Nearest Neighbor Dirichlet Mixtures
This work addresses computational bottlenecks in Bayesian density estimation for statisticians and data scientists, offering a more efficient alternative to traditional MCMC-based methods.
The authors tackled the computational inefficiency and instability of Bayesian density estimation methods by proposing nearest neighbor-Dirichlet mixtures, which maintain Bayesian strengths while enabling fast, parallel computation and achieving desirable asymptotic properties.
There is a rich literature on Bayesian methods for density estimation, which characterize the unknown density as a mixture of kernels. Such methods have advantages in terms of providing uncertainty quantification in estimation, while being adaptive to a rich variety of densities. However, relative to frequentist locally adaptive kernel methods, Bayesian approaches can be slow and unstable to implement in relying on Markov chain Monte Carlo algorithms. To maintain most of the strengths of Bayesian approaches without the computational disadvantages, we propose a class of nearest neighbor-Dirichlet mixtures. 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 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 data set in the context of classification.