The information decomposition problem requires an additive decomposition of the mutual information between the input and target variables into nonnegative terms. The recently introduced solution to this problem, Information Attribution, involves the Shapley-style value measuring the influence of predictors in the coalitional game associated with the joint probability distribution of the input random vector and the target variable. Motivated by the original problem, we consider a general setting of coalitional games where the players form a boolean algebra, and the coalitions are the corresponding down-sets. This enables us to study in detail various single-valued solution concepts, called values. Namely, we focus on the classes of values that can represent very general alternatives to the solution of the information decomposition problem, such as random-order values or sharing values. We extend the axiomatic characterization of some classes of values that were known only for the standard coalitional games.
翻译:信息分解问题要求将输入变量和目标变量之间的相互信息分解为非负值术语。 这个问题最近引入的解决方案“ 信息归属 ” ( Information Accreditation), 涉及到测量输入随机矢量和目标变量共同概率分布相关联玩游戏预测者影响的“ 沙普利式” 值。 受原始问题的驱动, 我们考虑联合游戏的总体设置, 玩家形成一个布林代数, 联盟是相应的下限。 这使我们能够详细研究各种单价解决方案概念, 叫做“ 价值 ” 。 也就是说, 我们侧重于能够代表信息分解问题解决方案的非常普通的替代方法的数值类别, 比如随机顺序值或共享值。 我们扩展了某些只有标准组合游戏才知道的数值类别的不言理特征 。