A useful reparameterisation to obtain samples from conditional inverse Wishart distributions

Abstract

A Bayesian joint analysis of normally distributed traits and binary traits, using the Gibbs sampler, requires the drawing of samples from a conditional inverse Wishart distribution. This is the fully conditional posterior distribution of the residual covariance matrix of the normally distributed traits and liabilities of the binary traits. Obtaining samples from the conditional inverse Wishart distribution is not straightforward. However, combining well-known matrix results and properties of the Wishart distribution, it is shown that this can be easily carried out by successively drawing from Wishart and normally distributed random variables.