Dempster–Shafer theory

As in Dempster–Shafer theory, a Bayesian belief function ${\displaystyle \operatorname {bel} :2^{X}\rightarrow [0,1]\,\!}$ has the properties ${\displaystyle \operatorname {bel} (\emptyset )=0}$ and ${\displaystyle \operatorname {bel} (X)=1}$. 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}


As an example of how the two approaches differ, a Bayesian could model the color of a car as a probability distribution over (red, green, blue), assigning one number to each color. Dempster–Shafer would assign numbers to each of (red, green, blue, (red or green), (red or blue), (green or blue), (red or green or blue)). These numbers do not have to be coherent; for example, Bel(red)+Bel(green) does not have to equal Bel(red or green).


Thus, Bayes' conditional probability can be considered as a special case of Dempster's rule of combination.^{[2]: p. 19f.} 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.^[20] For example, DS theory violates the requirements for Cox's theorem, which implies that it cannot be considered a coherent (contradiction-free) generalization of classical logic—specifically, DS theory violates the requirement that a statement be either true or false (but not both). 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.

Criticism[edit]

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." 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^[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. 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. 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.^[8] 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.