AI 中文总结
研究旨在证明评分规则等同于余弦相似性规则，基于基本最小二乘特征给出更简单证明，表明得分向量算术平均值是总欧几里得平方距离唯一最小值，为评分规则提供几何解释，阐明二者一致原因。
AI 中文摘要
川田（2018）证明每个评分规则都等同于相应的余弦相似性规则。原证明依赖于对余弦相似性优化问题的直接分析。本文给出了一个基于基本最小二乘特征的更简单的替代证明。论证表明得分向量的算术平均值是总欧几里得平方距离的唯一最小值，且余弦相似性公式是此优化性质的直接结果。该结果为评分规则提供了清晰的几何解释，阐明了余弦相似性规则与相应评分规则必然一致的原因。
英文摘要
Kawada (2018) proved that every scoring rule is equivalent to the corresponding cosine similarity rule. The original proof relies on a direct analysis of the cosine similarity optimization problem. In this note, we present an alternative, simpler proof based on a basic least-squares characterization. Our argument shows that the arithmetic mean of the score vectors is the unique minimizer of the total squared Euclidean distance and that the cosine similarity formulation is an immediate consequence of this optimization property. This result provides a transparent geometric interpretation of scoring rules and clarifies why the cosine similarity rule necessarily coincides with the corresponding scoring rule.
