Relative Uncertainty and Evidence Sets: A Constructivist Framework

L.M. Rocha
Computer Research Group, MS P990
Los Alamos National Laboratory
Los Alamos, NM 87545

International Journal of General Systems Vol. 26 (1-2), pp. 35-61.

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Several measures of uncertainty, in its various forms of nonspecificity, conflict, and fuzziness, valid both in finite and infinite domains are investigated. It is argued that dimensionless measures, relating uncertainty situations to the information content of their respective universal sets, can capture uncertainty efficiently both in finite and infinite domains. These measures are also considered more intuitive. To establish them, a more general approach to uncertainty measures is developed. After this, the utilization of these measures is exemplified in the measurement of the uncertainty content of evidence sets. These interval-based set structures, defined through evidence theory, are shown to possess ideal characteristics for the modeling of human cognitive categorization processes, within a constructivist framework.

INDEX TERMS: Uncertainty, Uncertainty in infinite domains, Interval Based Fuzzy Sets, Evidence Theory, Constructivism, Cognitive Categorization.

1. Introduction

Two aspects of George Klir's vast array of research achievements have influenced me greatly: his work on the measurement of uncertainty-based information [Klir and Folger, 1987; Klir 1993], and his constructivist approach to general systems theory [Klir, 1991]. My own research is tightly tied to George Klir's very important contributions to these areas. For that reason, I present here some developments in the measurement of uncertainty based information, relevant for a constructivist model of human cognitive categorization processes.

In the first part of the paper (sections 3 to 5), measures of uncertainty, for both countable and uncountable domains, are developed in the context of evidence theory. In particular, measures of nonspecificity are defined starting with a very general framework based on monotone functions and classic measure theory. This framework is shown to lead to a coherent methodology for measuring nonspecificity in countable, uncountable, and hybrid domains established as a combination (cross-product) of the two. It is defended that this framework 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. Relative uncertainty measures are dimensionless, since they are defined as information ratios.

In section 6 evidence sets are summarized with a brief overview of interval based set structures. Evidence sets are shown to capture the main forms of uncertainty recognized in generalized information theory. The uncertainty content of evidence sets is described with the measures of uncertainty defined in sections 3 through 5, which establish a three dimensional uncertainty space. Finally, in section 7, evidence sets are discussed as set structures capable of explicitly modeling subjective, contextual dependencies of cognitive and linguistic categories, thus offering a mathematical tool for a constructivist approach to approximate reasoning.