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1.132 .113(theory of evidence, theory of the certainty factor\) or non-numerical methods \(non-)J
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57 301 :M
(mutandis.)S
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1.589 .159(ignorance, uncertainty and vagueness are really different concepts. Random usage)J
57 397 :M
-.109(usually leads to inappropriate matching.)A
57 421 :M
.858 .086(This paper is a plea for the use of correct models. An understanding of the forms of)J
57 433 :M
-.029(ignorance and the nature and the foundations of each model are required. Before using a)A
57 445 :M
-.082(quantified model, we must)A
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.024 .002(1\) provide a meaning for the numbers, i.e. provide canonical examples where the origin)J
57 481 :M
-.109(of the numbers can be justified)A
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-.034(2\) understand the fundamental axioms of the model and their consequences. The choice)A
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-.074(of axioms should be justified by "natural" requirements.)A
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1.171 .117(3\) study the consequence of the derived models in practical contexts to check their)J
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-.122(validity and appropriateness)A
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.71 .071(A common error consists in accepting a model because it 'worked' nicely in the past.)J
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-.098(This property is not a proof that the model is correct. Experimental results can only prove)A
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-.075(that a model is wrong, not that it is correct. They only give hints about its value.)A
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.22 .022(To illustrate our message, we present a survey of certain forms of ignorance and of the)J
57 613 :M
-.075(mathematical models that have been suggested to quantify ignorance. We first present an)A
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-.067(example of an inadequate model. We present the survey, and finish with a plea for future)A
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-.093(studies covering the integration of the models.)A
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.1 .01( The following text presents research results of the Belgian National incentive-program for fundamental)J
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.517 .052(research in artificial intelligence initiated by the Belgian State, Prime Minister's Office, Science Policy)J
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.436 .044(Programming. Scientific responsibility is assumed by its author. Research has partly been supported by)J
57 727 :M
-.008(the DRUMS \(Defeasable reasoning and Uncertainty Management Systems\) project funded by EEC grants)A
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-.024(under the ESPRIT II Basic Research Project 3085.)A
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1.371 .137(to be used in an expert system to quantify uncertainty. Their use results from the)J
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-.046(excellent insight that probability models are too restrictive to model quantified beliefs as)A
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.19 .019(they appear in diagnostic contexts. A piece of evidence e could support a hypothesis h)J
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-.018(without necessarily supporting the complement of that hypothesis. The authors rejected)A
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-.192(the rule that)A
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-.061(where belief\(h)A
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.079 .008(for lack of alternative models they created an ad hoc model based on measure of belief,)J
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.264 .026(disbelief and what is known by now as certainty factors. The aim was great, the result)J
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.711 .071(was shaky.)J
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.106 .011(Not only was he meaning of the numbers not supported but what was more, the models)J
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.527 .053(did not satisfy some fundamental requirements. What does .7 mean in "my CF is .7"?)J
85 281 :M
.722 .072(Why not .6 or .8? Some yardstick is required. For subjective probability theory, urns)J
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.684 .068(provide a yardstick such as our belief that the next randomly selected ball is white is)J
85 305 :M
1.441 .144(equal to the proportion of white balls in the urn - an objective unassailable value.)J
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1.146 .115(Analogous canonical examples have also been developed for belief functions based)J
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.055 .005(models \(Shafer and Tversky, 1985\).)J
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.562 .056(As an example of the weakness of the model itself, suppose there are two rules IF E)J
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.648 .065(A natural requirement for any model for parallel combination is that the combination)J
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-.065(should be associative. But the rules \(*\) are not associative.)A
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-.036(Furthermore \(*\) should never be applied if E)A
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-.004(where all certainty factors are 1. Let the certainty factor of E)A
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.293 .029(This example shows the danger of using ad hoc models blindly. The correct way is to)J
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.442 .044(build a set of requirements and to build a model that satisfies these requirements as is)J
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-.039(done in Heckerman \(1986\).)A
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3.153 .315(3. Variety of ignorance.)J
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1.335 .133(Ignorance can be subdivided into 3 large categories : incompleteness, imprecision,)J
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-.016(uncertainty \(Bonissone and Tong, 1985\).)A
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.223 .022( covers cases where the value of a variable is missing.)J
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1.408 .141( covers cases where the value of a variable is given but not with the)J
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-.053(precision required.)A
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.495 .049( covers cases where an agent can construct a personal subjective opinion)J
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-.092(on a proposition that is not definitively established for him.)A
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1.994 .199(The distinctions between these categories are vague and it might be argued that)J
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.587 .059(incompleteness is just an extreme form of imprecision \(when nothing is known\). The)J
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-.101(following examples illustrate the three forms of ignorance.)A
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.244(Incompleteness)A
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.799 .08(. Suppose we have a database that should include marital status and)J
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.477 .048(the name of the spouse, and on which the information is error-free. Also suppose the)J
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.022 .002(database contains the information "John is married" but the value of the variable "name)J
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.28 .028(of spouse" is missing. This information is incomplete since we do not know the name)J
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-.084(of John's wife but what is available is precise \(he is married\) and certain \(the information)A
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.028 .003(is error-free\).)J
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.326(Imprecision)A
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.93 .093( . Suppose the value of the variable "name of spouse" is "Jill or Joan".)J
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.547 .055(This information is complete inasmuch as both marital status and the wife's name are)J
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.096 .01(known, but it is imprecise because there is some ambiguity as to whom his wife is. No)J
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-.098(uncertainty is present here since the information is error-free.)A
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.671(Uncertainty)A
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2.33 .233(. Suppose the information in the database were provided by some)J
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.495 .05(untrustworthy person who said that "John's wife is Jill". The information is complete,)J
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-.094(precise but uncertain since it might be wrong.)A
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.01 .001(A major difference between these three forms of ignorance is related to the objective or)J
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-.06(subjective component, as to whether the observer is involved or not. Incompleteness and)A
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2.544 .254(imprecision are )J
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.605(objective)A
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2.282 .228( forms of ignorance. They exist independently of the)J
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1.295 .129(observer: these properties belong to the data. Uncertainty is a )J
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.388(subjective)A
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1.1 .11( form of)J
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.097 .01(ignorance. It appears when the observer is taken into account. It is the observer that is)J
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-.033(not certain about the available information. This information only induces some form of)A
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-.089(partial knowledge or belief in the observer.)A
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.994 .099(Imprecision and incompleteness are )J
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1.476 .148(context dependent)J
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.588 .059(. When I invite guest to my)J
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.661 .066(party and I only know "John is married" and his wife is "Jill or Joan", this imprecise)J
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-.049(information is sufficient if I want to invite only married people, but insufficient if I want)A
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-.081(to assign seats at the dinner tablein such a way that John will be sitting on the right of his)A
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.586 .059(wife. Similarly, if I want to select people whose height is above 150 cm, information)J
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.328 .033(like 'Paul's height is >170 cm' or 'Paul is tall' are sufficient, whereas if I want to select)J
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-.099(only those taller than 175 cm, neither piece of information is sufficient.)A
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.97 .097(Within the three categories of ignorance, one can describe many subcategories. The)J
86 655 :M
.229 .023(following table presents the types of model, the types of ignorance and an example for)J
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-.059(each subcategory.)A
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-.109(Universal:)A
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-.145(All computer scientists like pizza,)A
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-.129(but their names are not available.)A
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.329 .033(John's wife is Jill or Joan.)J
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-.14(Negation)A
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.347 .035(Jill is not John's wife.)J
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-.14(Interval theory)A
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-.153(Interval valued )A
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.242 .024(Paul's height is)J
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-.15(information )A
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-.016(between 170 and 180.)A
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-.035(Fuzzy sets)A
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-.109(Fuzzy valued )A
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-.055(Paul is tall.)A
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-.067(Possibility)A
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-.024(the possibility for Paul's height)A
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-.086(\(physical form\) )A
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-.055(to be about 175 cm.)A
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-.078( Upper-Lower Probabilities)A
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(when tossing a coin.)S
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-.078(Possibility Theory)A
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-.067(Possibility)A
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-.053(the possibility that Paul's height)A
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-.153(\(epistemic form\))A
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-.021(is about 175 cm.)A
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-.144(Subjective Probabilities)A
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-.199(Credibility)A
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-.143(my degree of belief that cancer X)A
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-.131(Belief functions)A
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(is due to a virus.)S
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1.649 .165(Combinatorial models are not considered here. In practice, they cannot be solved)J
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-.044(through brute force solutions because of the combinatorial explosion. Default logics and)A
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-.071(other non monotonic logics have been proposed to solve these problems.)A
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-.03(Among the numerical models proposed to cope with the various forms of ignorance, the)A
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1.937 .194(most used are the fuzzy sets theory, the probability theory, the upper and lower)J
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-.085(probability theory, the possibility theory and the theory of evidence.)A
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3.6 .36(4.1. Fuzzy Sets Theory)J
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-.009(Fuzziness is the property related to the use of vague predicates like in 'John is tall'. The)A
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.667 .067(predicates are vague, fuzzy because the words used to define them are themselves ill)J
86 649 :M
.836 .084(defined, vague, fuzzy. The idea is that belonging to a set admits a degree that is not)J
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1.11 .111(necessarily just 0 or 1 as is the case in classical set theory. Intermediate values are)J
86 673 :M
1.099 .11(accepted in order to cope with borderline cases. It is different from probability and)J
86 685 :M
.181 .018(randomness. Randomness talks about the certainty of whether a given element belongs)J
86 697 :M
1.04 .104(or not to a well-defined set. Fuzziness talks about the imprecision derived from the)J
86 709 :M
-.117(partial membership of a given and well defined element to a set whose boundaries are not)A
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-.01(sharply defined. \(Zadeh 1965, 1975, Dubois and Prade 1980\))A
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3.818 .382(Probability theory.)J
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.076 .008(Probability theory is used to quantify the chance that an event might occur or the belief)J
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.538 .054(that a proposition is true. Events occur or do not occur, propositions are true or false.)J
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-.083(No vagueness is involved. Probability theory provides a metalanguage that quantifies the)A
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.794 .079(chance that some events might occur or some propositions are true. Its adequacy for)J
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.229 .023(random processes has been known for centuries. Its role for decision under risk is well)J
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.086 .009(established \(Degroot 1970\). Bayesians postulate that models like this should be used to)J
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-.026(quantify one's beliefs, but it is still an open question \(Fine 1973\).)A
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2.914 .291(4.3. Upper and Lower Probability)J
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1.185 .118(Among the models proposed to describe degrees of belief, Smith \(1961, 1965\) and)J
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.632 .063(Good \(1950, 1983\) have postulated that one can often only claim that the probability)J
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.718 .072(function that describes our degrees of belief belongs to a convex set P of probability)J
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.125 .012(functions. This set can be characterized by the so called upper and lower probabilities,)J
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1.597 .16(that is the maximal and minimal probability given to each proposition, where the)J
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-.098(extremes are taken on the probability functions belonging to the convex set P.)A
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.102 .01(A special case of upper and lower probabilities has been described by Dempster \(1967,)J
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.736 .074(1968\). He assumes the existence of a probability function on a space X and a one to)J
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.763 .076(many mapping M from X to Y. Then the lower probability of A in Y is equal to the)J
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.134 .013(probability of the largest subset of X such that its image under M is included in A. The)J
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.669 .067(upper probability of A in Y is the probability of the largest subset of X such that the)J
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-.125(images under M of all its elements have a non empty intersection with A.)A
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2(4.4.)A
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.802(Possibility-necessity)A
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.365 .037(Incomplete information such as "John's height is above 170" implies that any height h)J
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.899 .09(above 170 is possible and any height equal or below 170 is impossible. This can be)J
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-.002(represented by a possibility function defined on the height domain whose value is 0 if h)A
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.85 .085(< 170 and 1 if h is )J
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.85 .085( 170 \(with 0 = impossible and 1 = possible\). Ignorance results)J
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2.565 .256(from the lack of precision, of specificity of the information "above 170" . Its)J
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.574 .057(fundamental axiom is that the possibility of the disjunction of two propositions is the)J
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.789 .079(maximum of the possibility of the individual propositions. \(Zadeh 1978, Dubois and)J
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.768 .077(Prade, 1985\).)J
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.84 .084(When the predicate is vague like in )J
f214 sf
.344(")A
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.966 .097(John is tall', possibility can admit degrees, the)J
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1.573 .157(largest the degree, the largest the possibility. But even though possibility is often)J
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1.137 .114(associated with fuzziness, the fact that non fuzzy \(crisp\) events can admit different)J
86 559 :M
.834 .083(degrees of possibility is shown in the following example. Suppose there is a box in)J
86 571 :M
1.416 .142(which you try to put tennis balls. You can say: it is possible to put 20 balls in it,)J
86 583 :M
.453 .045(impossible to put 30 balls, quite possible to put 24 balls, but not so possible to put 26)J
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.509 .051(balls...These degrees of possibility are degrees of realizability)J
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.063 .006( )J
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.485 .049(and totally unrelated to)J
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-.023(any supposedly underlying random process.)A
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1.737 .174(Two forms of \(continuous valued\) possibility)J
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1.419 .142( have been described: the physical)J
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1.009 .101(and the epistemic. These 2 forms of possibility can be recognized by their different)J
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.681 .068(linguistic uses: it is )J
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1.258 .126(possible that)J
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.388 .039( and it is )J
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.845 .085(possible for )J
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.943 .094(\(Hacking 1975\). When I say)J
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.386 .039(it is possible that Paul's height is 170, it means that for all what I know, Paul's height)J
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1.778 .178(may be 170. When I say it is possible for Paul's height to be 170, it means that)J
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.865 .086(physically, Paul's height may be 170. The first form, 'possible that', is related to our)J
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.394 .039(state of knowledge and is called epistemic. The second form, 'possible for', deals with)J
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-.097(actual abilities independently of our knowledge about them. It is a degree of realizability.)A
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.083 .008(The distinction is not unrelated to the one between the epistemic concept of probability)J
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.536 .054(\(called here the credibility\) and the aleatory one \(called here chance\). These forms of)J
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1.136 .114(possibilities are evidently not independent concepts, but the exact structure of their)J
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-.093(interrelation)A
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-.097(s)A
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-.095( is not yet clearly established.)A
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1.473 .147( is defined by the dual property)J
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.361(:)A
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1.397 .14( the necessity of a proposition A is the)J
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-.095(complement \(to 1\) of the possibility of not-A.)A
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3.102 .31(4.5. Credibility: the transferable belief model.)J
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1.194 .119(Information can induce in us some subjective, personal credibility \(hereafter called)J
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.926 .093(belief\) that a proposition is true. Its origin lies either in the )J
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2.149 .215(random nature)J
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.75 .075( of the)J
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.032 .003(underlying event or in the )J
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.058 .006(partial reliability)J
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.032 .003( that we give to the source of information.)J
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.415 .042(In the first case, one ends up with a probability function if one accepts the )J
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.17(frequency)A
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.08(principle)A
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.269 .027( \(Hacking 1965\) that, given the chance that a random event X might occur is)J
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-.069(p, our degree of belief that it will occur is p.)A
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-.034(IF chance\(X\)=p THEN belief\(X\) = p)A
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.384 .038(This is fundamental for the )J
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.634 .063(classical bayesian model)J
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.363 .036(, as it relates chance and belief.)J
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2.148 .215(When randomness is not involved, there is no necessity that )J
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3.276 .328(credal states)J
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1.831 .183( \(the)J
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.367 .037(psychological level where beliefs are entertained\) have to be quantified by probability)J
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.717 .072(functions. \(Levi 1984\). The coherence principle advanced by the bayesians to justify)J
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-.106(probability functions is adequate in a context of decision \(Degroot 1970\), but it cannot be)A
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.497 .05(used when all one wants to describe is a cognitive process. Beliefs can be entertained)J
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1.611 .161(outside any decision context. In the )J
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2.569 .257(transferable belief model)J
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1.693 .169( \(Smets 1988\) we)J
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.284 .028(assume that beliefs at the credal level are quantified by belief functions \(Shafer 1976\).)J
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1.436 .144(When decisions must be made, our belief at the credal level induces a probability)J
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.327 .033(function at the so-called )J
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.38 .038('pignistic' level )J
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.342 .034(\(the level at which decisions are made\). This)J
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.52 .052(bona fide probability function will be used in order to make decisions using expected)J
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-.1(utilities theory. Relations between belief functions and pignistic probabilities are given in)A
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.31 .031(Smets \(1989\).)J
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3.337 .334(5. Combining models of ignorance.)J
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.337 .034(The various forms of ignorance can be encountered simultaneously and it is necessary)J
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.07 .007(that we be able to integrate them. In common-sense reasoning, two forms of ignorance,)J
86 511 :M
.05 .005(sometime three, are often encountered in the same statement. Just to give an idea of the)J
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-.073(problem, consider the following example of generalized modus ponens.)A
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.853 .085(I strongly believe that it is somehow possible that 'If a father is tall, then his son is)J
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-.008(usually quite tall'.)A
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-.055(I believe that it is more or less true that 'Paul is quite tall'.)A
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-.062(What can I say about the height of his son.)A
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-.093(This example is evidently too complex to be encountered in practice, but it includes most)A
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.183 .018(forms of ignorance..)J
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.621 .062(To deal with problems like this, beliefs, possibilities, fuzziness need to be combined,)J
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1.122 .112(and )J
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1.709 .171(a set of metalanguages)J
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1.405 .141( must be constructed. Care must be given however to)J
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2.028 .203(what are the domains of each operator. For instance, probability deals with two)J
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-.018(domains, the set of propositions \(as are usually mentioned\) and the truth domain \(that is)A
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-.029(usually disregarded as it contains only two elements, but must be considered once fuzzy)A
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-.074(propositions are accepted\).)A
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1.39 .139(The first problem is to see the )J
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.529(connections)A
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1.765 .177( between the probability theory in its)J
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.169 .017(frequency approach and the physical possibility theory. The next problem is to see the)J
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.888 .089(connections between subjective probability functions, belief functions and epistemic)J
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.157 .016(possibility functions. Finally, one must establish the connections between the physical)J
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-.035(properties and the epistemic properties. There is the further the problem of extending all)A
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-.071(these theories when the propositions involved are fuzzy.)A
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.723 .072(Almost no work has yet been done in this area. But its importance for )J
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.243(datafusion)A
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.413 .041( is)J
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.156 .016(obvious: when several sensors provide information, how do we recognize the nature of)J
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-.018(the ignorance involved and select the appropriate model, how do we collapse them into)A
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.097 .01(more compact forms, how do we combine them, how do we take into consideration the)J
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1.421 .142(redundancies, the correlations and the contradictions. All these problems must be)J
86 175 :M
-.121(studied and the implementation of potential solutions tested.)A
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f214 sf
5.183 .518(Understanding of the meaning of statements)J
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3.852 .385( and their translation into)J
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1.219 .122(appropriate models is delicate, if not hazardous. For example, how do we translate)J
86 223 :M
1.194 .119("usually bald men are old". Which of P\(bald|old\) or P\(old|bald\) is somehow large?)J
86 235 :M
.709 .071("When x shaves himself, usually x does not die". Which conditioning is appropriate:)J
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1.076 .108(Pl\(dead|shaving\) or Pl\(shaving|dead\)? Is it a problem of plausibility or possibility?)J
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-.095(These examples are just illustrative of the kind of problems that must be addressed.)A
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f214 sf
.721(Bibliography.)A
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f148 sf
1.045 .104(BONISSONE P.P. and TONG R.M. \(1985\) Editorial: reasoning with uncertainty in expert systems.)J
86 329 :M
.144 .014(I.J.Man Machine Studies 22:241-250.)J
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.05 .005(DEGROOT M. \(1970\) )J
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.06 .006(Optimal statistical decisions.)J
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.043 .004( McGraw Hill, New York.)J
86 351 :M
.847 .085(DEMPSTER A.P. \(1967\) Upper and lower probabilities induced by a multiplevalued mapping. Ann.)J
86 362 :M
.571 .057(Math. Statistics 38:325-339.)J
86 373 :M
.255 .026(DEMPSTER A.P. \(1968\) A generalization of Bayesian inference. J. Roy. Statist. Soc. B.30:205-247.)J
86 384 :M
.421 .042(DUBOIS D. and PRADE H. \(1980\) )J
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.47 .047(Fuzzy sets and systems: theory and applications.)J
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.515 .051( Academic Press,)J
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-.053(New York.)A
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.156 .016(DUBOIS D. and PRADE H. \(1985\) )J
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.208 .021(Theorie des possibilit\216s.)J
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.136 .014( Masson, Paris .)J
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.013 .001(FINE T. \(1973\) )J
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.02 .002(Theories of probability)J
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.016 .002(. Academic Press, New York.)J
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-.015(GOOD I.J. \(1950\) )A
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-.014(Probability and the weighting of evidence)A
f148 sf
-.015(. Hafner.)A
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.338 .034(GOOD I.J. \(1983\) )J
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.382 .038(Good thinking: the foundations of probability and its applications.)J
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.42 .042( Univ. Minnesota)J
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.502 .05(Press, Minneapolis.)J
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-.014(HACKING I. \(1965\) )A
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-.011(Logic of statistical inference)A
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-.013(. Cambridge UNiversity Press, Cambridge.)A
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-.02(HACKING I.\(1975\) )A
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-.017(The emergence of probability.)A
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-.018( Cambridge UNiversity Press, Cambridge.)A
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.592 .059(HECKERMAN D. \(1986\) Probabilistic interpretation for MYCIN's certainty factors, in KANAL L.N.)J
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(and LEMMER J.F. \(eds\) Uncertainty in Artificial Intelligence, North Holland, Amsterdam, pp.167-196.)S
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.866 .087(LEVI I. \(1984\) )J
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1.191 .119(Decisions and revisions: philosophical essays on knowledge and value.)J
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1.534 .153( Cambridge)J
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.054 .005(University Press, Cambridge.)J
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.096 .01(LOPEZ DE MANTARAS R. \(1990\) Approximate reasoning models. Ellis Horwood, Chichester.)J
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.238 .024(SHAFER G. \(1976\) )J
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.259 .026(A mathematical theory of evidence)J
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.238 .024(. Princeton Univ. Press, Princeton, NJ.)J
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.745 .074(SHAFER G. and TVERSKY A. \(1985\) Languages and designs for probability. Cognitive Sc. 9:309-)J
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.167(339.)A
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1.834 .183(SHORTLIFFE E.H. and BUCHANAN B.G. \(1975\) A model for inexact reasoning in medicine.)J
86 582 :M
-.038(Mathematical Biosciences 23:351-379.)A
86 593 :M
.416 .042(SMETS P. \(1988\) Belief functions. in Non standard logics for automated reasoning, edited by SMETS)J
86 604 :M
.235 .023(P., MAMDANI A., DUBOIS D. AND PRADE H., Academic Press, London, pp. 253-286.)J
86 615 :M
1.179 .118(SMETS P. \(1989\) Constructing the pignistic probability function in a context of uncertainty. Proc.)J
86 626 :M
.009 .001(Fifth Workshop on Uncertainty in AI, Windsor, Canada, 319-326)J
86 637 :M
.259 .026(SMITH C.A.B. \(1961\) Consistency in statistical inference and decision. J. Roy. Statist. Soc. B23:1-37.)J
86 648 :M
1.002 .1(SMITH C.A.B. \(1965\) Personal probability and statistical analysis. J. Roy. Statist. Soc. A128, 469-)J
86 659 :M
.167(499.)A
86 670 :M
.286 .029(ZADEH L.A. \(1965\) Fuzzy sets. Inform.Control. 8:338-353.)J
86 681 :M
.236 .024(ZADEH L.A. \(1978\) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems. 1:3-28.)J
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f289 sf
.675(Acknowledgemeents:)A
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1.276 .128(The author is indebted to Yen-Teh Hsia, Robert Kennes and Aleesandro Saffiotti for their help in)J
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.379 .038(completing this work.)J
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