# On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoising

arXiv: https://arxiv.org/abs/1712.02009 · HTML: https://mlsanity.com/paper/1712.02009/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Dec 2017.

## Built on by (most cited first, 19 of 19)

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- [Model-free Estimation of Latent Structure via Multiscale Nonparametric Maximum Likelihood](https://mlsanity.com/paper/2410.22248/) (2024)
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- [Empirical Bayes for correlated Gaussian sequence model](https://mlsanity.com/paper/2607.03596/) (2026)
- [Polylogarithmic Sparsity of Randomly Reweighted NPMLEs for Gaussian Mixtures](https://mlsanity.com/paper/2610.01088/) (2026)

## Abstract (arXiv)

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.