标题： 协同、抑制和不道德性：熵函数的前向差分 显示英文标题

标题： Synergy, suppression and immorality: forward differences of the entropy function

摘要： 条件互信息在图形模型的选择和解释中很重要。它的经验版本众所周知是一个广义似然比检验，并且可以表示为熵的差异。我们考虑在所研究变量的所有子集上定义的熵函数的前向差分展开。这个展开的元素对于其下标的排列是不变的，并将高阶互信息与低阶互信息联系起来。三阶差分可表示为一个边缘互信息和一个条件互信息之间的显然不对称的差分。它在解释信息分解中的作用为三个随机变量之间的协同作用提供了技术定义。当两个变量为第三个变量提供替代解释时出现正值；当解释信息的总和大于各部分之和时出现负值，这些负值称为协同作用。协同作用往往不常见；它们将回归中的抑制变量和贝叶斯网络中的未屏蔽碰撞器（不道德性）这两个看似无关的概念联系起来。我们给出了这些现象的新特征，这些特征可以推广到分类变量和更高维度。我们提出了一种系统地从给定图中计算低阶差分的算法。从小规模实际研究中得到的例子表明这些技术在经验统计分析中的潜力。 显示英文摘要

摘要： Conditional mutual information is important in the selection and interpretation of graphical models. Its empirical version is well known as a generalised likelihood ratio test and that it may be represented as a difference in entropy. We consider the forward difference expansion of the entropy function defined on all subsets of the variables under study. The elements of this expansion are invariant to permutation of their suffices and relate higher order mutual informations to lower order ones. The third order difference is expressible as an, apparently assymmetric, difference between a marginal and a conditional mutual information. Its role in the decomposition for explained information provides a technical definition for synergy between three random variables. Positive values occur when two variables provide alternative explanations for a third; negative values, termed synergies, occur when the sum of explained information is greater than the sum of its parts. Synergies tend to be infrequent; they connect the seemingly unrelated concepts of suppressor variables in regression, on the one hand, and unshielded colliders in Bayes networks (immoralities), on the other. We give novel characterizations of these phenomena that generalise to categorical variables and to higher dimensions. We propose an algorithm for systematically computing low order differences from a given graph. Examples from small scale real-life studies indicate the potential of these techniques for empirical statistical analysis.