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Modeling binary and categorical data is one of the most commonly encounteredtasks of applied statisticians and econometricians. While Bayesian methods inthis context have been available for decades now, they often require a highlevel of familiarity with Bayesian statistics or suffer from issues such as lowsampling efficiency. To contribute to the accessibility of Bayesian models forbinary and categorical data, we introduce novel latent variable representationsbased on P\'olya-Gamma random variables for a range of commonly encounteredlogistic regression models. From these latent variable representations, newGibbs sampling algorithms for binary, binomial, and multinomial logit modelsare derived. All models allow for a conditionally Gaussian likelihoodrepresentation, rendering extensions to more complex modeling frameworks suchas state space models straightforward. However, sampling efficiency may stillbe an issue in these data augmentation based estimation frameworks. Tocounteract this, novel marginal data augmentation strategies are developed anddiscussed in detail. The merits of our approach are illustrated throughextensive simulations and real data applications.

Ultimate P\'olya Gamma Samplers -- Efficient MCMC for possibly imbalanced binary and categorical data