Properties of Unique Information

12/28/2019

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by   Johannes Rauh, et al.

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We study the measure of unique information UI(T:X∖ Y) defined by Bertschinger et al. (2014) within the framework of information decompositions. We study uniqueness and support of the solutions to the optimization problem underlying the definition of UI. We give necessary conditions for non-uniqueness of solutions with full support in terms of the cardinalities of T, X and Y and in terms of conditional independence constraints. Our results help to speed up the computation of UI(T:X∖ Y), most notably in the case where T is binary. In the case that all variables are binary, we obtain a complete picture where the optimizing probability distributions lie.