Conjugate Bayesian analysis of compound-symmetric Gaussian models

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.