Motivated by recent advances on elicitability of risk measures and practical considerations of risk optimization, we introduce the notions of Bayes pairs and Bayes risk measures. Bayes risk measures are the counterpart of elicitable risk measures, extensively studied in the recent literature. The Expected Shortfall (ES) is the most important coherent risk measure in both industry practice and academic research in finance, insurance, risk management, and engineering. One of our central results is that under a continuity condition, ES is the only class of coherent Bayes risk measures. We further show that entropic risk measures are the only risk measures which are both elicitable and Bayes. Several other theoretical properties and open questions on Bayes risk measures are discussed.

中文翻译：

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贝叶斯风险、可诱导性和预期不足

受风险度量可引出性的最新进展和风险优化的实际考虑的启发，我们引入了贝叶斯对和贝叶斯风险度量的概念。贝叶斯风险度量是可引出风险度量的对应物，在最近的文献中进行了广泛研究。预期缺口 (ES) 是金融、保险、风险管理和工程领域的行业实践和学术研究中最重要的连贯风险度量。我们的核心结果之一是，在连续性条件下，ES 是唯一一类一致的贝叶斯风险度量。我们进一步表明，熵风险测度是唯一既可诱导又可贝叶斯的风险测度。讨论了贝叶斯风险度量的其他几个理论属性和开放性问题。