We describe the transferable belief model, a model for representing quantified beliefs based on belief functions. Beliefs can be held at two levels: (1) a credal level where beliefs are entertained and quantified by belief functions, (2) a pignistic level where beliefs can be used to make decisions and are quantified by probability functions. The relation between the belief function and the probability function when decisions must be made is derived and justified. Four paradigms are analyzed in order to compare Bayesian, upper and lower probability, and the transferable belief approaches.

We generalize the Bayes ’ theorem within the transferable belief model framework. The Generalized Bayesian Theorem (GBT) allows us to compute the belief over a space Θ givenanobservationx⊆Xwhen one knows only the beliefs over X for every θi ∈ Θ. We also discuss the Disjunctive Rule of Combination (DRC) for distinct pieces of evidence. This rule allows us to compute the belief over X from the beliefs induced by two distinct pieces of evidence when one knows only that one of the pieces of evidence holds. The properties of the DRC and GBT and their uses for belief propagation in directed belief networks are analysed. The use of the discounting factors is justfied. The application of these rules is illustrated by an example of medical diagnosis.

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.

This paper explains how multisensor data fusion and target identification can be performed within the transferable belief model, a model for the representation of quantified uncertainty based on belief functions. The paper is presented in two parts: methodology and application. In this part, we present the underlying theory, in particular the General Bayesian Theorem needed to transform likelihoods into beliefs and the pignistic transformation needed to build the probability measure required for decision making. We end with a simple example. More sophisticated examples and some comparative studies are presented in Part II. The results presented here can be extended directly to many problems of data fusion and diagnosis.

We generalize the TBM (transferable belief model) to the case where the frame of discernment is the extended set of real numbers R = [−∞, ∞], under the assumptions that ‘masses’ can only be given to intervals. Masses become densities, belief functions, plausibility functions and commonality functions become integrals of these densities and pignistic probabilities become pignistic densities. The mathematics of belief functions become essentially the mathematics of probability density functions on R².

The mathematic of belief functions can be handled by the use of the matrix notation. This representation helps greatly the user thanks to its notational simplicity and its efficiency for proving theorems. We show how to use them for several problems related to belief functions and the transferable belief model.

We provide a semantic for the values given to possibility measures. It is based on the semantic of the transferable belief model, itself based on the same approach as used for subjective probabilities. Besides we explain how the conjunctive combination of two possibility measures corresponds to the hyper-cautious conjunctive combination of the belief functions induced by the possibility measures.

Abstract not found

Abstract- The discounting operation is a well known operation on belief functions, which has proved to be useful in many applications. However, the discounting operation only allows one to weaken a source, whereas it is sometimes useful to strengthen it when it is deemed to be too cautious. For that purpose, the de-discounting operation was introduced as the inverse operation of the discounting operation by Denœux and Smets. From another point of view, Zhu and Basir introduced an extension of the classical discounting operation by allowing the discount rate to be out of the range [0,1]. This operation performs a discounting or a de-discounting of a belief function. A new interpretation of this scheme is presented in this paper. A more general form of reinforcement process, as well as a parameterized family of transformations encompassing all previous schemes, are also introduced. Keywords: Dempster-Shafer theory, Evidence theory, discounting, de-discounting.

Abstract – In a lot of operational situations, we have to deal with uncertain and inaccurate information. The theory of belief functions is a mathematical framework useful to handle this kind of imperfection. However, in most of the cases, uncertain data are modeled with a distribution of probability. We present in this paper different principles to induce belief functions from probabilities. Hence, we decide to use these functions in a pattern recognition problem. We discuss about the results we obtain according the way we generate the belief function. To illustrate our work, it will be applied to seabed characterization.