Bayesian density estimation and model selection using nonparametric hierarchical mixtures

A class of nonparametric hierarchical mixtures is considered for Bayesian density estimation. This class, namely mixtures of parametric densities on the positive reals with a normalized generalized gamma process as mixing measure, is very flexible in the detection of clusters in the data. With an almost sure approximation of the posterior trajectories of the mixing process a Markov chain Monte Carlo algorithm is run to estimate linear and nonlinear functionals of the predictive distributions. The best-fitting mixing measure is found by minimizing a Bayes factor for parametric against nonparametric alternatives. Simulated and historical data illustrate the method, finding a trade-off between the best-fitting model and the correct identification of the number of components in the mixture.