Purpose: To illustrate the use of normalized random measures as mixing measures in nonparametric hierarchical mixture models and point out how possible computational issues can be successfully addressed. Summary: Ever since its introduction the Dirichlet process mixture model is found to represent the most popular Bayesian nonparametric model. The Dirichlet process mixture model and more general mixtures based on discrete random probability measures have been shown to be flexible and accurate models for density estimation and clustering. The idea of this paper is to illustrate the use of normalized random measures as mixing measures in nonparametric hierarchical mixture models and point out how possible computational issues can be successfully addressed. In this regard the concept of normalized random measures is reviewed and their uses for Bayesian nonparametric inference are highlighted. Particular emphasis is given to the posterior representation since it plays a key role in the elaboration of the sampling scheme used in this article. A conditional algorithm for simulating from the posterior of mixtures based on normal random measures with independent increments (NRMI) is described in detail. A comparative analysis for location-scale mixtures that considers a set of alternatives for the mixture kernel and for the nonparametric component is presented. A comprehensive analysis using real time data sets is done highlighting the potential of NRMI mixtures and the outcomes are discussed. Simulation results indicate that normalized random measure mixtures potentially represent a valid default choice for density estimation problems. (49 refs.)
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