Dempster–Shafer theory

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.

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

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.

[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.

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.

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.

A hypothesis is represented by a subset of this frame of discernment, like "(Ming dynasty, China)", or "(19th century, Germany)".[2]: p.35f.

Shafer's framework allows for belief about such propositions to be represented as intervals, bounded by two values, belief (or support) and plausibility: In a first step, subjective probabilities (masses) are assigned to all subsets of the frame; usually, only a restricted number of sets will have non-zero mass (focal elements).[2]: 39f.

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.

Finally, the all-encompassing "Either" hypothesis (which simply acknowledges there is a cat in the box) picks up the slack so that the sum of the masses is 1.

[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.

However, any information contained in the missing priors and conditionals is not used in Dempster's rule of combination unless it can be obtained indirectly—and arguably is then available for calculation using Bayes equations.

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.

This sort of situation, and whether there is a real distinction between risk and ignorance, has been extensively discussed by statisticians and economists.

See, for example, the contrasting views of Daniel Ellsberg, Howard Raiffa, Kenneth Arrow and Frank Knight.

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

[11] The problem we now face is how to combine two independent sets of probability mass assignments in specific situations.

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

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.

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

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.

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.

to a (discrete) probability distribution, i.e. only singleton subsets of the frame of discernment are allowed to be focal elements of the approximated version

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."

Jøsang proved that Dempster's rule of combination actually is a method for fusing belief constraints.

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.