Divergence-based Robust Generalised Bayesian Inference for Directional Data via von Mises-Fisher models

TL;DR

This paper introduces robust Bayesian methods for estimating parameters of the von Mises-Fisher distribution in directional data, addressing outliers and data contamination with new divergence-based approaches and computational algorithms.

Contribution

It proposes novel divergence-based Bayesian inference methods for directional data, establishing their robustness and asymptotic properties, along with a new posterior computation algorithm.

Findings

Methods demonstrate robustness in simulations.

Reliable estimation in contaminated real datasets.

Effective uncertainty assessment with small samples.

Abstract

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 $\gamma$-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.