= Metropolis-within-Gibbs algorithm
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= Metropolis-within-Gibbs
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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.
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