Dempster-Shafer theory
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The Dempster-Shafer theory is a mathematical theory of evidence [SH76] based on belief functions and plausible reasoning, which is used to combine separate pieces of information (evidence) to calculate the probability of an event. The theory was developed by Arthur Dempster and Glenn Shafer.
It is a way of representing epistemic plausibilities but it can yield answers which contradict those arrived at using probability theory.
It is often used as a method of sensor fusion.
In this formalism a degree of belief is represented as a belief function rather than a Bayesian probability distribution. 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.
Shafer's framework allows for belief about propositions to be represented as intervals, taking on two values, support and plausibility, with
- support ≤ plausibility.
Support for a hypothesis indicates the probability mass given to sets of events that are enclosed by it. Or in other words, it gives the amount of belief that directly supports a given hypothesis. Plausibility is 1 minus the masses given to sets of events whose intersection with the hypothesis results in an empty set. Again, in other words, it gives an upper bound on the belief that the hypothesis could possibly happen, i.e. it "could possibly happen" up to that value, because there was not any evidence that would contradict that hypothesis.
For example, suppose we have a support of 0.5 and a plausibility of 0.8 for a proposition, say "the cat in the box is dead." This means that we have evidence that allows us to state strongly that the proposition is true with probability 0.5. However, the evidence contrary to that hypothesis (i.e. "the cat is alive") only has probability 0.2. This means that it is possible that the cat is alive, up to 0.8, since the remaining probability mass of 0.3 is essentially "indeterminate," meaning that the cat could either be dead or alive.
Essentially this interval represents the level of uncertainty based on the evidence in your system.
Beliefs corresponding to independent pieces of information are combined using Dempster's rule of combination which is a generalisation of the special case of Bayes' theorem where events are independent. (There is as yet no method of combining non-independent pieces of information.) Note that the probability masses from propositions that contradict each other can also be used to obtain a measure of how much conflict there is in a system. This measure has been used before as a criterion for clustering multiple pieces of seemingly conflicting evidence around competing hypotheses.
In addition, one of the advantages of the Dempster-Shafer framework is that priors and conditionals need not be specified, unlike Bayesian methods which often use symmetry arguments to assign prior probabilities to random variables (e.g. assigning 0.5 to binary values in which no information is available about which is more likely).
Consider two possible gambles.
Gamble A is that we bet on a head turning up when we toss a coin that is known to be fair.
Now consider Gamble B, in which we bet on the outcome of a fight between the world's greatest boxer and the world's greatest wrestler. (Assume we are fairly ignorant about martial arts and would have great difficulty making a choice of who to bet on.)
Many people would feel more unsure about taking Gamble B in which the probabilities are unknown, rather than Gamble A, in which the probabilities are easily seen to be one half for each outcome. In the case of Gamble B a Bayesian would still assign one-half to each outcome, provided that no information was available which makes one outcome more likely than the other.
Dempster-Shafer theory allows one to specify a degree of ignorance in this situation instead of being forced to supply prior probabilities which 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 Ellsberg and Howard Raiffa.
Dezert-Smarandanche theory is a generalised form of Dempster-Shafer theory.
References
- [SH76] Shafer, Glenn, A Mathematical Theory of Evidence, Princeton University Press 1976
See also
Dezert-Smaradanche theory: http://www.gallup.unm.edu/~smarandache/DSmT.htm
