The large sample behaviour is examined for upper and lower probabilities generated by precise priors (on a finite set) and imprecise likelihoods, for equally independently distributed variables taking values in a finite set. It can easily happen that the posterior upper probability of $\theta$ will tend to $1$ in situations where the relative frequencies are tending to a measure far removed from the set of conditional measures associated with $\theta$. A number of different modifications to upper and lower probability are made in such a way that the posterior probabilities will, with probability one, become or tend to $0$ in such situations; these all involve rejecting measures which are extremely implausible given the data, and hence considering more restricted sets of likelihoods.
Keywords. Upper and lower probability, imprecise likelihoods, large sample behaviour, Bernouilli trials
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School of Computing and Mathematical Sciences
Oxford Brookes University