Calculating the normalising constant of the Bingham distribution on the sphere using the holonomic gradient method
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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. (Adv. Appl. Math. 47:639–658, 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.
KeywordsBingham distributions Directional statistics Holonomic functions
The first author is supported by JSPS Institutional Program for Young Researcher Overseas Visits.
- Koyama, T.: A holonomic ideal annihilating the Fisher–Bingham integral (2011). arXiv:1104.1411
- Koyama, T., Nakayama, H., Nishiyama, K., Takayama, N.: Holonomic gradient descent for the Fisher–Bingham distribution on the n-dimensional sphere (2012a). arXiv:1201.3239
- Koyama, T., Nakayama, H., Nishiyama, K., Takayama, N.: The holonomic rank of the Fisher–Bingham system of differential equations (2012b). arXiv:1205.6144