Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE GEOMETRY OF THE DIRICHLET MANIFOLD

  • Zhong, Fengwei (Department of Mathematics Beijing Institute of Technology) ;
  • Sun, Huafei (Department of Mathematics Beijing Institute of Technology) ;
  • Zhang, Zhenning (Department of Mathematics Beijing Institute of Technology)
  • Published : 2008.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In the present paper, we investigate the geometric structures of the Dirichlet manifold composed of the Dirichlet distribution. We show that the Dirichlet distribution is an exponential family distribution. We consider its dual structures and give its geometric metrics, and obtain the geometric structures of the lower dimension cases of the Dirichlet manifold. In particularly, the Beta distribution is a 2-dimensional Dirich-let distribution. Also, we construct an affine immersion of the Dirichlet manifold. At last, we give the e-flat hierarchical structures and the orthogonal foliations of the Dirichlet manifold. All these work will enrich the theoretical work of the Dirichlet distribution and will be great help for its further applications.

Keywords

References

  1. S. Amari, Differential-Geometrical Methods in Statistics, Lecture Notes in Statistics, 28. Springer-Verlag, New York, 1985
  2. S. Amari, Information geometry on hierarchy of probability distributions, IEEE Trans. Inform. Theory 47 (2001), no. 5, 1701-1711 https://doi.org/10.1109/18.930911
  3. S. Amari and H. Nagaoka, Methods of Information Geometry, Translated from the 1993 Japanese original by Daishi Harada. Translations of Mathematical Monographs, 191. American Mathematical Society, Providence, RI; Oxford University Press, Oxford, 2000
  4. C. T. J Dodson and H. Matsuzoe, An affine embedding of the gamma manifold, Appl. Sci. 5 (2003), no. 1, 7-12
  5. A. M. Li, U. Simon, and G. S. Zhao, Global Affine Differential Geometry of Hypersurfaces, de Gruyter Expositions in Mathematics, 11. Walter de Gruyter & Co., Berlin, 1993
  6. F. Zhong, H. Sun, and Z. Zhang, Information geometry of the Beta distribution, preprint