The role of vagueness in the numerical translation of verbal probabilities: A fuzzy approach

Abstract

The Role of Vagueness in the Numerical Translation of Verbal Probabilities: A Fuzzy Approach Franziska Bocklisch 1 (franziska.bocklisch@psychologie.tu-chemnitz.de) Steffen F. Bocklisch 1 (steffen.bocklisch@etit.tu-chemnitz.de) Martin R. K. Baumann 2 (martin.baumann@dlr.de) Agnes Scholz 1 (agnes.scholz@psychologie.tu-chemnitz.de) Josef F. Krems 1 (josef.krems@psychologie.tu-chemnitz.de) Wilhelm-Raabe-Str. 43, Chemnitz University of Technology, Germany Lilienthalplatz 7, German Aerospace Center Braunschweig, Germany Abstract The paper describes a general two-step procedure for the numerical translation of linguistic terms using parametric fuzzy potential membership functions. In an empirical study 121 participants estimated numerical values that correspond to 13 verbal probability expressions. Among the estimates are the most typical numerical equivalent and the minimal and maximal values that just correspond to the given linguistic terms. These values serve as foundation for the proposed fuzzy approach. Positions and shapes of the resulting membership functions suggest that the verbal probability expressions are not distributed equidistantly along the probability scale and vary considerably in symmetry, vagueness and overlap. The role of vagueness for further investigations in reasoning and decision making is discussed and relations to knowledge representation and working memory are highlighted. Keywords: verbal probability expressions; vagueness; fuzzy potential membership functions; knowledge representation; diagnostic reasoning; working memory Introduction Since the 1960s up to the present time researchers of different scientific areas have sustained an interest in studying the relationship between verbal and numerical probability expressions (Lichtenstein & Newman, 1967; Teigen & Brun, 2003; Smits & Hoorens, 2005). Among these are cognitive psychologists that inquire about the influence of uncertainty expressions on basic cognitive processes such as reasoning and decision making (Windschitl & Wells, 1996) as well as engineers, computer scientists and others that focus on the characterization (Zadeh, 1978, 2002) or on the treatment of uncertainty in applications such as medical decision support systems (Boegl, Adlassnig, Hayashi, Rothenfluh & Leitich, 2004). This broad interdisciplinary interest may be motivated by the essential role language plays in our daily life. Verbal probability terms, such as probably or thinkable are very widely used to express uncertainty about the occurrence of future events or about the degree of belief in hypotheses. For example, a typical statement that illustrates the use of linguistic terms in the conversation of stock market traders could be: “It is very unlikely that there will be a significant increase in the price of oil in the next month vice future.”. Several studies consistently show that people prefer words over numbers to express uncertainty (e.g. Wallsten, Budescu, Zwick & Kemp, 1993). This preference may be explained by the possibility of saying something about two different kinds of subjective uncertainty by using only one word. First, the stochastic uncertainty about the occurrence of an event (e.g. the probability of an increase of the oil price) and second, the vagueness of the event (e.g. what is meant by “a significant increase”). The understanding of these two kinds of uncertainty, their relations to each other and the way in which they influence human reasoning and decision making is crucial for any application that aims to support decision makers for example in medicine, business, risk management, marketing or politics. In our view, in order to contribute to the understanding of uncertainty, it is essential to first uncover the underlying relationship between word meaning and mathematical concepts such as subjective probability or fuzzy membership. Therefore, we propose a general two- step procedure for the numerical translation of verbal probability expressions based on (1) empirical estimates modelled by (2) fuzzy membership functions (Zadeh, 1965, Bocklisch & Bitterlich, 1994). The paper is structured as follows: first, we compare verbal and numerical probability expressions and discuss existing translation approaches. Second, we present our proposal that goes beyond other methodical issues and the results of an empirical investigation. Thereafter, the results are discussed and conclusions (e.g. for the construction of verbal probability scales for questionnaires) are highlighted. Further, potentialities of the fuzzy pattern classification method for reasoning and decision processes are pointed out. Verbal and Numerical Probabilities There is broad agreement concerning the different features of verbal and numerical expressions (see Teigen & Brun, 2003 for an overview). Numerical probabilities are commonly described as precise, unambiguous and especially useful for calculations. Additionally, the quality of numerical expressions can be evaluated and compared to predictions of normative models such as Bayes nets. Currently many researchers in the area of cognitive