Maximum Likelihood Estimation and Integration Algorithm for Modeling Complex Systems

  • Yoshinao ShirakiEmail author
Part of the Emergence, Complexity and Computation book series (ECC, volume 14)


The holonomic gradient descent (HGD) method has been proposed as a means for calculating the maximum likelihood estimate (MLE), and its effectiveness has, in recent years, been reported within the statistics community. The purpose of HGD calculations is to reduce the calculation of the maximum likelihood estimate (MLE) of particular types of functions to calculating the minimum value of the holonomic function. As is well known, the maximum likelihood estimate (MLE) plays an important role in complex systems theory. In the complex systems community, however, little is known about the holonomic gradient descent (HGD) method. In this article, we introduce this method to the complex systems community and review the calculation mechanism of HGD.


Maximum Likelihood Estimate (MLE) Groebner Basis Holonomic Gradient Descent (HGD) Integration algorithm 


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  1. 1.
    Hashiguchi, H., Numata, Y., Takayama, N., Takemura, A.: The holonomic gradient method for the distribution function of the largest root of a Wishart matrix. Journal of Multivariate Analysis 117, 296–312 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Hibi, T.: Harmony of Groebner basis and modern industrial society. Sugaku 63(3) (July 2013) (in Japanese)Google Scholar
  3. 3.
    Nakayama, H., Nishiyama, K., Noro, M., Ohara, K., Sei, T., Takayama, N., Takemura, A.: Holonomic Gradient Descent and its Application to the Fisher-Bingham Integral. Advances in Applied Mathematics 47(3), 639–658 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Mardia, K.V., Jupp, P.E.: Directional statistics. Wiley, New York (2000)zbMATHGoogle Scholar
  5. 5.
    Milstein, G.N., Nussbaum, M.: Maximum likelihood estimate for nonparametric signal in white noise by optimal control, WIAS-Preprint No. 596, Berlin (2000)Google Scholar
  6. 6.
    Oaku, T.: Algorithms for b-functions, restrictions, and algebraic local cohomology groups of D-modules. Adv. Appl. Math. 19, 61–105 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Oaku, T., Shiraki, Y., Takayama, N.: Algebraic algorithms for D-modules and numerical analysis. In: Li, Z.M., Sit, W. (eds.) Proceedings of the Sixth Asian Symposium on Computer Mathematics, pp. 23–39. World scientific (2003)Google Scholar
  8. 8.
    Sato, M., Kawai, T., Kashiwara, M.: Microfunctions and Pseudo-differential Equations. Lecture Note in Mathematics, vol. 287, pp. 265–529. Springer (1973)Google Scholar
  9. 9.
    Saito, M., Sturmfels, B., Takayama, N.: Gröbner Deformations of Hypergeometric Differential Equations. Springer (2000)Google Scholar
  10. 10.
    Takayama, N.: MAP estimation based on the holonomic gradient method, Harmony of Groebner basis and modern industrial society. Jst Crest (September 8, 2011) (in Japanese)Google Scholar
  11. 11.
    Takayama, N.: Integration algorithm for D-modules and estimation theory. Sugaku Seminar, 41–46 (February 2012) (in Japanese)Google Scholar
  12. 12.
    Edited by JST CREST Hibi team, Gröbner Dojo, Kyoritsu (2011) (in Japanese)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Toho UniversityChibaJapan

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