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 principled, practical and systematic framework for selecting priors that effectively prevents overfitting 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 priors 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：计算能力和方法的进步使得对海量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（如风向和顾客在24小时制下的到达时间）的工具仍然不够完善。尽管统计学家已为此类分析提出了圆分布，但在构建圆统计模型，尤其是在贝叶斯方法背景下，仍存在重大挑战。这些挑战源于有限的理论发展以及对圆分布参数先验选择缺乏历史研究。本文提出一个原则性、实用且系统化的先验选择框架，能有效防止圆场景中的过拟合，特别是在缺乏足够信息指导先验选择时。我们为最广泛使用的圆分布引入了经过充分检验的惩罚复杂性（PC）先验。通过模拟研究和实际案例研究，与文献中现有先验进行了全面比较。最后，我们讨论了本工作的贡献和意义，为构建贝叶斯圆统计模型的进一步发展奠定了基础。