Imprecise Probability

1 Overview Quantification of uncertainty is mostly done by the use of precise probabilities: for each event A, a single (classical, precise) probability P (A) is used, typically satisfying Kolmogorov's axioms [4]. Whilst this has been very successful in many applications, it has long been recognized to have severe limitations. Classical probability requires a very high level of precision and consistency of information, and thus it is often too restrictive to cope carefully with the multi-dimensional nature of uncertainty. Perhaps the most straightforward restriction is that the quality of underlying knowledge cannot be adequately represented using a single probability measure. An increasingly popular and successful generalization is available through the use of lower and upper probabilities, denoted by P (A) and P (A) respectively, with 0 ≤ P (A) ≤ P (A) ≤ 1, or, more generally, by lower and upper expectations (previsions) [33, 36, 41]. The special case with P (A) = P (A) for all events A provides precise probability, whilst P (A) = 0 and P (A) = 1 represents complete lack of knowledge about A, with a flexible continuum in between. Some approaches, summarized under the name nonadditive probabilities [18], directly use one of these set-functions, assuming the other one to be naturally defined such that P (A c) = 1 − P (A) , with A c the complement of A. Other related concepts understand the corresponding intervals [P (A), P (A)] for all events as the basic entity [38, 39]. Informally, P (A) can be interpreted as reflecting the evidence certainly in favour of event A, and 1 − P (A) as reflecting the evidence against A hence in favour of A c. The idea to use imprecise probability, and related concepts, is quite natural and has a long history (see [22] for an extensive historical overview of nonadditive probabilities), and the first formal treatment dates back at least to the middle of the nineteenth century [9]. In the last twenty years the theory has gathered strong momentum, initiated by comprehensive foundations put forward by Walley [36] (see [30] for a recent survey), who coined the term imprecise probability, by Kuznetsov [27], and by Weichselberger [38, 39], 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 …