Bayesian Ensembles of Binary-Event Forecasts: When Is It Appropriate to Extremize or Anti-Extremize?

Many organizations face critical decisions that rely on forecasts of binary events. In these situations, organizations often gather forecasts from multiple experts or models and average those forecasts to produce a single aggregate forecast. Because the average forecast is known to be under-confident, methods have been proposed that create an aggregate forecast more extreme than the average forecast. But is it always appropriate to extremize the average forecast? And if not, when is it appropriate to anti-extremize (i.e., to make the aggregate forecast less extreme)? To answer these questions, we introduce a class of optimal aggregators. These aggregators are Bayesian ensembles because they follow from a Bayesian model of the underlying information experts have. Each ensemble is a generalized additive model of experts' probabilities that first transforms the experts' probabilities into their corresponding information states, then linearly combines these information states, and finally transforms the combined information states back into the probability space. Analytically, we find that these optimal aggregators do not always extremize the average forecast, and when they do, they can run counter to existing methods. On two publicly available datasets, we demonstrate that these new ensembles are easily fit to real forecast data and are more accurate than existing methods.