Summary

A property, or statistical functional, is said to be elicitable if it minimizes the expected loss for some loss function. The study of which properties are elicitable sheds light on the capabilities and limitations of point estimation and empirical risk minimization. While recent work has sought to identify which properties are elicitable, here we investigate a more nuanced question: how many dimensions are required to indirectly elicit a given property? This number is called the elicitation complexity of the property. We lay the foundation for a general theory of elicitation complexity, which includes several basic results on how elicitation complexity behaves and the complexity of standard properties of interest. Building on this foundation, our main result gives tight complexity bounds for the broad class of Bayes risks. We apply these results to several properties of interest, including variance, entropy, norms and several classes of financial risk measures. The article concludes with a discussion and open questions.

中文翻译：

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统计特性的引出复杂度

概括

如果一个属性或统计函数最小化了某些损失函数的预期损失，则称该属性或统计函数是可引出的。对哪些属性是可引出的研究揭示了点估计和经验风险最小化的能力和局限性。虽然最近的工作试图确定哪些属性是可引出的，但在这里我们研究了一个更细微的问题：间接引出给定属性需要多少维度？这个数字称为属性的引出复杂度。我们为诱导复杂性的一般理论奠定了基础，其中包括关于诱导复杂性如何表现的几个基本结果以及感兴趣的标准属性的复杂性。在此基础上，我们的主要结果为广泛的贝叶斯风险类别提供了严格的复杂性界限。我们将这些结果应用于几个感兴趣的属性，包括方差、熵、规范和几类金融风险度量。文章以讨论和开放式问题结束。