Using probability boxes to model elicited information: a case study

Governments are responsible for making policy decisions, often in the face of severe uncertainty about the factors involved. Expert elicitation can be used to fill information gaps where data are not available, cannot be obtained, or where there is no time for a full-scale study and analysis. Various features of distributions for variables may be elicited, for example, the mean, standard deviation, and quantiles, but uncertainty about these values is not always recorded. Distributional and dependence assumptions often have to be made in models and although these are sometimes elicited from experts, modelers may also make assumptions for mathematical convenience (e.g., assuming independence between variables). Probability boxes (p-boxes) provide a flexible methodology to analyze elicited quantities without having to make assumptions about the distribution shape. If information about distribution shape(s) is available, p-boxes can provide bounds around the results given these possible input distributions. P-boxes can also be used to combine variables without making dependence assumptions. This article aims to illustrate how p-boxes may help to improve the representation of uncertainty for analyses based on elicited information. We focus on modeling elicited quantiles with nonparametric p-boxes, modeling elicited quantiles with parametric p-boxes where the elicited quantiles do not match the elicited distribution shape, and modeling elicited interval information.