Modified upper and lower probabilities based on imprecise likelihoods

The large sample behaviour is examined for upper and lower probabilities generated by precise priors on a nite set and imprecise likelihoods for equally in dependently distributed variables taking values in a nite set It can easily happen that the posterior up per probability of will tend to in situations where the relative frequencies are tending to a measure far removed from the set of conditional measures associ ated with A number of di erent modi cations to upper and lower probability are made in such a way that the posterior probabilities will with probability one become or tend to in such situations these all involve rejecting measures which are extremely im plausible given the data and hence considering more restricted sets of likelihoods