🤖 AI Summary
This work uncovers the intrinsic equivalence between scoring rules and cosine similarity by establishing a direct geometric connection through vector geometry and Euclidean distance optimization. From a least-squares perspective, it demonstrates that the arithmetic mean of score vectors uniquely minimizes the sum of squared Euclidean distances to all individual score vectors, naturally yielding the cosine similarity formulation. This approach not only provides a clear geometric interpretation of scoring rules but also simplifies existing proofs by revealing that the alignment between scoring rules and cosine similarity arises inevitably from this underlying optimization principle. The result clarifies why these two seemingly distinct formulations must coincide, grounding their equivalence in fundamental properties of vector space geometry.
📝 Abstract
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.