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Consider a graph $G =(V,E)$ and let the family of random variables $X=(X_v)_{v\in V}$ be a Markov Random Field with respect to $G$. Then we know that for any two vertices $\{u,v\}\subset V$ that are not adjacent, the pairwise Markov property holds, that is $$ X_u \perp \!\!\! \perp X_v | X_{V\setminus\{u,v\}}. $$ For any vertex $w \in V\setminus\{u,v\}$ do we have that $$ X_u \perp \!\!\! \perp X_v | X_w $$ always holds?

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This is not necessarily true. Consider a graph with $V=\{u,v,w,o\}$ with edges $E=\{(u,o), (v,o), (w,o)\}$.

u--o--v
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   w

If $X_u=X_v=X_o$ is independent of $X_w$, then this satisfies the Markov Random Field assumptions. But $X_u$ and $X_v$ are not conditionally independent given $X_w$.

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  • $\begingroup$ Could you explain your answer a bit more please? $\endgroup$ Jul 14, 2021 at 22:29

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