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Keywords:

  • Asymptotic behaviour;
  • Bayesian methods;
  • Mixture models;
  • Overfitting;
  • Posterior concentration

Summary.  We study the asymptotic behaviour of the posterior distribution in a mixture model when the number of components in the mixture is larger than the true number of components: a situation which is commonly referred to as an overfitted mixture. We prove in particular that quite generally the posterior distribution has a stable and interesting behaviour, since it tends to empty the extra components. This stability is achieved under some restriction on the prior, which can be used as a guideline for choosing the prior. Some simulations are presented to illustrate this behaviour.