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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". enter image description here

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:

enter image description here

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?

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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.

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  • $\begingroup$ Thank you clarifying things out! I was indeed missing something on the definition of a backdoor path: that path must "go into" the treatment and not "go from" it (and influence the outcome). $\endgroup$
    – Tanguy
    Jul 29 at 22:05
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    $\begingroup$ @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! $\endgroup$ Jul 29 at 22:07
  • $\begingroup$ OK, though I understand that a path with a collider can still be counted as a backdoor path (even though it is blocked unless the collider is conditioned on). $\endgroup$
    – Tanguy
    Jul 29 at 22:11
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    $\begingroup$ Perhaps. At that point, it's more semantics. The crucial thing to know is whether you need to condition on various variables. $\endgroup$ Jul 29 at 22:15
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    $\begingroup$ I think you're right, by the way: backdoor paths can be blocked or not. The first arrow into the treatment is the critical piece. $\endgroup$ Jul 29 at 22:32

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