  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.
