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.     

Distributed combination of belief functions

We consider the problem of combining belief functions in a situation where pieces of evidence are held by agents at the node of a communication network, and each agent can only exchange information with its neighbors. Using the concept of weight of evidence, we propose distributed implementations of Dempster’s rule and the cautious rule based, respectively, on average and maximum consensus algorithms. We also describe distributed procedures whereby the agents can agree on a frame of discernment and a list of supported hypotheses, thus reducing the amount of data to be exchanged in the network. Finally, we show the feasibility of a robust combination procedure based on a distributed implementation of the random sample consensus (RANSAC) algorithm.     

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.     

Analyzing the monotonicity of belief interval based uncertainty measures in belief function theory

Measuring the uncertainty of pieces of evidence is an open issue in belief function theory. A rational uncertainty measure for belief functions should meet some desirable properties, where monotonicity is a very important property. Recently, measuring the total uncertainty of a belief function based on its associated belief intervals becomes a new research idea and has attracted increasing interest. Several belief interval based uncertainty measures have been proposed for belief functions. In this paper, we summarize the properties of these uncertainty measures and especially investigate whether the monotonicity is satisfied by the measures. This study provide a comprehensive comparison to these belief interval based uncertainty measures and is very useful for choosing the appropriate uncertainty measure in the practical applications.     

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.     

Representation of interrelationships among binary variables under dempster–shafer theory of belief functions

This paper presents an algorithm for developing models under Dempster–Shafer theory of belief functions for categorical and “uncertain” logical relationships among binary variables. We illustrate the use of the algorithm by developing belief‐function representations of the following categorical relationships: “AND,” “OR,” “exclusive OR (EOR)” and “not exclusive OR (NEOR),” and “AND‐NEOR” and of the following uncertain relationships: “discounted AND,” “conditional OR,” and “weighted average.” Such representations are needed to fully model and analyze a problem with a network of interrelated variables under Dempster–Shafer theory of belief functions. In addition, we compare our belief‐function representation of the “weighted average” relationship with the “weighted average” representation developed and used by Shenoy and Shenoy (in Belief Functions in Business Decisions, edited by R. P. Srivastava and T. Mock; Heidelberg, Germany: Physica‐Verlag; 2002; pp 316–332). We find that Shenoy and Shenoy representation of the weighted average relationship is an approximation and yields significantly different values under certain conditions. © 2009 Wiley Periodicals, Inc.     

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.