I spent my research time today writing in a paper Rix (MPIA) and I are preparing about selecting sources for a catalog or target selection. The fundamental story is that you need to make a likelihood function at the end of the day. And this, in turn, means that you need a tractable and relatively accurate selection function. This all took me down old paths I have traveled with Bovy (Toronto) and Foreman-Mackey (Flatiron).
In email correspondence, Foreman-Mackey reminded me of past correspondence with Loredo (Cornell), who disagrees with our work on these things for very technical reasons. His (very nice) explanation of his point is around equations (8) through (10) in this paper: It has to do with how to factorize a probability distribution for a collection of objects obtained in a censored, variable-rate Poisson process. But our historical view of this (and my restored view after a day of struggling) is that the form of the likelihood depends on fine details of how you believe the objects of study were selected for the catalog, or censored. If they were censored only by your detector, I think Loredo's form is correct. But if they were censored for physical reasons over which you have no dominion (for example a planet transit obscured by a tiny fluctuation in a star's brightness), the selection can come in to the likelihood function differently. That is, it depends on the causal chain involved in the source censoring.
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