When multiple forecasts are available for a probability distribution, forecast combining enables a pragmatic synthesis of the available information to extract the wisdom of the crowd. A linear opinion pool has been widely used, whereby the combining is applied to the probability predictions of the distributional forecasts. However, it has been argued that this will tend to deliver overdispersed distributional forecasts, prompting the combination to be applied, instead, to the quantile predictions 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 probability forecasts can be viewed as vertical combining, with quantile forecast combining seen as horizontal combining. Our alternative approach 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. 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 support for angular combining using weekly distributional forecasts of COVID-19 mortality at the national and state level in the U.S.

翻译：当存在多个概率分布预测时，预测组合能够对可用信息进行实用综合，以提取群体智慧。线性意见池已被广泛使用，其中组合应用于分布预测的概率预测。然而，有观点认为这往往会产生过度分散的分布预测，从而促使组合转而应用于分布预测的分位数预测。不同应用的结果参差不齐，使得究竟组合概率还是分位数成为一个经验性问题。本文提出了一种替代方法。审视分布预测时，将概率预测组合视为垂直组合，而分位数预测组合则视为水平组合。我们的替代方法是允许组合在垂直组合与水平组合这两种极端情况之间的角度上进行。我们将此称为角度组合。该角度是一个参数，可通过适当的评分规则进行优化。我们证明，与垂直平均和水平平均类似，角度平均得到的分布均值等于被组合分布均值的平均值。我们还证明，角度平均产生的分布方差低于垂直平均，而在特定假设下，方差高于水平平均。我们利用美国全国及州级新冠肺炎死亡率的每周分布预测，为角度组合提供了实证支持。