MCMC algorithms for structured multivariate normal models

In this paper we look at the use of MCMC methods for the general class of structured multivariate Normal (SMVN) models. These models consider a response vector as following a multivariate normal distribution with a limited number of parameters representing the mean vector and variance matrix. We in particular look at how multilevel models can be represented in this framework and show that a simple random walk Metropolis algorithm for such models can be very quick and outperform MCMC algorithms for the equivalent model in the standard ‘random effects’ framework on some simple examples. We show however that this computational advantage only holds for very simple models. We discuss model comparison which is more natural in this framework. We finish with discussion of other models that fit naturally in the framework and link to other work on these models.