Identifiability of parameters in latent structure models with many observed variables

TL;DR

This paper develops algebraic methods to establish the identifiability of parameters in complex latent structure models, including mixtures and hidden Markov models, showing that generic identifiability often suffices for meaningful inference.

Contribution

It introduces a general algebraic approach for proving identifiability in latent models, extending Kruskal's theorem to diverse models and clarifying conditions for parametric and nonparametric cases.

Findings

Identifiability can be established using algebraic arguments for various latent models.

Generic identifiability typically holds, meaning nonidentifiable parameters are rare.

New results improve understanding of finite mixture models and hidden Markov models.

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.