Bingham distribution

In statistics, the Bingham distribution, named after Christopher Bingham, is an antipodally symmetric probability distribution on the n-sphere.^[1] 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,^[2] and has been used in the field of computer vision.^[3][4][5]

Its probability density function is given by

${\displaystyle f(\mathbf {x} \,;\,M,Z)\;dS^{n-1}={}_{1}F_{1}\left({\tfrac {1}{2}};{\tfrac {n}{2}};Z\right)^{-1}\cdot \exp \left(\operatorname {tr} ZM^{T}\mathbf {x} \mathbf {x} ^{T}M\right)\;dS^{n-1}}$

which may also be written

${\displaystyle f(\mathbf {x} \,;\,M,Z)\;dS^{n-1}\;=\;{}_{1}F_{1}\left({\tfrac {1}{2}};{\tfrac {n}{2}};Z\right)^{-1}\cdot \exp \left(\mathbf {x} ^{T}MZM^{T}\mathbf {x} \right)\;dS^{n-1}}$

where x is an axis (i.e., a unit vector), M is an orthogonal orientation matrix, Z is a diagonal concentration matrix, and ${\displaystyle {}_{1}F_{1}(\cdot ;\cdot ,\cdot )}$ 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.