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.
Covarion models of character evolution describe inhomogeneities in substitution processes through time. In phylogenetics, such models are used to describe changing functional constraints or selection regimes during the evolution of biological sequenc
es. In this work the identifiability of such models for generic parameters on a known phylogenetic tree is established, provided the number of covarion classes does not exceed the size of the observable state space. `Generic parameters as used here means all parameters except possibly those in a set of measure zero within the parameter space. Combined with earlier results, this implies both the tree and generic numerical parameters are identifiable if the number of classes is strictly smaller than the number of observable states.
Inference of evolutionary trees and rates from biological sequences is commonly performed using continuous-time Markov models of character change. The Markov process evolves along an unknown tree while observations arise only from the tips of the tre
e. Rate heterogeneity is present in most real data sets and is accounted for by the use of flexible mixture models where each site is allowed its own rate. Very little has been rigorously established concerning the identifiability of the models currently in common use in data analysis, although non-identifiability was proven for a semi-parametric model and an incorrect proof of identifiability was published for a general parametric model (GTR+Gamma+I). Here we prove that one of the most widely used models (GTR+Gamma) is identifiable for generic parameters, and for all parameter choices in the case of 4-state (DNA) models. This is the first proof of identifiability of a phylogenetic model with a continuous distribution of rates.
