Intuition and meaning of a “discriminating path” in a causal DAG

causal-diagramcausalitydaggraph theory

In Ali, Richardson and Spirtes (2009) (open copy here) and many other papers in the causal DAG literature, there is the notion of a "discriminating path". The definition is:

A path $\pi = ⟨x,q_1,…,q_p,b,y⟩$ $(p \ge 1)$ is a discriminating path for $b$ in a DAG $G$ if:

  1. $x$ is not adjacent to $y$,
  2. $b$ is not an end-point and is adjacent to $y$ on $\pi$ and,
  3. every vertex $q_i$ $(1\le i \le p)$ is a collider on $\pi$, and a parent of $y$.

Question: What is the intuition of a discriminating path? Why is it called a "discriminating path"?

Best Answer

The content of this answer is mainly taken from the paper:

Zhang, Jiji. "On the completeness of orientation rules for causal discovery in the presence of latent confounders and selection bias." Artificial Intelligence 172.16-17 (2008): 1873-1896.

For DAGs we have the two nice facts that

  1. The node $Z$ is a collider between $X$ and $Y$, i.e.: $$ X \to Z \leftarrow Y $$ iff any set $d$-separating $X$ and $Y$ does not contain $Z$ and that
  2. $Z$ is not a collider between $X$ and $Y$, i.e.: $$ X \to Z \to Y \quad \mbox{or}\quad X \leftarrow Z \leftarrow Y \quad\mbox{or}\quad X\leftarrow Z \to Y, $$ iff any set $d$-separating $X$ and $Y$ does contain $Z$.

Those facts are often used for orientation in discovery algorithms like FCI and friends. Now, if you want to extend those algorithms to the case where you include hidden confounders and selection bias, you have to consider MAGs. And while in MAGs the above two facts don't hold anymore, they have pendants obtained by substituting colliders with discriminating paths:

If you have a discriminating path between $X$ and $Y$ with $Z$ being discriminated on, then you have again two facts:

  1. The node $Z$ is a collider on the discriminating path iff any set m-separating $X$ and $Y$ does not contain $Z$, and
  2. the node $Z$ is not a collider on the discriminating path iff any set m-separating $X$ and $Y$ contains $Z$.

This can then again be used to orient edges in discovery algorithms, this time in the more demanding case of considering hidden confounders and selection bias.

Thus, the intuition should be that discriminating paths replace the role of confounders in edge-orientation methods when switching to MAGs.

I don't know why they named it "discriminating path", maybe because it helps identify edge heads, i.e. discriminating between the possibilities contained in PAGs.

Note that a discriminating path between two nodes $X$ and $Y$ does not decide about whether they are m-separable. If such a discriminating path exists, it is still possible that $X$ and $Y$ are separable and it is still possible that they are not separable. And if such a discriminating path does not exist, it is again not excluding neither separability or nonseparability of $X$ and $Y$.

Further intuition can be built by noting that the node $Z$ is the only reason that prevents a discriminating path from being an inducing path. Those inducing paths are very important, they are the reason why there are ancestral graphs that contain pairs of nodes that are not adjacent, yet cannot be separated by any set, something impossible in DAGs.

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