to Z \to Y \quad \mbox{or}\quad X \leftarrow Z \leftarrow Y \quad\mbox{or}\quad X\leftarrow Z \to Y,
$$
iff any set $d$-separating $X$ and $Y$ does contain $Z$.
Those facts are often used for orientation in discovery algorithms like FCI and friends. Now, if you want to extend those algorithms to the case where you include hidden confounders and selection bias, you have to consider MAGs. And while in MAGs the above two facts don't hold anymore, they have pendants obtained by substituting colliders with discriminating paths:
If you have a discriminating path between $X$ and $Y$ with $Z$ being discriminated on, then you have again two facts:
- The node $Z$ is a collider on the discriminating path iff any set m-separating $X$ and $Y$ does not contain $Z$, and
- the node $Z$ is not a collider on the discriminating path iff any set m-separating $X$ and $Y$ contains $Z$.
This can then again be used to orient edges in discovery algorithms, this time in the more demanding case of considering hidden confounders and selection bias.
Thus, the intuition should be that discriminating paths replace the role of confounders in edge-orientation methods when switching to MAGs.
I don't know why they named it "discriminating path", maybe because it helps identify edge heads, i.e. discriminating between the possibilities contained in PAGs.
Note that a discriminating path between two nodes $X$ and $Y$ does not decide about whether they are m-separable. If such a discriminating path exists, it is still possible that $X$ and $Y$ are separable and it is still possible that they are not separable. And if such a discriminating path does not exist, it is again not excluding neither separability or nonseparability of $X$ and $Y$.
Further intuition can be built by noting that the node $Z$ is the only reason that prevents a discriminating path from being an inducing path. Those inducing paths are very important, they are the reason why there are ancestral graphs that contain pairs of nodes that are not adjacent, yet cannot be separated by any set, something impossible in DAGs.
I haven't read Pearl's book, but have learned my share of graph theory.
A path between two nodes is a directed path if it can be traced along the arrows, that is, if no node on the path has two edges on the path directed into it, or two edges directed out of it.
The "that is" is inconsistent. A path that "can be traced along the arrows" would include loops, "if no node on the path has two edges on the path directed into it, or two edges directed out of it" excludes loops. These are two different things.
In the example, $X\to Z\to Y$ would be a directed path under both definitions.
However, assume there was an additional arrow from $Z$ back to $X$. Then $X\to Z\to X\to Z\to Y$ would be a directed path under the first definition ("it can be traced along the arrows"), but not under the second ("no node on the path has two edges on the path directed into it, or two edges directed out of it": $Z$ has two edges on the path directed out).
So the answer is clear for the example you are given. For the rest of the book, you should be a little careful if you encounter DAGs that contain potential loops. I would assume that Pearl really meant the second definition: a "directed path" for him probably is a "loopless directed path".
Best Answer
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.