Astronomical selection function ($S(d)\in [0,1]$)

An astronomical selection function specifies the probability that an object or its measured data enter a catalogue. It can depend on sky position, flux, colour, angular extent, time sampling and follow-up success. A generative calculation combines the parent population, instrument response, measurement noise and inclusion rule. A selection cut on latent properties must be propagated through the measurement likelihood rather than substituted for a cut on observed quantities.

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.

survey completeness is the fraction of a specified true source population recovered by a survey, conditional on properties such as flux, size and sky position. Injecting simulated sources into realistic data and applying the actual detection pipeline measures recovery probability. Completeness differs from purity: recovered objects can include false detections or incorrectly identified sources.