Using Sufficient Statistics in MCMC Solutions of the Bayesian Hierarchical Linear Model

The Bayesian approach to hierarchical linear models has many advantages when compared with likelihood-based methods. Initially, the clear advantage has been with robust inference in small sample settings. But more recent approaches to Bayesian computations, based on Markov Chain Monte Carlo (MCMC) simulation, have vastly improved the viability of Bayesian models in practice. Many of the newer advantages relate to the relative ease with which robustness issues, such as entertaining alternative families of residual errors, can be accommodated. Additionally, inference for arbitrary functions of parameters is now direct. The one significant problem that has remained pertains to the very heavy computations required for larger samples. This paper details a straightforward procedure for MCMC estimation in sizable samples. Results from a simple example in a school accountability context will be used to illustrate the issue.