This paper addresses the label switching problem in Bayesian mixture models, where the symmetry of the likelihood function leads to multimodal and symmetric posterior distributions, making parameter estimation and clustering difficult. The common approach of using artificial identifiability constraints to break this symmetry is shown to be insufficient. Instead, the authors propose relabelling algorithms, which aim to minimize the posterior expected loss under a class of loss functions. These algorithms are described in detail and demonstrated on two examples. The label switching problem arises because the likelihood is invariant under permutations of the mixture components. This symmetry causes the posterior distribution to be symmetric and multimodal, making it challenging to summarize. The authors show that relabelling algorithms can effectively address this issue by permuting the MCMC samples to align with the true component labels, thereby removing the symmetry and allowing for accurate estimation of parameters and clustering. In the first example, the authors apply relabelling algorithms to a dataset of galaxy velocities, which are modeled as a mixture of six normal distributions. The relabelling algorithm successfully removes the label switching problem, allowing for accurate estimation of the component means and variances. In the second example, the authors fit a mixture of three t-distributions to the same dataset, demonstrating that the relabelling algorithm can also handle genuine multimodality in the posterior distribution. The authors argue that relabelling algorithms are more general and effective than artificial identifiability constraints, as they do not rely on specific assumptions about the parameter space. They also note that relabelling algorithms can be applied to a wide range of mixture models, including those with high-dimensional component densities. The paper concludes that relabelling algorithms provide a more satisfactory solution to the label switching problem than traditional methods.This paper addresses the label switching problem in Bayesian mixture models, where the symmetry of the likelihood function leads to multimodal and symmetric posterior distributions, making parameter estimation and clustering difficult. The common approach of using artificial identifiability constraints to break this symmetry is shown to be insufficient. Instead, the authors propose relabelling algorithms, which aim to minimize the posterior expected loss under a class of loss functions. These algorithms are described in detail and demonstrated on two examples. The label switching problem arises because the likelihood is invariant under permutations of the mixture components. This symmetry causes the posterior distribution to be symmetric and multimodal, making it challenging to summarize. The authors show that relabelling algorithms can effectively address this issue by permuting the MCMC samples to align with the true component labels, thereby removing the symmetry and allowing for accurate estimation of parameters and clustering. In the first example, the authors apply relabelling algorithms to a dataset of galaxy velocities, which are modeled as a mixture of six normal distributions. The relabelling algorithm successfully removes the label switching problem, allowing for accurate estimation of the component means and variances. In the second example, the authors fit a mixture of three t-distributions to the same dataset, demonstrating that the relabelling algorithm can also handle genuine multimodality in the posterior distribution. The authors argue that relabelling algorithms are more general and effective than artificial identifiability constraints, as they do not rely on specific assumptions about the parameter space. They also note that relabelling algorithms can be applied to a wide range of mixture models, including those with high-dimensional component densities. The paper concludes that relabelling algorithms provide a more satisfactory solution to the label switching problem than traditional methods.