Philippe Smets

The transferable belief model

Abstract 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.

The combination of evidence in the transferable belief model

A description of the transferable belief model, which is used to quantify degrees of belief based on belief functions, is given. The impact of open- and closed-world assumption on conditioning is discussed. The nature of the frame of discernment on which a degree of belief will be established is discussed. A set of axioms justifying Dempsters rule for the combination of belief functions induced by two distinct evidences is presented. >

Belief Functions: The Disjunctive Rule of Combination and the Generalized Bayesian Theorem

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 given an observation x ⊆ X when 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.

Decision making in the TBM: the necessity of the pignistic transformation

In the transferable belief model (TBM), pignistic probabilities are used for decision making. The nature of the pignistic transformation is justified by a linearity requirement. We justify the origin of this requirement showing it is not ad hoc but unavoidable provides one accepts expected utility theory.

Constructing the Pignistic Probability Function in a Context of Uncertainty

Summary We derive axiomatically the probability function that should be used to make decisions given any form of underlying uncertainty.

The Transferable Belief Model for Quantified Belief Representation

We present the transferable belief model (TBM), a model for the representation of quantified beliefs. The model aims in representing the same concept as the Bayesian model, i.e., the graded dispositions that guide ‘our’ behaviour. We use the word ‘belief’ in a broad sense. It could be replaced by quantified credibility, subjective support, strength of opinion... These beliefs are not categorical as in modal logic, but admits degrees as in probability theory. Our approach is normative. The beliefs are held by an idealized rational agent, denoted by You. This ‘You’ can be a human, but also a robot, a computer program...

The paradoxical success of fuzzy logic

Fuzzy logic methods have been used successfully in many real-world applications, but the foundations of fuzzy logic remain under attack. Taken together, these two facts constitute a paradox. A second paradox is that almost all of the successful fuzzy logic applications are embedded controllers, while most of the theoretical papers on fuzzy methods deal with knowledge representation and reasoning. I hope to resolve these paradoxes by identifying which aspects of fuzzy logic render it useful in practice, and which aspects are inessential. My conclusions are based on a mathematical result, on a survey of literature on the use of fuzzy logic in heuristic control and in expert systems, and on practical experience in developing expert systems.<<ETX>>

Data fusion in the transferable belief model

When Shafer introduced his theory of evidence based on the use of belief functions, he proposed a rule to combine belief functions induced by distinct pieces of evidence. Since then, theoretical justifications of this so-called Dempsters rule of combination have been produced and the meaning of distinctness has been assessed. The author presents practical applications where the fusion of uncertain data is well achieved by Dempsters rule of combination. It is essential that the meaning of the belief functions used to represent uncertainty be well fixed, as the adequacy of the rule depends strongly on a correct understanding of the context in which they are applied. Missing to distinguish between the upper and lower probabilities theory and the transferable belief model can lead to serious confusion, as Dempsters rule of combination is central in the transferable belief model whereas it hardly fits with the upper and lower probabilities theory.

Postoperative radiation therapy in lung cancer: A controlled trial after resection of curative design

Abstract Postoperative supervoltage radiotherapy was tested in a controlled clinical trial in an attempt to improve the survival for patients with bronchogenic carcinoma. Radiation therapy began 3 to 4 weeks after surgery; three fields were used giving a dose of 6,000 rad in six weeks to the mediastinum from a Co 60 unit. Between 1966 and 1975, 224 patients were included in this study. All had a resection of curative design. 175 patients were evaluable. No increase in survival time was noticed in the irradiated group. The 5 year survival rate was lower in this group (24% versus 43% for the control group) but the difference was not statistically significant. For squamous cell carcinoma, the analysis showed a detrimental effect of radiation therapy in the T 2 group (p

Implication in fuzzy logic

Abstract Intermediate truth values and the order relation “as true as” are interpreted. The material implication A → B quantifies the degree by which “B is at least as true as A.” Axioms for the → operator lead to a representation of → by the pseudo-Lukasiewicz model. A canonical scale for the truth value of a fuzzy proposition is selected such that the → operator is the Lukasiewicz operator and the negation is the classical 1−. operator. The mathematical structure of some conjunction and disjunction operators related to → are derived.