Upper and lower probabilities are representations of

imprecise probability Imprecise probability generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify. There ...

. Whereas

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

uses a single number, the

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...

, 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 Frequentist inference is a type of statistical inference based in frequentist probability, which treats “probability” in equivalent terms to “frequency” and draws conclusions from sample-data by means of emphasizing the frequency or pro ...

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 The theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty, with understood connections to other frameworks such as probability, possibility and ...

or Choquet (1953). More precisely, in the work of these authors one considers in a

power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...

P(S)\,\!

, a ''mass'' function

m : P(S)\rightarrow R

satisfying the conditions :

m(\varnothing) = 0 \,\,\,\,\,\,\! ; \,\,\,\,\,\, m(A) \ge 0 \,\,\,\,\,\,\! ; \,\,\,\,\,\, \sum_ m(A) = 1. \,\!

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

\operatorname(A) = \sum_ m(B)\,\,\,\,;\,\,\,\,
\operatorname(A) = \sum_ m(B)

In the case where

S

is infinite there can be

\operatorname

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(A) = \inf_ p(A)\,\,\,\,;\,\,\,\,
\operatorname(A) = \sup_ p(A)

The upper and lower probabilities are also related with

probabilistic logic Probabilistic logic (also probability logic and probabilistic reasoning) involves the use of probability and logic to deal with uncertain situations. Probabilistic logic extends traditional logic truth tables with probabilistic expressions. A diffic ...

: 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

* * * * * * * * {{cite journal , first=P. , last=Walley , first2=T. L. , last2=Fine , title=Towards a frequentist theory of upper and lower probability , journal=

Annals of Statistics The ''Annals of Statistics'' is a peer-reviewed statistics journal published by the Institute of Mathematical Statistics. It was started in 1973 as a continuation in part of the ''Annals of Mathematical Statistics (1930)'', which was split into th ...

, volume=10 , issue=3 , pages=741–761 , year=1982 , jstor=2240901 , doi=10.1214/aos/1176345868, doi-access=free Exotic probabilities Probability bounds analysis Dempster–Shafer theory

See also

References