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I'm currently learning about Bayesian Networks and came across this question.

Prove or give a counterexample: Given random variables $A$, $B$, $C$, and $D$, if $A \perp B | \{C, D\}$, then $C \perp D | \{A, B\}$

I've created some simple tables where the first statement holds, and it seems that second statement follows.

How would one come up with a mathematical proof for this? I imagine there would need to be some substitution via Bayes' Rule.

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1 Answer

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Counterexample : If $C=D\sim\mathcal U[0,1]$ and $A\bot B \sim \mathcal U[0,C]$ conditionally to $C,D$, then $A\bot B\mid\{C,D\}$ but $C\not\bot D\mid\{A,B\}$.

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