A semantics for possibility theory based on likelihoods
The first section of this paper relates everything to everything: possibility measures are represented by possibility distributions, which in turn are represented as upper probability envelopes; given a mass function, as in Shafer's theory, this can be construed so as to give rise to a membership function in Zadeh's sense, and thus a possibility function; likelihoods can be used to obtain membership functions; and, as Cheeseman has observed, we can give a Bayesian twist to the membership function. The authors use convex sets of probabilities and conditional probabilities in a quite classical framework to provide a semantics for the possibility theory calculus. The basic semantical entity is a triple (\(\mathcal{O},\mathcal{P}\),\(O)\) where \(\mathcal O\) represents a space of observations, and \(\mathcal P\) = \(\mathcal{J} \times\)\( P_{1|2}\) is a convex set of joint probabilities. \(\mathcal J\) is the set of all possible probability measures on the hypothesis space \(U\) (it is assumed that such a set exists) and \(\mathcal P_{2|1}\) is a convex set of probabilities on \(\mathcal O\), given an element of \(U\). A possibility measure \(\Pi(A)\) is identified with \(\sup\{P(O|A):P \in \mathcal P\}\). The authors give some results on the combination of possibility measures when these measures are exclusive, or independent, and on the complement of a possibility measure. The main import of the paper is that the authors provide a semantical justification of the possibility measure calculus in terms of upper and lower probabilities.
scientific article; zbMATH DE number 1005661
scientific article