Imprecise Probability

Abstract

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 for-ward 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 gam-bles, 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 classical probability distributions. Therefore, as a welcome by-product, the theory also provides a formal framework for models used in frequentist robust statistics [6] and robust Bayesian approaches [31]. Included are also concepts based on so-called two-monotone [24] and totally monotone capacities, which have become very popular in artificial intelligence under the name (Dempster-Shafer) belief functions [17, 32]. Moreover, there is a strong connection [14] to Shafer and Vovk's notion of game-theoretic probability [20]. The term 'imprecise probability'—although an unfortunate misnomer as lower and upper prob-ability enable more accurate quantification of uncertainty than precise probability—appears to have been established over the last two decades, and actually brings together a variety of different theories. In applications, clear advantages over the established theory of precise probability have been demonstrated (see Section 2). This justifies the further development of imprecise probability, particularly towards building a complete methodological framework for applications in statistics, decision support, and related fields. Imprecise probability provides important new methods that promise greater flexibility for uncertainty quantification. Its advantages include the possibility to deal with conflicting evidence, to base inferences on weaker assumptions than needed for precise probabilistic methods, and to allow for simpler and more realistic elicitation of subjective informa-tion, as imprecise probability does not require experts to represent their judgements through a full probability distribution, which often does not reflect their beliefs appropriately. The Society for Imprecise Probability: Theories and Applications (www.sipta.org) organises conferences, workshops and summer schools, and provides useful introductory information sources and contacts through its web-page.