Summary
This arXiv paper (2610.30109) by Shayan Hundrieser and Tudor Manole studies the nonparametric maximum likelihood estimator (NPMLE) for univariate Gaussian mixture densities estimated from n i.i.d. samples. The authors prove that the unconstrained NPMLE attains the minimax optimal rate of log(n)/n in squared Hellinger distance when the true mixing measure is sub-Gaussian. When the mixing measure has compact support, an NPMLE constrained to a compact interval achieves a slightly faster minimax optimal rate of order log(n)/(n log log n). The proofs sharply characterize the local Hellinger bracketing entropy of finite Gaussian mixture classes, with a key technical innovation being sharp bounds on the L^2 bracketing entropy of balls of bounded-degree polynomials, which may be of independent interest. As an application, the paper establishes regret optimality of NPMLE-based g-modeling procedures for Gaussian empirical Bayes problems with bounded priors.
We analyze the problem of estimating univariate Gaussian mixture densities using the nonparametric maximum likelihood estimator (NPMLE) based on n independent and identically distributed samples.
We show that the unconstrained NPMLE achieves the minimax optimal rate log(n)/n in squared Hellinger distance, when the true mixing measure is sub-Gaussian. In addition, we show that when the true mixing measure has compact support, the NPMLE constrained to a compact interval also achieves the minimax optimal rate, which is slightly faster and of order log(n)/(n log log n).
Our proof consists of sharply characterizing the local Hellinger bracketing entropy of the class of finite Gaussian mixtures. One of our key technical innovations is to sharply bound the L^2 bracketing entropy of balls of bounded-degree polynomials, which may be of independent interest.
As an application, we establish the regret optimality of NPMLE-based g-modeling procedures for Gaussian empirical Bayes problems with bounded priors.
Paper details
- Field: Machine Learning
- Authors: Shayan Hundrieser, Tudor Manole
- arXiv: 2610.30109
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