The Inverse-Wishart (IW) distribution is a standard and popular choice of priors for covariance matrices and has attractive properties such as conditional conjugacy. However, the IW family of priors has crucial drawbacks, including the lack of effective choices for non-informative priors. Several classes of priors for covariance matrices that alleviate these drawbacks, while preserving computational tractability, have been proposed in the literature. These priors can be obtained through appropriate scale mixtures of IW priors. However, the high-dimensional posterior consistency of models which incorporate such priors has not been investigated. We address this issue for the multi-response regression setting ($q$ responses, $n$ samples) under a wide variety of IW scale mixture priors for the error covariance matrix. Posterior consistency and contraction rates for both the regression coefficient matrix and the error covariance matrix are established in the ``large $q$, large $n$" setting under mild assumptions on the true data-generating covariance matrix and relevant hyperparameters. In particular, the number of responses $q_n$ is allowed to grow with $n$, but with $q_n = o(n)$. Also, some results related to the inconsistency of posterior mean for $q_n/n \to \gamma$, where $\gamma \in (0,\infty)$ are provided.

翻译：逆威沙特（IW）分布是协方差矩阵先验的常用标准选择，具有条件共轭性等优良性质。然而，IW先验族存在关键缺陷，包括缺乏有效的无信息先验选择。文献中已提出多类可缓解这些缺陷同时保持计算可行性的协方差矩阵先验，这些先验可通过IW先验的适当尺度混合获得。但纳入此类先验模型的高维后验相合性尚未得到研究。本文针对多元响应回归场景（$q$个响应变量，$n$个样本），在误差协方差矩阵采用多种IW尺度混合先验的条件下解决该问题。在"大$q$，大$n$"设定下，通过对真实数据生成协方差矩阵及相关超参数施加温和假设，建立了回归系数矩阵和误差协方差矩阵的后验相合性与收缩速率。特别地，响应变量数$q_n$允许随$n$增长，但满足$q_n = o(n)$。此外，本文提供了当$q_n/n \to \gamma$（$\gamma \in (0,\infty)$）时后验均值不一致性的相关结论。