可能性规则的不确定性形式化模型
Formal Model of Uncertainty for Possibilistic Rules
摘要：
给定一个讨论X的宇宙（一个可能结果的领域），一个实验可能包括选择其中一个元素，服从偶然性的操作，或者观察元素，服从不精确性。关于实验实际结果的先验不确定性可以被量化，表示选择r_X的可能性或任何这样的X适合作为结果描述的程度。前一种情况对应于一个概率分布，而后一种情况给出了X上的一个可能性分配。这种分配及其性质的研究属于可能性理论的范畴[DP 88，Y 80，Z 783。它和概率论一样，分配0到1之间的值来表示结果的可能性。然而，相似之处到此为止。可能性理论使用最大值和最小值函数来组合不确定性，而概率理论使用加号和乘法运算。这导致理论在分析框架方面非常不同，尽管它们共享一些语义概念。其中一个共同的概念是定量地表达与给定分布相关的不确定性。在概率论中，它的值对应于进行实验和确定实际结果所获得的信息。这种信息的获得同样可以被看作是实验结果不确定性的减少。在这种情况下，信息的标准度量以及不确定性是香农熵[AD 75，G77]。它有几个优点--它的特点是只有几个非常自然的性质，而且它可以方便地用于决策过程。这个应用程序是基于最大熵的原则，它已成为一个流行的方法，有关决策的不确定性。本文论证了在可能性理论的基础上可以建立一个同等完整的理论。我们首先展示了如何定义的信息和不确定性的可能性分配措施。接下来，我们构建一个基于信息的度量空间上的所有可能性分布定义在一个给定的域。它使我们能够捕捉到分布之间信息内容的接近性的概念。最后，我们证明了上述所有构造都可以在任意可测域上对连续分布-可能性指派进行。我们认为这一步是非常重要的有限域的话语，但近似的现实生活中的无限域。如果可能性理论是代表现实世界的情况下，它必须处理连续分布直接和通过有限的近似。在最后一节中，我们讨论了可能性分布的最大不确定性原理。我们将展示如何将这样的原则形式化为一个推理规则。我们还认为，它可以作为关于组合信息的简单假设的结果而导出。我们要提到的是，可能性分配可以被视为模糊集，每个模糊集都会产生可能性的分配。这种对应关系在逻辑学和控制理论中具有深远的影响。我们在这里的处理与任何特殊的解释无关;特别是我们谈到可能性分布和可能性测度，将它们定义为到区间[0，1]的可测映射。我们的介绍是一个自成一体的，虽然简洁的总结。所讨论的主题都是精心挑选的，以证明该理论的完整性和一定的优雅性。不包括证明;我们只提供说明性的例子。
英文摘要：
Given a universe of discourse X-a domain of possible outcomes-an experiment may consist of selecting one of its elements, subject to the operation of chance, or of observing the elements, subject to imprecision. A priori uncertainty about the actual result of the experiment may be quantified, representing either the likelihood of the choice of :r_X or the degree to which any such X would be suitable as a description of the outcome. The former case corresponds to a probability distribution, while the latter gives a possibility assignment on X. The study of such assignments and their properties falls within the purview of possibility theory [DP88, Y80, Z783. It, like probability theory, assigns values between 0 and 1 to express likelihoods of outcomes. Here, however, the similarity ends. Possibility theory uses the maximum and minimum functions to combine uncertainties, whereas probability theory uses the plus and times operations. This leads to very dissimilar theories in terms of analytical framework, even though they share several semantic concepts. One of the shared concepts consists of expressing quantitatively the uncertainty associated with a given distribution. In probability theory its value corresponds to the gain of information that would result from conducting an experiment and ascertaining an actual result. This gain of information can equally well be viewed as a decrease in uncertainty about the outcome of an experiment. In this case the standard measure of information, and thus uncertainty, is Shannon entropy [AD75, G77]. It enjoys several advantages-it is characterized uniquely by a few, very natural properties, and it can be conveniently used in decision processes. This application is based on the principle of maximum entropy; it has become a popular method of relating decisions to uncertainty. This paper demonstrates that an equally integrated theory can be built on the foundation of possibility theory. We first show how to define measures of in formation and uncertainty for possibility assignments. Next we construct an information-based metric on the space of all possibility distributions defined on a given domain. It allows us to capture the notion of proximity in information content among the distributions. Lastly, we show that all the above constructions can be carried out for continuous distributions-possibility assignments on arbitrary measurable domains. We consider this step very significant-finite domains of discourse are but approximations of the real-life infinite domains. If possibility theory is to represent real world situations, it must handle continuous distributions both directly and through finite approximations. In the last section we discuss a principle of maximum uncertainty for possibility distributions. We show how such a principle could be formalized as an inference rule. We also suggest it could be derived as a consequence of simple assumptions about combining information. We would like to mention that possibility assignments can be viewed as fuzzy sets and that every fuzzy set gives rise to an assignment of possibilities. This correspondence has far reaching consequences in logic and in control theory. Our treatment here is independent of any special interpretation; in particular we speak of possibility distributions and possibility measures, defining them as measurable mappings into the interval [0, 1]. Our presentation is intended as a self-contained, albeit terse summary. Topics discussed were selected with care, to demonstrate both the completeness and a certain elegance of the theory. Proofs are not included; we only offer illustrative examples.
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