A Dual Measure of Uncertainty: The Deng Extropy

The extropy has recently been introduced as the dual concept of entropy. Moreover, in the context of the Dempster–Shafer evidence theory, Deng studied a new measure of discrimination, named the Deng entropy. In this paper, we define the Deng extropy and study its relation with Deng entropy, and examples are proposed in order to compare them. The behaviour of Deng extropy is studied under changes of focal elements. A characterization result is given for the maximum Deng extropy and, finally, a numerical example in pattern recognition is discussed in order to highlight the relevance of the new measure

How far we are from the complete knowledge: Complexity of knowledge acquisition in Dempster-Shafer approach

When a knowledge base represents the experts' uncertainty, then it is reasonable to ask how far we are from the complete knowledge, that is, how many more questions do we have to ask (to these experts, to nature by means of experimenting, etc) in order to attain the complete knowledge. Of course, since we do not know what the real world is, we cannot get the precise number of questions from the very beginning: it is quite possible, for example, that we ask the right question first and thus guess the real state of the world after the first question. So we have to estimate this number and use this estimate as a natural measure of completeness for a given knowledge base. We give such estimates for Dempster-Shafer formalism. Namely, we show that this average number of questions can be obtained by solving a simple mathematical optimization problem. In principle this characteristic is not always sufficient to express the fact that sometimes we have more knowledge. For example, it has the same value if we have an event with two possible outcomes and nothing else is known, and if there is an additional knowledge that the probability of every outcome is 0.5. We'll show that from the practical viewpoint this is not a problem, because the difference between the necessary number of questions in both cases is practically negligible

Completing an uncertainty criterion of classification

We present a variation of a method of classification based in uncertainty on credal set. Similarly to its origin it use the imprecise Dirichlet model to create the credal set and the same uncertainty measures. It take into account sets of two variables to reduce the uncertainty and to seek the direct relations between the variables in the data base and the variable to be classified. The success are equivalent to the success of the first method except in those where there are a direct relations between some variables that decide the value of the variable to be classified where we have a notable improvement

On the Informational Comparison of Qualitative Fuzzy Measures

International audienceFuzzy measures or capacities are the most general representation of uncertainty functions. However, this general class has been little explored from the point of view of its information content, when degrees of uncertainty are not supposed to be numerical, and belong to a finite qualitative scale, except in the case of possibility or necessity measures. The thrust of the paper is to define an ordering relation on the set of qualitative capacities expressing the idea that one is more informative than another, in agreement with the possibilistic notion of relative specificity. To this aim, we show that the class of qualitative capacities can be partitioned into equivalence classes of functions containing the same amount of information. They only differ by the underlying epistemic attitude such as pessimism or optimism. A meaningful information ordering between capacities can be defined on the basis of the most pessimistic (resp. optimistic) representatives of their equivalence classes. It is shown that, while qualitative capacities bear strong similarities to belief functions, such an analogy can be misleading when it comes to information content

Fractional Deng Entropy and Extropy and Some Applications

Deng entropy and extropy are two measures useful in the Dempster–Shafer evidence theory (DST) to study uncertainty, following the idea that extropy is the dual concept of entropy. In this paper, we present their fractional versions named fractional Deng entropy and extropy and compare them to other measures in the framework of DST. Here, we study the maximum for both of them and give several examples. Finally, we analyze a problem of classification in pattern recognition in order to highlight the importance of these new measures

Contradiction measures and specificity degrees of basic belief assignments

In the theory of belief functions, many measures of uncertainty have been introduced. However, it is not always easy to understand what these measures really try to represent. In this paper, we re-interpret some measures of uncertainty in the theory of belief functions. We present some interests and drawbacks of the existing measures. On these observations, we introduce a measure of contradiction. Therefore, we present some degrees of non-specificity and Bayesianity of a mass. We propose a degree of specificity based on the distance between a mass and its most specific associated mass. We also show how to use the degree of specificity to measure the specificity of a fusion rule. Illustrations on simple examples are given