Imprecise probability

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: 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: Uncertainty is traditionally modelled by a probability distribution, as argued 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 straightforward generalization is to replace a single probability specification with an interval specification. Lower and upper probabilities, denoted by P _ ( A ) {displaystyle {underline {P}}(A)} and P ¯ ( A ) {displaystyle {overline {P}}(A)} , or more generally, lower and upper expectations (previsions), aim to fill this gap: