Interpretation of Imprecise Probabilities According To Walley
A unification of many of the above mentioned imprecise probability theories was proposed by Walley, although this is in no way the first attempt to formalize imprecise probabilities. In terms of probability interpretations, Walley’s formulation of imprecise probabilities is based on the subjective variant of the Bayesian interpretation of probability. Walley defines upper and lower probabilities as special cases of upper and lower previsions and the gambling framework advanced by Bruno de Finetti. In simple terms, a decision maker’s lower prevision is the highest price at which the decision maker is sure he or she would buy a gamble, and the upper prevision is the lowest price at which the decision maker is sure he or she would buy the opposite of the gamble (which is equivalent to selling the original gamble). If the upper and lower previsions are equal, then they jointly represent the decision maker’s fair price for the gamble, the price at which the decision maker is willing to take either side of the gamble. The existence of a fair price leads to precise probabilities.
The allowance for imprecision, or a gap between a decision maker's upper and lower previsions, is the primary difference between precise and imprecise probability theories. Interestingly, such gaps arise naturally in betting markets which happen to be financially illiquid due to asymmetric information.
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