A marginal DAG model is the model induced by the observed variables in a DAG with latent nodes, where we make no assumption about the state-space of the unobserved variables. A question that arises in this context is when such a model is reducible to another DAG model. We show that this occurs if and only if the model does not induce any non-trivial inequality constraints. This has a few interesting consequences: for instance, both maximal ancestral graph models and nested Markov models (that are not Markov equivalent to a DAG model) are shown to be strictly larger than the corresponding marginal model. The proof relies on a series of reductions to smaller classes, including so-called arid and ancestral graphs, which we will define and present as part of the talk.