Dr. Jeroen P. van der Sluijs

1. UFP workshop 25/26 August 2008 Utrecht Quantification of uncertain quantities using subjective probability distributions Dr. Jeroen P. van der Sluijs Copernicus Institute for Sustainable Development and InnovationUtrecht University & Centre d'Economie et d'Ethique pour l'Environnement et le Développement, Université de Versailles Saint-Quentin-en-Yvelines, France

2. Do we know enough to quantify? Risbey & Kandlikar (2007): What format is in accordance with the level of knowledge on the quantity? • Full probability density function • Robust, well defended distribution • Bounds • Well defended percentile bounds • First order estimates • Order of magnitude assessment • Expected sign or trend • Well defended trend expectation • Ambiguous sign or trend • Equally plausible contrary trend expectations • Effective ignorance • Lacking or weakly plausible expectations

3. PDF / CDF • Probability Density Function represents relative likelihood with which values of a variable may be obtained • Cumulative Density Function shows probability fractiles on y-axis and the corresponding value of the distribution on the x-axes

4. Reliability intervals in case of normal distributions   = 68 %  2 = 95 %  3 = 99.7 %

5. Percentiles illustrated for a normal distribution A percentile (e.g. the 90th percentile) indicates the value where that percent (i.c. 90%) of the distribution is below that value

6. PDF properties • Mean • Median (50th percentile) • Mode (peak) • Skewness (asymmetry) • Kurtosis (“ peakedness”)

7. Skewness A distribution is said to be skewed if it is not symmetrical A: positive skewness B: negative skewness

8. Kurtosis (peakedness) of distribution Normal distribution: Kurtosis = 3 Platykurtic (flat): Kurtosis < 3 lepokurtic (peaked): Kurtosis > 3

9. Most commonly used distributions • Normal • Uniform • Triangular • Lognormal

12. Steps in quantifying subjective probability • Clarify the definition of the uncertain quantity • Choose scale and unit which is most familiar to the expert • Specify all assumptions on which the expert should condition his response as well as those factors that the expert is expected to integrate in his judgement

13. Steps in quantifying subjective probability - II • What information do you take into account making your quantitative estimates on uncertainties in these quantities? • First provide estimates for the extreme upper and lower values of the distribution for this quantity • Explain why these values for the extremes were chosen

14. Steps in quantifying subjective probability III • Imagine a new, perfectly reliable method is developed to establish the same quantity and it turns out that the 'true' value of the quantity lies outside the range you just provided. Could you think of a plausible reason why your method had misestimated the quantity? Would you like to revise your prior estimates of min and max values?

15. Steps in quantifying subjective probability IV • Is the distribution uniform? • Is it symmetric or asymmetric? • Is it flat or peaked? • Interval Method. Partition the probability distribution into ranges of equal probability. • Specify the median (50th percentile) • Then, assess the quartiles (25 and 75 percentiles), as the point where it is equally likely that the value is within the interval bounded by the extreme or the median. • Tails of distributions can be very long, therefore specify the 5th and the 95th percentile as well (or the 2.5th and the 97.5th to have a 95% interval)

16. Example • Travel time from this meeting room to Schiphol Airport department hall.