Résumé (EN)

While mixtures of Gaussian distributions have been studied for more than a century (Pearson, 1894), the construction of a reference Bayesian analysis of those models still remains unsolved, with a general prohibition of the usage of improper priors (Fruwirth-Schnatter, 2006) due to the ill-posed nature of such statistical objects. This difficulty is usually bypassed by an empirical Bayes resolution (Richardson and Green, 1997). By creating a new parameterisation cantered on the mean and possibly the variance of the mixture distribution itself, we are able to develop here a weakly informative prior for a wide class of mixtures with an arbitrary number of components. We demonstrate that some posterior distributions associated with these priors is almost surely proper and we provide MCMC implementations that exhibit the expected exchangeability. We only study here the univariate case, the extension to multivariate location-scale mixtures being currently under study. An R package called Ultimixt is attached to this paper.