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 ( p − 1 ) {displaystyle (p-1)} -dimensional sphere in R p {displaystyle mathbb {R} ^{p}} . If p = 2 {displaystyle p=2} the distribution reduces to the von Mises distribution on the circle. In directional statistics, the von Mises–Fisher distribution (named after Ronald Fisher and Richard von Mises), is a probability distribution on the ( p − 1 ) {displaystyle (p-1)} -dimensional sphere in R p {displaystyle mathbb {R} ^{p}} . If p = 2 {displaystyle p=2} 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 x {displaystyle mathbf {x} ,} is given by: where κ ≥ 0 , ‖ μ ‖ = 1 {displaystyle kappa geq 0,leftVert {oldsymbol {mu }} ightVert =1,} and the normalization constant C p ( κ ) {displaystyle C_{p}(kappa ),} is equal to where I v {displaystyle I_{v}} denotes the modified Bessel function of the first kind at order v {displaystyle v} . If p = 3 {displaystyle p=3} , the normalization constant reduces to The parameters μ {displaystyle mu ,} and κ {displaystyle kappa ,} are called the mean direction and concentration parameter, respectively. The greater the value of κ {displaystyle kappa ,} , the higher the concentration of the distribution around the mean direction μ {displaystyle mu ,} . The distribution is unimodal for κ > 0 {displaystyle kappa >0,} , and is uniform on the sphere for κ = 0 {displaystyle kappa =0,} . The von Mises–Fisher distribution for p = 3 {displaystyle p=3} , 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. A series of N independent measurements x i {displaystyle x_{i}} are drawn from a von Mises–Fisher distribution. Define Then (Sra, 2011) the maximum likelihood estimates of μ {displaystyle mu ,} and κ {displaystyle kappa ,} are given by Thus κ {displaystyle kappa ,} is the solution to