A blocked Gibbs sampler draws groups of coordinates jointly from their full conditional distributions. Each update preserves the joint probability distribution; dependence between blocks can still cause a long mixing time.
For a Markov random field whose positive probability density function factors into single-vertex and pairwise edge factors on a bipartite graph, variables in one color class are conditionally independent given the other color. A checkerboard Gibbs sampler therefore draws every black variable from its full conditional distribution in parallel, then every white variable conditional on the newly drawn blacks. Each color update is a blocked Gibbs sampler preserving the joint target, and their composition preserves it too.
Bipartiteness of a displayed pixel-neighbour graph is insufficient if the actual likelihood introduces factors coupling several same-color variables. For example, an observation depending jointly on two black neighbours can destroy their conditional independence. The conditional factorization of the posterior, rather than the picture alone, justifies the parallel updates.
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