The Poisson-Gamma model, as described by Clayton and Kaldor,
is a two-layers Bayesian Hierarchical model:$$O_i|\theta_i \sim Po(\theta_i E_i)$$
$$\theta_i \sim Ga(\nu, \alpha)$$
The posterior distribution of $O_i$,unconditioned to
$\theta_i$, is Negative Binomial with size $\nu$ and
probability $\alpha/(\alpha+E_i)$.
The estimators of relative risks are
$\widehat{\theta}_i=\frac{O_i+\nu}{E_i+\alpha}$.
Estimators of $\nu$ and $\alpha$
($\widehat{\nu}$ and $\widehat{\alpha}$,respectively)
are calculated by means of an iterative procedure using these two equations
(based on mean and variance estimations):
$$\frac{\widehat{\nu}}{\widehat{\alpha}}=\frac{1}{n}\sum_{i=1}^n
\widehat{\theta}_i$$
$$\frac{\widehat{\nu}}{\widehat{\alpha}^2}=\frac{1}{n-1}\sum_{i=1}^n(1+\frac{\widehat{\alpha}}{E_i})(\widehat{\theta}_i-\frac{\widehat{\nu}}{\widehat{\alpha}})^2$$