Consider the belief network

belief network

The question now is to list all nodes which are d-seperated from E given I, G. The formal definition is as follows: any node in $X$ is d-seperated from any node in $Y$ given all nodes in $C$ (conditioning set) when all paths from any element of $X$ to any element of $Y$ are blocked by $C$.

A path is blocked by $C$ if at least one of the following conditions is satisfied

  1. There is a collider in the path $P$ such that neither the collider nor any of its descendants is in the conditioning set $C$.

  2. There is a non-collider in the path $P$ that is in the conditioning set $C$.

When I apply this definition, all I can find is that $H$ is d-seperated from $E$ given $I$ and $G$ (the paths are {$HGBE, HGDCBE, HIE$}. But this just seems intuitively wrong that given these two, nearly nothing is independent. I double checked for each variable but there's always a non blocking path. Why would $D$ for example be dependant since we know that it has a parent ($G$) that is in $C$.

The definition at point 1 talks about a collider (in this case, most of the time it is $B$, but this collider gets an arrow from $A$ which has nothing to do with the rest of the path. Should I reinterpret this definition? And what if there are multiple colliders like in the path $H-G-B-E$ where $H$, $G$ and $B$ are colliders. Should the rule apply to every one of them?


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