Mixture models are widely used in modeling heterogeneous data populations. A standard approach of mixture modeling is to assume that the mixture component takes a parametric kernel form, while the flexibility of the model can be obtained by using a large or possibly unbounded number of such parametric kernels. In many applications, making parametric assumptions on the latent subpopulation distributions may be unrealistic, which motivates the need for nonparametric modeling of the mixture components themselves. In this paper we study finite mixtures with nonparametric mixture components, using a Bayesian nonparametric modeling approach. In particular, it is assumed that the data population is generated according to a finite mixture of latent component distributions, where each component is endowed with a Bayesian nonparametric prior such as the Dirichlet process mixture. We present conditions under which the individual mixture component's distributions can be identified, and establish posterior contraction behavior for the data population's density, as well as densities of the latent mixture components. We develop an efficient MCMC algorithm for posterior inference and demonstrate via simulation studies and real-world data illustrations that it is possible to efficiently learn complex distributions for the latent subpopulations. In theory, the posterior contraction rate of the component densities is nearly polynomial, which is a significant improvement over the logarithm convergence rate of estimating mixing measures via deconvolution.

When a population exhibits heterogeneity, we often model it via a finite mixture: decompose it into several different but homogeneous subpopulations. Contemporary practice favors learning the mixtures by maximizing the likelihood for statistical efficiency and the convenient EM-algorithm for numerical computation. Yet the maximum likelihood estimate (MLE) is not well defined for the most widely used finite normal mixture in particular and for finite location-scale mixture in general. We hence investigate feasible alternatives to MLE such as minimum distance estimators. Recently, the Wasserstein distance has drawn increased attention in the machine learning community. It has intuitive geometric interpretation and is successfully employed in many new applications. Do we gain anything by learning finite location-scale mixtures via a minimum Wasserstein distance estimator (MWDE)? This paper investigates this possibility in several respects. We find that the MWDE is consistent and derive a numerical solution under finite location-scale mixtures. We study its robustness against outliers and mild model mis-specifications. Our moderate scaled simulation study shows the MWDE suffers some efficiency loss against a penalized version of MLE in general without noticeable gain in robustness. We reaffirm the general superiority of the likelihood based learning strategies even for the non-regular finite location-scale mixtures.