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.

Sensitivity analysis and estimation of extreme tail behavior in two-dimensional Monte Carlo simulation.

Two-dimensional Monte Carlo simulation is frequently used to implement probabilistic risk models, as it allows for uncertainty and variability to be quantified separately. In many cases, we are interested in the proportion of individuals from a variable population exceeding a critical threshold, together with uncertainty about this proportion. In this article we introduce a new method that can accurately estimate these quantities much more efficiently than conventional algorithms. We also show how those model parameters having the greatest impact on the probabilities of rare events can be quickly identified via this method. The algorithm combines elements from well-established statistical techniques in extreme value theory and Bayesian analysis of computer models. We demonstrate the practical application of these methods with a simple example, in which the true distributions are known exactly, and also with a more realistic model of microbial contamination of milk with seven parameters. For the latter, sensitivity analysis (SA) is shown to identify the two inputs explaining the majority of variation in distribution tail behavior. In the subsequent prediction of probabilities of large contamination events, similar results are obtained using the new approach taking 43 seconds or the conventional simulation that requires more than 3 days.

Nonparametric predictive inference for exposure assessment.

Exposure assessment for food and drink consumption requires the combining of information about people's consumption of products with concentration data sets to provide predictions for chemical intake by humans. In this article, we present a method called nonparametric predictive inference (NPI) for exposure assessment. NPI is a distribution-free method relying only on Hill's assumption A(n). Effectively, A(n) is a postdata exchangeability assumption, which is a natural starting point for nonparametric statistics. For further discussion we refer to works by Hill and Coolen. We illustrate how NPI can be implemented to produce predictions for an individual's exposure based on consumption, body weight, and concentration data. NPI has the advantage that we do not have to assume a distribution to implement it. There may, however, be information available to suggest a distribution for a random quantity. Therefore, we present an NPI-Bayes hybrid method where this information can be taken into account by using Bayesian methods while using NPI for the other random quantities in the model.