This paper aims to examine the characteristics of the posterior distribution of covariance/precision matrices in a "large $p$, large $n$" scenario, where $p$ represents the number of variables and $n$ is the sample size. Our analysis focuses on establishing asymptotic normality of the posterior distribution of the entire covariance/precision matrices under specific growth restrictions on $p_n$ and other mild assumptions. In particular, the limiting distribution turns out to be a symmetric matrix variate normal distribution whose parameters depend on the maximum likelihood estimate. Our results hold for a wide class of prior distributions which includes standard choices used by practitioners. Next, we consider Gaussian graphical models which induce sparsity in the precision matrix. Asymptotic normality of the corresponding posterior distribution is established under mild assumptions on the prior and true data-generating mechanism.

翻译：本文旨在研究“大p、大n”情景下协方差/精度矩阵后验分布的性质，其中p表示变量数，n表示样本量。我们的分析重点是在p_n特定增长限制及其他温和假设下，建立整个协方差/精度矩阵后验分布的渐近正态性。特别地，极限分布被证明是对称矩阵变量正态分布，其参数依赖于极大似然估计。我们的结果适用于一大类先验分布，包括从业者常用的标准选择。接下来，我们考虑在精度矩阵中引入稀疏性的高斯图模型。在关于先验和真实数据生成机制的温和假设下，建立了相应后验分布的渐近正态性。