Back-door path

A back-door path in a causal graph is an indirect path with an arrow into the treatment/causal variable.

Formally, given a graph with nodes including $X$ (treatment/causal variable) and $Y$ (outcome variable), a direct path from $X$ to $Y$ is just a directed edge $X\to Y$. An indirect path from $X$ to $Y$ is some traversal of edges from $X$ to $Y$ that isn't a direct path. Importantly, the edges don't have to be pointed in the "right way", so $X\to Z\leftarrow W\to Y$ is an indirect path. A back-door path is an indirect path with an arrow into $X$.

"a back-door path is defined as any path between the causal variable and the outcome variable that begins with an arrow that points to the causal variable" (p. 30). So this definition doesn't mention indirect paths, but I guess it's implied, because a direct path from the outcome variable to the causal variable would mean that the causal variable isn't really a causal variable?

Examples

• $X\leftarrow Z\to Y$ is a back-door path from $X$ to $Y$: it is indirect (involves $Z$) and has an arrow into $X$ (from $Z$).
• $X\to Z\leftarrow W\to Y$ is not a back-door path from $X$ to $Y$: it is an indirect path but there is no arrow into $X$.