An Analytic Method for Evaluating the Performance of Aggregation Rules for Probability Densities

It is shown how infinite sequences of densities with defined properties can be used to evaluate the expected performance of mathematical aggregation rules for elicited densities. The performance of these rules is measured through the average variance, calibration, and average Brier score of the aggregates. A general result for the calibration of the arithmetic average of densities from well-calibrated independent experts is given. Arithmetic and geometric aggregation rules are compared using sequences of normal densities. Sequences are developed that exhibit dependence among experts and lack of calibration. The impact of correlation, number of experts, and degree of calibration on the performance of the aggregation is demonstrated.