## Classic Works of the Dempster-Shafer Theory of Belief Functions (Google eBook)Ronald R. Yager, Liping Liu This volume is a welcome addition to the literature on the Dempster-Shafer theory. It mayhelp turn the theory, which now enjoys a lively but fragmented existence, into a more coherent and better understood set of tools for pro- bilistic thinking in science and technology. Thevolume’stitlesuggeststhatthetheoryhadaclassicalperiodextending from the 1960s through the 1980s. In its ?rst two decades, it consisted of theoretical writings by the two of us: Dempster’s work on upper and lower probabilities in the 1960s and Shafer’s work on belief functions in the 1970s. Then interestinapplications suddenly ?owered.After Je?Barnettintroduced thename“Dempster-Shafer”in1981[1],thetheoryquicklyacquiredtextbook statusinarti?cialintelligence.Bytheendoftheclassicalperiod,around1990, the theory had acquired powerful computational tools, remarkably diverse applications, and the attention of many researchers interested in variations and generalizations. By many measures, the theory continues to ?ourish in the 21st century. Internet searches for “Dempster-Shafer” produce ever more hits. The theory is used in many branches of technology,only a few of which are representedin thisvolume.Articlesonthetheoryanditsapplicationsappearinaremarkable number of journals and recurring conferences. Books on the theory continue to appear. In other important respects, however, the theory has not been moving forward.Westillhearquestionsthatwereaskedinthe1980s:Howdowetellif bodiesofevidenceareindependent?Whatdowedoiftheyaredependent?We still encounter confusion and disagreement about how to interpret the theory. And we still ?nd little acceptance of the theory in mathematical statistics, where it ?rst began 40 years ago. We have come to believe that three things are needed to move the theory forward. |

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### Contents

1 | |

New Methods for Reasoning Towards Posterior Distributions Based on Sample Data | 35 |

Upper and Lower Probabilities Induced by a Multivalued Mapping | 57 |

A Generalization of Bayesian Inference | 73 |

On Random Sets and Belief Functions | 105 |

NonAdditive Probabilities in the Work of Bernoulli and Lambert | 117 |

Allocations of Probability | 182 |

Computational Methods for A Mathematical Theory of Evidence | 197 |

Some Characterizations of Lower Probabilities and Other Monotone Capacities through the use of Möbius Inversion | 477 |

Axioms for Probability and BeliefFunction Propagation | 499 |

Generalizing the DempsterShafer Theory to Fuzzy Sets | 529 |

Bayesian Updating and Belief Functions | 555 |

BeliefFunction Formulas for Audit Risk | 577 |

Decision Making Under DempsterShafer Uncertainties | 619 |

Belief Functions The Disjunctive Rule of Combination and the Generalized Bayesian Theorem | 633 |

Representation of Evidence by Hints | 665 |

Constructive Probability | 217 |

Belief Functions and Parametric Models | 265 |

Entropy and Speciﬁcity in a Mathematical Theory of Evidence | 291 |

A Method for Managing Evidential Reasoning in a Hierarchical Hypothesis Space | 310 |

Languages and Designs for Probability Judgment | 345 |

A SetTheoretic View of Belief Functions | 375 |

Weights of Evidence and Internal Conﬂict for Support Functions | 411 |

A Framework for EvidentialReasoning Systems | 418 |

Epistemic Logics Probability and the Calculus of Evidence | 435 |

Implementing Dempsters Rule for Hierarchical Evidence | 449 |

Combining the Results of Several Neural Network Classiﬁers | 682 |

The Transferable Belief Model | 693 |

A kNearest Neighbor Classiﬁcation Rule Based on DempsterShafer Theory | 737 |

Logicist Statistics II Inference | 761 |

About Editors | 786 |

About Authors | 788 |

797 | |

798 | |

### Common terms and phrases

applied argument Artiﬁcial Intelligence audit objective audit risk basic probability assignment Bayesian inference Bel(A Bel1 Bel2 belief function belief function Bel belief-function Bernoulli betting bodies of evidence classiﬁers computation conditional probabilities conﬂict Conjectandi corresponding decision deﬁned deﬁnition degrees of belief Dempster-Shafer Dempster-Shafer theory Dempster’s rule denote diﬀerent eﬀect epistemic probability evidential example ﬁducial ﬁnancial statement ﬁnd ﬁnite ﬁrst focal elements formula frame of discernment fuzzy set given Glenn Shafer hyperedges hypertree hypotheses independent inference interpretation intersection items of evidence logic lower probabilities m-values mapping mycin nodes normalization observations obtained parameter plausibility functions possible probability distribution probability judgments probability measure problem proposition random represented result rule of combination sample Sect simple support function speciﬁc statistical subset Suppose theorem theory of belief Theory of Evidence tion transferable belief model uncertainty upper and lower vacuous values