Holonomic gradient descent for the Fisher–Bingham distribution on the $$d$$d-dimensional sphere

We propose an accelerated version of the holonomic gradient descent and apply it to calculating the maximum likelihood estimate (MLE) of the Fisher–Bingham distribution on a $$d$$d-dimensional sphere. We derive a Pfaffian system (an integrable connection) and a series expansion associated with the normalizing constant with an error estimation. These enable us to solve some MLE problems up to dimension $$d=7$$d=7 with a specified accuracy.