Dempster–Shafer belief structures for decision making under uncertainty
Introduction
An important task in decision-making is the modeling and representation of uncertain information. While probability is the most common form of uncertainty representation the variety of types information available from modern technological sources makes it imperative that we have at our disposal mathematical formalisms to model various types of uncertain knowledge. The Dempster–Shafer belief structure provides a rather general formal framework in which we can represent various types of knowledge about uncertain variables [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. In addition to probabilistic knowledge we can represent the type of possibilistic knowledge that arises from human sourced linguistic expression of information [18]. It also allows for the representation of imprecise probabilistic information, situations in which the probabilities are only available as values within ranges rather as precise values. This been recently referred to as type two probabilities [19], [20], [21], [22], [23], [24]. Our objective here is to first introduce some ideas from the theory of belief structures and then to look at the task of decision making under Dempster–Shafer uncertain information.
Section snippets
Dempster–Shafer belief structures
A Dempster–Shafer belief structure can be viewed as a generalization of a probability distribution that can provide the ability to model sophisticated information about the uncertainty associated with a variable. Formally a Dempster–Shafer belief structure m on a space X is defined via a collection of non-empty subsets of X, , called focal elements, and a mapping such that and . Here is known as the weight associated with .
While there are various
Further aspects of belief structures
A useful process associated with a belief structure is discounting [2]. Assume m is a belief structure with q focal element, with weights . Discounting m by results in a new belief structure having the same focal elements as m plus the addition of a focal element X. Here however and the additional focal element X has . Discounting is often used to include confidence information associated with provided belief structure information. Thus we can represent the
Measures and belief structures
Since a D–S belief is essentially providing imprecise information about a probability distribution it would be nice if we could infer a probability distribution from a given D–S structure. Let m be a belief structure for which for some A, where . Assume now we try to entail some belief structure for which the lower and upper bounds are equal, . This requires that p < a and p > b, which is impossible. The implication here is that we cannot infer a
Fusing belief structures
An important issue in the use of belief function is the combination of multiple belief functions as it allows us to fuse information from multiple sources [33]. The pioneering approach to combining multiple belief functions is Dempster’s rule [16]. Assume and are two belief functions on the space X providing information about a variable V. Assume their focal elements are respectively and . Under the Dempster rule the combination or conjunction of these two belief structures
Decision making uncertainty
We now turn to some tasks in that can arise in the process of decision-making. We particularly look at this for the case where information about uncertainty is carried by a belief structure. One task is the comparison of alternative courses of action in the face of uncertain information. Fig. 3 will be useful in this discussion.
In the situation depicted in Fig. 3 we have a number of courses of action for i = 1 to r. In addition we have some variable V whose domain is . Furthermore,
Decisions made using other features
In the preceding we focused on the use of expected value as a means of selecting an alternative in the face of the uncertainty carried by a belief structure. In some cases the decision maker may be interested in making his decision based on other features of the uncertain payoffs associated with an alternative. An example of this may be a decision maker’s concern with the occurrence of a “large loss”, he may want to avoid an alternative that has a significant probability of resulting in large
Conclusion
We discussed the need for tools for representing various type of uncertain information in decision-making. We introduced the Dempster–Shafer belief structure and discussed how it provides a formal mathematical framework for representing various types uncertain information. We provided some fundamental ideas and mechanisms related to these Dempster–Shafer structures. We then investigated their role in the important task of decision-making under uncertainty.
Acknowledgement
Ronald Yager’s was supported by a Multidisciplinary University Research Initiative (MURI) grant and by an ONR grant. The authors would also like to acknowledge the support from the Distinguished Scientist Fellowship Program at King Saud University.
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