Calculating the normalising constant of the Bingham distribution on the sphere using the holonomic gradient method

TL;DR

This paper applies the holonomic gradient method to precisely compute the normalising constant of Bingham distributions on the sphere, providing explicit formulas, implementation, and adjustments for degenerate cases.

Contribution

It explicitly derives the Pfaffian system for Bingham distributions and implements the holonomic gradient method for exact normalising constant calculation.

Findings

Exact computation of the normalising constant achieved.

Implementation of the method for maximum likelihood estimation.

Adjusted approach for cases with parameter multiplicities.

Abstract

In this paper we implement the holonomic gradient method to exactly compute the normalising constant of Bingham distributions. This idea is originally applied for general Fisher-Bingham distributions in Nakayama et al. (2011). In this paper we explicitly apply this algorithm to show the exact calculation of the normalising constant; derive explicitly the Pfaffian system for this parametric case; implement the general approach for the maximum likelihood solution search and finally adjust the method for degenerate cases, namely when the parameter values have multiplicities.