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Dealing with label-switching in mixture models

Abstract

In a Bayesian analysis of finite mixture models, parameter estimation and clustering are sometimes less straightforward that might be expected. In particular, the common practice of estimating parameters by their posterior mean, and summarising joint posterior distributions by marginal distributions, often leads to nonsensical answers. This is due to the so-called ``label-switching'' problem, which is caused by symmetry in the likelihood of the model parameters. A frequent response to this problem is to remove the symmetry using artificial identifiability constraints. We demonstrate that this fails in general to solve the problem, and describe an alternative class of approaches, {\it relabelling algorithms}, which arise from attempting to minimise the posterior expected loss under a class of loss functions. We describe in detail one particularly simple and general relabelling algorithm, and illustrate its success in dealing with the label-switching problem on two examples.

KEYWORDS: Bayesian, Classification, Clustering, Identifiability, MCMC, Mixture model, Multimodal posterior

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