The likelihood is
$$y_i \sim \text{N}(\mu_i, \xi^2 / w_i)$$
where
\(y_i\) is a scaled value for an, such of the log of income, for some
combination \(i\) of classifying variables,
such as age, sex, and region;
\(\mu_i\) is a mean;
\(\xi\) is a standard deviation parameter; and
\(w_i\) is a weight.
The scaling of the outcome variable is done internally.
If \(y_i^*\) is the original, then \(y_i = (y_i^* - m)/s\)
where \(m\) and \(s\) are the sample mean and standard
deviation of \(y_i^*\).
In some applications, \(w_i\) is set to 1
for all \(i\).
The means \(\mu_i\) equal the sum of terms formed
from classifying variables,
$$\mu_i = \sum_{m=0}^{M} \beta_{j_i^m}^{(m)}$$
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().