all the catalogs consistent with an image

I got up exceedingly early in the morning, highly motivated to write a short theory paper—that's theory of data analysis, of course—about the posterior probability distribution over catalogs. I have become motivated to write something like this because I have started to become concerned about various bits of wrongness in my old paper with Turner on faint source photometry. The principal results or conclusions of the paper are not wrong, but the language is all wrong (the terms likelihood and measurement are used wrongly) and the priors are improper. I asked Phil Marshall what you do about a paper that is wrong; he said: Write a new paper correcting it!

One of the key things that has to be fixed in the problem is that the explanation of an image in terms of a catalog is—or should be—properly probabilistic. That means that if the image is not high in signal to noise, there are necessarily many even qualitatively different catalogs that could explain the image. This means describing or sampling a posterior distribution over models with varying complexity (or number of parameters or number of sources). That's not a trivial problem, of course.

The nice thing, if I can do it all, is that the new paper will not just resolve the issues with Hogg & Turner, it will also generalize the result to include positional and confusion noise, all in one consistent framework. The key idea is that for any source population you care about (faint galaxies, say, or Milky-Way stars), it is very easy to write down a proper and informative prior over catalog space (because, as Marshall often reminds me, we can simulate imaging data and the catalogs they imply very accurately).