  Predictive distributions need to be aggregated when probabilistic forecasts
are merged, or when expert opinions expressed in terms of probability
distributions are fused. We take a prediction space approach that applies to
discrete, mixed discrete-continuous and continuous predictive distributions
alike, and study combination formulas for cumulative distribution functions
from the perspectives of coherence, probabilistic and conditional calibration,
and dispersion. Both linear and non-linear aggregation methods are
investigated, including generalized, spread-adjusted and beta-transformed
linear pools. The effects and techniques are demonstrated theoretically, in
simulation examples, and in case studies on density forecasts for S&P 500
returns and daily maximum temperature at Seattle-Tacoma Airport.
