Open Access
December 2009 Identifiability of parameters in latent structure models with many observed variables
Elizabeth S. Allman, Catherine Matias, John A. Rhodes
Ann. Statist. 37(6A): 3099-3132 (December 2009). DOI: 10.1214/09-AOS689


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.


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Elizabeth S. Allman. Catherine Matias. John A. Rhodes. "Identifiability of parameters in latent structure models with many observed variables." Ann. Statist. 37 (6A) 3099 - 3132, December 2009.


Published: December 2009
First available in Project Euclid: 17 August 2009

zbMATH: 1191.62003
MathSciNet: MR2549554
Digital Object Identifier: 10.1214/09-AOS689

Primary: 62E10
Secondary: 62F99 , 62G99

Keywords: Algebraic statistics , Conditional independence , Contingency table , finite mixture , Identifiability , latent structure , multivariate Bernoulli mixture , nonparametric mixture

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 6A • December 2009
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