 Metropolis-within-Gibbs algorithm

A Metropolis-within-Gibbs sweep updates one coordinate or block at a time by a Metropolis–Hastings algorithm transition whose invariant law is that block's full conditional distribution. For a proposal changing only block $j$, the acceptance ratio uses the full joint target with the other coordinates fixed, together with the reverse/forward proposal ratio. Each block kernel preserves the joint posterior, so their composition does as well. Conjugate blocks can instead use exact Gibbs sampling draws. Systematic composition need not be reversible, even when every block kernel is reversible.