  The importance of accurately quantifying forecast uncertainty has motivated
much recent research on probabilistic forecasting. In particular, a variety of
deep learning approaches has been proposed, with forecast distributions
obtained as output of neural networks. These neural network-based methods are
often used in the form of an ensemble based on multiple model runs from
different random initializations, resulting in a collection of forecast
distributions that need to be aggregated into a final probabilistic prediction.
With the aim of consolidating findings from the machine learning literature on
ensemble methods and the statistical literature on forecast combination, we
address the question of how to aggregate distribution forecasts based on such
deep ensembles. Using theoretical arguments, simulation experiments and a case
study on wind gust forecasting, we systematically compare probability- and
quantile-based aggregation methods for three neural network-based approaches
with different forecast distribution types as output. Our results show that
combining forecast distributions can substantially improve the predictive
performance. We propose a general quantile aggregation framework for deep
ensembles that shows superior performance compared to a linear combination of
the forecast densities. Finally, we investigate the effects of the ensemble
size and derive recommendations of aggregating distribution forecasts from deep
ensembles in practice.
