Idempotent conjunctive combination of belief functions: Extending the minimum rule of possibility theory

When conjunctively merging two belief functions concerning a single variable but coming from different sources, Dempster rule of combination is justified only when information sources can be considered as independent. When dependencies between sources are ill-known, it is usual to require the property of idempotence for the merging of belief functions, as this property captures the possible redundancy of dependent sources. To study idempotent merging, different strategies can be followed. One strategy is to rely on idempotent rules used in either more general or more specific frameworks and to study, respectively, their particularization or extension to belief functions. In this paper, we study the feasibility of extending the idempotent fusion rule of possibility theory (the minimum) to belief functions. We first investigate how comparisons of information content, in the form of inclusion and least-commitment, can be exploited to relate idempotent merging in possibility theory to evidence theory. We reach the conclusion that unless we accept the idea that the result of the fusion process can be a family of belief functions, such an extension is not always possible. As handling such families seems impractical, we then turn our attention to a more quantitative criterion and consider those combinations that maximize the expected cardinality of the joint belief functions, among the least committed ones, taking advantage of the fact that the expected cardinality of a belief function only depends on its contour function.     

Analyzing the combination of conflicting belief functions

We consider uncertain data which uncertainty is represented by belief functions and that must be combined. The result of the combination of the belief functions can be partially conflictual. Initially Shafer proposed Dempster’s rule of combination where the conflict is reallocated proportionally among the other masses. Then Zadeh presented an example where Dempster’s rule of combination produces unsatisfactory results. Several solutions were proposed: the TBM solution where masses are not renormalized and conflict is stored in the mass given to the empty set, Yager’s solution where the conflict is transferred to the universe and Dubois and Prade’s solution where the masses resulting from pairs of conflictual focal elements are transferred to the union of these subsets. Many other suggestions have then been made, creating a ‘jungle’ of combination rules. We discuss the nature of the combinations (conjunctive versus disjunctive, revision versus updating, static versus dynamic data fusion), argue about the need for a normalization, examine the possible origins of the conflicts, determine if a combination is justified and analyze many of the proposed solutions.     

On the fusion of imprecise uncertainty measures using belief structures

Our interest is in the fusion of information from multiple sources when the information provided by the individual sources is expressed in terms of an imprecise uncertainty measure. We observe that the Dempster-Shafer belief structure provides a framework for the representation of a wide class of imprecise uncertainty measures. We then discuss the fusion of multiple Dempster-Shafer belief structures using the Dempster rule and note the problems that can arise when using this fusion method because of the required normalization in the face of conflicting focal elements. We then suggest some alternative approaches fusing multiple belief structures that avoid the need for normalization.     

Associativity in combination of belief functions; a derivation of minC combination

 The nature of a contradiction (conflict) between two belief functions is investigated. Alternative ways of distributing the contradiction among nonempty subsets of frame of discernment are studied. The paper employes a new approach to understanding contradictions and introduces an original notion of potential contradiction. A method of an associative combination of generalized belief functions – minC combination and its derivation – is presented as part of the new approach. A proportionalization of generalized results is suggested as well. RID="*" ID="*" Support by Grant No. 1030803 of the GA AV ČR is acknowledged. I am grateful to Philippe Smets for a fruitful discussion on the topic.     

Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence

Dempster's rule plays a central role in the theory of belief functions. However, it assumes the combined bodies of evidence to be distinct, an assumption which is not always verified in practice. In this paper, a new operator, the cautious rule of combination, is introduced. This operator is commutative, associative and idempotent. This latter property makes it suitable to combine belief functions induced by reliable, but possibly overlapping bodies of evidence. A dual operator, the bold disjunctive rule, is also introduced. This operator is also commutative, associative and idempotent, and can be used to combine belief functions issues from possibly overlapping and unreliable sources. Finally, the cautious and bold rules are shown to be particular members of infinite families of conjunctive and disjunctive combination rules based on triangular norms and conorms.     

Study of interval belief combination

In this paper, a new mathematical procedure is presented for combining different pieces of evidence which are represented in the interval form to reflect our knowledge about the truth of a hypothesis. Evidence may be correlated to each other (dependent evidence) or conflicting in support (conflicting evidence). First, assuming independent evidence, we propose a methodology to construct combination rules which obey a set of essential properties. The method is based on a geometric model. We compare results obtained from the Dempster—Shafer rule, interval Bayes rule, and the proposed combination rules with both conflicting and nonconflicting data and show that the values generated by the proposed combining rules are in tune with our intuition in both cases. Secondly, in the case that evidence is known to be dependent, we consider extensions of the rules derived for handling conflicting evidence. The performance of proposed rules are shown by different examples. The results show that the proposed rules reasonably make decisions under dependent evidence.     

Object association with belief functions, an application with vehicles

The problem tackled in this article consists in associating perceived objects detected at a certain time with known objects previously detected, knowing uncertain and imprecise information regarding the association of each perceived objects with each known objects. For instance, this problem can occur during the association step of an obstacle tracking process, especially in the context of vehicle driving aid. A contribution in the modeling of this association problem in the belief function framework is introduced. By interpreting belief functions as weighted opinions according to the Transferable Belief Model semantics, pieces of information regarding the association of known objects and perceived objects can be expressed in a common global space of association to be combined by the conjunctive rule of combination, and a decision making process using the pignistic transformation can be made. This approach is validated on real data.     

A new uncertainty measure for belief networks with applications tooptimal evidential inferencing

We are concerned with the problem of measuring the uncertainty in a broad class of belief networks, as encountered in evidential reasoning applications. In our discussion, we give an explicit account of the networks concerned, and call them the Dempster-Shafer (D-S) belief networks. We examine the essence and the requirement of such an uncertainty measure based on well-defined discrete event dynamical systems concepts. Furthermore, we extend the notion of entropy for the D-S belief networks in order to obtain an improved optimal dynamical observer. The significance and generality of the proposed dynamical observer of measuring uncertainty for the D-S belief networks lie in that it can serve as a performance estimator as well as a feedback for improving both the efficiency and the quality of the D-S belief network-based evidential inferencing. We demonstrate, with Monte Carlo simulation, the implementation and the effectiveness of the proposed dynamical observer in solving the problem of evidential inferencing with optimal evidence node selection