伪布尔函数的影响和相互作用指数：统一的最小二乘方法
Influence and interaction indexes for pseudo-Boolean functions: a unified least squares approach
摘要：
伪布尔函数（或合作对策）的Banzhaf幂指数和交互指数自然地出现在该函数的标准最小二乘逼近中，作为指定次数的伪布尔函数的首项系数。我们首先观察到，如果我们考虑仅依赖于指定变量的伪布尔函数的近似，则该性质仍然成立。然后，我们表明，Banzhaf影响指数也可以从后者的近似问题。考虑到这个近似问题的某些加权版本，我们引入了一类加权Banzhaf影响指数，分析其最重要的性质，并指出加权Banzhaf影响指数和相应的加权Banzhaf相互作用指数之间的相似之处。我们还讨论了从规定的影响重构伪布尔函数的问题，并指出加权和非加权情况下非常不同的行为。
英文摘要：
The Banzhaf power and interaction indexes for a pseudo-Boolean function (or a cooperative game) appear naturally as leading coefficients in the standard least squares approximation of the function by a pseudo-Boolean function of a specified degree. We first observe that this property still holds if we consider approximations by pseudo-Boolean functions depending only on specified variables. We then show that the Banzhaf influence index can also be obtained from the latter approximation problem. Considering certain weighted versions of this approximation problem, we introduce a class of weighted Banzhaf influence indexes, analyze their most important properties, and point out similarities between the weighted Banzhaf influence index and the corresponding weighted Banzhaf interaction index. We also discuss the issue of reconstructing a pseudo-Boolean function from prescribed influences and point out very different behaviors in the weighted and non-weighted cases.
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