Selecting Uncertainty Calculi and Granularity: An Experiment in Trading-Off Precision and Complexity

TL;DR

This paper establishes a theoretical framework for uncertainty calculi in expert systems, analyzing the trade-offs between precision and complexity through experiments with different operators and term sets.

Contribution

It introduces a formalism for defining various uncertainty calculi based on triangular norms and conorms, and proposes a context-dependent rule set for selecting the most suitable calculus.

Findings

Nine and eleven different calculi tested with varying term set sizes

Fewer calculi produce significantly different results, indicating limited effective options

A small rule set can effectively select appropriate calculus based on context

Abstract

The management of uncertainty in expert systems has usually been left to ad hoc representations and rules of combinations lacking either a sound theory or clear semantics. The objective of this paper is to establish a theoretical basis for defining the syntax and semantics of a small subset of calculi of uncertainty operating on a given term set of linguistic statements of likelihood. Each calculus is defined by specifying a negation, a conjunction and a disjunction operator. Families of Triangular norms and conorms constitute the most general representations of conjunction and disjunction operators. These families provide us with a formalism for defining an infinite number of different calculi of uncertainty. The term set will define the uncertainty granularity, i.e. the finest level of distinction among different quantifications of uncertainty. This granularity will limit the ability to differentiate between two similar operators. Therefore, only a small finite subset of the infinite number of calculi will produce notably different results. This result is illustrated by two experiments where nine and eleven different calculi of uncertainty are used with three term sets containing five, nine, and thirteen elements, respectively. Finally, the use of context dependent rule set is proposed to select the most appropriate calculus for any given situation. Such a rule set will be relatively small since it must only describe the selection policies for a small number of calculi (resulting from the analyzed trade-off between complexity and precision).