Pattern Recognition and Machine Learning

B. PROBABILITY DISTRIBUTIONS 693

Von Mises

The von Mises distribution, also known as the circular normal or the circular Gaus-
sian, is a univariate Gaussian-like periodic distribution for a variableθ∈[0, 2 π).

p(θ|θ 0 ,m)=

1

2 πI 0 (m)

exp{mcos(θ−θ 0 )} (B.77)

whereI 0 (m)is the zeroth-order Bessel function of the first kind. The distribution
has period 2 πso thatp(θ+2π)=p(θ)for allθ. Care must be taken in interpret-
ing this distribution because simple expectations will be dependent on the (arbitrary)
choice of origin for the variableθ. The parameterθ 0 is analogous to the mean of a
univariate Gaussian, and the parameterm> 0 , known as theconcentrationparam-
eter, is analogous to the precision (inverse variance). For largem, the von Mises
distribution is approximately a Gaussian centred onθ 0.

Wishart

The Wishart distribution is the conjugate prior for the precision matrix of a multi-
variate Gaussian.

W(Λ|W,ν)=B(W,ν)|Λ|(ν−D−1)/^2 exp

(

−

1

2

Tr(W−^1 Λ)

)

(B.78)

where

B(W,ν) ≡|W|−ν/^2

(

2 νD/^2 πD(D−1)/^4

∏D

i=1

Γ

(

ν+1−i

2

))−^1

(B.79)

E[Λ]=νW (B.80)

E[ln|Λ|]=

∑D

i=1

ψ

(

ν+1−i

2

)

+Dln 2 + ln|W| (B.81)

H[Λ]=−lnB(W,ν)−

(ν−D−1)

2

E[ln|Λ|]+

νD

2

(B.82)

whereWis aD×Dsymmetric, positive definite matrix, andψ(·)is the digamma
function defined by (B.25). The parameterνis called thenumber of degrees of
freedomof the distribution and is restricted toν>D− 1 to ensure that the Gamma
function in the normalization factor is well-defined. In one dimension, the Wishart
reduces to the gamma distributionGam(λ|a, b)given by (B.26) with parameters
a=ν/ 2 andb=1/ 2 W.