The likelihood is
$$y_i \sim \text{binomial}(\gamma_i; w_i)$$
where
\(y_i\) is a count, such of number of births, for some
combination \(i\) of classifying variables,
such as age, sex, and region;
\(\gamma_i\) is a probability of 'success'; and
\(w_i\) is the number of trials.
The probabilities \(\gamma_i\) are assumed to be drawn
a beta distribution
$$y_i \sim \text{Beta}(\xi^{-1} \mu_i, \xi^{-1} (1 - \mu_i))$$
where
Expected value \(\mu_i\) equals, on a logit scale,
the sum of terms formed from classifying variables,
$$\text{logit} \mu_i = \sum_{m=0}^{M} \beta_{j_i^m}^{(m)}$$
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.
The prior for \(\xi\) is described in set_disp().