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Concept
Imprecise probability
Summary
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:
People have a limited ability to determine their own subjective probabilities and might find that they can only provide an interval.
As an interval is compatible with a range of opinions, the analysis ought to be more convincing to a range of different people.
Uncertainty is traditionally modelled by a probability distribution, as developed 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 common generalization is to replace a single probability specification with an interval specification. Lower and upper probabilities, denoted by and , or more generally, lower and upper expectations (previsions), aim to fill this gap.
A lower probability function is superadditive but not necessarily additive, whereas an upper probability is subadditive.
To get a general understanding of the theory, consider:
the special case with for all events is equivalent to a precise probability
and for all non-trivial events represents no constraint at all on the specification of
We then have a flexible continuum of more or less precise models in between.
Official source
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