This paper generalizes several results on linear pooling from squared error loss to all 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. Its members include the Continuous Ranked Probability Score for univariate distribution forecasting and the Energy Score for multivariate distribution forecasting. Our results indicate that forecast disagreement (measured as the average pairwise divergence of all component distributions) has important implications for the linear pool's performance. The results are useful for understanding and designing linear pools in general combination settings. In particular, they motivate using the linear pool (as opposed to other combination formulas) and yield a novel condition under which equal combination weights are optimal under a given kernel scoring rule.

翻译：本文将从平方误差损失到所有核评分，推广了关于线性池的若干结论。核评分是一类丰富的评分规则，涵盖单变量与多变量、离散与连续情形下的点预测和分布预测。其成员包括用于单变量分布预测的连续排秩概率评分和用于多变量分布预测的能量评分。我们的结果表明，预测分歧（即所有分量分布的平均成对散度）对线性池的性能具有重要影响。这些结论有助于在一般组合情形下理解和设计线性池。特别是，它们为使用线性池（而非其他组合公式）提供了理论依据，并给出了一个新颖条件，在该条件下，给定核评分规则时等权重组合是最优的。