A Scalable Blocked Gibbs Sampling Algorithm For Gaussian And Poisson   Regression Models
  Markov Chain Monte Carlo (MCMC) methods are a popular technique in Bayesian statistical modeling. They have long been used to obtain samples from posterior distributions, but recent research has focused on the scalability of these techniques for large problems. We do not develop new sampling methods but instead describe a blocked Gibbs sampler which is sufficiently scalable to accomodate many interesting problems. The sampler we describe applies to a restricted subset of the Generalized Linear Mixed-effects Models (GLMM's); this subset includes Poisson and Gaussian regression models. The blocked Gibbs sampling steps jointly update a prior variance parameter along with all of the random effects underneath it. We also discuss extensions such as flexible prior distributions. 
Nicholas A. Johnson, Frank O. Kuehnel, Ali Nasiri Amini
stat.ME