Euler method

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,Template:Cn 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 $m:P(S)\to R$ 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:

$\mathrm{bel}(A)=\sum _{B\mid B\subseteq A}m(B);\mathrm{pl}(A)=\sum _{B\mid B\cap A\ne \varnothing }m(B)$

In the case where $S$ is infinite there can be $\mathrm{bel}$ 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

$\mathrm{e}\mathrm{n}{\mathrm{v}}_{\mathrm{1}}(A)=\underset{p\in C}{inf}p(A);\mathrm{e}\mathrm{n}{\mathrm{v}}_{\mathrm{2}}(A)=\underset{p\in C}{sup}p(A)$

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

• Possibility theory
• Fuzzy measure theory
• Interval finite element

References

• 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.