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## The choice of generalized Dempster–Shafer rules for aggregating belief functions.

In the paper we investigate the criteria of choosing generalized Dempster–Shafer rules for aggregating sources whose information is represented by belief functions. The approach is based on measuring various types of uncertainty in information and we use for this purpose in particular linear imprecision indices. Some results concerning properties of such rules are also presented.

The aim of this paper is to show that the Kantorovich problem, well known in models of economics and very intensively studied in probability theory in recent years, can be viewed as the basis of some constructions in the theory of belief functions. We demonstrate this by analyzing specialization relation for finitely defined belief functions and belief functions defined on reals. In addition, for such belief functions, we consider the Wasserstein metric and study its connections to disjunctions of belief functions.

In the paper we investigate the criteria of choosing generalized Dempster-Shafer rules for aggregating sources of information presented by belief functions. The approach is based on measuring various types of uncertainty in information and we use for this linear imprecision indices. Some results concerning properties of such rules are also presented.

In the paper, we formalize the notion of contradiction between belief functions: we argue that belief functions are not contradictory if they provide non-contradictory models for decision-making. To elaborate on this idea, we take the decision rule from imprecise probabilities and show that sources of information described by belief functions are not contradictory iff the intersection of corresponding credal sets is not empty. We demonstrate that evidential conjunctive and disjunctive rules fit with this idea and they are justified in a probabilistic setting. In the case of contradictory sources of information, we analyze possible conjunctions and show how the result can be described by generalized credal sets. Based on generalized credal sets, we propose a measure of contradiction between information sources and find its axiomatics. We show how the contradiction correction can be produced based on generalized credal sets and how it can be done on sets of surely desirable gambles.

In the paper we argue that aggregation rules in the theory of belief functions should be in accordance with underlying decision models, i.e. aggregation produced in conjunctive manner has to produce the order embedded to the union of partial orders constructed in each source of information; and if we take models based on imprecise probabilities, then such aggregation exists if the intersection of underlying credal sets is not empty. In the opposite case there is contradiction in information and the justifiable functional to measure it is the functional giving the smallest contradiction by applying all possible conjunctive rules. We give also the axiomatics of this contradiction measure.

The paper is devoted to the investigation of imprecision indices. They are used for evaluating imprecision (or non-specificity) contained in information described by monotone (non-additive) measures. These indices can be considered as generalizations of the generalized Hartley measure. We argue that in some cases, for example in approximation problems, the application of imprecision indices is well justified comparing with well-known uncertainty measures because of their good sensitivity. In the paper, we investigate properties of so called linear imprecision indices; in particular, we introduce their various representations and describe connections to the theory of imprecise probabilities. We also study the algebraic structure of imprecision indices in the linear space and describe the extreme points of the convex set of all possible imprecision indices, in particular, of imprecision indices with symmetrical properties. We also show how to measure inconsistency in information by impression indices. At the end of the paper, we consider the application of imprecision indices in analysing the applicability of different aggregation rules in evidence theory.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.