Gibbs sampling for a finite hidden spin field

For a strictly positive posterior on finitely many binary spins, a Random-scan Gibbs sampler uses the two local posterior weights to redraw one spin. Positive full conditional distribution probabilities allow every finite sequence of spin changes, giving an irreducible Markov chain with positive self-transition probabilities.

When an observation mean depends on the neighboring spins, changing a spin affects observations centered at its neighbors. Its own observation need not involve that spin. Selecting precisely the affected factors gives a correct local full conditional distribution without recomputing the entire posterior.