Self-regularizing Property of Nonparametric Maximum Likelihood Estimator in Mixture Models
This provides a self-regularization property for mixture models, automatically selecting the right model complexity without explicit model selection, which is incremental but useful for statisticians and machine learning practitioners.
The paper tackles the problem of model complexity in nonparametric maximum likelihood estimation (NPMLE) for mixture models, showing that the NPMLE solution has only O(log n) atoms with high probability, which is a significant improvement over the previous deterministic bound of n.
Introduced by Kiefer and Wolfowitz \cite{KW56}, the nonparametric maximum likelihood estimator (NPMLE) is a widely used methodology for learning mixture odels and empirical Bayes estimation. Sidestepping the non-convexity in mixture likelihood, the NPMLE estimates the mixing distribution by maximizing the total likelihood over the space of probability measures, which can be viewed as an extreme 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 a subgaussian mixing distribution. Leveraging complex-analytic techniques, we show that with high probability the NPMLE based on a sample of size $n$ has $O(\log n)$ atoms (mass points), significantly improving the deterministic upper bound of $n$ due to Lindsay \cite{lindsay1983geometry1}. Notably, any such Gaussian mixture is statistically indistinguishable from a finite one with $O(\log n)$ components (and this is tight for certain mixtures). Thus, absent any explicit form of model selection, NPMLE automatically chooses the right model complexity, a property we term \emph{self-regularization}. Extensions to other exponential families are given. As a statistical application, we show that this structural property can be harnessed to bootstrap existing Hellinger risk bound of the (parametric) MLE for finite Gaussian mixtures to the NPMLE for general Gaussian mixtures, recovering a result of Zhang \cite{zhang2009generalized}.