Bayesian analysis for the covariance matrix of a multivariate normal distribution has received a lot of attention in the last two decades. In this paper, we propose a new class of priors for the covariance matrix, including both inverse Wishart and reference priors as special cases. The main motivation for the new class is to have available priors – both subjective and objective – that do not “force eigenvalues apart,” which is a criticism of inverse Wishart and Jeffreys priors. Extensive comparison of these ‘shrinkage priors’ with inverse Wishart and Jeffreys priors is undertaken, with the new priors seeming to have considerably better performance. A number of curious facts about the new priors are also observed, such as that the posterior distribution will be proper with just three vector observations from the multivariate normal distribution – regardless of the dimension of the covariance matrix – and that useful inference about features of the covariance matrix can be possible. Finally, a new MCMC algorithm is developed for this class of priors and is shown to be computationally effective for matrices of up to 100 dimensions.

中文翻译：

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具有一类新先验的多元正态分布协方差矩阵的贝叶斯分析

在过去的二十年中，对多元正态分布的协方差矩阵的贝叶斯分析受到了很多关注。在本文中，我们为协方差矩阵提出了一类新的先验，包括逆 Wishart 和参考先验作为特殊情况。新类的主要动机是拥有可用的先验 - 主观和客观 - 不会“强制特征值分开”，这是对逆 Wishart 和 Jeffreys 先验的批评。对这些“收缩先验”与逆 Wishart 和 Jeffreys 先验进行了广泛的比较，新的先验似乎具有更好的性能。还观察到了一些关于新先验的奇怪事实，例如，后验分布将适用于多元正态分布中的三个向量观测值——无论协方差矩阵的维数如何——并且可以对协方差矩阵的特征进行有用的推断。最后，为此类先验开发了一种新的 MCMC 算法，并证明对多达 100 维的矩阵具有计算效率。