Quantile regression, i.e. modelling conditional quantiles of some covariates and other effects through the linear predictor, has typically been carried out exploiting the asymmetric Laplace distribution (ALD) as a working «likelihood». In the Bayesian framework, this is highly questionable as the posterior variance is affected by the artificial ALD «likelihood». With continuous responses, we can reparameterize the likelihood in terms of a $\alpha$-quantile, and let the $\alpha$-quantile depend on the linear predictor. We can then do model based quantile regression with little effort using the R-INLA package doing approximate Bayesian inference for latent Gaussian models, and trust the quantile regression posterior in the same way as when doing parametric mean regression.
