Abstract:We consider the problem of quantifying the information shared by a pair of random variables X 1 , X 2 about another variable S. We propose a new measure of shared information, called extractable shared information, that is left monotonic; that is, the information shared about S is bounded from below by the information shared about f (S) for any function f . We show that our measure leads to a new nonnegative decomposition of the mutual information I(S; X 1 X 2 ) into shared, complementary and unique components. We study properties of this decomposition and show that a left monotonic shared information is not compatible with a Blackwell interpretation of unique information. We also discuss whether it is possible to have a decomposition in which both shared and unique information are left monotonic.
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