Neat little paper on astro-ph today by Mitchell-Wynne et al. presenting a maximum likelihood/Bayesian approach to inferring the intrinsic trend of radio source counts below a survey’s nominal detection threshold. A similar idea has occurred to me before as well in the context of fitting the galaxy luminosity function: since there is clearly information hidden in the extra-Poissonian noise below our ~3 sigma detection threshold, why not make use of it? [Particularly in the galaxy LF case where this analysis could help to distinguish between models with differing faint end slopes.] However, modelling the faint source population as viewed through a realistically noisy observational process is highly non-trivial so it’s going to take a lot of work to persuade a skeptical audience of the robustness of any proposed method.
In this sense I think Mitchell-Wynne’s study presents a solid ”first go” under some substantial simplifying assumptions: power-law flux density model; Gaussian background noise only; and no false-detections owing to their targeted search at the location of previously identified sources at other wavelengths. The latter condition in particular does raise some questions as to the appropriateness of the power-law model adopted: if we have information about the flux of these targets in other wavelengths then perhaps we should be looking rather at a hierarchical model for galaxy SED “colour”? The authors’ use of parallel tempering (even referencing Liu 2001!) for posterior sampling is encouraging though, since as we know from reverse logistic regression (etc.) this method allows easy access to estimates of the marginal likelihood (with prior-sensitivity analysis) should they extend their investigation in future to a model selection framework.
The problem remains though that the robustness of any such faint source analysis depends critically on a correct modelling of the observational process. [The authors’ suggest at least one improvement would be to allow for a locally variable background RMS.] Here I would advocate further testing of the artificial galaxy simulation variety; e.g. constructing a mock catalogue based on source injection of a population constructed according to the mode of their posterior power-law parameterisation into the real noisy images at random positions displaced from their target sightlines, reapplying the MCMC scheme, and confirming recovery of the known solution.