A semantics for possibility theory based on likelihoods
type: Item
instance of: scholarly article
title: A semantics for possibility theory based on likelihoods
zbMATH Open document ID: 0884.03017
author: Serafin Moral
author: Henri Prade
author: Didier Dubois
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-03-17T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03B48
Mathematics Subject Classification ID: 94D05
Mathematics Subject Classification ID: 60A05
Mathematics Subject Classification ID: 03B52
zbMATH DE Number: 1005661
zbMATH Keywords: possibility theory
zbMATH Keywords: likelihood
zbMATH Keywords: fuzzy sets
zbMATH Keywords: possibility distributions
zbMATH Keywords: upper probability envelopes
zbMATH Keywords: membership function
zbMATH Keywords: space of observations
zbMATH Keywords: hypothesis space
zbMATH Keywords: combination of possibility measures
zbMATH Keywords: complement
reviewed by: Henry E. jun. Kyburg
MaRDI profile type: Publication
OpenAlex ID: W2000850517
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DOI: 10.1006/JMAA.1997.5193
description: scientific article; zbMATH DE number 1005661