
[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 counterintuitive 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 counterintuitive 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 nonadditive 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 nonzero 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 counterintuitive results in case of low conflict[edit] The following example shows where Dempster’s rule produces a counterintuitive 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 welldefined belief and plausibility functions but no welldefined 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
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