A Bayesian approach for variable selection in mixture of logistic regressions with Pólya-Gamma data augmentation

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We present Bayesian methods for estimating and selecting variables in a mixture of logistic regression models. A common issue with the logistic model is its intractable likelihood, which prevents us from applying simpler Bayesian algorithms, such as Gibbs sampling, for estimating and selecting the model since there is no conjugacy for the regression coefficients. We propose to solve this problem by applying the data augmentation approach with Pólya-Gamma random variables to the logistic regression mixture model. For selecting covariates in this model, we investigate the performance of two prior distributions for the regression coefficients. A Gibbs sampling algorithm is then applied to perform variable selection and fit the model. The conjugacy obtained for the distribution of the regression coefficients allows us to analytically calculate the marginal likelihood and gain computational efficiency in the variable selection process. The methodologies are applied to both synthetic and real data.