The Overlooked Potential of Generalized Linear Models in Astronomy-III: Bayesian Negative Binomial Regression and Globular Cluster Populations

TL;DR

This paper introduces a Bayesian negative binomial regression model to analyze globular cluster populations in galaxies, effectively handling count data and measurement errors, and revealing differences among galaxy types.

Contribution

It develops a novel Bayesian negative binomial regression approach for galaxy globular cluster counts, accounting for heteroscedasticity, measurement errors, and galaxy types, improving modeling accuracy.

Findings

The model provides 99% prediction intervals that include the Milky Way.

S0 galaxies have approximately 35% fewer GCs than similar brightness other types.

The approach naturally handles measurement errors and intrinsic scatter.

Abstract

In this paper, the third in a series illustrating the power of generalized linear models (GLMs) for the astronomical community, we elucidate the potential of the class of GLMs which handles count data. The size of a galaxy's globular cluster population $N_{\rm GC}$ is a prolonged puzzle in the astronomical literature. It falls in the category of count data analysis, yet it is usually modelled as if it were a continuous response variable. We have developed a Bayesian negative binomial regression model to study the connection between $N_{\rm GC}$ and the following galaxy properties: central black hole mass, dynamical bulge mass, bulge velocity dispersion, and absolute visual magnitude. The methodology introduced herein naturally accounts for heteroscedasticity, intrinsic scatter, errors in measurements in both axes (either discrete or continuous), and allows modelling the population of globular clusters on their natural scale as a non-negative integer variable. Prediction intervals of 99% around the trend for expected $N_{\rm GC}$comfortably envelope the data, notably including the Milky Way, which has hitherto been considered a problematic outlier. Finally, we demonstrate how random intercept models can incorporate information of each particular galaxy morphological type. Bayesian variable selection methodology allows for automatically identifying galaxy types with different productions of GCs, suggesting that on average S0 galaxies have a GC population 35% smaller than other types with similar brightness.