Hierarchical Bayesian Model to Infer PL(Z) Relations Using Gaia Parallaxes

In a recent study we analysed period–luminosity–metallicity (PLZ) relations for RR Lyrae stars using the Gaia Data Release 2 (DR2) parallaxes. It built on a previous work that was based on the first Gaia Data Release (DR1), and also included period–luminosity (PL) relations for Cepheids and RR Lyrae stars. The method used to infer the relations from Gaia DR2 data and one of the methods used for Gaia DR1 data was based on a Bayesian model, the full description of which was deferred to a subsequent publication. This paper presents the Bayesian method for the inference of the parameters of PL(Z) relations used in those studies, the main feature of which is to manage the uncertainties on observables in a rigorous and well-founded way. The method encodes the probability relationships between the variables of the problem in a hierarchical Bayesian model and infers the posterior probability distributions of the PL(Z) relationship coefficients using Markov chain Monte Carlo simulation techniques. We evaluate the method with several semi-synthetic data sets and apply it to a sample of 200 fundamental and first-overtone RR Lyrae stars for which Gaia DR1 parallaxes and literature Ks-band mean magnitudes are available. We define and test several hyperprior probabilities to verify their adequacy and check the sensitivity of the solution with respect to the prior choice. The main conclusion of this work, based on the test with semi-synthetic Gaia DR1 parallaxes, is the absolute necessity of incorporating the existing correlations between the period, metallicity, and parallax measurements in the form of model priors in order to avoid systematically biased results, especially in the case of non-negligible uncertainties in the parallaxes. The relation coefficients obtained here have been superseded by those presented in our recent paper that incorporates the findings of this work and the more recent Gaia DR2 measurements.