Holonomic gradient descent for the Fisher–Bingham distribution on the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$d$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>d</mi> </math> </EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$d$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>d</mi> </math> </EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$d=7$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>d</mi> <mo>=</mo> <mn>7</mn> </mrow> </math> </EquationSource> </InlineEquation> with a specified accuracy. Copyright Springer-Verlag Berlin Heidelberg 2014