dempster-shafer theory


  • [2][3] The theory allows one to combine evidence from different sources and arrive at a degree of belief (represented by a mathematical object called belief function) that
    takes into account all the available evidence.

  • [4] It follows from the last two equations that, for a finite set X, one needs to know only one of the three (mass, belief, or plausibility) to deduce the other two; though
    one may need to know the values for many sets in order to calculate one of the other values for a particular set.

  • Probability values are assigned to sets of possibilities rather than single events: their appeal rests on the fact they naturally encode evidence in favor of propositions.

  • Often used as a method of sensor fusion, Dempster–Shafer theory is based on two ideas: obtaining degrees of belief for one question from subjective probabilities for a related
    question, and Dempster’s rule[7] for combining such degrees of belief when they are based on independent items of evidence.

  • Relational measures In considering preferences one might use the partial order of a lattice instead of the total order of the real line as found in Dempster–Schafer theory.

  • It is the amount of belief that directly supports either the given hypothesis or a more specific one, thus forming a lower bound on its probability.

  • Example producing correct results in case of high conflict[edit] The following example shows how Dempster’s rule produces intuitive results when applied in a preference fusion
    situation, even when there is high conflict.

  • In case different sources express their beliefs over the frame in terms of belief constraints such as in the case of giving hints or in the case of expressing preferences,
    then Dempster’s rule of combination is the appropriate fusion operator.

  • Belief and plausibility[edit] Shafer’s formalism starts from a set of possibilities under consideration, for instance numerical values of a variable, or pairs of linguistic
    variables like “date and place of origin of a relic” (asking whether it is antique or a recent fake).

  • [6] Put another way, it is a way of representing epistemic plausibilities, but it can yield answers that contradict those arrived at using probability theory.

  • Example producing counter-intuitive results in case of high conflict[edit] An example with exactly the same numerical values was introduced by Lotfi Zadeh in 1979,[12][13][14]
    to point out counter-intuitive results generated by Dempster’s rule when there is a high degree of conflict.

  • For example, if: then The elements of the power set can be taken to represent propositions concerning the actual state of the system, by containing all and only the states
    in which the proposition is true.

  • Also, in DST the empty set is considered to have zero mass, meaning here that the signal light system exists and we are examining its possible states, not speculating as to
    whether it exists at all.

  • This interval contains the precise probability of a set of interest (in the classical sense), and is bounded by two non-additive continuous measures called belief (or support)
    and plausibility: The belief bel(A) for a set A is defined as the sum of all the masses of subsets of the set of interest: The plausibility pl(A) is the sum of all the masses of the sets B that intersect the set of interest A: The two measures
    are related to each other as follows: And conversely, for finite A, given the belief measure bel(B) for all subsets B of A, we can find the masses m(A) with the following inverse function: where is the difference of the cardinalities of the
    two sets.

  • The degrees of belief themselves may or may not have the mathematical properties of probabilities; how much they differ depends on how closely the two questions are related.

  • Dempster–Shafer theory allows one to specify a degree of ignorance in this situation instead of being forced to supply prior probabilities that add to unity.

  • Use of that rule in other situations than that of combining belief constraints has come under serious criticism, such as in case of fusing separate belief estimates from multiple
    sources that are to be integrated in a cumulative manner, and not as constraints.

  • In either case, it would be reasonable to expect that: since the existence of non-zero belief probabilities for other diagnoses implies less than complete support for the
    brain tumour diagnosis.

  • First, the mass of the empty set is zero: Second, the masses of all the members of the power set add up to a total of 1: The mass m(A) of A, a given member of the power set,
    expresses the proportion of all relevant and available evidence that supports the claim that the actual state belongs to A but to no particular subset of A.

  • Dempster’s rule of combination always produces correct and intuitive results in situation of fusing belief constraints from different sources.

  • [11] Dempster’s rule of combination[edit] The problem we now face is how to combine two independent sets of probability mass assignments in specific situations.

  • [17][18][19] Example producing counter-intuitive results in case of low conflict[edit] The following example shows where Dempster’s rule produces a counter-intuitive result,
    even when there is low conflict.

  • It also ranges from 0 to 1 and measures the extent to which evidence in favor of leaves room for belief in p. For example, suppose we have a belief of 0.5 for a proposition,
    say “the cat in the box is dead.”

  • Kłopotek and Wierzchoń[26] proposed to interpret the Dempster–Shafer theory in terms of statistics of decision tables (of the rough set theory), whereby the operator of combining
    evidence should be seen as relational joining of decision tables.

  • For this system, any belief function assigns mass to the first state, the second, to both, and to neither.

  • Criticism Judea Pearl (1988a, chapter 9;[23] 1988b[24] and 1990)[25] has argued that it is misleading to interpret belief functions as representing either “probabilities of
    an event,” or “the confidence one has in the probabilities assigned to various outcomes,” or “degrees of belief (or confidence, or trust) in a proposition,” or “degree of ignorance in a situation.”

  • However, the semantics of interpreting preference as a probability is vague: if it is referring to the probability of seeing film X tonight, then we face the fallacy of the
    excluded middle: the event that actually occurs, seeing none of the films tonight, has a probability mass of 0.

  • Specifically, the combination (called the joint mass) is calculated from the two sets of masses in the following manner: where K is a measure of the amount of conflict between
    the two mass sets.

  • [citation needed] Formal definition Let X be the universe: the set representing all possible states of a system under consideration.

  • The theory of evidence assigns a belief mass to each element of the power set.

  • In another interpretation M. A. Kłopotek and S. T. Wierzchoń[27] propose to view this theory as describing destructive material processing (under loss of properties), e.g.

  • [28] Given a set of criteria C and a bounded lattice L with ordering ≤, Schmidt defines a relational measure to be a function from the power set of C into L that respects
    the order on (C): and such that takes the empty subset of (C) to the least element of L, and takes C to the greatest element of L. Schmidt compares with the belief function of Schafer, and he also considers a method of combining measures generalizing
    the approach of Dempster (when new evidence is combined with previously held evidence).

  • In particular, many authors have proposed different rules for combining evidence, often with a view to handling conflicts in evidence better.

  • However, it is more common to use the term in the wider sense of the same general approach, as adapted to specific kinds of situations.

  • [9] Note that the probability masses from propositions that contradict each other can be used to obtain a measure of conflict between the independent belief sources.

  • For instance, assume a situation where there are two possible states of a system.

  • In essence, the degree of belief in a proposition depends primarily upon the number of answers (to the related questions) containing the proposition, and the subjective probability
    of each answer.

  • He further demonstrated that, if partial knowledge is encoded and updated by belief function methods, the resulting beliefs cannot serve as a basis for rational decisions.

  • Belief functions base degrees of belief (or confidence, or trust) for one question on the subjective probabilities for a related question.

  • The third condition, however, is subsumed by, but relaxed in DS theory:[2]: p. 19  Either of the following conditions implies the Bayesian special case of the DS theory:[2]: p.
    37, 45  • • For finite X, all focal elements of the belief function are singletons.

  • As a result, DS theory is subject to the Dutch Book argument, implying that any agent using DS theory would agree to a series of bets that result in a guaranteed loss.

  • In the case of an infinite X, there can be well-defined belief and plausibility functions but no well-defined mass function.

  • Dempster’s rule of combination produces intuitive results even in case of totally conflicting beliefs when interpreted in this way.

  • [4] The early contributions have also been the starting points of many important developments, including the transferable belief model and the theory of hints.

  • However, it lacks many (if not most) of the properties that make Bayes’ rule intuitively desirable, leading some to argue that it cannot be considered a generalization in
    any meaningful sense.

  • This combination rule for evidence can therefore produce counterintuitive results, as we show next.

  • When combining the preferences with Dempster’s rule of combination it turns out that their combined preference results in probability 1.0 for film Y, because it is the only
    film that they both agree to see.

  • In DST the mass assigned to Any refers to the proportion of evidence that can not be assigned to any of the other states, which here means evidence that says there is a light
    but does not say anything about what color it is.

  • The value of m(A) pertains only to the set A and makes no additional claims about any subsets of A, each of which have, by definition, their own mass.

  • This interval represents the level of uncertainty based on the evidence in the system.


Works Cited

[‘Dempster, A. P. (1967). “Upper and lower probabilities induced by a multivalued mapping”. The Annals of Mathematical Statistics. 38 (2): 325–339. doi:10.1214/aoms/1177698950.
2. ^ Jump up to:a b c d e f Shafer, Glenn; A Mathematical Theory of Evidence,
Princeton University Press, 1976, ISBN 0-608-02508-9
3. ^ Fine, Terrence L. (1977). “Review: Glenn Shafer, A mathematical theory of evidence”. Bull. Amer. Math. Soc. 83 (4): 667–672. doi:10.1090/s0002-9904-1977-14338-3.
4. ^ Jump up to:a b Kari
Sentz and Scott Ferson (2002); Combination of Evidence in Dempster–Shafer Theory, Sandia National Laboratories SAND 2002-0835
5. ^ Jump up to:a b Kohlas, J., and Monney, P.A., 1995. A Mathematical Theory of Hints. An Approach to the Dempster–Shafer
Theory of Evidence. Vol. 425 in Lecture Notes in Economics and Mathematical Systems. Springer Verlag.
6. ^ Shafer, Glenn; Dempster–Shafer theory, 2002
7. ^ Dempster, Arthur P.; A generalization of Bayesian inference, Journal of the Royal Statistical
Society, Series B, Vol. 30, pp. 205–247, 1968
8. ^ Jump up to:a b Jøsang, A.; Simon, P. (2012). “Dempster’s Rule as Seen by Little Colored Balls”. Computational Intelligence. 28 (4): 453–474. doi:10.1111/j.1467-8640.2012.00421.x. S2CID 5143692.
9. ^
Jøsang, A., and Hankin, R., 2012. Interpretation and Fusion of Hyper Opinions in Subjective Logic. 15th International Conference on Information Fusion (FUSION) 2012. E-ISBN 978-0-9824438-4-2, IEEE.|url=
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Jøsang, A.; Diaz, J. & Rifqi, M. (2010). “Cumulative and averaging fusion of beliefs”. Information Fusion. 11 (2): 192–200. CiteSeerX doi:10.1016/j.inffus.2009.05.005. S2CID 205432025.
11. ^ J.Y. Halpern (2017) Reasoning about Uncertainty
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12. ^ L. Zadeh, On the validity of Dempster’s rule of combination, Memo M79/24, Univ. of California, Berkeley, USA, 1979
13. ^ L. Zadeh, Book review: A mathematical theory of evidence, The Al Magazine, Vol. 5, No. 3, pp. 81–83, 1984
14. ^
L. Zadeh, A simple view of the Dempster–Shafer Theory of Evidence and its implication for the rule of combination, The Al Magazine, Vol. 7, No. 2, pp. 85–90, Summer 1986.
15. ^ E. Ruspini, “The logical foundations of evidential reasoning”, SRI Technical
Note 408, December 20, 1986 (revised April 27, 1987)
16. ^ N. Wilson, “The assumptions behind Dempster’s rule”, in Proceedings of the 9th Conference on Uncertainty in Artificial Intelligence, pages 527–534, Morgan Kaufmann Publishers, San Mateo,
CA, USA, 1993
17. ^ F. Voorbraak, “On the justification of Dempster’s rule of combination”, Artificial Intelligence, Vol. 48, pp. 171–197, 1991
18. ^ Pei Wang, “A Defect in Dempster–Shafer Theory”, in Proceedings of the 10th Conference on Uncertainty
in Artificial Intelligence, pages 560–566, Morgan Kaufmann Publishers, San Mateo, CA, USA, 1994
19. ^ P. Walley, “Statistical Reasoning with Imprecise Probabilities”, Chapman and Hall, London, pp. 278–281, 1991
20. ^ Dezert J., Tchamova A., Han
D., Tacnet J.-M., Why Dempster’s fusion rule is not a generalization of Bayes fusion rule, Proc. Of Fusion 2013 Int. Conference on Information Fusion, Istanbul, Turkey, July 9–12, 2013
21. ^ Bauer; Mathias (1996). Proceedings of the Twelfth international
conference on Uncertainty in artificial intelligence. pp. 73–80.
22. ^ Voorbraak, Frans (1989-05-01). “A computationally efficient approximation of Dempster-Shafer theory”. International Journal of Man-Machine Studies. 30 (5): 525–536. doi:10.1016/S0020-7373(89)80032-X.
hdl:1874/26317. ISSN 0020-7373.
23. ^ Pearl, J. (1988a), Probabilistic Reasoning in Intelligent Systems, (Revised Second Printing) San Mateo, CA: Morgan Kaufmann.
24. ^ Pearl, J. (1988b). “On Probability Intervals”. International Journal of Approximate
Reasoning. 2 (3): 211–216. doi:10.1016/0888-613X(88)90117-X.
25. ^ Pearl, J. (1990). “Reasoning with Belief Functions: An Analysis of Compatibility”. The International Journal of Approximate Reasoning. 4 (5/6): 363–389. doi:10.1016/0888-613X(90)90013-R.
26. ^
M. A. Kłopotek, S. T. Wierzchoń’: “A New Qualitative Rough-Set Approach to Modeling Belief Functions.” [in:] L. Polkowski, A, Skowron eds: Rough Sets And Current Trends In Computing. Proc. 1st International Conference RSCTC’98, Warsaw, June 22–26,
1998, Lecture Notes in Artificial Intelligence 1424, Springer-Verlag, pp. 346–353.
27. ^ M. A. Kłopotek and S. T. Wierzchoń, “Empirical Models for the Dempster–Shafer Theory”. in: Srivastava, R. P., Mock, T. J., (Eds.). Belief Functions in Business
Decisions. Series: Studies in Fuzziness and Soft Computing. Vol. 88 Springer-Verlag. March 2002. ISBN 3-7908-1451-2, pp. 62–112
28. ^ Gunther Schmidt (2006) Relational measures and integration, Lecture Notes in Computer Science # 4136, pages 343−57,
Springer books
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