= Solution
Tune the proposal scale using pilot runs on a grid of candidate $\sigma_q$ 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 $\operatorname{ESS}\approx n/(1+2\sum_{k\ge1}\operatorname{Corr}(h(Z_0),h(Z_k)))$. 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 $\sigma_q\downarrow0$, 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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