BAYESIAN INFERENCE ON MULTIVARIATE MEDIANS AND QUANTILES

We consider Bayesian inferences on a type of multivariate median and the multivariate quantile functionals of a joint distribution using a Dirichlet process prior. Unlike univariate quantiles, the exact posterior distribution of multivariate median and multivariate quantiles are not obtainable explicitly; thus we study these distributions asymptotically. We derive a Bernstein-von Mises theorem for the multivariate l(1)-median with respect to a general l(p)-norm, showing that its posterior concentrates around its true value at the n(-1/2)-rate, and that its credible sets have asymptotically correct frequentist coverages. In particular, the asymptotic normality results for the empirical multivariate median with a general l(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 using a Bayesian bootstrap process and deriving a conditional Donsker theorem. We also obtain analogous results for an affine equivariant version of the multivariate l(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 using a simulation study. We also use the results to obtain Bayesian credible regions for the multivariate medians for Fisher's iris data, which consist of four features measured for each of three plant species.