Upper and Lower Probabilities
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Upper and Lower Probabilities

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,[] 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 or Choquet (1953). More precisely, in the work of these authors one considers in a power set, ${\displaystyle P(S)\,\!}$, a mass function ${\displaystyle m:P(S)\rightarrow R}$ satisfying the conditions

${\displaystyle m(\varnothing )=0\,\,\,\,\,\,\!;\,\,\,\,\,\,\sum _{A\in P(X)}m(A)=1.\,\!}$

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

${\displaystyle \operatorname {bel} (A)=\sum _{B\mid B\subseteq A}m(B)\,\,\,\,;\,\,\,\,\operatorname {pl} (A)=\sum _{B\mid B\cap A\neq \varnothing }m(B)}$

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

${\displaystyle \operatorname {env_{1}} (A)=\inf _{p\in C}p(A)\,\,\,\,;\,\,\,\,\operatorname {env_{2}} (A)=\sup _{p\in C}p(A)}$

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.

## References

• Choquet, G. (1953). "Theory of Capacities". Annales de l'Institut Fourier. 5: 131-295. doi:10.5802/aif.53.
• Gerla, G. (1994). "Inferences in Probability Logic". Artificial Intelligence. 70 (1-2): 33-52. doi:10.1016/0004-3702(94)90102-3.
• Halpern, J. Y. (2003). Reasoning about Uncertainty. MIT Press. ISBN 978-0-262-08320-1.
• Halpern, J. Y.; Fagin, R. (1992). "Two views of belief: Belief as generalized probability and belief as evidence". Artificial Intelligence. 54 (3): 275-317. CiteSeerX 10.1.1.70.6130. doi:10.1016/0004-3702(92)90048-3.
• Huber, P. J. (1980). Robust Statistics. New York: Wiley. ISBN 978-0-471-41805-4.
• Saffiotti, A. (1992). "A Belief-Function Logic". Procs of the 10h AAAI Conference. San Jose, CA. pp. 642-647. ISBN 978-0-262-51063-9.
• Shafer, G. (1976). A Mathematical Theory of Evidence. Princeton: Princeton University Press. ISBN 978-0-691-08175-5.
• Walley, P.; Fine, T. L. (1982). "Towards a frequentist theory of upper and lower probability". Annals of Statistics. 10 (3): 741-761. doi:10.1214/aos/1176345868. JSTOR 2240901.

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