COPULA MODELS FOR AGGREGATING EXPERT OPINIONS

This paper discusses the use of multivariate distributions that are fu nctions of their marginals for aggregating information from various so urces. The function that links the marginals is called a copula. The i nformation to be aggregated can be point estimates of an unknown quant ity theta or, with suitable modeling assumptions, probability distribu tions for theta. This approach allows the Bayesian decision maker perf orming the aggregation to separate two difficult aspects of the model- construction procedure. Qualities of the individual sources, such as b ias and precision, are incorporated into the marginal distributions. D ependence 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 f or situations in which the decision maker must use subjective judgment s as a basis for constructing the aggregation model. We review propert ies of copulas pertinent to the information-aggregation problem. A sub jectively assessable measure of dependence is developed that allows th e decision maker to choose from a one-parameter family of copulas a sp ecific member that is appropriate for the level of dependence among th e information sources. The discussion then focuses on the class of Arc himedean copulas and Frank's family of copulas in particular, showing the specific relationship between the family and our measure of depend ence. A realistic example demonstrates the approach.