Higher order probabilities and intervals

AbstractMany researchers have felt uncomfortable with the precision of degrees of belief that seems to be demanded by the subjective Bayesian treatment of uncertainty. Various responses have been suggested. The most common one has been to incorporate higher order probabilities in systems that reason in beliefs. These probabilities concern statements of first-order probability. Thus a first-order probability (e.g., the probability of heads on the next toss of this coin is 12) is the subject of a second-order probability; for example, the probability is .9 that the probability of heads on the next toss of this coin is 12. This approach is explored and is found to be epistemologically wanting, although there are important intuitions about beliefs that are captured by it. Furthermore, this approach may, in some circumstances, be computationally attractive. We also briefly explore a number of other approaches, including taking probabilities to be intervals and construing probability values as fuzzy sets