Upper and lower probabilities

{{Short description|Representations of imprecise probability}}

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,{{cn|date=May 2012}} 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, $P(S)\backslash ,\backslash !$, a mass function $m\; :\; P(S)\backslash rightarrow\; R$ satisfying the conditions

:$m(\backslash varnothing)\; =\; 0\; \backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash !\; ;\; \backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\; m(A)\; \backslash ge\; 0\; \backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash !\; ;\; \backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\; \backslash sum\_\{A\; \backslash in\; P(S)\}\; m(A)\; =\; 1.\; \backslash ,\backslash !$

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

:$\backslash operatorname\{bel\}(A)\; =\; \backslash sum\_\{B\; \backslash mid\; B\; \backslash subseteq\; A\}\; m(B)\backslash ,\backslash ,\backslash ,\backslash ,;\backslash ,\backslash ,\backslash ,\backslash ,$

\operatorname{pl}(A) = \sum_{B \mid B \cap A \ne \varnothing} m(B)

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

:$\backslash operatorname\{env\_1\}(A)\; =\; \backslash inf\_\{p\; \backslash in\; C\}\; p(A)\backslash ,\backslash ,\backslash ,\backslash ,;\backslash ,\backslash ,\backslash ,\backslash ,$

\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.