Solution
= Solution
Multiplying the likelihood in part (d) by the <Haldane prior> gives $p(\phi\mid y)\propto\phi^{s-1}(1-\phi)^{n-1}$. Consequently
$$
\boxed{\phi\mid y\sim\operatorname{Beta}(s,n),\qquad s=\sum_i y_i>0.}
$$
For at least one hospital, $n>0$, this posterior is proper exactly when $s>0$. If every count is zero, the kernel diverges at $\phi=0$ and does not define a <posterior distribution>; the <improper prior> cannot then be used for the subsequent expectation.