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.Comment: 12 pages, 5 figures; to appear in Bayesian Inference And Maximum
Entropy Methods In Science And Engineering: 24th International Workshop,
Garching, Germany, 2004; ed. Volker Dose et al. (AIP Conference Proceedings
Series