Formal Model of Uncertainty for Possibilistic Rules
  Given a universe of discourse X-a domain of possible outcomes-an experiment
may consist of selecting one of its elements, subject to the operation of
chance, or of observing the elements, subject to imprecision. A priori
uncertainty about the actual result of the experiment may be quantified,
representing either the likelihood of the choice of :r_X or the degree to which
any such X would be suitable as a description of the outcome. The former case
corresponds to a probability distribution, while the latter gives a possibility
assignment on X. The study of such assignments and their properties falls
within the purview of possibility theory [DP88, Y80, Z783. It, like probability
theory, assigns values between 0 and 1 to express likelihoods of outcomes.
Here, however, the similarity ends. Possibility theory uses the maximum and
minimum functions to combine uncertainties, whereas probability theory uses the
plus and times operations. This leads to very dissimilar theories in terms of
analytical framework, even though they share several semantic concepts. One of
the shared concepts consists of expressing quantitatively the uncertainty
associated with a given distribution. In probability theory its value
corresponds to the gain of information that would result from conducting an
experiment and ascertaining an actual result. This gain of information can
equally well be viewed as a decrease in uncertainty about the outcome of an
experiment. In this case the standard measure of information, and thus
uncertainty, is Shannon entropy [AD75, G77]. It enjoys several advantages-it is
characterized uniquely by a few, very natural properties, and it can be
conveniently used in decision processes. This application is based on the
principle of maximum entropy; it has become a popular method of relating
decisions to uncertainty. This paper demonstrates that an equally integrated
theory can be built on the foundation of possibility theory. We first show how
to define measures of in formation and uncertainty for possibility assignments.
Next we construct an information-based metric on the space of all possibility
distributions defined on a given domain. It allows us to capture the notion of
proximity in information content among the distributions. Lastly, we show that
all the above constructions can be carried out for continuous
distributions-possibility assignments on arbitrary measurable domains. We
consider this step very significant-finite domains of discourse are but
approximations of the real-life infinite domains. If possibility theory is to
represent real world situations, it must handle continuous distributions both
directly and through finite approximations. In the last section we discuss a
principle of maximum uncertainty for possibility distributions. We show how
such a principle could be formalized as an inference rule. We also suggest it
could be derived as a consequence of simple assumptions about combining
information. We would like to mention that possibility assignments can be
viewed as fuzzy sets and that every fuzzy set gives rise to an assignment of
possibilities. This correspondence has far reaching consequences in logic and
in control theory. Our treatment here is independent of any special
interpretation; in particular we speak of possibility distributions and
possibility measures, defining them as measurable mappings into the interval
[0, 1]. Our presentation is intended as a self-contained, albeit terse summary.
Topics discussed were selected with care, to demonstrate both the completeness
and a certain elegance of the theory. Proofs are not included; we only offer
illustrative examples.
