#StackBounty: #probability #inference #terminology #causality #causal-diagram Determining direct cause with some "realization&quot…

Bounty: 100

I’m bouncing around "Causality" by Judea Pearl.

On page 222 it offers this definition of a direct cause:

"$X$ is a direct cause of $Y$" if there exist two values $x$ and $x’$ of $X$ and a value $u$ of $U$ such that $Y_{xr}(u) not= Y_{x’r}(u)$ where $r$ is some realization of $V setminus {X, Y}$.

My questions are:

  1. What is a "realization", is it the same as Wikipedia’s Realization (probability) definition?
  2. What does the $setminus$ symbol mean in the context of the two functions $X$ and $Y$? Can you give me an example?
  3. Finally, how do I use this definition in practice?

Let’s say I have two structural causal models:

  1. $X rightarrow Y$
  2. $X rightarrow Q rightarrow Y$

How does this definition of direct cause allow me to discover that $X$ is a direct cause in the first case, and $X$ is not a direct cause in the second case?

Get this bounty!!!

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