Editing Dempster–Shafer theory (section)

==Dempster's rule of combination==
The problem we now face is how to combine two independent sets of probability mass assignments in specific situations. 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. This rule derives common shared belief between multiple sources and ignores ''all'' the conflicting (non-shared) belief through a normalization factor. 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. Cumulative fusion means that all probability masses from the different sources are reflected in the derived belief, so no probability mass is ignored.

Specifically, the combination (called the '''joint mass''') is calculated from the two sets of masses ''m''<sub>1</sub> and ''m''<sub>2</sub> in the following manner:

:<math>m_{1,2}(\emptyset) = 0 \, </math>

:<math>m_{1,2}(A) = (m_1 \oplus m_2) (A) = \frac 1 {1 - K} \sum_{B \cap C = A \ne \emptyset} m_1(B) m_2(C) \,\!</math>

where

:<math>K = \sum_{B \cap C = \emptyset} m_1(B) m_2(C). \, </math>

''K'' is a measure of the amount of conflict between the two mass sets.

===Effects of conflict===
The normalization factor above, 1&nbsp;−&nbsp;''K'', has the effect of completely ignoring conflict and attributing ''any'' mass associated with conflict to the empty set. This combination rule for evidence can therefore produce counterintuitive results, as we show next.

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

:Suppose that two friends, Alice and Bob, want to see a film at the cinema one evening, and that there are only three films showing: X, Y and Z. Alice expresses her preference for film X with probability 0.99, and her preference for film Y with a probability of only 0.01. Bob expresses his preference for film Z with probability 0.99, and his preference for film Y with a probability of only 0.01. 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.

:Dempster's rule of combination produces intuitive results even in case of totally conflicting beliefs when interpreted in this way. Assume that Alice prefers film X with probability 1.0, and that Bob prefers film Z with probability 1.0. When trying to combine their preferences with Dempster's rule it turns out that it is undefined in this case, which means that there is no solution. This would mean that they can not agree on seeing any film together, so they do not go to the cinema together that evening. 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 [[False dilemma|fallacy of the excluded middle]]: the event that actually occurs, seeing none of the films tonight, has a probability mass of 0.

====Example producing counter-intuitive results in case of high conflict====
An example with exactly the same numerical values was introduced by [[Lotfi Zadeh]] in 1979,<ref name="Zadeh79">L. Zadeh, On the validity of Dempster's rule of combination, Memo M79/24, Univ. of California, Berkeley, USA, 1979</ref><ref name="Zadeh84">L. Zadeh, Book review: A mathematical theory of evidence, The Al Magazine, Vol. 5, No. 3, pp. 81–83, 1984</ref><ref name="Zadeh86">L. Zadeh, [https://wvvw.aaai.org/ojs/index.php/aimagazine/article/download/542/478 A simple view of the Dempster–Shafer Theory of Evidence and its implication for the rule of combination] {{Webarchive|url=https://web.archive.org/web/20190728221641/https://wvvw.aaai.org/ojs/index.php/aimagazine/article/download/542/478 |date=2019-07-28 }}, The Al Magazine, Vol. 7, No. 2, pp. 85–90, Summer 1986.</ref>
to point out counter-intuitive results generated by Dempster's rule when there is a high degree of conflict. The example goes as follows:

:Suppose that one has two equi-reliable doctors and one doctor believes a patient has either a brain tumor, with a probability (i.e. a basic belief assignment—bba's, or mass of belief) of 0.99; or meningitis, with a probability of only 0.01. A second doctor believes the patient has a concussion, with a probability of 0.99, and believes the patient suffers from meningitis, with a probability of only 0.01. Applying  Dempster's rule to combine these two sets of masses of belief, one gets finally ''m''(meningitis)=1 (the meningitis is diagnosed with 100 percent of confidence).

Such result goes against common sense since both doctors agree that there is a little chance that the patient has a meningitis. This example has been the starting point of many research works for trying to find a solid justification for Dempster's rule and for foundations of Dempster–Shafer theory<ref name="Ruspini88">E. Ruspini, "[https://www.researchgate.net/profile/Enrique_Ruspini/publication/243627217_The_Logical_Foundations_of_Evidential_Reasoning/links/55ce0d6408ae502646a73338/The-Logical-Foundations-of-Evidential-Reasoning.pdf The logical foundations of evidential reasoning]", ''SRI Technical Note'' '''408''', December 20, 1986 (revised April 27, 1987)</ref><ref name="Wilson93">N. Wilson, "[https://arxiv.org/abs/1303.1518 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</ref> or to show the inconsistencies of this theory.<ref name="Voorbraak88">F. Voorbraak, "[https://dspace.library.uu.nl/bitstream/handle/1874/26413/Preprint_no_42.pdf?sequence=1 On the justification of Dempster's rule of combination]", ''Artificial Intelligence'', Vol. '''48''', pp. 171–197, 1991</ref><ref name="Wang1994">Pei Wang, "[https://arxiv.org/abs/1302.6849 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</ref><ref name="Walley91">P. Walley, "[https://philarchive.org/rec/WALSRW?fId=&eId=WALSRW&gId=&cId=&tSort=ct+desc Statistical Reasoning with Imprecise Probabilities]{{Dead link|date=January 2024 |bot=InternetArchiveBot |fix-attempted=yes }}", Chapman and Hall, London, pp. 278–281, 1991</ref>

====Example producing counter-intuitive results in case of low conflict====
The following example shows where Dempster's rule produces a counter-intuitive result, even when there is low conflict.

:Suppose that one doctor believes a patient has either a brain tumor, with a probability of 0.99, or meningitis, with a probability of only 0.01. A second doctor also believes the patient has a brain tumor, with a probability of 0.99, and believes the patient suffers from concussion, with a probability of only 0.01. If we calculate m (brain tumor) with Dempster's rule, we obtain

::<math>m(\text{brain tumor}) = \operatorname{Bel}(\text{brain tumor}) = 1. \, </math>

This result implies ''complete support'' for the diagnosis of a brain tumor, which both doctors believed ''very likely''. The agreement arises from the low degree of conflict between the two sets of evidence comprised by the two doctors' opinions.

In either case, it would be reasonable to expect that:

:<math>m(\text{brain tumor}) < 1\text{ and } \operatorname{Bel}(\text{brain tumor}) < 1,\,</math>

since the existence of non-zero belief probabilities for other diagnoses implies ''less than complete support'' for the brain tumour diagnosis.