There is a rich literature on Bayesian nonparametric methods for unknown densities. The most popular approach relies on Dirichlet process mixture models. These models characterize the unknown density as a kernel convolution with an unknown almost surely discrete mixing measure, which is given a Dirichlet process prior. Such models are very flexible and have good performance in many settings, but posterior computation relies on Markov chain Monte Carlo algorithms that can be complex and inefficient. As a simple and general alternative, we propose a class of nearest neighbor-Dirichlet processes. The approach starts by grouping the data into neighborhoods based on standard algorithms. Within each neighborhood, the density is characterized via a Bayesian parametric model, such as a Gaussian with unknown parameters. Assigning a Dirichlet prior to the weights on these local kernels, we obtain a simple pseudo-posterior for the weights and kernel parameters. A simple and embarrassingly parallel Monte Carlo algorithm is proposed to sample from the resulting pseudo-posterior for the unknown density. Desirable asymptotic properties are shown, and the methods are evaluated in simulation studies and applied to a motivating dataset in the context of classification.

In population genetics, determining the "population structure" is an important step in the analysis of sampled data. As an illustrative example, consider the impala, a species of antelope in southern Africa. Impalas are divided into two subspecies: the common impala occupying much of the eastern half of the region, and the black-faced impala inhabiting a small area in the west. While common impalas are abundant, the number of black-faced impalas has been decimated by drought, poaching, and declining resources due to human and livestock expansion. To assist conservation efforts, Lorenzen, Arctander and Siegismund (2006) collected samples from 216 impalas, and analyzed the genetic variation between/within the two subspecies. A key part of their analysis consisted of inferring the population structure -- that is, partitioning the data into distinct populations, and in particular, determining how many such populations there are. To infer the impala population structure, Lorenzen et al. employed a widely-used tool called Structure (Pritchard, Stephens and Donnelly, 2000) which, in the simplest version, models the data as a finite mixture, with each component in the mixture corresponding to a dis-Supported in part by NSF grant DMS-1007593 and DARPA contract FA8650-11-1-715.