On a multivariate implementation of the Gibbs sampler

It is well established that when the parameters in a model are correlated, the rate of convergence of Gibbs chains to the appropriate stationary distributions is faster and Monte-Carlo variances of features of these distributions are lower for a given chain length, when the Gibbs sampler is implemented by blocking the correlated parameters and sampling from the respective conditional posterior distributions takes place in a multivariate rather than in a scalar fashion. This block sampling strategy often requires knowledge of the inverse of large matrices. In this note a block sampling strategy is implemented which circumvents the use of these inverses. The algorithm applies in the context of the Gaussian model and is illustrated with a small simulated data set.