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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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].