 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.