Bingham distribution

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title: "Bingham distribution" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["directional-statistics", "continuous-distributions"] description: "Antipodally symmetric probability distribution on the n-sphere" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Bingham_distribution" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Antipodally symmetric probability distribution on the n-sphere ::

In statistics, the Bingham distribution, named after Christopher Bingham, is an antipodally symmetric probability distribution on the n-sphere. It is a generalization of the Watson distribution and a special case of the Kent and Fisher–Bingham distributions.

The Bingham distribution is widely used in paleomagnetic data analysis, and has been used in the field of computer vision. S. Teller and M. Antone (2000). Automatic recovery of camera positions in Urban Scenes

Its probability density function is given by

: f(\mathbf{x},;,M,Z) ; dS^{n-1} = {}_1 F_1 \left( \tfrac12 ; \tfrac n2 ; Z \right)^{-1} \cdot \exp \left( \operatorname{tr} Z M^T \mathbf{x} \mathbf{x}^T M \right); dS^{n-1} which may also be written : f(\mathbf{x},;,M,Z); dS^{n-1} ;=; {}_1 F_1 \left( \tfrac12 ; \tfrac n2 ;Z \right)^{-1} \cdot \exp\left( \mathbf{x}^T M Z M^T \mathbf{x} \right); dS^{n-1}

where x is an axis (i.e., a unit vector), M is an orthogonal orientation matrix, Z is a diagonal concentration matrix, and {}{1}F{1}(\cdot;\cdot,\cdot) is a confluent hypergeometric function of matrix argument. The matrices M and Z are the result of diagonalizing the positive-definite covariance matrix of the Gaussian distribution that underlies the Bingham distribution.

References

References

1. Bingham, Ch. (1974) "[https://projecteuclid.org/download/pdf_1/euclid.aos/1176342874 An antipodally symmetric distribution on the sphere]". ''Annals of Statistics'', 2(6):1201–1225.
2. Onstott, T.C. (1980) "[https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JB085iB03p01500 Application of the Bingham distribution function in paleomagnetic studies]{{Dead link. (February 2022)
3. (2008). "Computer Vision – ECCV 2008". Springer.
4. (October 7, 2013). "Better robot vision: A neglected statistical tool could help robots better understand the objects in the world around them.". MIT News.

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