We consider two ways to aggregate expert opinions using simple averages: averaging probabilities and averaging quantiles. We examine analytical properties of these forecasts and compare their ability to harness the wisdom of the crowd. In terms of location, the two average forecasts have the same mean. The average quantile forecast is always sharper: it has lower variance than the average probability forecast. Even when the average probability forecast is overconfident, the shape of the average quantile forecast still offers the possibility of a better forecast. Using probability forecasts for gross domestic product growth and inflation from the Survey of Professional Forecasters, we present evidence that both when the average probability forecast is overconfident and when it is underconfident, it is outperformed by the average quantile forecast. Our results show that averaging quantiles is a viable alternative and indicate some conditions under which it is likely to be more useful than averaging probabilities. This paper was accepted by Peter Wakker, decision analysis.

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W e introduce an alternative to the popular linear opinion pool for combining individual probability forecasts. One of the well-known problems with the linear opinion pool is that it can be poorly calibrated. It tends toward underconfidence as the crowd's diversity increases, i.e., as the variance in the individuals' means increases. To address this calibration problem, we propose the exterior-trimmed opinion pool. To form this pool, forecasts with low and high means, or cumulative distribution function (cdf) values, are trimmed away from a linear opinion pool. Exterior trimming decreases the pool's variance and improves its calibration. A linear opinion pool, however, will remain overconfident when individuals are overconfident and not very diverse. For these situations, we suggest trimming away forecasts with moderate means or cdf values. This interior trimming increases variance and reduces overconfidence. Using probability forecast data from U.S. and European Surveys of Professional Forecasters, we present empirical evidence that trimmed opinion pools can outperform the linear opinion pool.

We explore some recent, and not so recent, developments concerning the use of probability forecasts and their combination in decision making. Despite these advances, challenges still exist. We expand on some important challenges influencing the “goodness” of combined probability forecasts such as miscalibration, dependence among forecasters, and selection of an appropriate evaluation measure while connecting the processes of aggregating and evaluating forecasts to decision making. Through three important applications from the domains of meteorology, economics, and political science, we illustrate state-of-the-art usage of probability forecasts: how they are combined, evaluated, and communicated to stakeholders. We expect to see greater use and aggregation of probability forecasts, especially given developments in statistical modeling, machine learning, and expert forecasting; the popularity of forecasting competitions; and the increased reporting of probabilities in the media. Our vision is that increased exposure to and improved visualizations of probability forecasts will enhance the public’s understanding of probabilities and how they can contribute to better decisions.

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