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:我们讨论了一个已知均值高斯模型的贝叶斯推断,其方差协方差矩阵是复合对称的。由于这样的矩阵空间是正定矩阵空间的线性子空间,我们利用Pisano(2022)的方法来分解通常的Wishart共轭先验,并为复合对称半精度矩阵导出了一个闭合的、三参数的双变量共轭先验分布。对于对角线条件,发现非对角线元素服从带有非中心Kummer-Beta分布的概率分布函数,而对角线元素则表现为广义的带有Gauss超几何函数的Gamma分布。通过这些考虑,可构建高斯设定下的最大后验估计的处理方式,包括Rougier和Priebe(2019)提到的贝叶斯证据和灵活性惩罚。我们还展示了如何利用该先验自然地检验随机斜率模型中公共类内相关性的积极性,并举了两个数据驱动的例子。