I will consider two approaches to vector encodings of causal graphs. In the 'direct' approach binary variables are used to indicate the presence/absence of certain graphical structures (e.g. edges, parent sets). In the 'imset' approach we have a vector encoding of *any* conditional independence (CI) model, and we focus attention on those CI models representable by some class of graphs (DAGs, MAGs, etc). The "characteristic imset" approach (Studeny, Hemmecke, Lindner) lies at the intersection of the 'direct' and 'imset' viewpoints.

There has been a reasonable amount of work on causal discovery using the 'direct' approach and I will give a high-level account of the associated pros and cons. Most (not all) of this work has been for causally sufficient DAGs. The 'imset' alternative is attractive (due to its universality) but I think it is still an open question whether it is practicable for causal discovery. As part of the discussion on this point I'll describe a CI representation that is 'dual' to that of imsets, namely supermodular functions.