Measures of conflict, basic axioms and their application to the clusterization of a body of evidence☆
Introduction
This paper is devoted to an axiomatic approach for constructing measures of conflict in the theory of belief functions. In probability theory, we have several system of axioms [47], [42] that define the Shannon entropy uniquely. If we drop the subadditivity axiom, then we can get the parametrical family of Rényi's entropies [41]. In the theory of belief functions, there are several attempts to develop such an approach [30], [25], [3]. In the book [30], a reader can find a number of requirements that every uncertainty measure has to possess and the author argues that a measure of uncertainty should be additive on independent (non-interactive) pieces of information and subadditive otherwise. In [25], Harmanec proposes a system of axioms for a measure of total uncertainty (aggregating uncertainty) on belief functions, which implies that a measure of total uncertainty should be the Shannon entropy on probability measures and the Hartley measure on categorical belief functions. He also shows that the maximal entropy satisfies all necessary requirements and this is the least measure of total uncertainty among possible ones. In [3], authors propose a system of axioms for a measure of total uncertainty and for its parts - a measure of conflict and a measure of non-specificity. They show that the proposed system of axioms can be extended to credal sets, and give examples of possible uncertainty measures satisfying this system of axioms. They prove the uniqueness of a total uncertainty measure under the law of conflict-non-specificity transformation. Under this law every measure of total uncertainty can be represented as linear combination of the maximal entropy and the generalized Hartley measure. In this case, the measure of conflict is the minimal entropy, and the measure of non-specificity has two parts: the first part is the difference between maximal and minimal entropies, and the second part is the generalized Hartley measure. In [3], there is also the following important result: every measure of conflict satisfying a system of axioms on the set of belief functions cannot be subadditive. Note that in probability theory [36], [12], we can find functionals that can be conceived as entropies that do not have the additivity and subadditivity properties like the Gini index [2]. Therefore, we think that it is of interest to formulate a weaker system of axioms that admits a wider class of possible entropies on the set of probability measures, and then to extend it to the set of all belief functions. This question is studied in our paper: we propose a system of axioms for belief functions without the additivity axiom and describe the corresponding class of possible conflict measures. As one can see from below, such measures of conflict are constructed like the minimal entropy: we define first a conflict measure on probability measures and then find its extension for each belief function by taking the minimum of this functional on the corresponding credal set.
We show that the evaluation of conflict in such a way can be viewed as the clusterization of a body of evidence. As a result, we can represent the initial belief function as a convex sum of non-conflicting belief functions. This allows us to use this decomposition in data processing as we show in the application. This inspired us to consider another problem. When a belief function is derived from the statistical data, there are many focal elements in a body of evidence which do not bring the useful information. Thus, we need to simplify the body of evidence keeping approximately the same basic characteristics of the initial belief function. This can be done by the algorithms that are very similar to algorithms used in data processing. In this case, the set of focal elements with their masses looks like the histogram, and the belief function looks like the cumulative distribution function with respect to the inclusion relation. We also need a metric between focal elements that should take into account the difference between the corresponding sets and their significance in the sample. This can be modeled by simple belief functions, in which the first focal element has the same mass as in the initial belief function and the second focal element correspond to the total ignorance. In computations, we use the recently introduced Wasserstein metric [6], [11] on simple belief functions. The clusterization procedure consists of two stages: the first stage looks like finding centers of clusters, in which we find the simplified body of evidence; at the second stage, we compute values of the mass function on focal elements by finding the approximation of the initial belief function using a metric on the belief functions.
The paper has the following structure. In Section 2, we recall some basic definitions and constructions from the theory of belief functions. In Section 3, we give an overview of known conflict measures and analyze the main approaches to construct them. In Section 4, we propose a system of axioms for conflict measures and describe the class of conflict measures under the proposed system of axioms. In Section 5, we study the optimization problem linked with computing conflict measures, and show how this problem can be viewed through the clusterization of the body of evidence. In Section 6, we consider the class of conflict measures constructed by the conflict density and check their properties. In Section 7, we study the problem of simplifying the body of evidence and propose several algorithms for realizing such simplification. In Section 8, we illustrate the application of the proposed characteristics of belief functions and algorithms on analyzing political preferences of parties in Germany and their influence on popularity of such parties. The paper finishes with the discussion of obtained results and some conclusions.
Section snippets
Belief functions: basic constructions and definitions
Let be a finite reference set and let be the powerset of X. The set function is called a belief function [15], [46] if there is a set function with and called the basic belief assignment (bba) such that With the help of bba m, we introduce also the plausibility function defined by It is well known that Bel and Pl are connected by the duality relation: , where and is
Basic approaches to evaluating conflict within belief functions
The concept of conflict is one of the key concepts in the theory of belief functions. The concepts of external and internal conflict are distinguished. The external conflict characterizes the contradiction between bodies of evidence representing different sources of information [17]. The internal conflict characterizes the conflict within a body of evidence. Because these two types of conflict are strongly linked together, we will firstly describe possible interpretations of external conflict
Measures of conflict and their axiomatics
Formally, as we emphasize above we should formulate the condition when a measure of conflict is equal to zero. Because we take the third interpretation of conflict-free information, we have
Axiom 1 for with a body of evidence iff .
Axiom 2 for if .
Therefore, Axiom 2 says that if the non-specificity is higher, then, as a rule, the conflict is lower.
For formulating the next axiom, we introduce some definitions.
Conflict measures in view of the body of evidence disaggregation
Let be a measure of conflict on satisfying Axiom 1, Axiom 4, and . Then we can disaggregate Bel on pieces of evidence as where , , , . We can postulate that is the amount of conflict within the pieces of evidence and is the amount of conflict among the pieces of evidence called also the external conflict or contradiction. In the literature [32], [33], [45], there are several approaches
Density of conflict
In the theory of belief functions [4], [8], [17] there are several approaches for defining conflict among sources of information, but in special cases most of them give us the same result. Consider this special case, when the first source of information is described by an arbitrary with the bba m and the second source by the categorical belief function , . Then the amount of external conflict (or contradiction) between them can be defined by We
Simplifying a body of evidence based on clusterization
In real problems, the number of focal elements in a body of evidence can be very large to produce all necessary computations. Therefore, we tackle the problem of how the body of evidence can be simplified with the minimal loss of information.
One way of formalizing this problem consists in the following. Let have a body of evidence , then we should find with a body of evidence with the minimal such that , where d is a metric on and is
Methodology for the formation of the body of evidence
The study of the influence of party positions on the voting result is an important problem in political sciences [50]. In some countries, there are services designed to assess the positions of parties before elections. For this purpose, voters choose the list of valuable questions and parties should answer them before elections. We use the data given by a service in Germany [24] collected before elections to Bundestag in 2013 [23]. To reduce computational problems, we take only 8 questions (the
Conclusion
There are many functionals for evaluating conflict in the theory of belief functions and the main way to classify them is to analyze their properties. In this paper, we propose a system of axioms for the conflict measures that are justifiable in probability theory with realizable extensions to more general models of imprecise probabilities like credal sets. There is another question - how to use them in real applications? What is their meaning? In our paper, we show that conflict measures can
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
We express our sincere thanks to the anonymous reviewers for their detailed and helpful remarks.
The article was prepared within the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE University) and supported within the framework of a subsidy by the Russian Academic Excellence Project x925-100x92.
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