Determining direct cause with some "realization" of `V \ {X, Y}` 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:

*

*What is a "realization", is it the same as Wikipedia's Realization (probability) definition?

*What does the $\setminus$ symbol mean in the context of the two functions $X$ and $Y$? Can you give me an example?

*Finally, how do I use this definition in practice?

Let's say I have two structural causal models:

*

*$X \rightarrow Y$

*$X \rightarrow Q \rightarrow Y$
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?

Here Pearl is defining under what condition $X$ is a direct cause of $Y$.
In his notation $U$ refers to exogenous variables (variables coming from outside the model) and $V$ refers to endogenous variables (variables inside the model). See wikipedia for a bit more detail.
Q2. The \ symbol is typically used in reference to sets to mean "exclusion". So $V\backslash \{X,Y\}$ means the set of endogonous variables (not including $X$ and $Y$).
Q1. By realization, he is refering to the statistical interpretation to which you linked. He means, let's imagine that the variables in $V$ (not including $X$ and $Y$) take some values which we call $r$.
Q3. This question is a little more vague to answer, but the definition is (effectively) saying that "all other things being equal", changes in the value of $X$ will change the observed value of $Y$.
Q4. The definition let's you distinguish these two use cases since in the second case $V\backslash \{X,Y\}=Q$ and so a different value of $X$ will create a different realized value of $r$. I'm not 100% certain on this point though so perhaps someone else can explain this better.
