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See Ancestral Graph Markov Models by Richardson and Spirtes for a complete description. Motivated by the property that Gaussians are closed under marginalization and conditioning, we ask what graphs are closed under both operations?

Consider an undirected graph \(x_1 - L - x_2 - x_3\) and we marginalize out \(L\). Is there another undirected graph which is a P-map for the original graph? Yes: \(x_1 - x_2 - x_3\). What if we instead condition on \(L\)? Again, there is a P-map for the original graph: \(x_1 \phantom{-} x_2 - x_3\). We conclude that undirected graphs are closed under conditioning and marginalization.

We now ask the same two questions of directed graphs. The answer to both will be no. Consider \(x_1 \rightarrow x_2\), \(x_3 \rightarrow x_4\), \(L \rightarrow x_2\) and \(L \rightarrow x_4\). For a concrete example, \(x_1, x_3\) might be treatments, \(x_2, x_4\) outcomes and \(L\) the general health state of the study participant. If we try to marginalize out \(L\), no directed graph will be a P-map. For another graph, too complicated for me to type, marginalization can create a new V-structure, meaning the original graph’s independencies are affected. Consequently, directed graphs are not closed under conditioning nor marginalization.

We can rectify this shortcoming by introducing bidirectional edges e.g. \(x_i \leftrightarrow x_j\) and undirected edges \(x_i - x_j\). This class of graphs is called Ancestral Graphs.

**Ancestral Graph**: An ancestral graph \(G=(v, (U, D, B))\) is a graph with three
sets of edges: undirected \(U\), directed \(D\) and bidirectional \(B\).

TODO: why is this?