Normal-normal conjugacy gives
$$
M_s\mid M_0,\tau^2,D_s
\sim N\left(
\frac{\tau^2D_s+\sigma^2M_0}{\tau^2+\sigma^2},
\frac{\tau^2\sigma^2}{\tau^2+\sigma^2}\right).
$$
With $b=\tau^2/(\tau^2+\sigma^2)$,
$$
\widetilde M_s=bD_s+(1-b)M_0,
\qquad
\operatorname{Var}(M_s\mid D_s)=b\sigma^2.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 219 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-219/3/a/solution