Solution
= Solution
Write $D_s=M_s+\varepsilon_s$. Then
$$
\widetilde M_s-M_s=b\varepsilon_s+(1-b)(M_0-M_s).
$$
The terms are independent and centered, so
$$
\mathbb E(\widetilde M_s-M_s)^2
=b^2\sigma^2+(1-b)^2\tau^2
=\frac{\tau^2\sigma^2}{\tau^2+\sigma^2}
=b\sigma^2<\sigma^2.
$$
The unpooled estimate $D_s$ has mean squared error $\sigma^2$, so population shrinkage improves it.