Bayesian variable selection for logistic mixture models with Pólya-Gamma data augmentation

Abstract

In this work, Bayesian methods for estimating and selecting variables in a mixture of logistic regressions model are presented. In order to simplify its Bayesian estimation, we extend the data augmentation approach with Pólya-Gamma random variables to the mixture of logistic regression models. Through the data augmentation approach, we present a Gibbs sampling algorithm for estimating the full model, and the number of components in the mixture is identified by Bayesian model selection criteria. In the model with variable selection, we investigate the performance of two prior distributions for the regression coefficients, adding a second set of latent variables to indicate the presence and non-presence of the predictor variables at each component of the mixture. Analogously to the full model, a Gibbs sampling algorithm is applied to the model with variable selection and the conjugation obtained for the distribution of the regression coefficients, through the inclusion of Pólya-Gamma variables, allows us to analytically calculate the marginal likelihood and gain computational efficiency in the variable selection process. To analyse the performance, the presented methodologies are applied in simulated and real data.