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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 givenanobservationx⊆Xwhen one knows only the beliefs over X for every θi ∈ Θ. We also discuss the Disjunctive Rule of Combination ( ..."
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Cited by 170 (8 self)
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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.
Decision Making in the TBM: the Necessity of the Pignistic Transformation
, 2004
"... 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 th ..."
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Cited by 95 (1 self)
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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.
Conditional Reasoning with Subjective Logic
- JOURNAL OF MULTIPLE-VALUED LOGIC AND SOFT COMPUTING 15(1):PP. 5-38
, 2008
"... Conditional inference plays a central role in logical and Bayesian reasoning, and is used in a wide range of applications. It basically consists of expressing conditional relationship between parent and child propositions, and then to combine those conditionals with evidence about the parent proposi ..."
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Cited by 63 (16 self)
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Conditional inference plays a central role in logical and Bayesian reasoning, and is used in a wide range of applications. It basically consists of expressing conditional relationship between parent and child propositions, and then to combine those conditionals with evidence about the parent propositions in order to infer conclusions about the child propositions. While conditional reasoning is a well established part of classical binary logic and probability calculus, its extension to belief theory has only recently been proposed. Subjective opinions represent a special type of general belief functions. This article focuses on conditional reasoning in subjective logic where beliefs are represented in the form of binomial or multinomial subjective opinions. Binomial conditional reasoning operators for subjective logic have been defined in previous contributions. We extend this approach to multinomial opinions, thereby making it possible to represent conditional and evidence opinions on frames of arbitrary size. This makes subjective logic a powerful tool for conditional reasoning in situations involving ignorance and partial information, and makes it possible to analyse Bayesian network models with uncertain probabilities.
The Consensus Operator for Combining Beliefs
- ARTIFICIAL INTELLIGENCE JOURNAL
, 2002
"... The consensus operator provides a method for combining possibly conflicting beliefs within the Dempster-Shafer belief theory, and represents an alternative to the traditional Dempster 's rule. This paper describes how the consensus operator can be applied to dogmatic conflicting opinions, i.e. ..."
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Cited by 59 (18 self)
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The consensus operator provides a method for combining possibly conflicting beliefs within the Dempster-Shafer belief theory, and represents an alternative to the traditional Dempster 's rule. This paper describes how the consensus operator can be applied to dogmatic conflicting opinions, i.e. when the degree of conflict is very high. It overcomes shortcomings of Dempster's rule and other operators that have been proposed for combining possibly conflicting beliefs.
Possibility theory and statistical reasoning
- Computational Statistics & Data Analysis Vol
, 2006
"... Numerical possibility distributions can encode special convex families of probability measures. The connection between possibility theory and probability theory is potentially fruitful in the scope of statistical reasoning when uncertainty due to variability of observations should be distinguished f ..."
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Cited by 57 (4 self)
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Numerical possibility distributions can encode special convex families of probability measures. The connection between possibility theory and probability theory is potentially fruitful in the scope of statistical reasoning when uncertainty due to variability of observations should be distinguished from uncertainty due to incomplete information. This paper proposes an overview of numerical possibility theory. Its aim is to show that some notions in statistics are naturally interpreted in the language of this theory. First, probabilistic inequalites (like Chebychev’s) offer a natural setting for devising possibility distributions from poor probabilistic information. Moreover, likelihood functions obey the laws of possibility theory when no prior probability is available. Possibility distributions also generalize the notion of confidence or prediction intervals, shedding some light on the role of the mode of asymmetric probability densities in the derivation of maximally informative interval substitutes of probabilistic information. Finally, the simulation of fuzzy sets comes down to selecting a probabilistic representation of a possibility distribution, which coincides with the Shapley value of the corresponding consonant capacity. This selection process is in agreement with Laplace indifference principle and is closely connected with the mean interval of a fuzzy interval. It sheds light on the “defuzzification ” process in fuzzy set theory and provides a natural definition of a subjective possibility distribution that sticks to the Bayesian framework of exchangeable bets. Potential applications to risk assessment are pointed out. 1
Analyzing the degree of conflict among belief functions
, 2006
"... The study of alternative combination rules in DS theory when evidence is in conflict has emerged again recently as an interesting topic, especially in data/information fusion applications. These studies have mainly focused on investigating which alternative would be appropriate for which conflictin ..."
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Cited by 53 (7 self)
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The study of alternative combination rules in DS theory when evidence is in conflict has emerged again recently as an interesting topic, especially in data/information fusion applications. These studies have mainly focused on investigating which alternative would be appropriate for which conflicting situation, under the assumption that a conflict is identified. The issue of detection (or identification) of conflict among evidence has been ignored. In this paper, we formally define when two basic belief assignments are in conflict. This definition deploys quantitative measures of both the mass of the combined belief assigned to the emptyset before normalization and the distance between betting commitments of beliefs. We argue that only when both measures are high, it is safe to say the evidence is in conflict. This definition can be served as a prerequisite for selecting appropriate combination rules.
Data Fusion in the Transferable Belief Model.
, 2000
"... 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 socalled Dempster's rule of combination have been produced and the meaning of ..."
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Cited by 52 (0 self)
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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 socalled Dempster's rule of combination have been produced and the meaning of distinctness has been assessed. We will present practical applications where the fusion of uncertain data is well achieved by Dempster's 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 Dempster's rule of combination is central in the transferable belief model whereas it hardly fits with the upper and lower probabilities theory. Keywords: belief function, transferable beli...
Practical Representation of Incomplete Probabilistic Information
- Advances in Soft Computing:Soft Methods of Probability and Statistics conference
, 2004
"... This article deals with the compact representation of incomplete probabilistic knowledge which can be encountered in risk evaluation problems, for instance in environmental studies. Various kinds of knowledge are considered such as expert opinions about characteristics of distributions or poor stati ..."
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Cited by 50 (13 self)
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This article deals with the compact representation of incomplete probabilistic knowledge which can be encountered in risk evaluation problems, for instance in environmental studies. Various kinds of knowledge are considered such as expert opinions about characteristics of distributions or poor statistical information. Our approach is based on probability families encoded by possibility distributions and belief functions. In each case, a technique for representing the available imprecise probabilistic information faithfully is proposed, using different uncertainty frameworks (possibility theory, probability theory, belief functions...). Moreover the use of probability-possibility transformations enables confidence intervals to be encompassed by cuts of possibility distributions, thus making the representation stronger. The respective appropriateness of pairs of cumulative distributions, continuous possibility distributions or discrete random sets for representing information about the mean value, the mode, the median and other fractiles of ill-known probability distributions is discussed in detail.
ON THE PLAUSIBILITY TRANSFORMATION METHOD FOR TRANSLATING BELIEF FUNCTION MODELS TO PROBABILITY MODELS
, 2006
"... In this paper, we propose the plausibility transformation method for translating Dempster-Shafer (D-S) belief function models to probability models, and describe some of its properties. There are many other transformation methods used in the literature for translating belief function models to proba ..."
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Cited by 47 (1 self)
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In this paper, we propose the plausibility transformation method for translating Dempster-Shafer (D-S) belief function models to probability models, and describe some of its properties. There are many other transformation methods used in the literature for translating belief function models to probability models. We argue that the plausibility transformation method produces probability models that are consistent with D-S semantics of belief function models, and that, in some examples, the pignistic transformation method produces results that appear to be inconsistent with Dempster’s rule of combination.
