Convergence of Posterior Distribution in the Mixture of Regressions

Mixture models provide a method of modeling a complex probability distribution in terms of simpler structures. In particular, the method of mixture of regressions has received considerable attention due to its modeling flexibility and availability of convenient computational algorithms. While the theoretical justification has been successfully worked out from the frequentist point of view, its Bayesian counterpart has not been fully investigated. This paper aims to contribute to theoretical justification for the mixtures of regression model from the Bayesian perspective. In particular, we establish strong consistency of posterior distribution and determine how fast posterior distribution converges to the true value of the parameter in the context of mixture of binary regressions, Poisson regressions and Gaussian regressions.