Median Aggregation of Distribution Functions

When multiple redundant probabilistic judgments are obtained from subject matter experts, it is common practice to aggregate their differing views into a single probability or distribution. Although many methods have been proposed for mathematical aggregation, no single procedure has gained universal acceptance. The most widely used procedure is simple arithmetic averaging, which has both desirable and undesirable properties. Here we propose an alternative for aggregating distribution functions that is based on the median cumulative probabilities at fixed values of the variable. It is shown that aggregating cumulative probabilities by medians is equivalent, under certain conditions, to aggregating quantiles. Moreover, the median aggregate has better calibration than mean aggregation of probabilities when the experts are independent and well calibrated and produces sharper aggregate distributions for well-calibrated and independent experts when they report a common location-scale distribution. We also compare median aggregation to mean aggregation of quantiles.