Comment on "Blessings of Multiple Causes"

This scenario is dir ectly analogous to longitudinal causal inference problems with multiple time-varying treatments that conta in time-varying confounders, variables that serve as confounders for some treatments and as mediators for othe r treatments. If there is an unmeasured con-founder for the R -Y relationship (represented by V and the dashed arrows in Figure 1 (a)), then conditioning on R fails to identify the direct effects of A on Y, because it opens a confounding pathway through V . See Hernan and Robins (2020) for an overview of these issues. The answer to the question posed in Appendix B of WB, "Can the c auses be causally dependent among themselves?" is therefore "no." If they are causally depend ent then the deconfounder, by dint of rendering the causes independent, breaks some of the structure among t he causes A, and as was originally established in the time-varying treatment setting, this undermines the identification of joint effects of A on Y by covariate adjustment. Analysis of Lemma 4. This simple argument also serves as a counterexample to Lemm a 4, which states that the deconfounder does not pick up any post-treatment va riables and can be treated as a pre-treatment covariate. This is necessarily false whenever the causes ar e causally dependent among themselves, but it need not hold even if the causes are not causally dependent, s ee below. The proof of Lemma 4 in Appendix I states that "Inferring the s ubstitute confounder Z