Computing Posterior Distributions for Covariance Matrices

Diiculties in computing the posterior distribution of a covariance matrix when using nonconjugate priors has been discussed by several authors. Typically, the posterior distribution for the covariance matrix is computed via the Gibbs sampler and when using a Wishart prior for the inverse of the covariance matrix, one obtains conditional conjugacy (the full conditional distribution of the inverse of the covariance matrix will be Wishart). However, when using a nonconjugate prior, the conditional conjugacy is lost and sampling from the full conditional becomes non-trivial. In light of this, we will rst examine which parameterization of the covariance matrix results in the best approximations to the posterior. Second, we will discuss various approaches to simulate from the posterior distribution of the covariance matrix including various Metropolis Hastings algorithms and a new approach based on Hybrid Monte Carlo (see e.g., Gustafson (1997)). A simple example will serve to illustrate the problems with the various MCMC approaches.