Journal Article

IDENTIFIABILITY OF PARAMETERS IN LATENT STRUCTURE MODELS WITH MANY OBSERVED VARIABLES

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

2009

Overview

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.

• Publisher: Institute of Mathematical Statistics,The Institute of Mathematical Statistics
• Subject: 62E10 / 62F99 / 62G99 / Algebra / algebraic statistics / Bernoulli Hypothesis / Combinatorics / Combinatorics. Ordered structures / conditional independence / contingency table / Distribution theory / Exact sciences and technology / finite mixture / General topics / Graph theory / Identifiability / latent structure / Markov analysis / Markov models / Mathematical models / Mathematics / Matrices / multivariate Bernoulli mixture / nonparametric mixture / Parameter identification / Parameterization / Parametric models / Polynomials / Probability and statistics / Probability distributions / Probability theory and stochastic processes / Random variables / Sciences and techniques of general use / Statistics / Statistics Theory / Studies / Tensors