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Suppose we have a graphical model defined by a directed acyclic graph $G$. Suppose that a node $a$ divides $G$ into two connected components. By this I mean that:

$G=G_1 \cup G_2 \cup \{a\}$

$G_1 \cap G_2 = \emptyset$

and there are no edges between $G_1$ and $G_2$. ( ok technically by $G$ I mean the set of vertices of $G$, I hope the question is clear anyway... )

Is it true that, given $g_1 \in G_1$ and $g_2 \in G_2$:

$$ p(g_1|g_2,a)=p(g_1|a) ? [1]$$

I conjecture this from how graphical models are build, but is it true ? Of course it is simple to generalize [1] but I wanted to write a simple statement.

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Yes it is true.

For example if you follow the d-separation procedure, at step 4, you will delete $a$ which necessarily disconnects $g_1$ and $g_2$, $\forall g_1\in\mathcal{G}_1$, $\forall g_2\in\mathcal{G}_2$, leading to the conditional independence.

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  • $\begingroup$ Thanks. Do you have also a reference where such statements, like the one reported in bold in the notes you linked, is discussed ? $\endgroup$
    – Thomas
    Feb 2 at 12:18
  • $\begingroup$ The statement in bold is somehow the way Bayes Net are defined. Maybe you can look here for more details, implications etc. $\endgroup$
    – TheCG
    Feb 2 at 12:36

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