The fundamental problem of gibbs sampler in mixture models

The mixture models were firstly studied by Pearson in 1894. These models are strong tools, through which the complicated systems can be analyzed in a wide range of disciplines such as As-tronomy, Economics, Mechanics, etc. although the structure of these models is apparently simple, it is very complicated to obtain maximum likelihood estimators and Bayesian ones in particular and it needs to be approximated in most cases. In this paper, we apply the Gibbs Sampling in order to approximate the Bayesian Estimator in Mixture models, present the Gibbs algorithms for the family of exponential distributions and finally, we would show the disadvantage of this algorithm through an example.