Jeffreys prior for mixture estimation

Mixture models may be a useful and flexible tool to describe data with a complicated structure, for instance characterized by multimodality or asymmetry. The literature about a Bayesian analysis of mixture models is huge, nevertheless an “objective” Bayesian approach for these models is not widespread, because it is a well established fact that one needs to be careful in using improper prior distributions, since the posterior distribution may not be proper, but noninformative priors are often improper. In this work a preliminary analysis based on the use of a dependent Jeffreys prior in the setting of mixture models will be presented. The Jeffreys prior which assumes the parameters of a Gaussian mixture model is shown to be improper and the conditional Jeffreys prior for each group of parameters is studied. The Jeffreys prior for the complete set of parameters is then used to approximate the derived posterior distribution via a Metropolis-Hastings algorithm and the behavior of the simulated chains is investigated to reach evidence in favor of the properness of the posterior distribution.