Gamma–Poisson hierarchical model

A Gamma–Poisson hierarchical model uses $Y_i\mid {\theta }_i\sim \mathrm{Poisson}(t_i{\theta }_i)$, ${\theta }_i\mid b\sim \mathrm{Gamma}(a,b)$, and $b\sim \mathrm{Gamma}(c,r)$, with gamma distributions in shape-rate convention. Conditional independence gives full conditional distributions ${\theta }_i\mid b,y\sim \mathrm{Gamma}(a+y_i,b+t_i)$ and $b\mid \theta ,y\sim \mathrm{Gamma}(c+na,r+{\sum }_i{\theta }_i)$.

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.

For independent Poisson distributions with means $u{e}^{{\eta }_k}$ and a shape–rate Gamma distribution prior on $u$, integrating $u$ adds $a$ to the total-count exponent and $b$ to the sum of relative rates. This distinguishes gamma mixing from conditioning on the total count, which gives a multinomial distribution.