计算排序评价指标AP@k的期望与方差，为推荐系统提供随机基线参考。
Average Precision at Cutoff k under Random Rankings: Expectation and Variance
- 推导了AP@k在离线与在线评估模型下的期望与方差公式。
- 揭示了纯随机排序下MAP@k的理论基准值及波动范围。
- 适用于评估推荐系统排名质量是否显著优于随机水平。
推荐系统与信息检索平台依赖排序算法向用户呈现最相关的内容，以提升参与度与满意度。评估排序质量需依赖可靠的评价指标。其中，截断点k处的平均精度均值（MAP@k）被广泛采用，因其同时考虑项目相关性与位置。本文推导了平均精度在截断点k（AP@k）的期望与方差，二者可作为MAP@k的基准。研究涵盖两种常见评估场景：离线与在线。期望值建立基准，反映完全随机排序下MAP@k所能达到的水平；方差则量化随机波动程度，有助于更可靠地解读实际观测得分。
原文摘要 · Abstract (English)
Recommender systems and information retrieval platforms rely on ranking algorithms to present the most relevant items to users, thereby improving engagement and satisfaction. Assessing the quality of these rankings requires reliable evaluation metrics. Among them, Mean Average Precision at cutoff k (MAP@k) is widely used, as it accounts for both the relevance of items and their positions in the list. In this paper, the expectation and variance of Average Precision at k (AP@k) are derived since they can be used as biselines for MAP@k. Here, we covered two widely used evaluation models: offline and online. The expectation establishes the baseline, indicating the level of MAP@k that can be achieved by pure chance. The variance complements this baseline by quantifying the extent of random fluctuations, enabling a more reliable interpretation of observed scores.
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