Von Mises–Fisher distribution

In directional statistics, the von Mises–Fisher distribution (named after Ronald Fisher and Richard von Mises), is a probability distribution on the -dimensional 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, 2000). Other applications are found in geology, bioinformatics, and text mining.

Relation to normal distribution

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

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

Then (Sra, 2011) the maximum likelihood estimates of and are given by

Thus is the solution to

A simple approximation to is

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


  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. Wiley. 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.
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