I am going through Causal Inference In Statistics by Pearl and I have come across the definition of path and directed path (section 1.4, page 25).
Path: A path between two nodes $X$ and $Y$ is a sequence of nodes beginning with $X$ and ending with $Y$, in which each node is connected to the next by an edge.
Directed path: 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.
I have confusion understanding the phrasing of the highlighted portion in the definition of "directed path". This clarity becomes important in the definition of variables satisfying front-door criterion (definition 3.4.1, section 3.4, page 69) -
A set of variables $Z$ is said to satisfy the front-door criterion relative to an ordered pair of variables $(X,Y)$ if
- $Z$ intercepts all directed paths from $X$ to $Y$.
- There is no backdoor path from $X$ to $Z$.
- All backdoor paths from $Z$ to $Y$ are blocked by $X$.
Clarification for the doubt can be provided via counterexamples as well. For example, does $X \rightarrow Z \rightarrow Y$ in the below figure qualify as a directed path, or does it violate the "two edges directed in/out" criterion?
Note: I could not refer to the Wikipedia definition, or other sources, as, I feel, there is no consensus about the definition of directed path across all literature, and is context dependent.