Imprecise Probabilities

A. P. Dempster (1968) and Peter Walley (1991), perhaps the most influential innovators in this domain, both proposed generalizations of Bayesian inference.

"For example, de Finetti assumes that for each event of interest, there is some betting rate that you regard as fair, in the sense that you are willing to accept either side of a bet on the event at that rate. This fair betting rate is your personal probability for the event. More generally, we take your lower probability to be the maximum rate at which you are prepared to bet on the event, and your upper probability to be the minimum rate at which you are prepared to bet against the event. It is not irrational for you to assess an upper probability that is strictly greater than your lower probability. Indeed, you ought to do so when you have little information on which to base your assessments. In that case we say that your beliefs about the event are indeterminate, and that (for you) the event has imprecise probability."
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P(lower probability < P(H | data, background information) < upper probability | data, background information) = 1