Properties of Unique Information

We study the measure of unique information $UI(T:X\setminus Y)$ 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 $UI(T:X\setminus Y)$, 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.