Generalized inverse Gaussian
Probability density function
a > 0, b > 0, p real|
x > 0|
In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function
where Kp is a modified Bessel function of the second kind, a > 0, b > 0 and p a real parameter. It is used extensively in geostatistics, statistical linguistics, finance, etc. This distribution was first proposed by Étienne Halphen.
It was rediscovered and popularised by Ole Barndorff-Nielsen, who called it the generalized inverse Gaussian distribution. Its statistical properties are discussed in Bent Jørgensen's lecture notes.
By setting and , we can alternatively express the GIG distribution as
where is the concentration parameter while is the scaling parameter.
Barndorff-Nielsen and Halgreen proved that the GIG distribution is infinitely divisible.
The entropy of the generalized inverse Gaussian distribution is given as
where is a derivative of the modified Bessel function of the second kind with respect to the order evaluated at
The characteristic of a random variable is given as(for a derivation of the characteristic function, see supplementary materials of  )
for where denotes the imaginary number.
The inverse Gaussian and gamma distributions are special cases of the generalized inverse Gaussian distribution for p = −1/2 and b = 0, respectively. Specifically, an inverse Gaussian distribution of the form
is a GIG with , , and . A Gamma distribution of the form
is a GIG with , , and .
Other special cases include the inverse-gamma distribution, for a = 0.
Conjugate prior for Gaussian
The GIG distribution is conjugate to the normal distribution when serving as the mixing distribution in a normal variance-mean mixture. Let the prior distribution for some hidden variable, say , be GIG:
and let there be observed data points, , with normal likelihood function, conditioned on
where is the normal distribution, with mean and variance . Then the posterior for , given the data is also GIG:
where .[note 1]
The Sichel distribution results when the GIG is used as the mixing distribution for the Poisson parameter .