Classification of matrix-variate Fisher–Bingham distribution via Maximum Likelihood Estimation using manifold valued data

Abstract

Abstract In the existing literature, the numerical estimation of multivariate normalizing constant is considered highly non-trivial, intractable or a big barrier to Bayesian approach, and hence non-parametric and many other techniques are adopted as an alternative for inferential problems, i.e., without the use of normalizing constant. This problem is a substantial one, and is mainly due to the difficult normalizing constant requiring accurate and efficient calculation for parametric models. The core objective of this paper is to estimate the numerical value of multivariate normalizing constant in Fisher–Bingham parametric model of distributions in the most general context. The ultimate goal is then the practicability of multivariate Fisher–Bingham model with the inclusion of normalizing constant for Bayesian inference via the standard Maximum Likelihood Estimation (MLE) algorithm. The normalizing constant is evaluated with the method of Saddle-Point Approximation (SPA) via Taylor Series expansion, where the first four terms of the series are considered the most crucial for obtaining the threshold value of the multivariate normalizing constant. Furthermore, it plays the most important role for the practicability of multivariate parametric models with manifold valued data being more useful for statistical inferences. We evaluate the numerical value of multivariate normalizing constant with best accuracy estimates. The numerical value is then used for statistical inferences with a simple platform of MLE for experiments on one synthetic dataset on Stiefel manifolds and three publicly available real world datasets with consideration of Grassmannian points on the Fisher–Bingham model. The results obtained via numerical experiments are compared with results of existing related techniques, where our algorithm outperforms or at least is best comparable to the results of existing techniques currently available in literature.