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Bob predicts a future observation based on a sample of size one. Alice candraw a sample of any size before issuing her prediction. How much better canshe 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 halveBob's risk. In this sense, half the information in an infinite sample iscontained in a sample of size one. The Cover-Hart family is a convex cone thatincludes metrics and negative definite functions, subject to slight regularityconditions. These results may help explain the small relative differences inempirical performance measures in applied classification and forecastingproblems, as well as the success of reasoning and learning by analogy ingeneral, and nearest neighbor techniques in particular.

On the Cover-Hart Inequality: What's a Sample of Size One Worth?