Generalised Bayesian Inference using Robust divergences for von Mises-Fisher distribution

TL;DR

The paper tackles robust Bayesian estimation for the von Mises–Fisher distribution on the unit sphere by introducing generalized posteriors based on density power-divergence and γ-divergence. It develops the theoretical framework, showing consistency and asymptotic normality of the robust posterior means, and provides a practical posterior computation method via the weighted Bayesian bootstrap. Through simulations and real-data examples (wind direction and gene expression), the authors demonstrate that the robust posteriors resist outlier contamination and yield stable mean directions and concentration estimates. The work enables principled uncertainty quantification under robustness, with avenues for adaptive tuning and extensions to more complex directional models.

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 density power-divergence and $γ$-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.

Figures (9)

• Figure 1: Histograms of WBB and Gibbs sampler in the case of $p = 3$.
• Figure 2: The figures display the SIF for these posterior means. The arrow length is value of the norm of SIF, and the direction of arrow indicates the direction of $\bm{y}$. The setting of this figures are that distribution $G$ is von Mises-Fisher distribution $vM(\bm{\mu}_0, \kappa_0)$ with $\bm{\mu}_0 = (1, 0)^{\top}$, $\kappa_0 = 5$, $\alpha = 0.15$, and $\gamma = 0.15$.
• Figure 3: These figures show the mean squared errors (MSEs) for the parameters of the von Mises-Fisher distribution with $p = 2$. Rows correspond to the sample sizes $n$, while columns list the parameters $\bm{\xi}$, $\bm{\mu}$, $\kappa$. The red solid line (KL), the green dotted line (DPD) and the blue dash line (Gam) represent the MSEs of the ordinary posterior, DPD posterior, and $\gamma$-D posterior means using a uniform prior for $\bm{\xi}$, respectively. Outliers are generated from the uniform distribution on the unit sphere.
• Figure 4: These figures show the mean squared errors (MSEs) for the parameters of the von Mises-Fisher distribution with $p = 3$. Rows correspond to the sample sizes $n$, while columns list the parameters $\bm{\xi}$, $\bm{\mu}$, $\kappa$. The red solid line (KL), the green dotted line (DPD) and the blue dash line (Gam) represent the MSEs of the ordinary posterior, DPD posterior, and $\gamma$-D posterior means using a uniform prior for $\bm{\xi}$, respectively. Outliers are generated from the uniform distribution on the unit sphere.
• Figure 5: These figures show the mean squared errors (MSEs) for the parameters of the von Mises-Fisher distribution with $p = 5$. Rows correspond to the sample sizes $n$, while columns list the parameters $\bm{\xi}$, $\bm{\mu}$, $\kappa$. The red solid line (KL), the green dotted line (DPD) and the blue dash line (Gam) represent the MSEs of the ordinary posterior, DPD posterior, and $\gamma$-D posterior means using a uniform prior for $\bm{\xi}$, respectively. Outliers are generated from the uniform distribution on the unit sphere.

Theorems & Definitions (3)

• Remark : Selection of Tuning Parameters
• Theorem 1: ghosh2016robustnakagawa2020robust
• Theorem 2: ghosh2016robustnakagawa2020robust