Probabilities of judgments provided by unknown experts by using the imprecise Dirichlet model

Abstract

Most models of aggregating expert judgments assume that there is available some information characterizing the experts. This information may be incorporated into the so-called hierarchical uncertainty models (second-order models). However, we often do not know anything about experts or it is difficult to evaluate their quality. In this case, beliefs to experts may be in the interval [0, 1] and the resulting assessments become to be non-informative. Moreover, attempts to assign some weights or beliefs to experts were not crowned with success because the behavior of experts may be distinguished in different circumstances. Therefore, this paper proposes to estimate expert judgm ents instead of experts themselves and studies how to assign interval probabilities of expert judgments by using a set of multinomial models.

Keywords:

• expert judgments
• imprecise probabilities
• hierarchical uncertainty
• linear programming
• multinomial model
• Dirichlet distribution

Acknowledgment

I am very grateful to George Klir (Binghamton State University, USA), Igor Kozine (Risoe National Laboratory, Denmark), Gert de Cooman, Matthias Troffaes (Ghent University, Belgium) for numerous comments and stimulating discussions during the St.Petersburg Mini-Workshop ‘Imprecise Reasonong in Uncertainty Quantification.’

Notes

¹It is worth noticing that the Dirichlet model should be regarded as one of the possible multinomial models that can be applied to the considered approach.

²Generally, the proposed method for analyzing expert judgments works with expert judgements related to some random variable X itself (not its expectation) in the same way. However, more often experts provide judgments about characteristics of random variables, e.g., probabilities, expectations, moments. Therefore, expectations of functions of X are considered here.

³The sample space Ω may be discrete. All expressions for probabilities of events in this case are the same.