math lessons - Scale-inverse-chi-square distribution

The scale-inverse-chi-square distribution arises in Bayesian data analysis (spam filtering in particular). It is a more general distribution than the inverse-chi-square distribution. Its probability density function over the domain $x > 0$ is

$p(x; \nu, \sigma^2) = \frac{(\nu/2)^{-\nu/2}}{\Gamma(\nu/2)} (\sigma^2)^\nu x^{-\nu/2+1} \exp \left( \frac{-\nu \sigma^2}{2 x} \right)$

Related distributions

$Y \sim \mbox{Scale-inv-}\chi^2(\nu, \sigma^2)$ if for the chi-square distribution $Y = \frac{\sigma^2 \nu}{X}$ and $X \sim \chi^2(\nu)$
$X \sim \mbox{Scale-inv-}\chi^2(\nu, \sigma^2)$ if for the gamma distribution $X \sim Gamma\left(\frac{\nu}{2}, \frac{\nu \sigma^2}{2}\right)$