A Kernel Score Perspective on Forecast Disagreement and the Linear Pool

The variance of a linearly combined forecast distribution (or linear pool) consists of two components: The average variance of the component distributions (`average uncertainty'), and the average squared difference between the components' means and the pool's mean (`disagreement'). This paper shows that similar decompositions hold for a class of uncertainty measures that can be constructed as entropy functions of kernel scores. The latter are a rich family of scoring rules that covers point and distribution forecasts for univariate and multivariate, discrete and continuous settings. The results in this paper are useful for two reasons. First, they provide a generic description of the uncertainty implicit in the linear pool. Second, they suggest principled measures of forecast disagreement in a wide range of applied settings.