Properties of Unique Information
Johannes Rauh, Maik Schünemann, and Jürgen Jost
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Submission date: 06. Jan. 2020 (revised version: May 2021)
MSC-Numbers: 94A15, 94A17
Keywords and phrases: information decomposition, Unique Information
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We study the unique information UI(T : X ∖ Y ) deﬁned by Bertschinger et al. (2014) within the framework of information decompositions. In particular, we study uniqueness and support of the solutions to the convex optimization problem underlying the deﬁnition of UI. We identify suﬃcient conditions for non-uniqueness of solutions with full support in terms of conditional independence constraints and in terms of the cardinalities of T, X and Y . Our results are based on a reformulation of the ﬁrst order conditions on the objective function as rank constraints on a matrix of conditional probabilities. These results help to speed up the computation of UI(T : X ∖ Y ), most notably when T is binary. Optima in the relative interior of the optimization domain are solutions of linear equations if T is binary. In the all binary case, we obtain a complete picture of where the optimizing probability distributions lie.