We present a hierarchical Bayesian method for estimating the total mass and mass profile of the Milky Way Galaxy. The new hierarchical Bayesian approach further improves the framework presented by Eadie et al. (2015b); Eadie & Harris (2016) and builds upon the preliminary reports by Eadie et al. (2015a,c). The method uses a distribution function f (E, L) to model the galaxy and kinematic data from satellite objects such as globular clusters (GCs) to trace the Galaxy's gravitational potential. A major advantage of the method is that it not only includes complete and incomplete data simultaneously in the analysis, but also incorporates measurement uncertainties in a coherent and meaningful way. We first test the hierarchical Bayesian framework, which includes measurement uncertainties, using the same data and power-law model assumed in Eadie & Harris (2016), and find the results are similar but more strongly constrained. Next, we take advantage of the new statistical framework and incorporate all possible GC data, finding a cumulative mass profile with Bayesian credible regions. This profile implies a mass within 125kpc of 4.8 × 1011 M with a 95% Bayesian credible region of (4.0 − 5.8) × 10 11 M . Our results also provide estimates of the true specific energies of all the GCs. By comparing these estimated energies to the measured energies of GCs with complete velocity measurements, we observe that (the few) remote tracers with complete measurements may play a large role in determining a total mass estimate of the Galaxy. Thus, our study stresses the need for more remote tracers with complete velocity measurements.
A common approach for characterizing the properties of time-series data that are evenly sampled in time is to estimate the power spectrum of the data using the periodogram. The periodogram as an estimator of the spectrum is (1) statistically inconsistent (i.e., its variance does not go to zero as infinite data are collected), (2) biased for finite samples, and (3) suffers from spectral leakage. In astronomy, time-series data are often unevenly sampled in time, and it is popular to use the Lomb–Scargle (LS) periodogram to estimate the spectrum. Unfortunately, from a statistical standpoint, the LS periodogram suffers from the same issues as the classical periodogram and has even worse spectral leakage. Here, we present an improvement on the LS periodogram by combining it with the Thomson multitaper approach. The multitaper spectral estimator is well established in the statistics and engineering literature for evenly sampled time series. It is attractive because it directly trades off bias and variance for frequency resolution, and is fast to compute: compared to an untapered spectral estimator, the multitaper adds no more than a couple of seconds for a time series with a million data points on a current desktop computer. Here, we describe an estimator that combines the multitaper with the LS periodogram. We show examples in which this new approach has improved properties compared to traditional approaches in the case of unevenly sampled time series. Finally, we demonstrate an application of the method to astronomy with an application to Kepler data.
Current observational evidence suggests that all large galaxies contain globular clusters (GCs), while the smallest galaxies do not. Over what galaxy mass range does the transition from GCs to no GCs occur? We investigate this question using galaxies in the Local Group (LG), nearby dwarf galaxies, and galaxies in the Virgo Cluster Survey. We consider four types of statistical model: (1) logistic regression to model the probability that a galaxy of stellar mass M * has any number of GCs; (2) Poisson regression to model the number of GCs versus M *; (3) linear regression to model the relation between GC system mass ( log M gcs ) and host galaxy mass ( log M ⋆ ); and (4) a Bayesian lognormal hurdle model of the GC system mass as a function of galaxy stellar mass for the entire data sample. From the logistic regression, we find that the 50% probability point for a galaxy to contain GCs is M * = 106.8 M ⊙. From postfit diagnostics, we find that Poisson regression is an inappropriate description of the data. Ultimately, we find that the Bayesian lognormal hurdle model, which is able to describe how the mass of the GC system varies with M * even in the presence of many galaxies with no GCs, is the most appropriate model over the range of our data. In an Appendix, we also present photometry for the little-known GC in the LG dwarf Ursa Major II.
We present an active-learning strategy for undergraduates that applies Bayesian analysis to candy-covered chocolate m&m's R . The exercise is best suited for small class sizes and tutorial settings, after students have been introduced to the concepts of Bayesian statistics. The exercise takes advantage of the non-uniform distribution of m&m's R colours, and the difference in distributions made at two different factories. In this paper, we provide the intended learning outcomes, lesson plan and step-by-step guide for instruction, and open-source teaching materials. We also suggest an extension to the exercise for the graduate-level, which incorporates hierarchical Bayesian analysis.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations –citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.