Angular Combining of Forecasts of Probability Distributions

When multiple forecasts are available for a probability distribution, forecast combining enables a pragmatic synthesis of the information to extract the wisdom of the crowd. The linear opinion pool has been widely used, whereby the combining is applied to the probabilities of the distributional forecasts. However, it has been argued that this will tend to deliver overdispersed distributions, prompting the combination to be applied, instead, to the quantiles of the distributional forecasts. Results from different applications are mixed, leaving it as an empirical question whether to combine probabilities or quantiles. In this paper, we present an alternative approach. Looking at the distributional forecasts, combining the probabilities can be viewed as vertical combining, with quantile combining seen as horizontal combining. Our proposal is to allow combining to take place on an angle between the extreme cases of vertical and horizontal combining. We term this angular combining. The angle is a parameter that can be optimized using a proper scoring rule. For implementation, we provide a pragmatic numerical approach and a simulation algorithm. Among our theoretical results, we show that, as with vertical and horizontal averaging, angular averaging results in a distribution with mean equal to the average of the means of the distributions that are being combined. We also show that angular averaging produces a distribution with lower variance than vertical averaging, and, under certain assumptions, greater variance than horizontal averaging. We provide empirical results for distributional forecasts of Covid mortality, macroeconomic survey data, and electricity prices.