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):

$$\frac{\widehat{\nu}}{\widehat{\alpha}}=\frac{1}{n}\sum_{i=1}^n
\widehat{\theta}_i$$

$$\frac{\widehat{\nu}}{\widehat{\alpha}^2}=\frac{1}{n-1}\sum_{i=1}^n(1+\frac{\widehat{\alpha}}{E_i})(\widehat{\theta}_i-\frac{\widehat{\nu}}{\widehat{\alpha}})^2$$