Modelling of directional data using Kent distributions

Abstract: The modelling of data on a spherical surface requires the consideration of directional probability distributions. To model asymmetrically distributed data on a three-dimensional sphere, Kent distributions are often used. The moment estimates of the parameters are typically used in modelling tasks involving Kent distributions. However, these lack a rigorous statistical treatment. The focus of the paper is to introduce a Bayesian estimation of the parameters of the Kent distribution which has not been carried out in the literature, partly because of its complex mathematical form. We employ the Bayesian information-theoretic paradigm of Minimum Message Length (MML) to bridge this gap and derive reliable estimators. The inferred parameters are subsequently used in mixture modelling of Kent distributions. The problem of inferring the suitable number of mixture components is also addressed using the MML criterion. We demonstrate the superior performance of the derived MML-based parameter estimates against the traditional estimators. We apply the MML principle to infer mixtures of Kent distributions to model empirical data corresponding to protein conformations. We demonstrate the effectiveness of Kent models to act as improved descriptors of protein structural data as compared to commonly used von Mises-Fisher distributions.