Why doesn't this work as a backdoor?

Question

In Pearl's book "Causality" on page 124 (http://bayes.cs.ucla.edu/BOOK-2K/ch3-3.pdf) he says:

A set of variables $Z$ satisfies the back-door criterion relative to an ordered pair $(X_i,X_j)$ in a DAG G if:

(i) no node in $Z$ is a descendant of $X_i$; and

(ii) $Z$ blocks every path between $X_i$ and $X_j$ that contains an arrow into $X_i$

Then, he gives the example graph:

And he says that $Z_1=\{X_3,X_4\}$ and $Z_2=\{X_4,X_5\}$ satisfy the back-door criterion but $Z_3=\{X_4\}$ does not because it does not block $(X_i,X_3,X_1,X_4,X_2,X_5,X_j)$.

But, this does not make sense to me. The path Pearl mentions goes through $X_4$ which is in $Z_3$ so this confused me. In fact, it seems to that $Z$ satisfies the back-door criterion relative to $(X_i,X_j)$ IFF $X_4\in Z$.

Am I totally missing something? It almost feels like I've gotten the definition totally wrong in my head.

Answer 1

Your assumption is that conditioning on a variable (i.e., $X_4$) blocks all paths through that variable, but that is not so. Conditioning on a variable opens a path between the antecedents of the variable. $X_1$ and $X_2$ are d-connected after conditioning on $X_4$. $X_4$ is a collider of $X_1$ and $X_2$.