  Modeling binary and categorical data is one of the most commonly encountered
tasks of applied statisticians and econometricians. While Bayesian methods in
this context have been available for decades now, they often require a high
level of familiarity with Bayesian statistics or suffer from issues such as low
sampling efficiency. To contribute to the accessibility of Bayesian models for
binary and categorical data, we introduce novel latent variable representations
based on P\'olya-Gamma random variables for a range of commonly encountered
logistic regression models. From these latent variable representations, new
Gibbs sampling algorithms for binary, binomial, and multinomial logit models
are derived. All models allow for a conditionally Gaussian likelihood
representation, rendering extensions to more complex modeling frameworks such
as state space models straightforward. However, sampling efficiency may still
be an issue in these data augmentation based estimation frameworks. To
counteract this, novel marginal data augmentation strategies are developed and
discussed in detail. The merits of our approach are illustrated through
extensive simulations and real data applications.
