chapter  4
20 Pages

## Bounding Chains

Monotonicity is a valuable property to have, but unfortunately, not every Markov chain update function has it. From among our examples, the Ising model when β < 0, the Potts model with k≥ 4, the hard-core gas model on nonbipartite graphs, weighted permutations, sink-free orientations, the antivoter model, and self-organizing lists all have nonmonotonic Gibbs samplers. (While the slice sampler is monotonic, for these examples it cannot be employed efficiently in its pure form.) While Gibbs samplers are often monotonic, Metropolis-Hastings, chains on permutations, and auxiliary variable chains (like Swendsen-Wang) are often not.