History
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, who aimed to reconcile the theories of logic (which can express complete ignorance) and probability. In the 1920s, in A Treatise on Probability, Keynes formulated and applied an explicit interval estimate approach to probability.
Since the 1990s, the theory has gathered strong momentum, initiated by comprehensive foundations put forward by Walley, who coined the term imprecise probability, by Kuznetsov, and by Weichselberger, who uses 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.
Usually assumed consistency conditions relate imprecise probability assignments to non-empty closed convex sets of probability distributions. Therefore, as a welcome by-product, the theory also provides a formal framework for models used in robust statistics 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.
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