Dempster-Shafer Theory (DST) generalizes Bayesian probability theory, offering useful additional information, but suffers from a much higher computational burden. A lot of work has been done to reduce the time complexity of information fusion with Dempster’s rule, which is a pointwise multiplication of two zeta transforms, and optimal general algorithms have been found to get the complete definition of these transforms. Yet, it is shown in this paper that the zeta transform and its inverse, the Möbius transform, can be exactly simplified, fitting the quantity of information contained in belief functions. Beyond that, this simplification actually works for any function on any partially ordered set. It relies on a new notion that we call focal point and that constitutes the smallest domain on which both the zeta and Möbius transforms can be defined. We demonstrate the interest of these general results for DST, not only for the reduction in complexity of most transformations between belief representations and their fusion, but also for theoretical purposes. Indeed, we provide a new generalization of the conjunctive decomposition of evidence and formulas uncovering how each decomposition weight is tied to the corresponding mass function.

Dempster-Shafer理论（DST）概括了贝叶斯概率理论，提供了有用的附加信息，但计算量却大得多。为了减少信息融合的时间复杂度，已经做了很多工作，这是使用两个zeta变换的逐点乘法的Dempster规则，并且找到了最佳通用算法来获得这些变换的完整定义。然而，本文表明，可以精确简化zeta变换及其逆Möbius变换，以适应信念函数中包含的信息量。除此之外，这种简化实际上适用于任何部分排序集上的任何函数。它依赖于我们称为联络人的新概念并且这是可以定义zeta和Möbius变换的最小域。我们证明了DST的这些一般结果的意义，不仅是为了降低信念表示及其融合之间大多数转换的复杂性，而且还出于理论目的。实际上，我们提供了证据和公式的联合分解的新概括，揭示了每个分解权重如何与相应的质量函数联系在一起。