Finite-sample investigation of likelihood and Bayes inference for the symmetric von Mises–Fisher distribution

We consider likelihood and Bayes analyses for the symmetric matrix von Mises–Fisher (matrix Fisher) distribution, which is a common model for three-dimensional orientations (represented by 3×33×3 orthogonal matrices with a positive determinant). One important characteristic of this model is a 3×33×3 rotation matrix representing the modal rotation, and an important challenge is to establish accurate confidence regions for it with an interpretable geometry for practical implementation. While we provide some extensions of one-sample likelihood theory (e.g., Euler angle parametrizations of modal rotation), our main contribution is the development of MCMC-based Bayes inference through non-informative priors. In one-sample problems, the Bayes methods allow the construction of inference regions with transparent geometry and accurate frequentist coverages in a way that standard likelihood inference cannot. Simulation is used to evaluate the performance of Bayes and likelihood inference regions. Furthermore, we illustrate how the Bayes framework extends inference from one-sample problems to more complicated one-way random effects models based on the symmetric matrix Fisher model in a computationally straightforward manner. The inference methods are then applied to a human kinematics example for illustration.