Commentary on Bayesian coincidence assessment (cross-matching)

This paper is an invited commentary on Tamas Budavari's presentation, "On statistical cross-identification in astronomy," for the Statistical Challenges in Modern Astronomy V conference held at Pennsylvania State University in June 2011. I begin with a brief review of previous work on probabilistic (Bayesian) assessment of directional and spatio-temporal coincidences in astronomy (e.g., cross-matching or cross-identification of objects across multiple catalogs). Then I discuss an open issue in the recent innovative work of Budavari and his colleagues on large-scale probabilistic cross-identification: how to assign prior probabilities that play an important role in the analysis. With a simple toy problem, I show how Bayesian multilevel modeling (hierarchical Bayes) provides a principled framework that justifies and generalizes pragmatic rules of thumb that have been successfully used by Budavari's team to assign priors.