What is the state-of-the-art regarding sampling from discrete distribution?

Question

After struggling with auto-Poisson model (a.k.a. Random Markov Network with conditional Poisson distributions) trying to force Gibbs sampler to obtain discrete sample of the network (since I know conditional distributions) I realized that I have never seen discrete version of this sampler.

I wonder what is the state-of-the-art regarding sampling from discrete distribution (not confined to Bayesian posterior sampling)? Is there something like Gibbs sampler to sample from conditional discrete distributions?

Answer 1

This is a very puzzling question: the Gibbs sampler draws its name from Gibbs random fields where it was originally used by the Geman brothers to generate realisations from such discrete models. For instance the Ising model is made of a vector of random variables each taking values in $\{-1,1\}$...

So indeed the Gibbs can be used for discrete models. And I do not understand why you have to force your "Gibbs sampler to obtain discrete samples". If you are using the proper full conditionals they should be on a discrete support.