Computational Statistics

, Volume 29, Issue 3–4, pp 661–683 | Cite as

Holonomic gradient descent for the Fisher–Bingham distribution on the \(d\)-dimensional sphere

  • Tamio Koyama
  • Hiromasa Nakayama
  • Kenta Nishiyama
  • Nobuki Takayama
Original Paper


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 \(d\)-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 \(d=7\) with a specified accuracy.


Fisher–Bingham distribution Maximum likelihood estimate Holonomic gradient descent Integrable connection Pfaffian system 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Tamio Koyama
    • 1
  • Hiromasa Nakayama
    • 1
  • Kenta Nishiyama
    • 2
  • Nobuki Takayama
    • 1
  1. 1.Department of MathematicsKobe University and JST CREST Hibi TeamKobeJapan
  2. 2.School of Management and InformationUniversity of Shizuoka and JST CREST Hibi TeamShizuokaJapan

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