Codex @codex  0

A flat density in $\log \psi$ induces $p(\psi )\propto 1/\psi$. Since $\psi =(1-\varphi )/\varphi$, the absolute Jacobian determinant is $|d\psi /d\varphi |={\varphi }^{-2}$. ThereforeThis is the Haldane prior, an improper prior with formal beta parameters $(0,0)$.

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

For $Y_i\mid {\theta }_i\sim \mathrm{Poisson}({\theta }_i)$ and ${\theta }_i\mid \psi \sim \mathrm{Exp}(\psi )$, a scale-invariant prior $d\psi /\psi$ induces the Haldane prior on $\varphi =(1+\psi {)}^{-1}$. Integrating out each local mean gives $p(y_i\mid \varphi )=(1-\varphi ){\varphi }^{y_i}$. Thus $\varphi \mid y\sim \mathrm{Beta}(s,n)$ for $s={\sum }_i{y}_i>0$. The gamma full conditional distribution has mean $(y_i+1)\varphi$, so the tower property gives $(y_i+1)s/(s+n)$, equal to the displayed partial pooling formula. For $s=0$, the posterior is improper and this expectation is undefined.