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Conjunctive and Disjunctive Combination of Belief Functions Induced by Non Distinct Bodies of Evidence
 ARTIFICIAL INTELLIGENCE
, 2007
"... 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 commuta ..."
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Cited by 34 (10 self)
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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.
Unifying practical uncertainty representations: I. Generalized pboxes
 International Journal of Approximate Reasoning
"... Abstract. There exist several simple representations of uncertainty that are easier to handle than more general ones. Among them are random sets, possibility distributions, probability intervals, and more recently Ferson’s pboxes and Neumaier’s clouds. Both for theoretical and practical considerati ..."
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Cited by 22 (10 self)
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Abstract. There exist several simple representations of uncertainty that are easier to handle than more general ones. Among them are random sets, possibility distributions, probability intervals, and more recently Ferson’s pboxes and Neumaier’s clouds. Both for theoretical and practical considerations, it is very useful to know whether one representation is equivalent to or can be approximated by other ones. In this paper, we define a generalized form of usual pboxes. These generalized pboxes have interesting connections with other previously known representations. In particular, we show that they are equivalent to pairs of possibility distributions, and that they are special kinds of random sets. They are also the missing link between pboxes and clouds, which are the topic of the second part of this study. 1.
Relating practical representations of imprecise probabilities
 5TH INTERNATIONAL SYMPOSIUM ON IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS, PRAGUE, CZECH REPUBLIC, 2007
, 2007
"... There exist many practical representations of probability families that make them easier to handle. Among them are random sets, possibility distributions, probability intervals, Ferson's pboxes and Neumaier's clouds. Both for theoretical and practical considerations, it is important to k ..."
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Cited by 6 (3 self)
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There exist many practical representations of probability families that make them easier to handle. Among them are random sets, possibility distributions, probability intervals, Ferson's pboxes and Neumaier's clouds. Both for theoretical and practical considerations, it is important to know whether one representation has the same expressive power than other ones, or can be approximated by other ones. In this paper, we mainly study the relationships between the two latter representations and the three other ones.
Constructing belief functions from sample data using multinomial confidence regions
 International Journal of Approximate Reasoning (To
"... The Transferable Belief Model is a subjectivist model of uncertainty in which an agent’s beliefs at a given time are modeled using the formalism of belief functions. Belief functions that enter the model are usually either elicited from experts, or must be constructed from observation data. There ar ..."
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The Transferable Belief Model is a subjectivist model of uncertainty in which an agent’s beliefs at a given time are modeled using the formalism of belief functions. Belief functions that enter the model are usually either elicited from experts, or must be constructed from observation data. There are, however, few simple and operational methods available for building belief functions from data. Such a method is proposed in this paper. More precisely, we tackle the problem of quantifying beliefs held by an agent about the realization of a discrete random variable X with unknown probability distribution PX, having observed a realization of an independent, identically distributed random sample with the same distribution. The solution is obtained using simultaneous confidence intervals for multinomial proportions, several of which have been proposed in the statistical literature. The proposed solution verifies two “reasonable ” properties with respect to PX: it is less committed than PX with some userdefined probability, and it converges towards PX in probability as the size of the sample tends to infinity. A general formulation is given, and a useful approximation with a simple analytical expression is presented, in the important special case where the domain of X is ordered.
Consonant Belief Function induced by a Confidence Set of Pignistic Probabilities
"... Abstract. A new method is proposed for building a predictive belief function from statistical data in the Transferable Belief Model framework. The starting point of this method is the assumption that, if the probability distributionPX of a random variable X is known, then the belief function quantif ..."
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Abstract. A new method is proposed for building a predictive belief function from statistical data in the Transferable Belief Model framework. The starting point of this method is the assumption that, if the probability distributionPX of a random variable X is known, then the belief function quantifying our belief regarding a future realization of X should have its pignistic probability distribution equal toPX. WhenPX is unknown but a random sample of X is available, it is possible to build a set P of probability distributions containingPX with some confidence level. Following the Least Commitment Principle, we then look for a belief function less committed than all belief functions with pignistic probability distribution in P. Our method selects the most committed consonant belief function verifying this property. This general principle is applied to the case of the normal distribution.
Constructing predictive belief functions from continuous sample data using confidence bands
 In ISIPTA
, 2007
"... Abstract We consider the problem of quantifying our belief in future values of a random variable X with unknown distribution P X , based on the observation of a random sample from the same distribution. The adopted uncertainty representation framework is the Transferable Belief Model, a subjectivis ..."
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Abstract We consider the problem of quantifying our belief in future values of a random variable X with unknown distribution P X , based on the observation of a random sample from the same distribution. The adopted uncertainty representation framework is the Transferable Belief Model, a subjectivist interpretation of belief function theory. In a previous paper, the concept of predictive belief function at a given confidence level was introduced, and it was shown how to build such a function when X is discrete. This work is extended here to the case where X is a continuous random variable, based on step or continuous confidence bands.
Multisource data fusion for bandlimited signals: a Bayesian perspective.
 In 26th MaxEnt workshop, AIP Conference Proceedings,
, 2006
"... Abstract. We consider data fusion as the reconstruction of a single model from multiple data sources. The model is to be inferred from a number of blurred and noisy observations, possibly from different sensors under various conditions. It is all about recovering a compound object, signal+uncertain ..."
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Abstract. We consider data fusion as the reconstruction of a single model from multiple data sources. The model is to be inferred from a number of blurred and noisy observations, possibly from different sensors under various conditions. It is all about recovering a compound object, signal+uncertainties, that best relates to the observations and contains all the useful information from the initial data set. We wish to provide a flexible framework for bandlimited signal reconstruction from multiple data. In this paper, we focus on a general approach involving forward modeling (prior model, data acquisition) and Bayesian inference. The proposed method is valid for nD objects (signals, images or volumes) with multidimensional spatial elements. For the sake of clarity, both formalism and test results will be shown in 1D for single band signals. The main originality lies in seeking an object with a prescribed bandwidth, hence our choice of a BSpline representation. This ensures an optimal sampling in both signal and frequency spaces, and allows for a shift invariant processing. The model resolution, the geometric distortions, the blur and the regularity of the sampling grid can be arbitrary for each sensor. The method is designed to handle realistic Gauss+Poisson noise. We obtained promising results in reconstructing a superresolved signal from two blurred and noisy shifted observations, using a Gaussian Markov chain as a prior. Practical applications are under development within the SpaceFusion project. For instance, in astronomical imaging, we aim at a sharp, wellsampled, noisefree and possibly superresolved image. Virtual Observatories could benefit from such a way to combine large numbers of multispectral images from various sources. In planetary imaging or remote sensing, a 3D image formation model is needed; nevertheless, this can be addressed within the same framework.