= Solution
Single-component updates propose one coordinate while holding the others fixed, with acceptance based on the resulting target ratio. They are cheap and permit individual scale tuning, but can crawl along a narrow correlated ridge. Block updates propose several coordinates jointly and can follow <posterior> <covariance> directions, potentially improving mixing substantially.
A practical compromise uses small correlated blocks, or a full <Gaussian> block with <covariance> estimated in a pilot run and rescaled for acceptance. Blocking <independent> coordinates need not help, and a badly scaled high-dimensional block may be rejected frequently. Compare computing cost per effective draw, rather than acceptance rate alone. A <Metropolis-within-Gibbs> sweep of coordinate updates and a properly accepted block update both preserve the target.
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