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23
Perspectives on the Theory and Practice of Belief Functions
 International Journal of Approximate Reasoning
, 1990
"... The theory of belief functions provides one way to use mathematical probability in subjective judgment. It is a generalization of the Bayesian theory of subjective probability. When we use the Bayesian theory to quantify judgments about a question, we must assign probabilities to the possible answer ..."
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Cited by 101 (7 self)
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The theory of belief functions provides one way to use mathematical probability in subjective judgment. It is a generalization of the Bayesian theory of subjective probability. When we use the Bayesian theory to quantify judgments about a question, we must assign probabilities to the possible answers to that question. The theory of belief functions is more flexible; it allows us to derive degrees of belief for a question from probabilities for a related question. These degrees of belief may or may not have the mathematical properties of probabilities; how much they differ from probabilities will depend on how closely the two questions are related. Examples of what we would now call belieffunction reasoning can be found in the late seventeenth and early eighteenth centuries, well before Bayesian ideas were developed. In 1689, George Hooper gave rules for combining testimony that can be recognized as special cases of Dempster's rule for combining belief functions (Shafer 1986a). Similar rules were formulated by Jakob Bernoulli in his Ars Conjectandi, published posthumously in 1713, and by JohannHeinrich Lambert in his Neues Organon, published in 1764 (Shafer 1978). Examples of belieffunction reasoning can also be found in more recent work, by authors
Belief Functions on Real Numbers.
, 2005
"... 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 function ..."
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Cited by 29 (0 self)
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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.
An evidential reasoning framework for object tracking
 SPIE Y PHOTONICS EAST 99
, 1999
"... Object tracking consists of reconstructing the configuration of an articulated body from a sequence of images provided by one or more cameras. In this paper we present a general method for pose estimation based on the evidential reasoning. The proposed framework integrates different levels of descri ..."
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Cited by 6 (3 self)
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Object tracking consists of reconstructing the configuration of an articulated body from a sequence of images provided by one or more cameras. In this paper we present a general method for pose estimation based on the evidential reasoning. The proposed framework integrates different levels of description of the object to improve robustness and precision, overcoming the limitations of approaches using singlefeature representations. Several image descriptions extracted from a singlecamera view are fused together using the DempsterShafer ”theory of evidence”. 14 Feature data are expressed as belief functions over the set of their possibile values. There is no need of any apriori assumptions about the model of the object. Learned refinement maps between feature spaces and the parameter space Q describing the configuration of the object characterize the relationships among distinct representations of the pose and play the role of the model. During training the object follows a sample trajectory in Q. Each feature space is reduced to a discrete frame of discernment (FOD) and refinements are built by mapping these FODs into subsets of the sample trajectory. During tracking new sensor data are converted to belief functions which are projected and combined in the approximate state space. Resulting degrees of belief indicate the best pose estimate at the current time step. The choice of a sufficiently dense (in a topological sense) sample trajectory is a critical problem. Experimental results concerning a simple tracking system are shown.
Representation of Evidence by Hints
 Advances in the DempsterShafer Theory of Evidence
, 1994
"... This paper introduces a mathematical model of a hint as a body of imprecise and uncertain information. Hints are used to judge hypotheses: the degree to which a hint supports a hypothesis and the degree to which a hypothesis appears as plausible in the light of a hint are defined. This leads in turn ..."
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Cited by 4 (2 self)
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This paper introduces a mathematical model of a hint as a body of imprecise and uncertain information. Hints are used to judge hypotheses: the degree to which a hint supports a hypothesis and the degree to which a hypothesis appears as plausible in the light of a hint are defined. This leads in turn to support and plausibility functions. Those functions are characterized as set functions which are normalized and monotone or alternating of order 1. This relates the present work to G. Shafer's mathematical theory of evidence. However, whereas Shafer starts out with an axiomatic definition of belief functions, the notion of a hint is considered here as the basic element of the theory. It is shown that a hint contains more information than is conveyed by its support function alone. Also hints allow for a straightforward and logical derivation of Dempster's rule for combining independent and dependent bodies of information. This paper presents the mathematical theory of evidence for general, infinite frames of discernment from the point of view of a theory of hints. KEYWORDS Hints, evidence, support functions, plausibility functions, Dempster's rule 1. HINTS  AN INTUITIVE
Fusion of Vegetation Indices Using Continuous Belief Functions and CautiousAdaptive Combination Rule
 IEEE Transactions on Geoscience and Remote Sensing
"... Abstract—The goal of this paper is to propose a methodology based on vegetation index fusion to provide an accurate estimation of the fraction of vegetation cover (fCover). Because of the partial and imprecise nature of remotesensing data, we opt for the evidential framework that allows us to hand ..."
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Abstract—The goal of this paper is to propose a methodology based on vegetation index fusion to provide an accurate estimation of the fraction of vegetation cover (fCover). Because of the partial and imprecise nature of remotesensing data, we opt for the evidential framework that allows us to handle such kind of information. The defined fCover belief functions are continuous with the interval [0, 1] as a discernment space. Since the vegetation indices are not independent (e.g., perpendicular vegetation index and weighted difference vegetation index are linearly linked), we define a new combination rule called “cautious adaptive ” to handle the partial “nondistinctness ” between the sources (vegetation indices). In this rule, the “nondistinctness ” is modeled by a factor varying from zero (distinct sources) to one (totally correlated sources), and the fusion rule varies accordingly from the conjunctive rule to the cautious one. In terms of results, both in the cases of simulated data and actual data, we show the interest of the combination of two or three vegetation indices to improve either the accuracy of fCover estimation or its robustness. Index Terms—Belief functions, combination rule, evidence theory, fraction of vegetation cover (fCover), vegetation index fusion. I.
Models of belief functions  Impacts for patterns recognitions
 in Proceedings of International Conference on Information Fusion 2010
, 2010
"... 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 probabilit ..."
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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.
Prepared for U.S. ARMY CORPS OF ENGINEERS0 ENGINEER TOPOGRAPHIC LABORATORIES
, 1985
"... a~ETL0381 ..."