The likelihood is
$$y_i \sim \text{Poisson}(\gamma_i w_i)$$
where
subscript \(i\) identifies some combination of the
classifying variables, such as age, sex, and time;
\(y_i\) is an outcome, such as deaths;
\(\gamma_i\) is rates; and
\(w_i\) is exposure.
In some applications, there is no obvious population at risk.
In these cases, exposure \(w_i\) can be set to 1
for all \(i\).
The rates \(\gamma_i\) are assumed to be drawn
a gamma distribution
$$y_i \sim \text{Gamma}(\xi^{-1}, (\xi \mu_i)^{-1})$$
where
\(\mu_i\) is the expected value for \(\gamma_i\); and
\(\xi\) governs dispersion (i.e. variation), with
lower values implying less dispersion.
Expected value \(\mu_i\) equals, on the log scale,
the sum of terms formed from classifying variables,
$$\log \mu_i = \sum_{m=0}^{M} \beta_{j_i^m}^{(m)}$$
where
\(\beta^{0}\) is an intercept;
\(\beta^{(m)}\), \(m = 1, \dots, M\), is a main effect
or interaction; and
\(j_i^m\) is the element of \(\beta^{(m)}\) associated with
cell \(i\).
The \(\beta^{(m)}\) are given priors, as described in priors.
\(\xi\) has an exponential prior with mean 1. Non-default
values for the mean can be specified with set_disp().
The model for \(\mu_i\)
can also include covariates,
as described in set_covariates().