Selection-normalized astronomical likelihood

For population parameters $\eta$ and an inclusion region $S$ in measured-data space, a selected datum has density $1_S(d)p(d\mid \eta )/P(S\mid \eta )$. The normalization integrates the noisy measurement model and population over the actual selection region. Inference must also specify what is conditioned on: deterministic selection information already contained in the conditioned data must not be counted twice.