Abstract: Multinomial logistic regression is one of the most popular models for modelling the effect of explanatory variables on a subject choice between a set of specified options. This model has found numerous applications in machine learning, psychology or economy. Bayesian inference in this model is non trivial and requires either to resort to a Metropolis-Hastings algorithm, or rejection sampling within a Gibbs sampler. In this paper, we propose an alternative model to multinomial logit. The model builds on the Plackett-Luce model, a popular model for multiple comparisons. We show that the introduction of a suitable set of auxiliary variables leads to an Expectation-Maximization algorithm to find Maximum A Posteriori estimates of the parameters. We further provide a full Bayesian treatment by deriving a Gibbs sampler, which only requires to sample from highly standard distributions, as well as a variational approximate inference scheme. All are very simple to implement. One property of our Plackett-Luce regression model is that it learns a sparse set of feature weights. We provide detailed comparisons of our method compared to sparse Bayesian multinomial logistic regression and show that it is competitive, especially in presence of polychotomous data.