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I'm trying to show that, given events $A,B,C,D$, such that $A,B$ are conditionally independent given $C$, whether or not $A,B$ are conditionally independent given $C\cap D$.

I spent a couple of hours trying to figure out whether it was true or not, but haven't made significant progress. Can anyone give me some hints?

Thanks for helping!

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Perhaps the quickest way to see that $A$ and $B$ aren't in general conditionally independent given $C\cap D$ is to take the entire space for $C$. Then $A$ and $B$ being conditionally independent given $C$ is equivalent to $A$ and $B$ being independent, and $C\cap D=D$. Certainly not all independent events are conditionally independent given arbitrary $D$.

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    $\begingroup$ I drew what you wrote and it became really clear. Thanks for helping!! $\endgroup$ – Guilherme Salomé Aug 26 '15 at 16:40

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