Accounting for Source Uncertainties in Analyses of Astronomical Survey Data

I discuss an issue arising in analyzing data from astronomical surveys: accounting for measurement uncertainties in the properties of individual sources detected in a survey when making inferences about the entire population of sources. Source uncertainties require the analyst to introduce unknown ``incidental'' parameters for each source. The number of parameters thus grows with the size of the sample, and standard theorems guaranteeing asymptotic convergence of maximum likelihood estimates fail in such settings. From the Bayesian point of view, the missing ingredient in such analyses is accounting for the volume in the incidental parameter space via marginalization. I use simple simulations, motivated by modeling the distribution of trans-Neptunian objects surveyed in the outer solar system, to study the effects of source uncertainties on inferences. The simulations show that current non-Bayesian methods for handling source uncertainties (ignoring them, or using an ad hoc incidental parameter integration) produce incorrect inferences, with errors that grow more severe with increasing sample size. In contrast, accounting for source uncertainty via marginalization leads to sound inferences for any sample size.