## A defect in Dempster-Shafer theory (1994)

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Venue: | InProceedings of the Tenth Conference on Uncertainty in Arti cial Intelligence |

Citations: | 15 - 12 self |

### BibTeX

@INPROCEEDINGS{Wang94adefect,

author = {Pei Wang},

title = {A defect in Dempster-Shafer theory},

booktitle = {InProceedings of the Tenth Conference on Uncertainty in Arti cial Intelligence},

year = {1994},

pages = {560--566},

publisher = {Morgan Kaufmann Publishers}

}

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### Abstract

By analyzing the relationships among chance, weight of evidence and degree ofbelief, it is shown that the assertion \chances are special cases of belief functions " and the assertion \Dempster's rule can be used to combine belief functions based on distinct bodies of evidence " together lead to an inconsistency in Dempster-Shafer theory. To solve this problem, some fundamental postulates of the theory must be rejected. A new approach for uncertainty management is introduced, which shares many intuitive ideas with D-S theory, while avoiding this problem. 1

### Citations

7092 |
Probabilistic reasoning in intelligent systems: networks of plausible inference
- Pearl
- 1988
(Show Context)
Citation Context ... except for some special points. Such a result is counter-intuitive [18] and inconsistent with Shafer's interpretation of chance. There are already many papers on the justification of Dempster's rule =-=[2, 5, 6, 11, 14, 19]-=-, but few of them addresses the relationships among degree of belief, weight of evidence, and chance. As a result, the mathematical properties of D-S theory are explored in detail, but its usage of no... |

2266 |
A mathematical theory of evidence
- Shafer
- 1976
(Show Context)
Citation Context ...s problem. 1 Introduction Evidence theory, or Dempster-Shafer (D-S) theory, is developed as an attempt to generalize probability theory by introducing a rule for combining distinct bodies of evidence =-=[1, 7]-=-. The most influential version of the theory is presented by Shafer in his book A Mathematical Theory of Evidence [7]. In the book, the following postulates are assumed, which form the foundation of D... |

591 |
Upper and lower probabilities induced by a multivalued mapping
- Dempster
- 1967
(Show Context)
Citation Context ...s problem. 1 Introduction Evidence theory, or Dempster-Shafer (D-S) theory, is developed as an attempt to generalize probability theory by introducing a rule for combining distinct bodies of evidence =-=[1, 7]-=-. The most influential version of the theory is presented by Shafer in his book A Mathematical Theory of Evidence [7]. In the book, the following postulates are assumed, which form the foundation of D... |

121 |
The combination of evidence in the transferable belief model
- Smets
- 1990
(Show Context)
Citation Context ... except for some special points. Such a result is counter-intuitive [18] and inconsistent with Shafer's interpretation of chance. There are already many papers on the justification of Dempster's rule =-=[2, 5, 6, 11, 14, 19]-=-, but few of them addresses the relationships among degree of belief, weight of evidence, and chance. As a result, the mathematical properties of D-S theory are explored in detail, but its usage of no... |

85 | Perspectives on the Theory and Practice of Belief Functions. Working Paper No
- Shafer
- 1990
(Show Context)
Citation Context ... Postulate 3 is to remove the concept of weight of evidence from D-S theory. Actually weight of evidence is seldom mentioned in the literature of D-S theory. Shafer, in his later papers (for example, =-=[9, 10]-=-), tends to relate belief functions to reliability of testimony and randomly coded message, rather than to weight of evidence. One problem of such a solution is the loss of the intuition in the notion... |

81 |
Bayesian and non-Bayesian evidential updating
- Kyburg
- 1987
(Show Context)
Citation Context ... except for some special points. Such a result is counter-intuitive [18] and inconsistent with Shafer's interpretation of chance. There are already many papers on the justification of Dempster's rule =-=[2, 5, 6, 11, 14, 19]-=-, but few of them addresses the relationships among degree of belief, weight of evidence, and chance. As a result, the mathematical properties of D-S theory are explored in detail, but its usage of no... |

47 | A new approach to updating beliefs
- Fagin, Halpern
- 1991
(Show Context)
Citation Context ... generalization of probability theory, and a chance can be used as a degree of belief. 2. The "lower-upper bounds of probability" interpretation for belief functions is still accepted by som=-=e authors [4]-=-. 3. Some other authors, including Shafer himself, reject the above interpretation, but they still refer to a probability function as a special type (or a limit) of belief functions [9]. 4. Though som... |

38 | Languages and designs for probability judgment
- SHAFER, TVERSKY
- 1985
(Show Context)
Citation Context ... Postulate 3 is to remove the concept of weight of evidence from D-S theory. Actually weight of evidence is seldom mentioned in the literature of D-S theory. Shafer, in his later papers (for example, =-=[9, 10]-=-), tends to relate belief functions to reliability of testimony and randomly coded message, rather than to weight of evidence. One problem of such a solution is the loss of the intuition in the notion... |

36 |
On the justification of Dempster's rule of combination
- Voorbraak
- 1991
(Show Context)
Citation Context |

33 | From Inheritance Relation to Non-Axiomatic Logic. International Journal of Approximate Reasoning 7, 1 74
- Wang
- 1994
(Show Context)
Citation Context ...ould be changed --- the absolute amount of evidence is absent in f . Though it is possible, in theory, to directly use w and w + as measurements of uncertainty, it is often unnatural and inconvenient =-=[17]-=-. Can we capture this kind of information without recording w and w + directly? Yes, we can. From the viewpoint of evidence combination, the influence of w appears in the stability of a frequence eval... |

21 | Belief revision in probability theory
- Wang
- 1993
(Show Context)
Citation Context ...nt probability or to combine evidence). Though the criticism of D-S theory to Bayes approach is justifiable, and the "lowerupper frequency" approach is motivated by similar theoretical consi=-=derations [15], the two -=-approaches solve the problem differently. The "lower-upper frequency" approach is not specially designed to replace D-S theory in general, but it does suggest a better way to represent and p... |

20 | Belief functions and parametric models - Shafer - 1982 |

18 |
The transferable belief model and other interpretations of Dempster-Shaferâ€™s model
- Smets
- 1991
(Show Context)
Citation Context ...elief functions do not generally correspond to Bayesian measures of belief, they still view a belief function as the lower bound of probability [18]. 5. In the transferable belief model of D-S theory =-=[11, 12, 13], Smets shows t-=-hat it is possible "for quantified beliefs developed independently of any underlying probabilistic model," though he still believes that "it seems reasonable to defend the idea that the... |

13 | Non-axiomatic reasoning system (version 2.2
- Wang
- 1993
(Show Context)
Citation Context ...oach is used in the Non-Axiomatic Reasoning System (NARS) project. As an intelligent reasoning system, NARS can adapt to its environment and answer questions with insufficient knowledge and resources =-=[16, 17]-=-. A complete comparison of NARS and D-S theory is beyond the scope of this paper. By introducing the approach here, we hope to show that the most promising solution for the previous inconsistency is t... |

11 |
Updating with belief functions, ordinal conditional functions and possibility measures. Uncertainty in
- Dubois, Prade
- 1991
(Show Context)
Citation Context |

9 |
Evidence, knowledge and belief functions
- Dubois, Prade
- 1992
(Show Context)
Citation Context ...99" will both be represented as Bel(fHg) = P l(fHg) = 1, and their difference will be lost. If Postulate 2 were rejected, it would be invalid to interpret Bel and P l as "lower and upper pro=-=bability" [1, 3, 4, 12]-=-. It is true that there are probability functions P (x) satisfying Bel(fxg)sP (x)sP l(fxg) , for all x 2 \Theta: However, as demonstrated above, these functions may be unrelated to P r(H). For the sam... |

7 |
Belief induced by the partial knowledge of the probabilities
- Smets
- 1994
(Show Context)
Citation Context ...e, according to Smets, "distinctness" is not satisfied in the present context because of the existence of a underlying probability function P r that create a link among the outcomes of the e=-=xperiment [13]. Acceptin-=-g such an opinion, however, means that Postulate 2 is rejected. How can we say that "chances are limits of belief functions," if it is always invalid to take this kind of limits (by repeated... |

6 | The Assumptions Behind Dempster's Rule
- Wilson
- 1993
(Show Context)
Citation Context |

6 |
Upper and lower probabilities induced by amultivalued mapping
- Dempster
- 1967
(Show Context)
Citation Context ...s problem. 1 Introduction Evidence theory, or Dempster-Shafer (D-S) theory, is developed as an attempt to generalize probability theory by introducing a rule for combining distinct bodies of evidence =-=[1, 7]-=-. The most in uential version of the theory is presented by Shafer in his book A Mathematical Theory of Evidence [7]. In the book, the following postulates are assumed, which form the foundation of D-... |

1 |
The combination of belief: when and how fast
- Wilson
- 1992
(Show Context)
Citation Context ...onvention made from different considerations. In D-S theory, however, they are combined to get a different Bayesian belief function, except for some special points. Such a result is counter-intuitive =-=[18]-=- and inconsistent with Shafer's interpretation of chance. There are already many papers on the justification of Dempster's rule [2, 5, 6, 11, 14, 19], but few of them addresses the relationships among... |

1 |
On the justi cation of Dempster's rule of combination. Arti cial Intelligence, 48:171{197
- Voorbraak
- 1991
(Show Context)
Citation Context ..., except for some special points. Such a result is counter-intuitive [18] and inconsistent with Shafer's interpretation of chance. There are already many papers on the justi cation of Dempster's rule =-=[2, 5, 6, 11, 14, 19]-=-, but few of them addresses the relationships among degree of belief, weight of evidence, and chance. As a result, the mathematical properties of D-S theory are explored in detail, but its usage of no... |