Second-order probabilities and belief functions

A second-order probability Q(P) may be understood as the probability that the true probability of something has the value P. True may be interpreted as the value that would be assigned if certain information were available, including information from reflection, calculation, other people, or ordinary evidence. A rule for combining evidence from two independent sources may be derived, if each source i provides a function Q_i(P). Belief functions of the sort proposed by Shafer (1976) also provide a formula for combining independent evidence, Dempster's rule, and a way of representing ignorance of the sort that makes us unsure about the value of P. Dempster's rule is shown to be at best a special case of the rule derived in connection with second-order probabilities. Belief functions thus represent a restriction of a full Bayesian analysis.