# d-seperation belief networks

Consider the 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?