We provide a novel Bayesian decision-theoretic motivation for Stein-type estimates, producing them as an adaptive choice between standard point estimation and estimation that rewards proximity to the origin. Unlike conventional approaches, our arguments provide shrunken estimates under any sampling model or prior. They also lead naturally to a form of credible interval, describing uncertainty about the underlying parameters yet focusing on the shrunken estimate. One specific method for constructing intervals provides a close Bayesian analogue of Samworth (2005)'s approximate confidence intervals around Stein estimates. Several examples are given, showing how shrinkage's focus on regions remote to the high-support areas of the posterior can lead to substantially larger credible sets.

中文翻译：

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Stein 型估计的贝叶斯最优性和区间

我们为 Stein 型估计提供了一种新颖的贝叶斯决策理论动机，将它们作为标准点估计和奖励接近原点的估计之间的自适应选择。与传统方法不同，我们的论点在任何采样模型或先验模型下都提供了缩小的估计。它们还自然地导致一种可信区间的形式，描述了基础参数的不确定性，但侧重于缩小的估计。一种构造区间的特定方法提供了 Samworth (2005) 围绕 Stein 估计的近似置信区间的贝叶斯类比。给出了几个例子，展示了收缩对远离后部高支撑区域的区域的关注如何导致更大的可信集。