Editing Dempster–Shafer theory (section)

==Criticism==
[[Judea Pearl]] (1988a, chapter 9;<ref name="Pearl-88">Pearl, J. (1988a), ''Probabilistic Reasoning in Intelligent Systems,'' (Revised Second Printing) San Mateo, CA: Morgan Kaufmann.</ref> 1988b<ref name="Pearl-1988b">{{Cite journal | doi = 10.1016/0888-613X(88)90117-X | last1 = Pearl | first1 = J. | year = 1988b | title = On Probability Intervals | journal = International Journal of Approximate Reasoning | volume = 2 | issue = 3| pages = 211–216 | doi-access = free }}</ref> and 1990)<ref name="Pearl-1990">{{Cite journal | doi = 10.1016/0888-613X(90)90013-R | last1 = Pearl | first1 = J. | year = 1990 | title = Reasoning with Belief Functions: An Analysis of Compatibility | journal = The International Journal of Approximate Reasoning | volume = 4 | issue = 5/6| pages = 363–389 | doi-access = free }}</ref> 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." Instead, belief
functions represent the probability that a given proposition is ''provable'' from a set of other propositions, to which probabilities are assigned. Confusing probabilities of ''truth'' with probabilities of ''provability'' may lead to counterintuitive results in reasoning tasks such as (1) representing incomplete knowledge, (2) belief-updating and (3) evidence pooling. 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.

Kłopotek and Wierzchoń<ref name="KW-98">M. A. Kłopotek, S. T. Wierzchoń': "[https://link.springer.com/chapter/10.1007/3-540-69115-4_47 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.</ref> 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. In another interpretation M. A. Kłopotek and S. T. Wierzchoń<ref name="KW-02">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
</ref> propose to view this theory as describing destructive material processing (under loss of properties), e.g. like in some semiconductor production processes. Under both interpretations reasoning in DST gives correct results, contrary to the earlier probabilistic interpretations, criticized by Pearl in the cited papers and by other researchers.

Jøsang proved that Dempster's rule of combination actually is a method for fusing belief constraints.<ref name="Jos12"/> It only represents an approximate fusion operator in other situations, such as cumulative fusion of beliefs, but generally produces incorrect results in such situations. The confusion around the validity of Dempster's rule therefore originates in the failure of correctly interpreting the nature of situations to be modeled. Dempster's rule of combination always produces correct and intuitive results in situation of fusing belief constraints from different sources.

Yang et al. <ref name="xu2025"/> prove that Dempster’s rule is inherently probabilistic, extends Bayes’ rule and reduces to Bayes’ rule when precise probabilities are available and fully reliable, regardless of whether prior is uniform. They further extend Bayes’ and Dempster’s rules to the Evidential Reasoning (ER) rule<ref>{{cite journal |last1=Yang |first1=Jian-Bo |last2=Xu |first2=Dong-Ling |title=Maximum Likelihood Evidential Reasoning |journal=Artificial Intelligence |date=2025 |pages=1-61 |doi=10.1016/j.artint.2025.104289 |url=https://pdf.sciencedirectassets.com/271585/1-s2.0-S0004370224X00132/1-s2.0-S0004370225000086/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEN7%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJHMEUCIH%2F4jyG7XhsHaa4reRvLipNDWMk32HMumIUZEBZ3bED%2BAiEAy3olW3aFISsJrrSux8J2ZVcANqaBDiqmPHoVl6LX9%2FwqvAUIp%2F%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARAFGgwwNTkwMDM1NDY4NjUiDGY%2BdATOPFf7HMI1GSqQBWtQydJqJ4XAYibzASNG%2BqHVsb6gooQ4jRKbbZTPLaoZM80uf7tqmUbFYY5ZAhkHB4g1AIETkjzllPRAi%2FnFb7NCk9YXKPLLdPl6jD69406x0fzX070Va5dcRD7k7b3SyqlbHpUgJ4Ew0dYzCSOWxC0uRrW6djIRRi07ZKeKHZOdQrxrfrgw0%2FGL6xJDe%2Fae08LV0LPm2a7Awp8e1kBnND3gz4t87l5UUzIywF%2BclTWoKUPo0Lt2uoJ0q8OzLhfAGZvHD3o0h93DV3YOkymQuKq0RT%2FYTOdOg5vcn8n4Nbrrh2HOkw8Vala5mG4foomzosRziGk1gDcaOmWciV8RTsu65ZOrjv%2FFrxBoJO4AIas1FKfaC6muY60HNxGklg8cHwGGNAPisyxz%2BYJITt%2BAzLdTsaBIVALnbHAtRXlEDUEFnx5pUr0hx63kCIjVhkwKhxyqGlM9SXp8pVCeuZAqFtx3vqd7QTmsKd%2Bc1Vl%2BdMaw%2FPxcp9VkGEBj40LgSZXuQerhTBPz%2FvT%2B8yMlmoH8mrxMcQdEJU8EilNek3sLpw1bDGty0nm4vDz1gUmFpz2TbJgRrpxjuTHFVMvzS35Hbmvu70xGdGSHnL0dEC2BRot%2BSHMJXZqdacpHSU5zm7cg0E2NeXaOiMSw60H%2FEg9fVmQy8hwTBY7QHPrKSGtg%2F23VD2RUsxR%2FkorTVCw8gtG4XlHaa9LslBnmluoYZcVsi8NRQ3RWosuNAP0Nb3tFQP3faGclV3upFKLgc7za6KCtX5hSSC8qfyUilbPxnqt0NZGpPEtb2RxvOGGSES0zemPInd5wK97Hyy1cyT4vvZuLqmAOF1DBEPcswXlSlyvKTtAV%2FAjjORiqSumQ4SvAYfWrMKDi5sEGOrEB1744exchj9qAJrb9u7yii7Bgx24HjHkl4vPJEVk8MTGpgi7HAYXTPMKHNVZS%2Fisoum3IJk9UXt4dH78foiOEb%2Fowfmuyyfmlg04nJ9tTnE4XZ%2Fb0sZ7I8zeJoksU%2Br6r7qmeQPH%2B5Pu5dDtcXBoOvw7EFN4KIatu1KuOnKSdyjwHA8R2bqnPR7qhx2P6pPzLlf%2Bja0RX9fbu3bjatv5Y5yzEU%2BGPe0KSFri410QDq7qn&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20250530T143733Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTYS6U2IEUN%2F20250530%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=69baa478f42a8479897ddaabd07230b5a3950851a4bf03c4763c8f1d9ce0d64e&hash=32bc7cbcc5c4aef4e9d3d70abd2868e2790dfea48de0c5810028898802154d8a&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=S0004370225000086&tid=spdf-2e746f64-bbb5-4709-83c1-d8456697b09d&sid=c68826f2359cc54d48383f1885a787824bdcgxrqb&type=client&tsoh=d3d3LnNjaWVuY2VkaXJlY3QuY29t&rh=d3d3LnNjaWVuY2VkaXJlY3QuY29t&ua=070558515700075f56&rr=947ef85daea13477&cc=gb}}</ref> which is probabilistic when probabilities are not precise or fully reliable. They address some criticisms of the behaviour of Dempster’s rule from a probabilistic perspective and explain the rationality of the behaviour.