Flat-prior elimination of a Gaussian common mean

For independent observations $y_s\sim N(\beta +d_s,V_s)$, put $a_s=V_s^{-1}$, $S={\sum }_s{a}_s$, $r_s=y_s-d_s$, $\bar{r}=S^{-1}{\sum }_s{a}_s{r}_s$ and $Q={\sum }_s{a}_s(r_s-\bar{r}{)}^2$. Integrating the likelihood function against a flat improper prior on $\beta$ gives, up to the arbitrary prior constant,Indeed ${\sum }_s{a}_s(r_s-\beta {)}^2=Q+S(\beta -\bar{r}{)}^2$, and the remaining one-dimensional Gaussian integral is $\sqrt{2\pi /S}$. The conditional Bayesian posterior of $\beta$ is $N(\bar{r},S^{-1})$.