Maximum Entropy Aggregation of Expert Predictions

Abstract

This paper presents a maximum entropy framework for the aggregation of expert opinions where the expert opinions concern the prediction of the outcome of an uncertain event. The event to be predicted and individual predictions rendered are assumed to be discrete random variables. A measure of expert competence is defined using a distance metric between the actual outcome of the event and each expert's predicted outcome. Following Levy and Delic (Levy, W. B., H. Delic. 1994. Maximum entropy aggregation of individual opinions. IEEE Trans. Sys. Man & Cybernetics 24 606–613.), we use Shannon's information measure (Shannon [Shannon, C. E. 1948. A mathematical theory of communication. Bell Syst. Tech. J. 27 379–423.], Jaynes [Jaynes, E. T. 1957. Information theory and statistical mechanics. Phys. Rev. 106 Part I: 620–630, 108 Part II: 171–190.]) to derive aggregation rules for combining two or more expert predictions into a single aggregated prediction that appropriately calibrates different degrees of expert competence and reflects any dependence that may exist among the expert predictions. The resulting maximum entropy aggregated prediction is least prejudiced in the sense that it utilizes all information available but remains maximally non committal with regard to information not available. Numerical examples to illuminate the implications of maximum entropy aggregation are also presented.