  We study the Nonparametric Maximum Likelihood Estimator (NPMLE) for
estimating Gaussian location mixture densities in $d$-dimensions from
independent observations. Unlike usual likelihood-based methods for fitting
mixtures, NPMLEs are based on convex optimization. We prove finite sample
results on the Hellinger accuracy of every NPMLE. Our results imply, in
particular, that every NPMLE achieves near parametric risk (up to logarithmic
multiplicative factors) when the true density is a discrete Gaussian mixture
without any prior information on the number of mixture components. NPMLEs can
naturally be used to yield empirical Bayes estimates of the Oracle Bayes
estimator in the Gaussian denoising problem. We prove bounds for the accuracy
of the empirical Bayes estimate as an approximation to the Oracle Bayes
estimator. Here our results imply that the empirical Bayes estimator performs
at nearly the optimal level (up to logarithmic multiplicative factors) for
denoising in clustering situations without any prior knowledge of the number of
clusters.
