 Scaled inverse chi-squared distribution

For parameters $\nu >0$ and $b^2>0$, the Scaled inverse chi-squared distribution is the law ofThe inverse change of variables $q=\nu b^2/x$ gives its normalized densityIt is an Inverse-gamma distribution with shape $\nu /2$ and scale $\nu b^2/2$. The label $b^2$ is a scale parameter, not the variance of $X$.

Its mean is $\nu b^2/(\nu -2)$ for $\nu >2$, and its variance is $2{\nu }^2{b}^4/[(\nu -2{)}^2(\nu -4)]$ for $\nu >4$. Below those respective thresholds the positive moments diverge. For example, $\nu =6,b^2=2$ gives mean three and variance nine.

This is a conjugate prior for the unknown variance of a Normal distribution with known mean $\mu$. With prior parameters ${\nu }_0,b_0^2$ and $n$ independent observations, the posterior has ${\nu }_n={\nu }_0+n$ and $b_n^2=[{\nu }_0{b}_0^2+{\sum }_i(x_i-\mu {)}^2]/{\nu }_n$. Multiplying prior and likelihood adds their powers and inverse-variance exponents, which proves the update. An unknown estimated mean requires a separate model and changes the update.