  In this paper, we consider Bayesian inference on a class of multivariate
median and the multivariate quantile functionals of a joint distribution using
a Dirichlet process prior. Since, unlike univariate quantiles, the exact
posterior distribution of multivariate median and multivariate quantiles are
not obtainable explicitly, we study these distributions asymptotically. We
derive a Bernstein-von Mises theorem for the multivariate $\ell_1$-median with
respect to general $\ell_p$-norm, which in particular shows that its posterior
concentrates around its true value at $n^{-1/2}$-rate and its credible sets
have asymptotically correct frequentist coverage. In particular, asymptotic
normality results for the empirical multivariate median with general
$\ell_p$-norm is also derived in the course of the proof which extends the
results from the case $p=2$ in the literature to a general $p$. The technique
involves approximating the posterior Dirichlet process by a Bayesian bootstrap
process and deriving a conditional Donsker theorem. We also obtain analogous
results for an affine equivariant version of the multivariate $\ell_1$-median
based on an adaptive transformation and re-transformation technique. The
results are extended to a joint distribution of multivariate quantiles. The
accuracy of the asymptotic result is confirmed by a simulation study. We also
use the results to obtain Bayesian credible regions for multivariate medians
for Fisher's iris data, which consists of four features measured for each of
three plant species.
