C14. Metropolis–Hastings cancellations

Topic: Markov chains, Gibbs sampling, and Metropolis–Hastings · more questions on this topic
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Why can Metropolis–Hastings use an unnormalised target? When do its proposal-density terms cancel?

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Definitions/notation. Let \pi(z)=k(z)/Z be the target and q(y\mid x) the proposal density from current state x to candidate y. The acceptance probability is \min\{1,k(y)q(x\mid y)/[k(x)q(y\mid x)]\} because Z cancels; the proposal terms cancel only when q(y\mid x)=q(x\mid y), as for a fixed-scale symmetric random walk.