# causal graph - counting the number of backdoor paths in a DAG

I am following "A Crash Course in Causality: Inferring Causal Effects from Observational Data" on Coursera.

I am struggling at correctly identifying backdoor paths in causal graphs (or DAG for Directed Acyclic Graph).

Example #1 :

The following DAG is given in example in week $$2$$'s video on the "backdoor path criterion". The course states that there are $$3$$ backdoor paths from $$A$$ to $$Y,$$ but I see $$4$$ of them:

\begin{align*} &A \leftarrow W \leftarrow Z \leftarrow V \to Y\\ &A \leftarrow W \to M \to Y\\ &A \leftarrow Z \leftarrow V \to Y\\ &A \leftarrow Z \to W \to M \to Y \quad (\text{not pointed out}) \end{align*}

Example #2 :

In the same week quiz, we are asked to count the number of backdoor paths on this DAG: Apparently, there is only $$1$$ but once again I see more of them (I count $$2$$ of them):

\begin{align*} &G \to A \to B\\ &G \leftarrow E \leftarrow D \to A \to B \end{align*}

Where am I wrong?

For Example 1, you are correct. $$A\leftarrow Z\to W\to M\to Y$$ is a valid backdoor path with no colliders in it (which would stop the backdoor path from being a problem).
In Example 2, you are incorrect. The definition of a backdoor path implies that the first arrow has to go into $$G$$ (in this case), or it's not a backdoor path. Only $$G\leftarrow E\leftarrow D\to A\to B$$ satisfies that criterion.
• @Tanguy Right, but that does only go for the first arrow. As long as you don't have a collider on the path, like this: $A\to B\leftarrow C,$ which blocks data flow from $A$ to $C$ unless you condition on $B,$ then the arrow direction is unimportant after that first arrow. You're welcome for the help! Jul 29 at 22:07