# Generalized inverse Gaussian distribution

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Parameters Probability density function a > 0, b > 0, p real x > 0 $f(x)={\frac {(a/b)^{p/2}}{2K_{p}({\sqrt {ab}})}}x^{(p-1)}e^{-(ax+b/x)/2}$ $\operatorname {E} [x]={\frac {{\sqrt {b}}\ K_{p+1}({\sqrt {ab}})}{{\sqrt {a}}\ K_{p}({\sqrt {ab}})}}$ $\operatorname {E} [x^{-1}]={\frac {{\sqrt {a}}\ K_{p+1}({\sqrt {ab}})}{{\sqrt {b}}\ K_{p}({\sqrt {ab}})}}-{\frac {2p}{b}}$ $\operatorname {E} [\ln x]=\ln {\frac {\sqrt {b}}{\sqrt {a}}}+{\frac {\partial }{\partial p}}\ln K_{p}({\sqrt {ab}})$ ${\frac {(p-1)+{\sqrt {(p-1)^{2}+ab}}}{a}}$ $\left({\frac {b}{a}}\right)\left[{\frac {K_{p+2}({\sqrt {ab}})}{K_{p}({\sqrt {ab}})}}-\left({\frac {K_{p+1}({\sqrt {ab}})}{K_{p}({\sqrt {ab}})}}\right)^{2}\right]$ $\left({\frac {a}{a-2t}}\right)^{\frac {p}{2}}{\frac {K_{p}({\sqrt {b(a-2t)}})}{K_{p}({\sqrt {ab}})}}$ $\left({\frac {a}{a-2it}}\right)^{\frac {p}{2}}{\frac {K_{p}({\sqrt {b(a-2it)}})}{K_{p}({\sqrt {ab}})}}$ In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function

$f(x)={\frac {(a/b)^{p/2}}{2K_{p}({\sqrt {ab}})}}x^{(p-1)}e^{-(ax+b/x)/2},\qquad x>0,$ 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. It is also known as the Sichel distribution, after Herbert Sichel. Its statistical properties are discussed in Bent Jørgensen's lecture notes.

## Properties

### Summation

Barndorff-Nielsen and Halgreen proved that the GIG distribution has Infinite divisibility

### Entropy

The entropy of the generalized inverse Gaussian distribution is given as[citation needed]

$H(f(x))={\frac {1}{2}}\log \left({\frac {b}{a}}\right)+\log \left(2K_{p}\left({\sqrt {ab}}\right)\right)-(p-1){\frac {\left[{\frac {d}{d\nu }}K_{\nu }\left({\sqrt {ab}}\right)\right]_{\nu =p}}{K_{p}\left({\sqrt {ab}}\right)}}+{\frac {\sqrt {ab}}{2K_{p}\left({\sqrt {ab}}\right)}}\left(K_{p+1}\left({\sqrt {ab}}\right)+K_{p-1}\left({\sqrt {ab}}\right)\right)$ where $\left[{\frac {d}{d\nu }}K_{\nu }\left({\sqrt {ab}}\right)\right]_{\nu =p}$ is a derivative of the modified Bessel function of the second kind with respect to the order $\nu$ evaluated at $\nu =p$ ### Differential equation

The pdf of the generalized inverse Gaussian distribution is a solution to the following differential equation:

$\left\{{\begin{array}{l}f(x)(x(ax-2p+2)-b)+2x^{2}f'(x)=0,\\f(1)={\frac {e^{{\frac {1}{2}}(-a-b)}\left({\frac {a}{b}}\right)^{p/2}}{2K_{p}\left({\sqrt {ab}}\right)}}\end{array}}\right\}$ ## Related distributions

### Special cases

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

$f(x;\mu ,\lambda )=\left[{\frac {\lambda }{2\pi x^{3}}}\right]^{1/2}\exp {\frac {-\lambda (x-\mu )^{2}}{2\mu ^{2}x}}$ is a GIG with $a=\lambda /\mu ^{2}$ , $b=\lambda$ , and $p=-1/2$ . A Gamma distribution of the form

$g(x;\alpha ,\beta )=\beta ^{\alpha }{\frac {1}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}$ is a GIG with $a=2\beta$ , $b=0$ , and $p=\alpha$ .

Other special cases include the inverse-gamma distribution, for a=0, and the hyperbolic distribution, for p=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 $z$ , be GIG:

$P(z|a,b,p)={\text{GIG}}(z|a,b,p)$ and let there be $T$ observed data points, $X=x_{1},\ldots ,x_{T}$ , with normal likelihood function, conditioned on $z$ :

$P(X|z,\alpha ,\beta )=\prod _{i=1}^{T}N(x_{i}|\alpha +\beta z,z)$ where $N(x|\mu ,v)$ is the normal distribution, with mean $\mu$ and variance $v$ . Then the posterior for $z$ , given the data is also GIG:

$P(z|X,a,b,p,\alpha ,\beta )={\text{GIG}}(z|a+T\beta ^{2},b+S,p-{\tfrac {T}{2}})$ where $\textstyle S=\sum _{i=1}^{T}(x_{i}-\alpha )^{2}$ .[note 1]