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Holonomic gradient descent for the Fisher–Bingham distribution on the $$d$$ d -dimensional sphere

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

Abstract

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 $$d$$ 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 $$d=7$$ d = 7 with a specified accuracy. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Tamio Koyama & Hiromasa Nakayama & Kenta Nishiyama & Nobuki Takayama, 2014. "Holonomic gradient descent for the Fisher–Bingham distribution on the $$d$$ d -dimensional sphere," Computational Statistics, Springer, vol. 29(3), pages 661-683, June.
    • Handle: RePEc:spr:compst:v:29:y:2014:i:3:p:661-683
    DOI: 10.1007/s00180-013-0456-z
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    References listed on IDEAS

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    1. A. Kume & Andrew T. A. Wood, 2005. "Saddlepoint approximations for the Bingham and Fisher–Bingham normalising constants," Biometrika, Biometrika Trust, vol. 92(2), pages 465-476, June.
    2. Hashiguchi, Hiroki & Numata, Yasuhide & Takayama, Nobuki & Takemura, Akimichi, 2013. "The holonomic gradient method for the distribution function of the largest root of a Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 296-312.
    3. Sei, Tomonari & Shibata, Hiroki & Takemura, Akimichi & Ohara, Katsuyoshi & Takayama, Nobuki, 2013. "Properties and applications of Fisher distribution on the rotation group," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 440-455.
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