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The marginal Metropolis–Hastings algorithm avoids storing allocation labels, but each component update changes every mixture likelihood term. Cache component contributions, calculate log-likelihoods stably and tune the proposal scale and random-walk Metropolis efficiency during a pilot run; then fix proposal scales or use a justified adaptive scheme. Means and variances may be strongly dependent, making separate random walks inefficient. Weight proposals must respect the simplex, variance proposals must respect positivity, and the displayed acceptance ratios must retain all relevant priors and proposal corrections.

The augmented Gibbs sampler uses elementary conditionals and typically costs order $nk$ for a label sweep, followed by sufficient-statistic updates. It avoids Metropolis rejections in these conditional draws, but does not remove slow mixing: ambiguous allocations can lock means, variances and labels into strongly coupled configurations. Empty components must be updated from their priors, not deleted automatically. The proper inverse-gamma scale $\beta >0$ suppresses arbitrarily small variances even though an unregularized Gaussian-mixture likelihood can be singular.

With exchangeable component priors, label switching creates $k!$ equivalent posterior modes. Ordering component means is an identification constraint, not an innocuous change unless the intended inferential model is made explicit. Predictive density and permutation-invariant summaries avoid arbitrary labelling; component-specific summaries require an appropriate relabelling convention. If the Dirichlet parameters differ between components, exact label symmetry may be broken, but substantial multimodality can remain.

Use dispersed initial states, multiple chains and diagnostics on several scientifically relevant functions. Report effective sample size of a Markov chain and Monte Carlo error estimates, not just a large iteration count. Burn-in can reduce initialization bias but does not prove convergence; thinning cannot by itself cure poor exploration. Closed-form Gibbs updates simplify computation, while reliable inference still requires adequate exploration of the posterior.

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 $\mathrm{ESS}\approx n/(1+2{\sum }_{k\ge 1}\mathrm{Corr}(h(Z_0),h(Z_k)))$. 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 ${\sigma }_q\downarrow 0$, 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.