Linguistic Probability Theory
Joe Halliwell · ERA · 2008
In recent years probabilistic knowledge-based systems such as Bayesian net¬ works and influence diagrams have come to the fore as a means of represent¬ ing and reasoning about complex real-world situations.Although some of the probabilities used in these models may be obtained statistically, where this is impossible or simply inconvenient, modellers rely on expert knowledge.Ex¬ perts, however, typically find it difficult to specify exact probabilities and con¬ ventional representations cannot reflect any uncertainty they may have.In this way, the use of conventional point probabilities can damage the accuracy, robustness and interpretability of acquired models.With these concerns in mind, psychometric researchers have demonstrated that fuzzy numbers are good candidates for representing the inherent vagueness of probability esti¬ mates, and the fuzzy community has responded with two distinct theories of fuzzy probabilities.This thesis, however, identifies formal and presentational problems with these theories which render them unable to represent even very simple scenarios.This analysis leads to the development of a novel and intuitively appealing alternativea theory of linguistic probabilities patterned after the standard Kolmogorov axioms of probability theory.Since fuzzy numbers lack algebraic inverses, the resulting theory is weaker than, but generalises its classical coun¬ terpart.Nevertheless, it is demonstrated that analogues for classical proba¬ bilistic concepts such as conditional probability and random variables can be constructed.In the classical theory, representation theorems mean that most of the time the distinction between mass/density distributions and probability measures can be ignored.Similar results are proven for linguistic probabilii Linguistic Probability Theory 57 4.1 Linguistic probability measures 58 4.1.1Conclusion 109 7.