# Why doesn't this work as a backdoor?

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.

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 $$X_2$$.