  Quantile regression relates the quantile of the response to a linear
predictor. For a discrete response distributions, like the Poission, Binomial
and the negative Binomial, this approach is not feasible as the quantile
function is not bijective. We argue to use a continuous model-aware
interpolation of the quantile function, allowing for proper quantile inference
while retaining model interpretation. This approach allows for proper
uncertainty quantification and mitigates the issue of quantile crossing. Our
reanalysis of hospitalisation data considered in Congdon (2017) shows the
advantages of our proposal as well as introducing a novel method to exploit
quantile regression in the context of disease mapping.
