Holonomic Gradient Descent for the Fisher-Bingham Distribution on the $d$-dimensional Sphere

We apply the holonomic gradient descent recently introduced in [10] to the maximum likelihood estimate (MLE) of the Fisher-Bingham distribution on the d-dimensional sphere. We derive a Pfaffian system (an integrable connection) and a series expansion associated to the normalizing constant. These enable us to solve some MLE problems up to dimension $d = 7$ with accuracy.