Bayesian supervised learning with non-Gaussian latent variables

We describe a Bayesian learning scheme for the hierarchal Bayesian linear model, which is based on the Gaussian scale mixture (GSM) modeling of the distribution of the latent variable. The proposed method takes advantage of the hierarchal Gaussian structure for a simple Monte-Carlo sampling algorithm. Particularly, with a single hidden scale parameter controlling the distribution of the latent variables, it leads to an efficient algorithm without explicit matrix inversion.