A Notion of Comparative Probabilistic Entropy Based on the Possibilistic Specificity Ordering

In this paper, we reconsider the problem of deciding whether one probability distribution is more informative (in the sense of representing a less indeterminate situation) than another one. Instead of using well-established information measures such as the Shannon entropy, however, we take up the idea of comparing probability distributions in a qualitative way. More specifically, we focus on a natural partial ordering induced by what is called the “peakedness” of a distribution. Moreover, there is a close connection between this ordering between probability distributions and the standard specificity ordering on possibility distributions that can be constructed from them. The main result of the paper is a proof showing that possibilistic specificity is consistent with probabilistic entropy in the sense that the (total) ordering defined by the latter refines the (partial) ordering defined by the former.