We propose a novel distributional regression model for a multivariate response vector based on a copula process over the covariate space. It uses the implicit copula of a Gaussian multivariate regression, which we call a ``regression copula''. To allow for large covariate vectors their coefficients are regularized using a novel multivariate extension of the horseshoe prior. Bayesian inference and distributional predictions are evaluated using efficient variational inference methods, allowing application to large datasets. An advantage of the approach is that the marginal distributions of the response vector can be estimated separately and accurately, resulting in predictive distributions that are marginally-calibrated. Two substantive applications of the methodology highlight its efficacy in multivariate modeling. The first is the econometric modeling and prediction of half-hourly regional Australian electricity prices. Here, our approach produces more accurate distributional forecasts than leading benchmark methods. The second is the evaluation of multivariate posteriors in likelihood-free inference (LFI) of a model for tree species abundance data, extending a previous univariate regression copula LFI method. In both applications, we demonstrate that our new approach exhibits a desirable marginal calibration property.

翻译：我们提出一种基于协变量空间上Copula过程的多元响应向量新型分布回归模型。该模型利用高斯多元回归的隐式Copula（称为"回归Copula"）。为了适应大规模协变量向量，我们采用马蹄先验的多元扩展对其系数进行正则化。通过高效变分推断方法评估贝叶斯推断与分布预测，使得该方法可应用于大规模数据集。该方法的优势在于能够独立且精确地估计响应向量的边际分布，从而得到具有边际校准特性的预测分布。两个实质性应用案例凸显了该方法在多元建模中的有效性。第一个应用是对澳大利亚半小时区域电价的计量经济学建模与预测——我们的方法比主流基准方法产生了更准确的分布预测；第二个应用是对树木物种丰度数据模型的似然自由推断（LFI）中多元后验的评估，扩展了先前单变量回归Copula LFI方法。在两个应用中，我们均证实新方法展现出理想的边际校准性质。