Extractable shared information and left monotonicity

The talk summarizes results of arxiv.org/abs/1701.07805 (joint work with P. Banerjee, E. Olbrich, J. Jost, N. Bertschinger).

We consider the problem of quantifying the information shared by a pair of random variables X1, X2 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;X1X2) 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.