Statistics and Computing

, Volume 25, Issue 2, pp 321–332 | Cite as

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



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.


Bingham distributions Directional statistics Holonomic functions 



The first author is supported by JSPS Institutional Program for Young Researcher Overseas Visits.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsKeio UniversityTokyoJapan
  2. 2.SMSASUniversity of KentCanterburyUK

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