Paper:
Ancestral Causal Inference
Constraint-based causal discovery from limited data is a notoriously difficult challenge due to the many borderline independence test decisions. Several approaches to improve the reliability of the predictions by exploiting redundancy in the independence information have been proposed recently. Though promising, existing approaches can still be greatly improved in terms of accuracy and scalability. We present a novel method that reduces the combinatorial explosion of the search space by using a more coarse-grained representation of causal information, drastically reducing computation time. Additionally, we propose a method to score causal predictions based on their confidence. Crucially, our implementation also allows one to easily combine observational and interventional data and to incorporate various types of available background knowledge. We prove soundness and asymptotic consistency of our method and demonstrate that it can outperform the state-ofthe-art on synthetic data, achieving a speedup of several orders of magnitude. We illustrate its practical feasibility by applying it to a challenging protein data set.

Review:
This paper considers learning a causal structure through constraint testing, representing that structure as an 'ancestral' object rather than the more common directed acyclic graphs.  I think this is essentially equivalent to representing each DAG by its transitive closure (perhaps the authors could comment on this). 
Overall this is a nice method to an important problem.  The setup seems quite flexible, so I can imagine incorporating other kinds of information relatively easily.  The use of ancestral models rather than Bayesian networks is sensible and novel to my knowledge - though see below for a comment about recovering a BN.    The paper is well written, seems technically solid and novel, and provides a method that could be very useful.  Some questions and suggestions for improvement are given below.  ----  COMMENTS  Page 2 line 46 - could you be a bit more precise when saying 'keep the computational burden low'.  Relatedly, it would be helpful to discuss how quickly the number of ancestral objects grows asymptotically, and whether Chickering's result about learning BNs being NP-hard applies to ancestral objects as well (is this is easy to check).  Lemma 2 - it would be useful to have here a signpost to the discussion after Theorem 1.  The list of rules immediately had me wondering: "how did they come up with this list and is it complete?"  In the simulations, I think it's a bit unreasonable to throw in the red curve (which simulates what happens if some interventional data are available), since none of the other methods claim to be able to deal with this.  This should be removed from the plot, as it is misleading.  What's going on with CFCI and FCI in the plots - they seem to have essentially no points on the plot.  Is this because it's too computationally hard to run FCI with different test sizes?  If so, perhaps using RFCI instead would help (I'm not sure if there is a CRFCI though!).  The description of the bootstrapping is a bit hard to follow - is this somewhat like stability selection?  Perhaps you could use the supplementary material to explain this better.  For the interventional setting, the method of Peters, Buhlmann and Meinshausen (2016) [invariant prediction] would seem like the most sensible comparison.  It can deal with a very flexible class of interventions, so would make a good counterpoint to the discussion at the end of Section 3.  Given the ancestral object, is it hard then just to recover a directed graph by doing some additional tests?  Perhaps this is harder than it seems because parts of the graph may not have an ordering at all, but I'd be interested to know the authors' thoughts.  ----  TYPOS ETC  Though not intentional, a naive reading of the second paragraph of the introduction gives the impression that 'hybrid methods' are a creation of this paper!  There are, of course, many other hybrid methods available for learning Bayesian networks and therefore causal structures.  p5 line 191 - "allow to deduce" -> "allow one to deduce"  Figure 3 means nothing to me - what is it supposed to represent?  Could you give the ancestral structure discovered in the appendix?  It would be good to compare to the 'ground truth' and the Sachs method.