 Normal mean-variance posterior with a log-uniform variance prior

For iid $N(\mu ,v)$ observations, the prior density $1/v$ relative to $d\mu dv$ gives posterior kernel $v^{-n/2-1}\exp [-\{S+n(\mu -\bar{x}{)}^2\}/(2v)]$. It is proper exactly when $n>1$ and $S=\sum (x_i-\bar{x}{)}^2>0$. The conditionals are $\mu \mid v,x\sim N(\bar{x},v/n)$ and $v\mid \mu ,x\sim \mathrm{IG}(n/2,{\sum }_i(x_i-\mu {)}^2/2)$. Integrating out $\mu$ changes the inverse-gamma shape to $(n-1)/2$.