Annals of Statistics
- Ann. Statist.
- Volume 48, Number 4 (2020), 2277-2302.
Identifiability of nonparametric mixture models and Bayes optimal clustering
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 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 infinite Gaussian mixtures. In contrast to the recent literature, 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.
Ann. Statist., Volume 48, Number 4 (2020), 2277-2302.
Received: November 2018
Revised: July 2019
First available in Project Euclid: 14 August 2020
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Primary: 62G05: Estimation 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]
Secondary: 62H12: Estimation
Aragam, Bryon; Dan, Chen; Xing, Eric P.; Ravikumar, Pradeep. Identifiability of nonparametric mixture models and Bayes optimal clustering. Ann. Statist. 48 (2020), no. 4, 2277--2302. doi:10.1214/19-AOS1887. https://projecteuclid.org/euclid.aos/1597370673
- Supplement to “Identifiability of nonparametric mixture models and Bayes optimal clustering”. This supplement contains proofs of all the main results along with various technical results and experiment details.