# Bingham distribution

In statistics, the Bingham distribution, named after Christopher Bingham, is an antipodally symmetric probability distribution on the n-sphere.[1]

It is widely used in paleomagnetic data analysis,[2] and has been reported as being of use in the field of computer vision.[3][4][5]

Its probability density function is given by

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

which may also be written

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

where x is an axis, M is an orthogonal orientation matrix, Z is a diagonal concentration matrix, ${}_{1}F_{1}(\cdot;\cdot,\cdot)$ is a confluent hypergeometric function of matrix argument.