The HERBAL Model: A Hierarchical Errors-in-variables Bayesian Lognormal Hurdle Model for Galactic Globular Cluster Populations

Galaxy stellar mass is known to be monotonically related to the size of the galaxy’s globular cluster (GC) population for Milky Way sized and larger galaxies. However, the relation becomes ambiguous for dwarf galaxies, where there is some evidence for a downturn in GC population size at low galaxy masses. Smaller dwarfs are increasingly likely to have no GCs, and these zeros cannot be easily incorporated into linear models. We introduce the Hierarchical Errors-in-variables ERrors-in-variables BAyesian Lognormal hurdle (HERBAL) model to represent the relationship between dwarf galaxies and their GC populations, and apply it to the sample of Local Group galaxies, where the luminosity range coverage is maximal. This bimodal model accurately represents the two populations of dwarf galaxies: those that have GCs and those that do not. Our model thoroughly accounts for all uncertainties, including measurement uncertainty, uncertainty in luminosity to stellar mass conversions, and intrinsic scatter. The hierarchical nature of our Bayesian model also allows us to estimate galaxy masses and individual mass-to-light ratios from luminosity data within the model. We find that 50% of galaxies are expected to host GC populations at a stellar mass of ${\mathrm{log}}_{10}({M}_{* })=6.996$ , and that the expected mass of GC populations remains linear down to the smallest galaxies. Our hierarchical model recovers an accurate estimate of the Milky Way stellar mass. Under our assumed error model, we find a nonzero intrinsic scatter of ${0.59}_{-0.21}^{+0.3}$ (95% credible interval) that should be accounted for in future models.