von Mises–Fisher distribution

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In directional statistics, the von Mises–Fisher distribution (named after Ronald Fisher and Richard von Mises), is a probability distribution on the -sphere in . If the distribution reduces to the von Mises distribution on the circle.

The probability density function of the von Mises–Fisher distribution for the random p-dimensional unit vector is given by:

where and the normalization constant is equal to

where denotes the modified Bessel function of the first kind at order . If , the normalization constant reduces to

The parameters and are called the mean direction and concentration parameter, respectively. The greater the value of , the higher the concentration of the distribution around the mean direction . The distribution is unimodal for , and is uniform on the sphere for .

The von Mises–Fisher distribution for , also called the Fisher distribution, was first used to model the interaction of electric dipoles in an electric field (Mardia&Jupp, 1999). Other applications are found in geology, bioinformatics, and text mining.

Relation to normal distribution[edit]

Starting from a normal distribution

the von Mises-Fisher distribution is obtained by expanding

using the fact that and are unit vectors, and recomputing the normalization constant by integrating over the unit sphere.

Estimation of parameters[edit]

A series of N independent measurements are drawn from a von Mises–Fisher distribution. Define

Then (Mardia&Jupp, 1999) the maximum likelihood estimates of and are given by the sufficient statistic



Thus is the solution to

A simple approximation to is (Sra, 2011)

but a more accurate measure can be obtained by iterating the Newton method a few times

For N ≥ 25, the estimated spherical standard error of the sample mean direction can be computed as[1]


It's then possible to approximate a confidence cone about with semi-vertical angle


For example, for a 95% confidence cone, and thus


The matrix von Mises-Fisher distribution has the density

supported on the Stiefel manifold of orthonormal p-frames , where is an arbitrary real matrix.[2][3]

See also[edit]


  1. ^ Embleton, N. I. Fisher, T. Lewis, B. J. J. (1993). Statistical analysis of spherical data (1st pbk. ed.). Cambridge: Cambridge University Press. pp. 115–116. ISBN 0-521-45699-1.
  2. ^ Jupp (1979). "Maximum likelihood estimators for the matrix von Mises-Fisher and Bingham distributions". The Annals of Statistics. 7 (3): 599–606. doi:10.1214/aos/1176344681.
  3. ^ Downs (1972). "Orientational statistics". Biometrika. 59: 665–676. doi:10.1093/biomet/59.3.665.
  • Dhillon, I., Sra, S. (2003) "Modeling Data using Directional Distributions". Tech. rep., University of Texas, Austin.
  • Banerjee, A., Dhillon, I. S., Ghosh, J., & Sra, S. (2005). "Clustering on the unit hypersphere using von Mises-Fisher distributions". Journal of Machine Learning Research, 6(Sep), 1345-1382.
  • Fisher, RA, "Dispersion on a sphere'". (1953) Proc. Roy. Soc. London Ser. A., 217: 295–305
  • Mardia, Kanti; Jupp, P. E. (1999). Directional Statistics. John Wiley & Sons Ltd. ISBN 978-0-471-95333-3.
  • Sra, S. (2011). "A short note on parameter approximation for von Mises-Fisher distributions: And a fast implementation of I s (x)". Computational Statistics. 27: 177–190. CiteSeerX doi:10.1007/s00180-011-0232-x.