唯一信息属性 (Properties of Unique Information)

from arxiv, 18 pages, one figure, 11 theorems. One author was added to acknowledge early contributions to the results. The text was improved, minor mistakes and typos were corrected. One figure was added to visualize the optimization domain

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 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 $UI(T:X\setminus Y)$, most notably when $T$ is binary. In the case that all variables are binary, we obtain a complete picture of where the optimizing probability distributions lie.

翻译：我们研究了Bertschinger等人(2014年)在信息分解框架内界定的独特信息 $UI(T:X\setminus Y) 的量度;我们研究了对美元定义所依据的优化问题的解决方案的独特性和支持;我们从有条件的独立限制和以美元、X美元和美元为主的角度,在全力支持下确定了非统一解决方案的充分条件;我们的结果基于重订目标功能的第一阶条件,即有条件概率矩阵的等级限制;这些结果有助于加速计算$UI(T:X\setminus Y) 美元,最显著的是当美元为二元时。如果所有变量都是二元的,我们就能全面了解最佳概率分布的位置。