莫比乌斯变换和Dempster-Shafer理论的焦点及其意义
Focal points and their implications for Möbius Transforms and Dempster-Shafer Theory
摘要：
Dempster-Shafer理论（DST）推广了贝叶斯概率理论，提供了有用的额外信息，但具有更高的计算负担。为了降低Dempster准则信息融合的时间复杂度，人们做了大量的工作。Dempster准则是两个zeta变换的逐点相乘，并且已经找到了最优的通用算法来得到这些变换的完整定义。然而，它是在本文中所示的zeta变换及其逆，莫比乌斯变换，可以精确地简化，拟合信度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偏序集上的任何函数。它依赖于一个新的概念，我们称之为焦点，它构成了最小的域，在这个域上可以定义zeta和莫比乌斯变换。我们证明了DST的这些一般结果的兴趣，不仅减少了复杂性之间的信仰表示和他们的融合，但也为理论目的的转换。事实上，我们提供了一个新的概括的证据和公式揭示每个分解权重是如何绑定到相应的质量函数的合取分解。
英文摘要：
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.
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