The 8-parameter Fisher-Bingham distribution on the sphere

Understand

The Fisher-Bingham distribution ($\mathrm{FB}_8$) is an eight-parameter family of probability density functions (PDF) on $S^2$ that, under certain conditions, reduce to spherical analogues of bivariate normal PDFs.

• Due to difficulties in computing its overall normalization constant, applications have been mainly restricted to subclasses of $\mathrm{FB}_8$, such as the Kent ($\mathrm{FB}_5$) or von Mises-Fisher (vMF) distributions.
• However, these subclasses often do not adequately describe directional data that are not symmetric along great circles.
• The normalizing constant of $\mathrm{FB}_8$ can be numerically integrated, and recently Kume and Sei showed that it can be computed using an adjusted holonomic gradient method.