- Upper and lower probabilities
**Upper and lower probabilities**are representations of imprecise probability. Whereasprobability theory uses a single number, theprobability , 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, 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 asDempster-Shafer theory (see also Choquet53).More precisely, in these authors one considers in apower set , $P(S),!$, a "mass" function $m\; :\; P(S)\; ightarrow\; R$ 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::$operatorname\{bel\}(A)\; =\; sum\_\{B\; mid\; B\; subseteq\; A\}\; m(B),,,,;,,,,operatorname\{pl\}(A)\; =\; sum\_\{B\; mid\; B\; cap\; A\; e\; varnothing\}\; m(B)$

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 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.

**References*** 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.

**ee also***

Possibility theory

*Probability theory

*Fuzzy measure theory

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