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Simultaneously applying a permutation to the mixture weights and mixture components leaves $f$ unchanged. Thus identifiability of named components requires a convention, such as ordering distinct component means; label switching describes the associated inferential issue.^[ref]

Even after choosing label names, identical mixture components reveal only their combined mixture weight, and an inactive mixture component has undetermined statistical parameters. Adding mixture components increases flexibility but can cause overfitting; judge predictions using cross-validation, rather than interpreting every fitted component as a discovered population.^[ref]

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.