Generalized inverse Gaussian distribution

Alternative parametrization By setting \theta = \sqrt{ab} and \eta = \sqrt{b/a}, we can alternatively express the GIG distribution as f(x) = \frac{1}{2\eta K_p(\theta)} \left(\frac{x}{\eta}\right)^{p-1} e^{-\theta(x/\eta + \eta/x)/2}, where \theta is the concentration parameter while \eta is the scaling parameter. Summation Barndorff-Nielsen and Halgreen proved that the GIG distribution is infinitely divisible. Entropy The entropy of the generalized inverse Gaussian distribution is given as \begin{align} H = \frac{1}{2} \log \left( \frac b a \right) & {} +\log \left(2 K_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{a b}\right)} \\ & {} + \frac{\sqrt{a b}}{2 K_p\left(\sqrt{a b}\right)}\left( K_{p+1}\left(\sqrt{ab}\right) + K_{p-1}\left(\sqrt{a b}\right)\right) \end{align} where \left[\frac{d}{d\nu}K_\nu\left(\sqrt{a b}\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 Characteristic Function The characteristic of a random variable X\sim GIG(p, a, b) is given as E(e^{itX}) = \left(\frac{a }{a-2it }\right)^{\frac{p}{2}} \frac{K_{p}\left( \sqrt{(a-2it)b} \right)}{ K_{p}\left( \sqrt{ab} \right) } for t \in \mathbb{R} where i denotes the imaginary unit. ==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. Let the prior distribution for some hidden variable, say z, be GIG: P(z\mid a,b,p) = \operatorname{GIG}(z\mid 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\mid z,\alpha,\beta) = \prod_{i=1}^T N(x_i\mid\alpha+\beta z,z) where N(x\mid\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\mid X,a,b,p,\alpha,\beta) = \text{GIG}\left(z\mid a+T\beta^2,b+S,p-\frac T 2 \right) where S = \sum_{i=1}^T (x_i-\alpha)^2. Sichel distribution The Sichel distribution results when the GIG is used as the mixing distribution for the Poisson parameter \lambda. ==Notes==