  The asymmetric Laplace density (ALD) is used as a working likelihood for
Bayesian quantile regression. Sriram et al.(2013) derived posterior consistency
for Bayesian linear quantile regression based on the misspecified ALD. While
their paper also argued for $\sqrt{n}-$consistency, Sriram and Ramamoorthi
(2017) highlighted that the argument was only valid for a rate less than
$\sqrt{n}$. So, the question of $\sqrt{n}-$rate has remained unaddressed, but
is necessary to carry out meaningful Bayesian inference based on the ALD. In
this paper, we derive results to obtain posterior consistency rates for
Bayesian quantile regression based on the misspecified ALD. We derive our
results in a slightly general setting where the quantile function can possibly
be non-linear. In particular, we give sufficient conditions for
$\sqrt{n}-$consistency for the Bayesian linear quantile regression using ALD.
We also provide examples of Bayesian non-linear quantile regression.
