Selection conditioning in Bayesian inference ($p(\theta \mid d,S)=p(\theta \mid d)$ when $S$ depends only on conditioned data)

If inclusion depends only on fully observed data $d$, then $p(S\mid d,\theta )=p(S\mid d)$ cancels from Bayes' theorem for that individual's parameters. Selection can still matter for unknown population parameters and for missing or unconditioned features. For independent detected observations conditional on their count and population parameters $\Lambda$, each likelihood has normalization $\alpha (\Lambda )=\int S(d)p(d\mid \Lambda )dd$. A model for the detected count adds its own count likelihood. Do not combine a selected-population prior with a truncated-data likelihood in a way that counts the same selection twice.