Annals of Statistics

Identifiability of parameters in latent structure models with many observed variables

Elizabeth S. Allman, Catherine Matias, and John A. Rhodes

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Abstract

While hidden class models of various types arise in many statistical applications, it is often difficult to establish the identifiability of their parameters. Focusing on models in which there is some structure of independence of some of the observed variables conditioned on hidden ones, we demonstrate a general approach for establishing identifiability utilizing algebraic arguments. A theorem of J. Kruskal for a simple latent-class model with finite state space lies at the core of our results, though we apply it to a diverse set of models. These include mixtures of both finite and nonparametric product distributions, hidden Markov models and random graph mixture models, and lead to a number of new results and improvements to old ones.

In the parametric setting, this approach indicates that for such models, the classical definition of identifiability is typically too strong. Instead generic identifiability holds, which implies that the set of nonidentifiable parameters has measure zero, so that parameter inference is still meaningful. In particular, this sheds light on the properties of finite mixtures of Bernoulli products, which have been used for decades despite being known to have nonidentifiable parameters. In the nonparametric setting, we again obtain identifiability only when certain restrictions are placed on the distributions that are mixed, but we explicitly describe the conditions.

Article information

Source
Ann. Statist., Volume 37, Number 6A (2009), 3099-3132.

Dates
First available in Project Euclid: 17 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1250515381

Digital Object Identifier
doi:10.1214/09-AOS689

Mathematical Reviews number (MathSciNet)
MR2549554

Zentralblatt MATH identifier
1191.62003

Subjects
Primary: 62E10: Characterization and structure theory
Secondary: 62F99: None of the above, but in this section 62G99: None of the above, but in this section

Keywords
Identifiability finite mixture latent structure conditional independence multivariate Bernoulli mixture nonparametric mixture contingency table algebraic statistics

Citation

Allman, Elizabeth S.; Matias, Catherine; Rhodes, John A. Identifiability of parameters in latent structure models with many observed variables. Ann. Statist. 37 (2009), no. 6A, 3099--3132. doi:10.1214/09-AOS689. https://projecteuclid.org/euclid.aos/1250515381


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