Tune the proposal scale using pilot runs on a grid of candidate values. After allowing for initial transients, compare the effective sample size of a Markov chain per unit computation time for representative functions, such as both coordinates and a quadratic function. Equivalently inspect their integrated autocorrelation time, using . Choose a useful scale and keep it fixed for the production run. Another pilot criterion is the average squared accepted displacement, possibly in coordinates standardized by the target covariance, which balances move size and rejection.
Acceptance alone is not an efficiency criterion. As , proposals are almost the current state, so acceptance tends to one while exploration becomes arbitrarily slow. Very large scales propose mostly low-density states and produce long sequences of identical values. The proposal scale and random-walk Metropolis efficiency trade-off requires both movement and acceptance; neither maximizing nor merely reporting the acceptance probability guarantees a useful sample. With strong correlation, an isotropic proposal also struggles with the target's narrow direction; covariance-scaled proposals can improve this geometry beyond tuning one scalar variance.
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