We discuss Bayesian inference for a known-mean Gaussian model with a compound symmetric variance-covariance matrix. Since the space of such matrices is a linear subspace of that of positive definite matrices, we utilize the methods of Pisano (2022) to decompose the usual Wishart conjugate prior and derive a closed-form, three-parameter, bivariate conjugate prior distribution for the compound-symmetric half-precision matrix. The off-diagonal entry is found to have a non-central Kummer-Beta distribution conditioned on the diagonal, which is shown to have a gamma distribution generalized with Gauss's hypergeometric function. Such considerations yield a treatment of maximum a posteriori estimation for such matrices in Gaussian settings, including the Bayesian evidence and flexibility penalty attributable to Rougier and Priebe (2019). We also demonstrate how the prior may be utilized to naturally test for the positivity of a common within-class correlation in a random-intercept model using two data-driven examples.

翻译：本文讨论已知均值、协方差矩阵具有复合对称结构的高斯模型的贝叶斯推断。由于该类矩阵空间是正定矩阵空间的线性子空间，我们利用Pisano（2022）的方法分解常见的Wishart共轭先验，推导出复合对称半精度矩阵的闭合形式三参数二元共轭先验分布。研究发现，非对角线元素服从以对角线元素为条件的非中心Kummer-Beta分布，而对角线元素表现为经高斯超几何函数广义化的伽马分布。这些结论为高斯背景下该类矩阵的最大后验估计提供了处理方案，包括Rougier和Priebe（2019）提出的贝叶斯证据与灵活性惩罚项。通过两个数据驱动实例，我们还展示了如何利用该先验自然检验随机截距模型中常见类内相关性的正定性。