On the identification of discrete graphical models with hidden nodes

We focus on the identification of discrete undirected graphical models with one unobserved binary variable and establish a necessary and sufficient condition for the rank of the transformation from the natural parameters to the parameters of the model to be full. This ensures local identification of this class of models. These models generalize the latent class model, by allowing associations between the observed variables conditionally on the latent one. The practical importance of this issue is witnessed by several applied papers. For non-full rank models, the obtained characterization allows us to find the expression of the (sub)space where the identifiability breaks down. Geometrically, this corresponds to the singularities in the parameter space. This in turn allows us (a) to derive a reparametrization that leads to an identified model and (b) to compute the correct dimension of the model. The condition is based on the topology of the undirected graph associated with the model and relies on the faithfulness assumption.