 Normal-normal conjugacy with known observation variance

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.