- Upper and lower probabilities
Upper and lower probabilities are representations of imprecise probability. Whereas
probability theoryuses a single number, the probability, to describe how likely an event is to occur, this method uses two numbers: the upper probability of the event and the lower probability of the event.
frequentist statisticsdisallows metaprobabilities, frequentists have had to propose new solutions. Cedric Smithand Arthur Dempster each developed a theory of upper and lower probabilities. Glenn Shaferdeveloped Dempster's theory further, and it is now known as Dempster-Shafer theory(see also Choquet53).More precisely, in these authors one considers in a power set, , a "mass" function satisfying the conditions
In turn, a mass is associated with two non-additive continuous measures called belief and plausibility defined as follows:
A different notion of upper and lower probabilities is obtained by the "lower and upper envelopes" obtained from a class "C" of probability distributions by setting :
The upper and lower probabilities are also related with probabilistic logic (see Gerla94).
Observe also that a necessity measure can be seen as a lower probability and a possibility measure can be seen as an upper probability.
* G. Gerla, Inferences in Probability Logic, "Artificial Intelligence" 70(1–2):33–52, 1994.
* J. Y. Halpern and R. Fagin, Two views of belief: Belief as generalized probability and belief as evidence. "Artificial Intelligence", 54:275-317, 1992.
* P. J. Huber, "Robust Statistics". Wiley, New York, 1980.
* Saffiotti, A., A Belief-Function Logic, in "Procs of the 10h AAAI Conference", San Jose, CA 642-647, 1992.
* Choquet, G., Theory of Capacities, "Annales de l'Institut Fourier" 5, 131-295, 1953.
* Shafer, G., "A Mathematical Theory of Evidence", (Princeton University Press, Princeton), 1976.
* P. Walley and T. L. Fine, Towards a frequentist theory of upper and lower probability. "Annals of Statistics", 10(3):741-761, 1982.
Fuzzy measure theory
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