  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.
