# 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,[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.