This paper deals with both exploration and interpretation problems related to posterior distributions for mixture models. The specification of mixture posterior distributions means that the presence of k! modes is known immediately. Standard Markov chain Monte Carlo techniques usually have difficulties with well-separated modes such as occur here; the Markov chain Monte Carlo sampler stays within a neighbourhood of a local mode and fails to visit other equally important modes. We show that exploration of these modes can be imposed on the Markov chain Monte Carlo sampler using tempered transitions based on Langevin algorithms. However, as the prior distribution does not distinguish between the different components, the posterior mixture distribution is symmetric and thus standard estimators such as posterior means cannot be used. Since this is also true for most non-symmetric priors, we propose alternatives for Bayesian inference for permutation invariant posteriors, including a cluster...

this paper. The reason for the paradoxical complexity of the mixture model is due to the product structure of the likelihood function, L(` 1 ; : : : ; ` k jx 1 ; : : : ; xn ) =

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Les modèles de mélange ont été beaucoup étudiés et ont suscités le développement de nombreuses méthodologies statistiques (méthode des moments, algorithme EM, gibbs sampling...). Malgré cela, de nombreuses difficultés de modélisation et d’estimation persistent. Cette note a pour objet de présenter le spectre des problèmes que l’on peut rencontrer lorsque l’on utilise les modèles de mélange dans le paradigme bayésien. Today’s data analysts and modellers are in the luxurious position of being able to more closely describe, estimate, predict and infer about complex systems of interest, thanks to ever more powerful computational methods but also wider ranges of modelling distributions. Mixture models constitute a fascinating illustration of these aspects: while within a parametric family, they offer malleable approximations in non-parametric settings; although based on standard distributions, they pose highly complex computational challenges; and they are both easy to constrain to meet identifiability requirements and fall within the class of ill-posed problems. This note aims to introduce the reader to Bayesian modelling and inference difficulties on mixtures of distributions. 1 The finite mixture framework The description of a mixture of distributions (Titterington et al. (1985)) is straightforward: any convex combination k� pifi(x), i=1 of other distributions fi is a mixture. While continuous mixtures g(x) = f(x|θ)h(θ)dθ

In Chib (1995), a method for approximating marginal densities in a Bayesian setting is proposed, with one proeminent application being the estimation of the number of components in a normal mixture. As pointed out in Neal (1999) and Frühwirth-Schnatter (2004), the approximation often fails short of providing a proper approximation to the true marginal densities because of the well-known label switching problem (Celeux et al., 2000). While there exist other alternatives to the derivation of approximate marginal densities, we reconsider the original proposal here and show as in Berkhof et al. (2003) and Lee et al. (2008) that it truly approximates the marginal densities once the label switching issue has been solved.