Graphical Models for Recovering Probabilistic and Causal Queries from Missing Data
We address the problem of deciding whether a causal or probabilistic query is estimable from data corrupted by missing entries, given a model of missingness process. We extend the results of  by presenting more general conditions for recovering probabilistic queries of the form P (y|x) and P (y, x) as well as causal queries of the form P (y|do(x)). We show that causal queries may be recoverable even when the factors in their identifying estimands are not recoverable. Specifically, we derive graphical conditions for recovering causal effects of the form P (y|do(x)) when Y and its missingness mechanism are not d-separable. Finally, we apply our results to problems of attrition and characterize the recovery of causal effects from data corrupted by attrition.
The paper deals with the problem of identifiability in the presence of various generated mechanisms for missing data, using a framework introduced by Mohan and others.  Some existing results are improved upon, and some nice examples are given (such as identifiability of causal distributions even when conditional distributions are unavailable).QualityTechnical quality seems high, the results are very involved but appear to be correct.  Results like Theorem 3 are clear and interpretable.  Some of the results are incremental improvements upon existing ones, and don't contribute (so far as I can see) to a bigger picture of what is really going on here.Section 3 seems orthogonal to the rest of the paper, and its main result (Theorem 1) is quite hard to comprehend.  It seems to be closely related to the Markov blanket results in Richardson (2009).  The quality improves later on, with section 5 giving some nice clear answers, and Section 6 some interesting results about causal queries.---ClarityThis is the paper's main weakness.  Firstly, there is probably too much material for a conference paper here (though the importance of some of it is unclear), as evidenced by the density of the text and the 13 page technical appendix.  The article does not flow clearly, and is hard to follow as a result - probably this would work better as a longer, more considered, journal paper.Whilst finding good notation for so many quantities is not easy, the notation could certainly be improved.  The use of the same capital 'V' to denote so many different objects makes it hard work for the reader, and in places the subscripts become illegible.  In addition, the fact that V_i is a vertex but V_o a set of vertices is particularly confusing.  The set equation in Corollary 1 is perhaps the most extreme example of poor presentation.---Originality and SignificanceSections 5 and 6 are clear and interesting, and potentially quite significant.  This paper deals with an important problem, and contains results which seem genuinely interesting and useful.  It would perhaps be better to step back and work out what is going on in the general case, and (try to) come up with a complete criterion for identifiability in this framework (as was achieved in the case of finding causal effects without missingness).  It is difficult to imagine a statistician trawling through a series of papers like this to find the precise result which shows (non)-identifiability of the causal effect she is interested in, especially if that result is as complicated as Theorem 1.Theorem 2 and Corollary 1 seem to be a form of inverse probability weighting.  The reason that these results work is presumably something to do with the conditioning sets being Markov blankets for R(i) in the mixed graphs the author is using.  Richardson (2003) and (2009) might be relevant.  ---Other Commentsp4, l214: R^part _is_ a partition of R, not "a set containing partitions".  The condition on line 214 should then be that Rx and Ry "belong to distinct elements of the partition".Language is sometimes a bit casual: e.g. 'oftentimes'p4 l213.  Use \leftrightarrow rather than <--> Is Garcia (2013) actually wrong, or just a different framework?A notation for graphs constructed by severing edges is used in Definition 4 and Theorem 4, but not defined anywhere.V_o and V_m are used in the sentence before they are defined (apparently unnecessarily).---ReferencesRichardson, T.S., Markov properties for acyclic directed mixed graphs, SJS 30(1) 145-157, 2003Richardson, T.S., A factorization criterion for acyclic directed mixed graphs, UAI 25, 2009.