# Relative Uncertainty: Measuring
Uncertainty in Discrete and Nondiscrete Domains

L.M. Rocha

Computer Research Group, MS P990

Los Alamos National Laboratory

Los Alamos, NM 87545

#### In: *Proceedings of the NAFIPS'96, UC Berkeley*. M. Smith, M. Lee, J. Keller, J.
Yen (Eds.).
IEEE Press, pp. 551-555.

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Abstract
** A general framework is proposed for the development of uncertainty measures in finite and
infinite domains in the context
of evidence theory. The proposed measures account uncertainty in the form of information ratios,
which relate an uncertainty
situation to the maximum uncertainty-based information present in its domain. They are shown
to be ideal for the
measurement of uncertainty of interval based structures such as evidence sets. **

INDEX TERMS: Uncertainty, Uncertainty in nondiscrete Domains, Uncertain Reasoning,
Evidence Theory.

** 1. Introduction**

**
**

To develop an extended theory of approximate reasoning whose mathematical
structures can capture all forms of uncertainty
recognized in generalized information theory, we need set structures with membership functions
that more than fuzzy are also
nonspecific and conflictive. Fuzzy sets provide a graded (fuzzy) membership. Interval valued
fuzzy sets (IVFS) add nonspecificity
to the membership degree. At previous NAFIPS meetings I have introduced an interval-based set
structure, named *evidence set*,
whose membership degrees are defined by the belief measures of the Dempster-Shafer theory of
evidence
[Rocha 1994,
1995a].
Evidence sets capture fuzziness, nonspecificity, and conflict in their membership degrees. I
believe they allow us to define a very
robust form of uncertain reasoning, able to deal more effectively with the simulation of linguistic
and cognitive categories [Rocha,
1994,
1995a, 1995b,
1996], as well as the more practical problems of reliable
computation
[Rocha et al, 1996]. In this paper, I develop
measures of nonspecificity and conflict for discrete (countable) and nondiscrete (uncountable,
infinite) domains. It is defended that
a general framework for measuring uncertainty should be built around the notion of *relative
uncertainty*, that is, when the uncertainty
content of some situation is related to the information content of the universal set on which the
situation is defined. The ideas here
summarized are developed in detail in Rocha[1996].