Bonferroni mean aggregates the interaction between all pairs of inputs from some n-dimensional input vector. Therefore, it is able to capture the dependency structure between the inputs. Weighted version of the Bonferroni mean then assumes that each input has a possibly different weight. Such an approach poses several constraints that decrease the modeling flexibility of the dependency structure on the input space. In this paper, we present an overview of the different approaches to weighted Bonferroni mean and introduce the Bonferroni mean with weighted interaction where interactions between inputs are weighted rather than the inputs themselves. We show several important properties of the Bonferroni mean with weighted interaction and its connection to concepts studied before. Finally, using the described approach we design a system for emergency management that is able to raise an alarm in the case where certain criteria are met.

中文翻译：

---

加权交互的 Bonferroni 均值


Bonferroni 均值聚合了某个 n 维输入向量的所有输入对之间的交互作用。因此，它能够捕获输入之间的依赖结构。 Bonferroni 均值的加权版本假设每个输入可能具有不同的权重。这种方法提出了一些限制，降低了输入空间上的依赖结构的建模灵活性。在本文中，我们概述了加权 Bonferroni 均值的不同方法，并介绍了带有加权交互的 Bonferroni 均值，其中对输入之间的交互而不是输入本身进行加权。我们展示了带有加权交互作用的 Bonferroni 均值的几个重要属性及其与之前研究的概念的联系。最后，使用所描述的方法，我们设计了一个应急管理系统，该系统能够在满足某些标准的情况下发出警报。