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The Dempster–Shafer theory of evidence (DST) has been widely used to handle uncertainty‐based information. It is based on the concept of basic probability assignment (BPA). Belief intervals are easier to manage than a BPA to represent uncertainty‐based information. For this reason, several uncertainty measures for DST recently proposed are based on belief intervals. In this study, we carry out a study about the crucial mathematical properties and behavioral requirements that must be verified by every uncertainty measure on belief intervals. We base on the study previously carried out for uncertainty measures on BPAs. Furthermore, we analyze which of these properties are satisfied by each one of the uncertainty measures on belief intervals proposed so far. Such a comparative analysis shows that, among these measures, the maximum of entropy on the belief intervals is the most suitable one to be employed in practical applications since it is the only one that satisfies all the required mathematical properties and behaviors. [ABSTRACT FROM AUTHOR]

The Non-Parametric Predictive Inference Model for Multinomial Data (NPI-M) is an imprecise probabilities model used to represent the available information about a categorical variable. It presents some advantages over another imprecise probabilities model frequently used in the literature called Imprecise Dirichlet Model (IDM), which assumes previous knowledge about the data through a parameter. The Approximate Non-Parametric Predictive Inference Model for Multinomial Data (A-NPI-M) is a model similar to the NPI-M that can be expressed by reachable sets of probability intervals, is easier to manage, and is non-parametric. As a novelty, in this work, we analyze the main properties of A-NPI-M credal sets, comparing them with the properties of credal sets associated with the IDM. Moreover, we present procedures to calculate the most important uncertainty measures on A-NPI-M credal sets. Those procedures represent useful tools to make the A-NPI-M very suitable to be used in practical applications. [ABSTRACT FROM AUTHOR]

Recently, an alternative measure of total uncertainty in Dempster-Shafer theory of evidence (DST) has been proposed in place of the maximum entropy measure. It is based on the pignistic probability of a basic probability assignment and it is proved that this measure verifies a set of needed properties for such a type of measure. The proposed measure is motivated by the problems that maximum (upper) entropy has. In this paper, we analyse the requirements, presented in the literature, for total uncertainty measures in DST and the shortcomings found on them. We extend the set of requirements, which we consider as a set of requirements of properties, and we use the set of shortcomings found on them to define a set of requirements of the behaviour for total uncertainty measures in DST. We present the differences of the principal total uncertainty measures presented in DST taking into account their properties and behaviour. Also, an experimental comparative study of the performance of total uncertainty measures in DST on a special type of belief decision trees is presented. [ABSTRACT FROM AUTHOR]

Abstract: When there is uncertainty about the values of a variable, it is possible to have different pieces of information about it. This can be due to the fact that they represent the points of view of different agents and these agents have followed different criteria when codify the pieces of information by means of numbers. The aim of this paper is to provide tools for the comparison of these representations. This is done in the framework of imprecise probabilities, more concretely the credal set theory is used. The reason is that credal sets are very general, including as particular cases possibility theory, probability theory and belief functions. So it can serve as basis to compare the representations of information obtained by following these formalisms. The measures used for comparison are: inconsistency measure, inclusion index and informative distance. [Copyright &y& Elsevier]

A general way of representing incomplete information is to use closed and convex sets of probability distributions, which are also called credal sets. Each credal set is associated with uncertainty, whose amount is quantified by an appropriate uncertainty measure. One of the requisite properties of uncertainty measures is the property of additivity, which is associated with the concept of independence. For credal sets, the concept of independence is not unique. This means that different definitions of independence lead to different definitions of additivity for uncertainty measures. In this paper, we compare the various definitions of independence, but our principal aim is to analyze those definitions that are employed in the most significant uncertainty measures established in the literature for credal sets. [ABSTRACT FROM AUTHOR]

This paper proposes the use of the maximum difference of entropies as a non-specificity measure for credal sets and studies its properties. The main advantage of the new measure is that it does not only take into account the absolute imprecision of the credal set, but also the position of the credal set with respect to the uniform distribution. The paper provides an algorithm to compute the most difficult part of the difference of entropies, the minimum of entropy. The algorithm computes the exact minimum of entropy for order-2 capacities and it is based on the branch and bound technique with some additional procedures to prune the search. [ABSTRACT FROM AUTHOR]