Parameter estimation for von Mises-Fisher distributions

  • Authors:

  • Akihiro Tanabe;Kenji Fukumizu;Shigeyuki Oba;Takashi Takenouchi;Shin Ishii

  • Affiliations:

  • Nara Institute of Science and Technology, Graduate School of Information Science, Nara, Japan 630-0192;The Institute of Statistical Mathematics, Tokyo, Japan;Nara Institute of Science and Technology, Graduate School of Information Science, Nara, Japan 630-0192;Nara Institute of Science and Technology, Graduate School of Information Science, Nara, Japan 630-0192;Nara Institute of Science and Technology, Graduate School of Information Science, Nara, Japan 630-0192

  • Venue:
  • Computational Statistics
  • Year:
  • 2007

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Abstract

When analyzing high-dimensional data, it is often appropriate to pay attention only to the direction of each datum, disregarding its norm. The von Mises---Fisher (vMF) distribution is a natural probability distribution for such data. When we estimate the parameters of vMF distributions, parameter 驴 which corresponds to the degree of concentration is difficult to obtain, and some approximations are necessary. In this article, we propose an iterative algorithm using fixed points to obtain the maximum likelihood estimate (m.l.e.) for 驴. We prove that there is a unique local maximum for 驴. Besides, using a specific function to calculate the m.l.e., we obtain the upper and lower bounds of the interval in which the exact m.l.e. exists. In addition, based on these bounds, a new and good approximation is derived. The results of numerical experiments demonstrate the new approximation exhibits higher precision than traditional ones.