Upper and lower probabilities

From Wikipedia, the free encyclopedia
Jump to: navigation, search

Upper and lower probabilities are representations of imprecise probability. Whereas probability theory uses 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.

Because frequentist statistics disallows metaprobabilities,[citation needed] frequentists have had to propose new solutions. Cedric Smith and Arthur Dempster each developed a theory of upper and lower probabilities. Glenn Shafer developed Dempster's theory further, and it is now known as Dempster–Shafer theory: see also Choquet(1953). More precisely, in the work of these authors one considers in a power set, P(S)\,\!, a mass function m : P(S)\rightarrow R satisfying the conditions

m(\varnothing) = 0 \,\,\,\,\,\,\! ; \,\,\,\,\,\, \sum_{A \in P(X)} m(A) = 1. \,\!

In turn, a mass is associated with two non-additive continuous measures called belief and plausibility defined as follows:

\operatorname{bel}(A) = \sum_{B \mid B \subseteq A} m(B)\,\,\,\,;\,\,\,\,
\operatorname{pl}(A) = \sum_{B \mid B \cap A \ne \varnothing} m(B)

In the case where S is infinite there can be \operatorname{bel} such that there is no associated mass function. See p. 36 of Halpern (2003). Probability measures are a special case of belief functions in which the mass function assigns positive mass to singletons of the event space only.

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

\operatorname{env_1}(A) = \inf_{p \in C} p(A)\,\,\,\,;\,\,\,\,
\operatorname{env_2}(A) = \sup_{p \in C} p(A)

The upper and lower probabilities are also related with probabilistic logic: see Gerla (1994).

Observe also that a necessity measure can be seen as a lower probability and a possibility measure can be seen as an upper probability.

See also[edit]


  • G. Gerla, Inferences in Probability Logic, Artificial Intelligence 70(1–2):33–52, 1994.
  • J.Y. Halpern 2003 Reasoning about Uncertainty MIT Press
  • 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.