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推广至所有核分数，对线性池的若干结论进行泛化。核分数是一类丰富的评分规则，涵盖单变量与多变量、离散与连续场景下的点预测与分布预测，其成员包括用于单变量分布预测的连续等级概率分数和用于多变量分布预测的能量分数。研究结果表明，预测不一致性（以所有分量分布的平均成对散度衡量）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场景中理解与设计线性池，具体而言，它们为使用线性池（而非其他组合公式）提供了理论依据，并推导出在给定核评分规则下等权重组合达到最优的新颖条件。