Belief structures, weight generating functions and decision-making

Abstract

We describe the Dempster–Shafer belief structure and provide some of its basic properties. We introduce the plausibility and belief measures associated with a belief structure. We note that these are not the only measures that can be associated with a belief structure. We describe a general approach for generating a class of measures that can be associated with a belief structure using a monotonic function on the unit interval, called a weight generating function. We study a number of these functions and the measures that result. We show how to use weight-generating functions to obtain dual measures from a belief structure. We show the role of belief structures in representing imprecise probability distributions. We describe the use of dual measures, other then plausibility and belief, to provide alternative bounding intervals for the imprecise probabilities associated with a belief structure. We investigate the problem of decision making under belief structure type uncertain. We discuss two approaches to this decision problem. One of which is based on an expected value of the OWA aggregation of the payoffs associated with the focal elements. The second approach is based on using the Choquet integral of a measure generated from the belief structure. We show the equivalence of these approaches.