  In this work we discuss the progress of Bayesian quantile regression models
since their first proposal and we discuss the importance of all parameters
involved in the inference process. Using a representation of the asymmetric
Laplace distribution as a mixture of a normal and an exponential distribution,
we discuss the relevance of the presence of a scale parameter to control for
the variance in the model. Besides that we consider the posterior distribution
of the latent variable present in the mixture representation to showcase
outlying observations given the Bayesian quantile regression fits, where we
compare the posterior distribution for each latent variable with the others. We
illustrate these results with simulation studies and also with data about Gini
indexes in Brazilian states from years with census information.
