Dempster-Shafer theory
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The Dempster-Shafer theory is a mathematical theory of evidence that was introduced in the late 1970s by Glenn Shafer.
It is a way of representing epistemic plausibilities. It developed from a sequence of works of Arthur Dempster, who was Shafer's advisor.
In this formalism the best representation of chance is 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 are combined using Dempster's rule of combination. 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 map unknown priors to random variables (i.e. assigning 0.5 to binary values).
Also see possibility theory, probability theory.