Identifiability of Nonparametric Mixture Models and Bayes Optimal Clustering

TL;DR

This paper develops a general framework for the identifiability of nonparametric mixture models, enabling Bayes optimal clustering with theoretical guarantees and practical algorithms, applicable to complex real-world data.

Contribution

It introduces new identifiability conditions for nonparametric mixtures using overfitted parametric models, extending classical clustering theory to nonparametric settings.

Findings

Established conditions for nonparametric mixture identifiability.

Generalized Bayes optimal clustering to nonparametric models.

Provided a practical algorithm with consistency guarantees.

Abstract

Motivated by problems in data clustering, we establish general conditions under which families of nonparametric mixture models are identifiable, by introducing a novel framework involving clustering overfitted \emph{parametric} (i.e. misspecified) mixture models. These identifiability conditions generalize existing conditions in the literature, and are flexible enough to include for example mixtures of Gaussian mixtures. In contrast to the recent literature on estimating nonparametric mixtures, we allow for general nonparametric mixture components, and instead impose regularity assumptions on the underlying mixing measure. As our primary application, we apply these results to partition-based clustering, generalizing the notion of a Bayes optimal partition from classical parametric model-based clustering to nonparametric settings. Furthermore, this framework is constructive so that it yields a practical algorithm for learning identified mixtures, which is illustrated through several examples on real data. The key conceptual device in the analysis is the convex, metric geometry of probability measures on metric spaces and its connection to the Wasserstein convergence of mixing measures. The result is a flexible framework for nonparametric clustering with formal consistency guarantees.