3

Imprecise Probabilities for Treating Uncertainty

As described in Chapter 1, it has been argued that in situations of poor knowledge, representations of uncertainty based on lower and upper probabilities are more appropriate than precise probabilities. This chapter presents imprecise probabilityes for treating uncertainty, largely taken from Aven (2011c).

The first theoretical foundation for imprecise probability was laid down by Boole (1854). More recently, Peter M. Williams developed a mathematical foundation for imprecise probabilities based on de Finetti's betting interpretation of probability (de Finetti, 1974). This foundation was further developed independently by Kuznetsov (1991) and Walley (1991).

The term “imprecise probability” brings together a variety of different theories (Coolen, Troffaes, and Augustin, 2010). It is used to refer to a generalization of probability theory based on the representation of the uncertainty about an event A through the use of a lower probability and an upper probability , where . The imprecision in the representation of event is defined as (Coolen, 2004)

(3.1)

The special case with (and hence ) for all events leads to conventional probability theory, whereas the case with and (and hence ) represents complete lack of knowledge, with a flexible continuum in between (Coolen, Troffaes, and Augustin, 2010).

Example 3.1

The uncertainty related to the occurrence of a catastrophic (CA) scenario given a failure event of the cooling system could be described by assigning, say, and . The associated imprecision would then be

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A range of generalizations of classical probability theory have been presented, in terms of axiom systems and further concepts and theorems (Coolen, 2004). Many researchers agree that the most complete framework of lower and upper probabilities is offered by Walley's (1991) theory of imprecise probability and the closely related theory of interval probability by Weichselberger (2000), where the former emphasizes developments via coherent subjective betting behavior (see the paragraph below) and the latter is formulated more as a generalization of Kolmogorov's axioms of classical probability (Coolen, 2004). Axioms and further concepts are far more complex for imprecise probability than for classical precise probability; for example, the concept of conditional probability does not have a unique generalization to imprecise probability (Coolen, 2004).

Following de Finetti's betting interpretation, Walley (1991) has proposed to interpret lower probability as the maximum price for which one would be willing to buy a bet which pays 1 if A occurs and 0 if not, and the upper probability as the minimum price for which one would be willing to sell the same bet.

Alternatively, lower and upper probabilities may be interpreted with reference to an uncertainty standard, as introduced in Section 2.3. Such an interpretation is indicated by Lindley (2006, p. 36). Consider the assignment of a subjective probability and suppose that the assigner states that his or her degree of belief is reflected by a probability greater than a comparable urn chance of 0.10 and less than an urn chance of 0.5. The analyst is not willing to make a more precise assignment. Then the interval can be considered an imprecision interval for the probability .

Of course, even if the assessor assigns a single probability , this can be understood as an imprecise probability interval equal to say (since a number in this interval is equal to 0.3 when only one digit is displayed), interpreted analogously as the interval above. Imprecision is always an issue in a practical context. This type of imprecision is typically seen to be a result of measurement problems (cf. the third bullet point in the list presented at the beginning of Part II). Lindley (2006) argues that the use of lower and upper probabilities confuses the concept of interpretation with the measurement procedures (in his terminology, the “concept” of measurement with the “practice” of measurement). The reference to the urn standard provides a norm, and measurement problems may make the assessor unable to behave according to it.

Other researchers and analysts are more positive with respect to the need for using imprecise/interval probabilities in practice; see the discussions in, for example, Dubois (2010), Aven and Zio (2011), Walley (1991), and Ferson and Ginzburg (1996). Imprecision intervals are required to reflect phenomena involved in uncertainty assessments, for example, experts that are not willing to represent their knowledge and uncertainty more precisely than by using probability intervals.

Imprecise probabilities can also be linked to the relative frequency interpretation (Coolen and Utkin, 2007). In the simplest case the “true” frequentist probability is in the specified interval with certainty (i.e., with subjective probability 1). More generally, and in line with a subjective interpretation of imprecision intervals, a two-level uncertainty characterization can be formulated (see, e.g., Kozine and Utkin, 2002). The interval is an imprecision interval for the subjective probability , where and are fixed quantities. In the special case that (say), we have a credibility interval (cf. Section 2.4) for p specifying that with a subjective probability the true value of is in the interval ; cf. Example 2.4 in Chapter 2.4.