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Register a new userIdentifiability of parameters in latent structure models with many observed variables
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John Rhodes
Publication date
2009
fields
Mathematical Statistics
and research's language is
English
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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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Latent class models with covariates are widely used for psychological, social, and educational researches. Yet the fundamental identifiability issue of these models has not been fully addressed. Among the previous researches on the identifiability of latent class models containing covariates, Huang and Bandeen-Roche (2004, Psychometrika, 69:5-32) studied the local identifiability conditions. However, motivated by recent advances in the identifiability of restricted latent class models, particularly the Cognitive Diagnosis Models (CDMs), we show in this work that the conditions in Huang and Bandeen-Roche (2004) are only necessary but not sufficient to determine the local identifiability of the model parameters. To address the open identifiability issue for latent class models with covariates, this work establishes conditions to ensure the global identifiability of the model parameters in both strict and generic sense. Moreover, our results extend to polytomous-response CDMs with covariates, which generalizes the existing identifiability results for CDMs.
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