Copula Models for Aggregating Expert Opinions

This paper discusses the use of multivariate distributions that are functions of their marginals for aggregating information from various sources. The function that links the marginals is called a copula. The information to be aggregated can be point estimates of an unknown quantity θ or, with suitable modeling assumptions, probability distributions for θ. This approach allows the Bayesian decision maker performing the aggregation to separate two difficult aspects of the model-construction procedure. Qualities of the individual sources, such as bias and precision, are incorporated into the marginal distributions. Dependence among sources is encoded into the copula, which serves as a dependence function and joins the marginal distributions into a single multivariate distribution. The procedure is designed to be suitable for situations in which the decision maker must use subjective judgments as a basis for constructing the aggregation model. We review properties of copulas pertinent to the information-aggregation problem. A subjectively assessable measure of dependence is developed that allows the decision maker to choose from a one-parameter family of copulas a specific member that is appropriate for the level of dependence among the information sources. The discussion then focuses on the class of Archimedean copulas and Frank's family of copulas in particular, showing the specific relationship between the family and our measure of dependence. A realistic example demonstrates the approach.