On the geometry of DAG models with hidden variables

We prove that many graphical models with hidden variables are not curved exponential families. This result, together with the fact that some graphical models are curved and not linear, implies that the hierarchy of graphical models, as linear, curved, and stratified, is non-collapsing; each level in the hierarchy is strictly contained in the larger levels. This result is discussed in the context of model selection of graphical models.