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Given a discrete random variable $(X,Y)$, one can consider the smallest entropy of a random variable $Z$ such that $X$ and $Y$ are independent conditioned to $Z$.

This quantity is akin to the mutual information, in fact one can see that it is always larger than half the mutual information.

Does this number have a name, and has it been studied?

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This is related to Wyner's common information which is always larger than the mutual information.

https://ieeexplore.ieee.org/document/1055346/

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  • $\begingroup$ thanks this is useful. This seems to be pretty close indeed, even though it's not clear to me if it's exactly the same $\endgroup$ – alesia May 13 '18 at 0:19
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    $\begingroup$ Starting from this reference, I found that the variant is called exact common information and was introduced in a 2014 paper by El Gamal et al. It is apparently an open problem whether it coincides with Wyner's common information. $\endgroup$ – alesia May 13 '18 at 1:27

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