I resolved (for now, to my current satisfaction) my issues from a few days ago, about likelihoods for catalogs. I showed that the likelihood that I advocate does not give biased inferences, and does permit inference of the selection function (censoring process) along with the inference of the world. I did this with my first ever use of the Google (tm) Colaboratory (tm). I wanted to see if it works, and it does. My notebook is here (subject to editing and changing, so no promises about its state when you go there). If your model includes the censoring process—that is, if you want to parameterize and learn the catalog censoring along with the model of the world—then (contra Loredo, 2004) you have to use a likelihood function that depends on the selection function at the individual-source level. And I think it is justified, because it is the assumption that the universe plus the censoring is the thing which is generating your catalog. That's a reasonable position to take.
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