PGM and Derivation some inference? I study my notes on Probabilistic Theory and notation and d-separation topics. we know PGM's are a rich framework for encoding probability distributions over complex domains. I couldn't get the point why these two ones are True according to this fig. how we can derived these two are conducted from this fig.

 A: d-separation allows us to infer conditional independences using the concept of active an inactive paths.
A path between two variables $C_1 - ... - C_n$ is active in a context $S$ if:

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*there is no fork $\leftarrow C_i \rightarrow $, chain $\rightarrow C_i \rightarrow$ or reverse chain $\leftarrow C_i\leftarrow$ such that $C_i \in S$


*for every collider $\rightarrow C_i \leftarrow$ either $C_i \in S$ or there exists a descendent $C_i \to D_1 \to ... \to D_n$ such that $D_n \in S$.
We say that two variables $A,B$ are d-separated in a context $S$ if there is no active path in $S$ between them.
d-separation is a useful concept because in a Bayesian network it implies conditional independence (though the reverse is not always true).
Now we are prepared to answer the question.
There are two paths between $X$ and $J$ in the diagram. But both paths include the collider $\rightarrow R \leftarrow$. Hence both paths will be inactive unless we control for $R$. In particular, $X\perp J$.
Between $L$ and $J$ there are also two paths. Let's focus on the path $L \to H \leftarrow W \rightarrow V \leftarrow J$. Here, $H,V$ are colliders. Thus, controlling for them without controlling for the fork $W$ opens up the path. Thus it is not necessarily true that $L\perp J | H,V$.
