Eliciting density ratio classes

  • Simon L. Rinderknechta, b,
  • Mark E. Borsukc,
  • Peter Reicherta, b, Corresponding author contact information, E-mail the corresponding author
  • a Eawag, Swiss Federal Institute of Aquatic Science and Technology, CH-8600 Dübendorf, Switzerland
  • b ETH Zürich, Institute of Biogeochemistry and Pollutant Dynamics (IBP), CH-8092 Zürich, Switzerland
  • c Dartmouth College, Thayer School of Engineering, Hanover, NH 03755-8000, USA
  • Received 22 April 2010. Revised 8 October 2010. Accepted 11 February 2011. Available online 16 February 2011.

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Abstract

The probability distributions of uncertain quantities needed for predictive modelling and decision support are frequently elicited from subject matter experts. However, experts are often uncertain about quantifying their beliefs using precise probability distributions. Therefore, it seems natural to describe their uncertain beliefs using sets of probability distributions. There are various possible structures, or classes, for defining set membership of continuous random variables. The Density Ratio Class has desirable properties, but there is no established procedure for eliciting this class. Thus, we propose a method for constructing Density Ratio Classes that builds on conventional quantile or probability elicitation, but allows the expert to state intervals for these quantities. Parametric shape functions, ideally also suggested by the expert, are then used to bound the nonparametric set of shapes of densities that belong to the class and are compatible with the stated intervals. This leads to a natural metric for the size of the class based on the ratio of the total areas under upper and lower bounding shape functions. This ratio will be determined by the characteristics of the shape functions, the scatter of the elicited values, and the explicit expert imprecision, as characterized by the width of the stated intervals. We provide some examples, both didactic and real, and conclude with recommendations for the further development and application of the Density Ratio Class.

Keywords

  • Probability assessment;
  • Expert elicitation of vague knowledge;
  • Imprecise probabilities;
  • Robust Bayesian statistics;
  • Quantile elicitation;
  • Density Ratio Class
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Corresponding author at: Eawag, Swiss Federal Institute of Aquatic Science and Technology, CH-8600 Dübendorf, Switzerland. Tel.: +41 587655281.

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