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.

中文翻译：

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多元中位数和分位数的贝叶斯推理

在本文中，我们考虑使用 Dirichlet 过程先验对一类多元中值和联合分布的多元分位数泛函进行贝叶斯推理。由于与单变量分位数不同，多元中位数和多元分位数的确切后验分布无法明确获得，因此我们渐近地研究这些分布。我们推导出多元 $\ell_1$-中值相对于一般 $\ell_p$-范数的 Bernstein-von Mises 定理，这特别表明它的后验集中在 $n^{-1/2} 处的真实值附近$-rate 及其可信集合具有渐近正确的频率论覆盖范围。特别是，在证明过程中也导出了具有一般 $\ell_p$-范数的经验多元中值的渐近正态性结果，将文献中的情况 $p=2$ 的结果扩展到一般 $p$。该技术涉及通过贝叶斯自举过程近似后验狄利克雷过程并推导出条件 Donsker 定理。我们还获得了基于自适应变换和重新变换技术的多元 $\ell_1$-中值的仿射等变版本的类似结果。结果扩展到多元分位数的联合分布。模拟研究证实了渐近结果的准确性。我们还使用这些结果来获得 Fisher 虹膜数据多元中值的贝叶斯可信区域，