Dual properties of the relative belief of singletons

In this paper we prove that a recent Bayesian approximation of belief functions, the relative belief of singletons, meets a number of properties with respect to Dempster’s rule of combination which mirrors those satisfied by the relative plausibility of singletons. In particular, its operator commutes with Dempster’s sum of plausibility functions, while perfectly representing a plausibility function when combined through Dempster’s rule. This suggests a classification of all Bayesian approximations into two families according to the operator they relate to

Alternative formulations of the theory of evidence based on basic plausibility and commonality assignments

In this paper we introduce indeed two alternative formulations of the theory of evidence by proving that both plausibility and commonality functions share the same combinatorial structure of sum function of belief functions, and computing their Moebius inverses called basic plausibility and commonality assignments. The equivalence of the associated formulations of the ToE is mirrored by the geometric congruence of the related simplices. Applications to the probabilistic approximation problem are briefly presented

Distances in evidence theory: Comprehensive survey and generalizations

AbstractThe purpose of the present work is to survey the dissimilarity measures defined so far in the mathematical framework of evidence theory, and to propose a classification of these measures based on their formal properties. This research is motivated by the fact that while dissimilarity measures have been widely studied and surveyed in the fields of probability theory and fuzzy set theory, no comprehensive survey is yet available for evidence theory. The main results presented herein include a synthesis of the properties of the measures defined so far in the scientific literature; the generalizations proposed naturally lead to additions to the body of the previously known measures, leading to the definition of numerous new measures. Building on this analysis, we have highlighted the fact that Dempster’s conflict cannot be considered as a genuine dissimilarity measure between two belief functions and have proposed an alternative based on a cosine function. Other original results include the justification of the use of two-dimensional indexes as (cosine; distance) couples and a general formulation for this class of new indexes. We base our exposition on a geometrical interpretation of evidence theory and show that most of the dissimilarity measures so far published are based on inner products, in some cases degenerated. Experimental results based on Monte Carlo simulations illustrate interesting relationships between existing measures

Belief Evolution Network-based Probability Transformation and Fusion

Smets proposes the Pignistic Probability Transformation (PPT) as the decision layer in the Transferable Belief Model (TBM), which argues when there is no more information, we have to make a decision using a Probability Mass Function (PMF). In this paper, the Belief Evolution Network (BEN) and the full causality function are proposed by introducing causality in Hierarchical Hypothesis Space (HHS). Based on BEN, we interpret the PPT from an information fusion view and propose a new Probability Transformation (PT) method called Full Causality Probability Transformation (FCPT), which has better performance under Bi-Criteria evaluation. Besides, we heuristically propose a new probability fusion method based on FCPT. Compared with Dempster Rule of Combination (DRC), the proposed method has more reasonable result when fusing same evidence

Advances and Applications of Dezert-Smarandache Theory (DSmT) for Information Fusion (Collected Works), Vol. 4

The fourth volume on Advances and Applications of Dezert-Smarandache Theory (DSmT) for information fusion collects theoretical and applied contributions of researchers working in different fields of applications and in mathematics. The contributions (see List of Articles published in this book, at the end of the volume) have been published or presented after disseminating the third volume (2009, http://fs.unm.edu/DSmT-book3.pdf) in international conferences, seminars, workshops and journals. First Part of this book presents the theoretical advancement of DSmT, dealing with Belief functions, conditioning and deconditioning, Analytic Hierarchy Process, Decision Making, Multi-Criteria, evidence theory, combination rule, evidence distance, conflicting belief, sources of evidences with different importance and reliabilities, importance of sources, pignistic probability transformation, Qualitative reasoning under uncertainty, Imprecise belief structures, 2-Tuple linguistic label, Electre Tri Method, hierarchical proportional redistribution, basic belief assignment, subjective probability measure, Smarandache codification, neutrosophic logic, Evidence theory, outranking methods, Dempster-Shafer Theory, Bayes fusion rule, frequentist probability, mean square error, controlling factor, optimal assignment solution, data association, Transferable Belief Model, and others. More applications of DSmT have emerged in the past years since the apparition of the third book of DSmT 2009. Subsequently, the second part of this volume is about applications of DSmT in correlation with Electronic Support Measures, belief function, sensor networks, Ground Moving Target and Multiple target tracking, Vehicle-Born Improvised Explosive Device, Belief Interacting Multiple Model filter, seismic and acoustic sensor, Support Vector Machines, Alarm classification, ability of human visual system, Uncertainty Representation and Reasoning Evaluation Framework, Threat Assessment, Handwritten Signature Verification, Automatic Aircraft Recognition, Dynamic Data-Driven Application System, adjustment of secure communication trust analysis, and so on. Finally, the third part presents a List of References related with DSmT published or presented along the years since its inception in 2004, chronologically ordered

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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

IATE : Axe 5 Application intégrée de la connaissance, de l’information et des technologies permettant d’accroître la qualité et la sécurité des aliments Contact : [email protected] (S. Destercke), [email protected] (D. Dubois) Fax: +33 0 4 9961 3076.International audienceWhen 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 particularisation 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 maximise 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