Multivariate regression with many correlated responses and predictors commonly violates Gaussian error assumptions due to heavy tails, outliers, and asymmetry. Gaussian procedures then lose efficiency in coefficient estimation and produce biased estimates of conditional dependence graphs. We develop a robust Bayesian framework using a scale-location mixture error distribution and horseshoe+ global-local priors on both the regression coefficients and off-diagonals of the error precision matrix, coupling sparsity in the regression map with sparsity in the residual dependence structure. Theoretical contributions include joint posterior contraction, selection consistency for both supports, a Kullback-Leibler risk bound showing the dominance of horseshoe+ over horseshoe, and bounded sensitivity, ensuring that a single large outlier has vanishing influence under t errors. Simulations across four error regimes, contamination, and varying dimensions show that our estimator matches Gaussian procedures under normality and dominates them under heavy tails and skewness. Applications to FRED-MD macroeconomic data and S&P 500 daily returns recover interpretable sparse coefficient maps and residual dependence graphs while automatically down-weighting crisis-period observations.

翻译：多元回归中常面临众多相关响应变量与预测变量，其误差项易因重尾、异常值和不对称性而违背高斯假设。此时高斯方法在系数估计中效率降低，且条件依赖图估计产生偏差。本文提出鲁棒贝叶斯框架，采用尺度-位置混合误差分布，对回归系数和误差精度矩阵非对角元素均施加蹄铁+全局-局部先验，实现回归映射稀疏性与残差依赖结构稀疏性的耦合。理论贡献包括：联合后验收缩性、两种支撑集的选择一致性、表明蹄铁+优于蹄铁的KL风险界，以及有界敏感性——确保在t误差下单个大异常值的影响趋近于零。针对四种误差机制、污染情况及不同维度的仿真表明，我们的估计量在正态性假设下与高斯方法相当，而在重尾和偏态分布下显著优于后者。对FRED-MD宏观经济数据与标普500日收益率数据的应用，在自动降低危机期间观测值权重的条件下，恢复了可解释的稀疏系数映射图和残差依赖图。