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：我们提出了一种基于协变量空间上的复制函数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复制函数，我们称之为"回归复制函数"。为处理大规模协变量向量，我们通过一种新颖的多元延拓马鞍先验对其系数进行正则化。采用高效变分推断方法进行贝叶斯推断与分布预测，使该方法可应用于大规模数据集。该方法的优势在于能够分别且精确地估计响应向量的边缘分布，从而产生具有边缘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。第一个案例是对澳大利亚区域半小时电力价格的经济计量建模与预测。在该案例中，我们的方法相较于主流基准方法产生了更准确的分布预测。第二个案例是在树种丰度数据模型的似然自由推断中评估多元后验，扩展了先前单变量回归复制函数似然自由推断方法。在两个应用中，我们证明了新方法具有理想的边缘校准特性。