Monte Carlo Experiments and a Few Observations on Bayesian Quantile Regression

In many simulation-based Bayesian approaches to quantile regression, Markov Chain Monte Carlo techniques are employed to generate draws from a posterior distribution based on an asymmetric Laplace “working” likelihood. Under flat improper priors, the mode of this posterior distribution is coincident with the desired quantile function. However, simulation-based approaches for estimation and inference commonly report a posterior mean as a point estimate and interpret that mean synonymously with the quantile. In this note, we analytically derive the exact posterior distribution of a quantile regression parameter in a simple univariate setting free of covariates. We note the non-uniqueness of the posterior mode in some cases and conduct a series of Monte Carlo experiments to compare the sampling performances of posterior means and modes. Interestingly, and perhaps surprisingly, the mean performs similarly to, if not favorably to, the mode under several standard metrics, even in very small samples.