| This dissertation is a major step toward the development of a well-founded theory of uncertainty in Dempster-Shafer theory. A new measure of uncertainty in Dempster-Shafer theory is proposed. The measure, denoted AU, is defined as the maximum of the Shannon entropy on the set of all probabilities dominating a given belief function. It is proven that this measure satisfies all the basic properties of a reasonable measure of uncertainty, most notably the property of subadditivity. Since the new measure is defined as a solution to a non-linear optimization problem, it was necessary for any potential applications of the measure to find an efficient algorithm for computing it. Such an algorithm is presented in the dissertation, both for the general case and for the special case of possibility theory. The correctness of the two versions of the algorithm is proven. As a step toward a proof of uniqueness, it is proven that the measure AU is the smallest measure among all measures (if any other exists) satisfying several intuitive axioms. As a development of the general principle of uncertainty invariance, uncertainty invariant and consistent transformations between belief functions, probability measures, and possibility measures are investigated. The requirement of uncertainty invariance generally does not guarantee a unique solution. The measure of nonspecificity is used as a useful secondary guidance criterion to resolve this indeterminacy. The relationship between uncertainty and combination of evidence by the Dempster rule of combination in Dempster-Shafer theory is studied. As a result of this study, new notions of conflict and information gain are proposed. Finally, directions for further research are discussed. |
