New semantics for numerical values given to possibility measures are provided. For epistemic possibilities, the new approach is based on the semantics of the Transferable Belief Model itself based on betting odds interpreted in a less drastic way than what subjective probabilities presupposes. It is shown that the least informative among the belief structures that are compatible with prescribed betting rates is nested, i.e. corresponds to a possibility measure. It is also proved that the idempotent conjunctive combination of two possibility measures corresponds to the hyper-cautious conjunctive combination of the belief functions induced by the possibility measures. This view differs from the subjective semantics first proposed by Giles and relying on upper and lower probability induced by non-exchangeable bets. For objective possibility degrees, the semantics is based on the most informative possibilistic approximation of a probability measure derived from a histogram. The motivation for this semantics is its capability to extend a well-known kind of confidence intervals around the mode of a distribution to a fuzzy confidence interval. We show how the idempotent disjunctive combination of possibility functions is related to the convex mixture of probability distributions.
Keywords. Quantitative possibility, belief functions,
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