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Introduced by Kiefer and Wolfowitz \cite{KW56}, the nonparametric maximumlikelihood estimator (NPMLE) is a widely used methodology for learning mixtureodels and empirical Bayes estimation. Sidestepping the non-convexity in mixturelikelihood, the NPMLE estimates the mixing distribution by maximizing the totallikelihood over the space of probability measures, which can be viewed as anextreme form of overparameterization.  In this paper we discover a surprising property of the NPMLE solution.Consider, for example, a Gaussian mixture model on the real line with asubgaussian mixing distribution. Leveraging complex-analytic techniques, weshow that with high probability the NPMLE based on a sample of size $n$ has$O(\log n)$ atoms (mass points), significantly improving the deterministicupper bound of $n$ due to Lindsay \cite{lindsay1983geometry1}. Notably, anysuch Gaussian mixture is statistically indistinguishable from a finite one with$O(\log n)$ components (and this is tight for certain mixtures). Thus, absentany explicit form of model selection, NPMLE automatically chooses the rightmodel complexity, a property we term \emph{self-regularization}. Extensions toother exponential families are given. As a statistical application, we showthat this structural property can be harnessed to bootstrap existing Hellingerrisk bound of the (parametric) MLE for finite Gaussian mixtures to the NPMLEfor general Gaussian mixtures, recovering a result of Zhang\cite{zhang2009generalized}.

Self-regularizing Property of Nonparametric Maximum Likelihood Estimator in Mixture Models