Hierarchical Bayesian model to infer PL(Z) relations using Gaia parallaxes

Context. (Gaia Collaboration et al. 2017) analysed exhaustively the Period-Luminosity (PL) and Period-Luminosity(-Metallicity) (PL(Z)) relations for Cepheids and RR Lyrae stars using the Gaia Data Release 1 (DR1) parallaxes in the Tycho-Gaia Astrometric Solution (TGAS). One of the methods used to infer the relations was based on a hierarchical Bayesian model, the description of which was deferred to a subsequent publication that is presented here. Aims. We aim at creating a Bayesian model to infer the coefficients of PL or PL(Z) relations that propagates uncertainties in the observables in a rigorous and well founded way. Methods. We propose a directed acyclic graph to encode the conditional probabilities of the inference model that will allow us to infer probability distributions for the PL and PL(Z) relations. We evaluate the model with several semi-synthetic data sets and apply it to a sample of 200 fundamental mode and first overtone mode 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. Results. We find that our Bayesian model successfully infers probability distributions for the RR Lyrae PLZ relation in the Ks-band when it is applied to semi-synthetic data. We find that our model systematically underestimates the slope corresponding to the period term of the PLZ relation when it is applied to the Gaia DR1 RR Lyrae sample. We demonstrate that this underestimation is due to the correlation between periods and TGAS parallaxes, which in turn is a consequence of the fact that period and metallicity are correlated in RR Lyrae stars (with shorter periods being characteristic of higher metallicities).