Bayesian normal model

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 $(n-1)/2$.

For independent $X_i\sim N(\theta ,{\sigma }^2)$ and prior $\theta \sim N(\mu ,{\nu }^2)$, the posterior has variance and meanAs $n\to \infty$, $m_n-\overline{X}=O_p(n^{-1})$ and $v_n^{1/2}\sim \sigma /\sqrt{n}$, so fixed-level posterior credible intervals agree asymptotically with the corresponding known-variance confidence intervals.