How to sample from the distribution of a Gaussian scale parameter

Question

I would like to be able to sample the standard deviation of a multidimensional Gaussian distribution of dimension $n$; that is, given some $\phi$, I would like to sample

$P(\sigma | \phi) \propto \frac{1}{\sigma^n} e^{-\frac{\phi}{2\sigma^2}}$

For high $n$, this distribution is sharply peaked; for my purposes $n$ will be on the order of 1000-3000. What is an ``efficient'' method of obtaining a single sample from this distribution? (A single sample as $\phi$ will change between each sample.)

Answer 1

Your problem, in my opinion, can be solved if you are willing to sample from $\sigma^2$ and not $\sigma$.

Look at the inverse Gamma distribution, with parameter $\alpha$ and $\beta$. $\sigma^2 \sim IG(\alpha, \beta)$ if its density is : $$ \frac{\beta^\alpha (\sigma^2)^{-(\alpha+1)} \exp(-\frac{\beta}{\sigma^2}) }{\Gamma(\alpha)} $$ If you set $\alpha = \frac{n}{2}$ and $\beta = \frac{\phi}{2}$, then your distribution is proportional to $IG(\alpha, \beta)$