Нақты емес ықтималдықтар теориясы: ақпараттың жетіспеуі мен белгісіздікті модельдеу

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. Thereby, the theory aims to represent the available knowledge more accurately. Imprecision is useful for dealing with expert elicitation, because:
People have a limited ability to determine their own subjective probabilities and might find that they can only provide an interval. As an interval is compatible with a range of opinions, the analysis ought to be more convincing to a range of different people.

Uncertainty is traditionally modelled by a probability distribution, as developed by Kolmogorov, Laplace, de Finetti, Ramsey, Cox, Lindley, and many others. However, this has not been unanimously accepted by scientists, statisticians, and probabilists: it has been argued that some modification or broadening of probability theory is required, because one may not always be able to provide a probability for every event, particularly when only little information or data is available—an early example of such criticism is Boole's critique of Laplace's work—, or when we wish to model probabilities that a group agrees with, rather than those of a single individual. Perhaps the most common generalization is to replace a single probability specification with an interval specification. Lower and upper probabilities, denoted by and , or more generally, lower and upper expectations (previsions), aim to fill this gap. A lower probability function is superadditive but not necessarily additive, whereas an upper probability is subadditive. To get a general understanding of the theory, consider:
the special case with for all events is equivalent to a precise probability
and for all non trivial events represents no constraint at all on the specification of
We then have a flexible continuum of more or less precise models in between. Some approaches, summarized under the name nonadditive probabilities, directly use one of these set functions, assuming the other one to be naturally defined such that , with the complement of Other related concepts understand the corresponding intervals for all events as the basic entity.

The idea to use imprecise probability has a long history. The first formal treatment dates back at least to the middle of the nineteenth century, by George Boole, formulated and applied an explicit interval estimate approach to probability. Work on imprecise probability models proceeded fitfully throughout the 20th century, with important contributions by Bernard Koopman, C. A. B. Smith, I. J. Good, Arthur Dempster, Glenn Shafer, Peter M. Williams, Henry Kyburg, Isaac Levi, and Teddy Seidenfeld. 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 and by Weichselberger, and non parametric statistics. Included are also concepts based on Choquet integration, and so called two monotone and totally monotone capacities, which have become very popular in artificial intelligence under the name (Dempster–Shafer) belief functions. Moreover, there is a strong connection to Shafer and Vovk's notion of game theoretic probability.

A unification of many of the above mentioned imprecise probability theories was proposed by Walley, For fuzzy measures, there is the work of Ronald R. Yager. For convex sets of distributions, Levi's works are instructive. Another approach asks whether the threshold controlling the boldness of the interval matters more to a decision than simply taking the average or using a Hurwicz decision rule. Other approaches appear in the literature.