Identification problems in mixture models: some general results

A structural model generating jointly a manifest variable y and a latent variable is considered. The object of this note is to examine, on the one hand, the relationships between the identification of the structural model and the component submodels, namely the conditional model generating y given and the marginal model generating, and, on the other hand, the relationships between the identification of the statistical model, i.e. a marginal model generating the manifest variable, and the component submodels. These relationships are obtained by using two concepts of identification, a weak one and a strong one.