A Characterization of the Orthocomplement of the Tangent Space of Semiparametric Markov Models

This paper studies the orthogonal complement of the tangent space for general semiparametric Markov models defined by conditional-independence restrictions. The authors note that this object is known for models Markov relative to DAGs, but has remained uncharacterized for broader classes such as undirected graphical models, chain graphs, and acyclic directed mixed graphs. They derive closed-form expressions for the orthogonal complement and use them to characterize classes of influence functions for conditional-mean parameters in several graphical models. The stated goal is to support efficient, regular, asymptotically linear inference beyond DAG-equivalent settings.