Imprecision is useful for dealing with expert elicitation, because: Uncertainty is traditionally modelled by a probability distribution, as developed by Kolmogorov,[1] Laplace, de Finetti,[2] Ramsey, Cox, Lindley, and many others.
, or more generally, lower and upper expectations (previsions),[4][5][6][7] aim to fill this gap.
Some approaches, summarized under the name nonadditive probabilities,[8] directly use one of these set functions, assuming the other one to be naturally defined such that
The first formal treatment dates back at least to the middle of the nineteenth century, by George Boole,[3] who aimed to reconcile the theories of logic and probability.
Work on imprecise probability models proceeded fitfully throughout the 20th century, with important contributions by Bernard Koopman, C.A.B.
Good, Arthur Dempster, Glenn Shafer, Peter M. Williams, Henry Kyburg, Isaac Levi, and Teddy Seidenfeld.
[12] At the start of the 1990s, the field started to gather some momentum, with the publication of Peter Walley's book Statistical Reasoning with Imprecise Probabilities[7] (which is also where the term "imprecise probability" originates).
The 1990s also saw important works by Kuznetsov,[13] and by Weichselberger,[9][10] who both use the term interval probability.
Walley's theory extends the traditional subjective probability theory via buying and selling prices for gambles, whereas Weichselberger's approach generalizes Kolmogorov's axioms without imposing an interpretation.
[15] Included are also concepts based on Choquet integration,[16] and so-called two-monotone and totally monotone capacities,[17] which have become very popular in artificial intelligence under the name (Dempster–Shafer) belief functions.
In simple terms, a decision maker's lower prevision is the highest price at which the decision maker is sure he or she would buy a gamble, and the upper prevision is the lowest price at which the decision maker is sure he or she would buy the opposite of the gamble (which is equivalent to selling the original gamble).
If the upper and lower previsions are equal, then they jointly represent the decision maker's fair price for the gamble, the price at which the decision maker is willing to take either side of the gamble.
One issue with imprecise probabilities is that there is often an independent degree of caution or boldness inherent in the use of one interval, rather than a wider or narrower one.
It does raise concerns about inappropriate claims of precision at endpoints, as well as for point values.
A more practical issue is what kind of decision theory can make use of imprecise probabilities.