Correlations and Copulas for Decision and Risk Analysis

The construction of a probabilistic model is a key step in most decision and risk analyses. Typically this is done by defining a joint distribution in terms of marginal and conditional distributions for the model's random variables. We describe an alternative approach that uses a copula to construct joint distributions and pairwise correlations to incorporate dependence among the variables. The approach is designed specifically to permit the use of an expert's subjective judgments of marginal distributions and correlations. The copula that underlies the multivariate normal distribution provides the basis for modeling dependence, but arbitrary marginals are allowed. We discuss how correlations can be assessed using techniques that are familiar to decision analysts, and we report the results of an empirical study of the accuracy of the assessment methods. The approach is demonstrated in the context of a simple example, including a study of the sensitivity of the results to the assessed correlations.