Posterior propriety of an objective prior for generalized hierarchical normal linear models

Bayesian Hierarchical models has been widely used in modern statistical application. To deal with the data having complex structures, we propose a generalized hierarchical normal linear (GHNL) model which accommodates arbitrarily many levels, usual design matrices and ‘vanilla’ covariance matrices. Objective hyperpriors can be employed for the GHNL model to express ignorance or match frequentist properties, yet the common objective Bayesian approaches are infeasible or fraught with danger in hierarchical modelling. To tackle this issue, [Berger, J., Sun, D., & Song, C. (2020b). An objective prior for hyperparameters in normal hierarchical models. Journal of Multivariate Analysis, 178, 104606. https://doi.org/10.1016/j.jmva.2020.104606] proposed a particular objective prior and investigated its properties comprehensively. Posterior propriety is important for the choice of priors to guarantee the convergence of MCMC samplers. James Berger conjectured that the resulting posterior is proper for a hierarchical normal model with arbitrarily many levels, a rigorous proof of which was not given, however. In this paper, we complete this story and provide an user-friendly guidance. One main contribution of this paper is to propose a new technique for deriving an elaborate upper bound on the integrated likelihood, but also one unified approach to checking the posterior propriety for linear models. An efficient Gibbs sampling method is also introduced and outperforms other sampling approaches considerably.