Bob predicts a future observation based on a sample of size one. Alice can draw a sample of any size before issuing her prediction. How much better can she do than Bob? Perhaps surprisingly, under a large class of loss functions, which we refer to as the Cover-Hart family, the best Alice can do is to halve Bob's risk. In this sense, half the information in an infinite sample is contained in a sample of size one. The Cover-Hart family is a convex cone that includes metrics and negative definite functions, subject to slight regularity conditions. These results may help explain the small relative differences in empirical performance measures in applied classification and forecasting problems, as well as the success of reasoning and learning by analogy in general, and nearest neighbor techniques in particular.

翻译：鲍伯预测了未来基于大小一样本的观测结果。爱丽丝可以在发布预测之前抽取任何大小的样本。 她能比鲍勃做得更好吗?也许令人惊讶的是,在我们称之为盖哈特家族的大规模损失功能下,爱丽丝所能做的最好就是将鲍勃的风险减半。从这个意义上讲,无限样本中的一半信息包含在规模一样本中。盖哈特家族是一个包括尺度和负确定功能的锥体,这取决于轻微的规律性条件。这些结果可能有助于解释应用分类和预测问题的经验性绩效衡量的细小的相对差异,以及一般性推理和学习的成功,特别是近邻技术的成功。