Gibbs Sampling for the Probit Regression Model with Gaussian Markov Random Field Latent Variables

We consider a binary probit model where the latent variable follow a Gaussian Markov Random Field (GMRF). Our main objective is to derive an efficient Gibbs sampler for the above model. For this purpose, we first review two Gibbs samplers available for the classical probit model with one latent variable. We find that the joint update of variables increases the rate of convergence. We use these results to derive Gibbs samplers for the probit model with GMRF latent variables. We discuss three different approaches to Gibbs sampling for the above model. The first two approaches are direct extensions of the Gibbs sampler for the classical probit model. The third approach involves a slight modification in the probit model and suggests that it may be possible to block sample all its variables at once.