# Bingham distribution

< List of probability distributions > Bingham distribution

The Bingham distribution can be used to model the distribution of data points on a sphere. It is especially useful for modeling the local behavior of any smooth distribution around a stationary point and is frequently used in directional statistics. Its applications include computer vision , differential privacy  and paleomagnetic data analysis .

## Bingham distribution properties

The Bingham distribution probability density function (PDF), with respect to the uniform (surface) measure is defined by a matrix A ∈ ℝd x d and given as 

The matrix A defines the direction of the distribution, and the uniform (surface) measure — which assigns equal weight to every point on the surface of a d-dimensional sphere — ensures that all directions are equally likely. This means that the probability of a point being chosen from the surface of the sphere is the same for all points on the surface.

The more general Fisher-Bingham distribution includes a linear term, which means it can locally model any smooth probability distribution. Any smooth probability distribution resembles a Bingham distribution in a sphere of small radius around a stationary point. Thus, the Bingham can be thought of as a “model” non-log-concave distribution . Non-log-concave distributions such as the Cauchy distribution and Pareto distribution can’t be represented as the product of a concave function and a non-negative function. They tend to be more challenging to work with than log-concave distributions, but they can model a wider range of phenomena such as multimodal distributions, including those that are multimodal and skewed.

## Bingham Distribution extensions

Various extensions of the Bingham have been formulated including the general eight-parameter Fisher-Bingham distribution family on the sphere, first proposed by Kent  .

The general form of the Fisher-Bingham (FB) distribution is given by the partial differential equation

C exp [κ0(x′ξ0) + κ1(x′ξ1)2 – κ2(x′ξ2)2] dS

Where ξ is a unit vector (ξ1, ξ2 are at right angles and involve five parameters) and κ is a shape parameter. The distribution can be unimodal or bimodal depending on the values of the shape parameter; it can also be concentrated around a closed curve.

• The FB4 is called the Bingham-Mardia distribution after Christopher Bingham and K.V. Mardia .
• The FB5 is usually called the Kent distribution, based on Kent .
• The FB6 distribution was first introduced in Rivest .

## References

 M Antone and Seth Teller. Automatic recovery of camera positions in urban scenes. Technical
Report, MIT LCS TR-814, 2000.

 Kamalika Chaudhuri, Anand D Sarwate, and Kaushik Sinha. A near-optimal algorithm for differentially-private principal components. The Journal of Machine Learning Research, 14(1): 2905–2943, 2013.

 Tullis C Onstott. Application of the Bingham distribution function in paleomagnetic studies. Journal of Geophysical Research: Solid Earth, 85(B3):1500–1510, 1980.

 Christopher Bingham. An antipodally symmetric distribution on the sphere. The Annals of Statistics,
pages 1201–1225, 1974.

 Ge, R. et al. Efficient sampling from the Bingham distribution. Proceedings of Machine Learning Research vol 132:1–13, 2021 32nd International Conference on Algorithmic Learning Theory

 Kent, J. (1982). The Fisher-Bingham Distribution on the Sphere. Journal of the Royal Statistical Society. Series B (Methodological). Vol 44, No.1, pp.71-80. Wiley.

 Bingham & Mardia. A small circle distribution on the sphere. Biometrika, Volume 65, Issue 2, August 1978, Pages 379–389

Scroll to Top