We study the problem of restricting Markov equivalence classes of maximal ancestral graphs (MAGs) containing certain edge marks, which we refer to as expert knowledge. MAGs forming a Markov equivalence class can be uniquely represented by an essential ancestral graph. We seek to learn the restriction of the essential ancestral graph containing the proposed expert knowledge. Our contributions are several-fold. First, we prove certain properties for the entire Markov equivalence class including a conjecture from Ali et al. (2009). Second, we present three sound graphical orientation rules, two of which generalize previously known rules, for adding expert knowledge to an essential graph. We also show that some orientation rules of Zhang (2008) are not needed for restricting the Markov equivalence class with expert knowledge. We provide an algorithm for including this expert knowledge and show that our algorithm is complete in certain settings i.e., in these settings, the output of our algorithm is a restricted essential ancestral graph. We conjecture this algorithm is complete generally. Outside of our specified settings, we provide an algorithm for checking whether a graph is a restricted essential graph and discuss its runtime. This work can be seen as a generalization of Meek (1995).

In this paper we discuss four problems regarding Markov equivalences for subclasses of loopless mixed graphs. We classify these four problems as finding conditions for internal Markov equivalence, which is Markov equivalence within a subclass, for external Markov equivalence, which is Markov equivalence between subclasses, for representational Markov equivalence, which is the possibility of a graph from a subclass being Markov equivalent to a graph from another subclass, and finding algorithms to generate a graph from a certain subclass that is Markov equivalent to a given graph. We particularly focus on the class of maximal ancestral graphs and its subclasses, namely regression graphs, bidirected graphs, undirected graphs, and directed acyclic graphs, and present novel results for representational Markov equivalence and algorithms.