# A hierarchical Bayesian model to infer PL(Z) relations using Gaia parallaxes

## TL;DR

This paper develops a Bayesian hierarchical model to accurately infer period-luminosity-metallicity relations for RR Lyrae stars using Gaia parallaxes, emphasizing the importance of modeling correlations to avoid bias.

## Contribution

It introduces a directed acyclic graph-based Bayesian model that incorporates correlations among observables, improving the inference of PL(Z) relations from Gaia data.

## Key findings

- Correlations between parallax, metallicity, and period are crucial for unbiased results.
- Incorporating correlations significantly impacts the inferred PL(Z) relation coefficients.
- Model sensitivity to prior choices highlights the importance of hyperprior testing.

## Abstract

Aims. We aim at creating a Bayesian model to infer the coefficients of PL or PLZ 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. The main conclusion of this work is the absolute necessity of incorporating the existing correlations between the observed variables (periods, metallicities and parallaxes) 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 tests with the semi-synthetic data based on the data set used in Gaia Collaboration et al. (2017) reveal the significant impact that the existing correlations between parallax, metallicity and periods have on the inferred parameters. The relation coefficients obtained here have been superseded by those presented in Muraveva et al. (2018a), that incorporates the findings of this work and the more recent Gaia DR2 measurements.