This paper focusses on robust estimation of location and concentration parameters of the von Mises-Fisher distribution in the Bayesian framework. The von Mises-Fisher (or Langevin) distribution has played a central role in directional statistics. Directional data have been investigated for many decades, and more recently, they have gained increasing attention in diverse areas such as bioinformatics and text data analysis. Although outliers can significantly affect the estimation results even for directional data, the treatment of outliers remains an unresolved and challenging problem. In the frequentist framework, numerous studies have developed robust estimation methods for directional data with outliers, but, in contrast, only a few robust estimation methods have been proposed in the Bayesian framework. In this paper, we propose Bayesian inference based on the density power divergence and the $γ$-divergence and establish their asymptotic properties and robustness. In addition, the Bayesian approach naturally provides a way to assess estimation uncertainty through the posterior distribution, which is particularly useful for small samples. Furthermore, to carry out the posterior computation, we develop the posterior computation algorithm based on the weighted Bayesian bootstrap for estimating parameters. The effectiveness of the proposed methods is demonstrated through simulation studies. Using two real datasets, we further show that the proposed method provides reliable and robust estimation even in the presence of outliers or data contamination.

翻译：本文聚焦于在贝叶斯框架下对von Mises-Fisher分布的位置参数与浓度参数进行稳健估计。von Mises-Fisher（或称Langevin）分布在方向统计学中具有核心地位。方向数据的研究已有数十年历史，近年来在生物信息学与文本数据分析等众多领域日益受到关注。尽管异常值会显著影响方向数据的估计结果，其处理仍是尚未解决且具有挑战性的问题。在频率学派框架下，已有大量研究针对含异常值的方向数据提出了稳健估计方法；然而相比之下，贝叶斯框架下的稳健估计方法仍鲜有提出。本文基于密度幂散度与$γ$-散度提出贝叶斯推断方法，并建立其渐近性质与稳健性理论。此外，贝叶斯方法天然地通过后验分布提供评估估计不确定性的途径，这对小样本情形尤为有益。为实施后验计算，我们进一步开发了基于加权贝叶斯自助法的后验计算算法以估计参数。通过模拟研究验证了所提方法的有效性。基于两个真实数据集，我们进一步证明所提方法即使在存在异常值或数据污染的情况下，仍能提供可靠且稳健的估计结果。