Advancements in computational power and methodologies have enabled research on massive datasets. However, tools for analyzing data with directional or periodic characteristics, such as wind directions and customers' arrival time in 24-hour clock, remain underdeveloped. While statisticians have proposed circular distributions for such analyses, significant challenges persist in constructing circular statistical models, particularly in the context of Bayesian methods. These challenges stem from limited theoretical development and a lack of historical studies on prior selection for circular distribution parameters. In this article, we propose a framework for selecting hyperpriors that contracts to a simpler model in circular scenarios, especially when there is insufficient information to guide prior selection. We introduce well-examined Penalized Complexity (PC) priors for the most widely used circular distributions. Comprehensive comparisons with existing hyperpriors in the literature are conducted through simulation studies and a practical case study. Finally, we discuss the contributions and implications of our work, providing a foundation for further advancements in constructing Bayesian circular statistical models.

翻译：计算能力与方法的进步使得大规模数据集的研究成为可能。然而，用于分析具有方向性或周期性特征数据（如风向、24小时制下的客户到达时间）的工具仍显不足。尽管统计学家已提出适用于此类分析的圆分布，但在构建圆统计模型时仍面临重大挑战，尤其在贝叶斯方法中。这些挑战源于理论发展的局限性以及圆分布参数先验选择方面历史研究的匮乏。本文提出一种在圆场景中选择超先验的框架，该框架在缺乏指导先验选择的充分信息时，能够收缩至更简单的模型。我们针对最常用的圆分布引入了经过充分检验的惩罚复杂度（PC）先验。通过模拟研究和实际案例分析，与文献中现有的超先验进行了全面比较。最后，我们讨论了本工作的贡献与意义，为构建贝叶斯圆统计模型的后续进展奠定了基础。