Paper:
Removing Hidden Confounding by Experimental Grounding
Observational data is increasingly used as a means for making individual-level causal predictions and intervention recommendations. The foremost challenge of causal inference from observational data is hidden confounding, whose presence cannot be tested in data and can invalidate any causal conclusion. Experimental data does not suffer from confounding but is usually limited in both scope and scale. We introduce a novel method of using limited experimental data to correct the hidden confounding in causal effect models trained on larger observational data, even if the observational data does not fully overlap with the experimental data. Our method makes strictly weaker assumptions than existing approaches, and we prove conditions under which it yields a consistent estimator. We demonstrate our method's efficacy using real-world data from a large educational experiment.

Review:
Summary: This paper proposes a new method for leveraging experimental data to remove the bias arising from potential confounders in a related observational study. I find the key idea of the paper to be simple but clever, interesting, and potentially useful. The simulation based on real data (Section 5.2) is in my opinion quite convincing. On the other hand, I find the general exposition to be imprecise and difficult to read. I think that this manuscript has the substance to be a good paper if major work is undertaken to tighten it up and clarify the exposition. Specific comments below:   Substance:  1) I believe that the method you propose is sensible, but I am not yet convinced by its practical significance. It would be helpful if you could point to a number of real world datasets on which such a method could be used. I am not asking you to actually apply your method to other datasets, nor am I asking you to point to datasets you have access to — what I am asking is whether there are any concrete datasets you know of that could benefit from your method?  2) There is a general sense in which when estimating causal effects in observational studies, bias is often more of an issue than variance. I agree with that view, overall. Still, it would be useful if you could briefly discuss the variance of your estimator, and propose a way to estimate that variance. Proving a CLT (or if you can’t prove it, at least argue heuristically for it) would be also helpful for obtaining confidence intervals, etc…  3) The precision of your estimator $\hat{\theta}$ is limited by the sample size of the UNCONF dataset. Now if you look at the mean squared error of the estimator $\hat{\tau}$ that you construct, it is quite possible that because of the potential large variance of $\hat{\theta}$, it exceeds that of a naive estimator that will be biased due to confounding but will have much lower variance thanks to the large sample size of CONF. In effect, I am pointing to the fact that there is a bias / variance tradeoff at play here, in the choice between your estimator, and a naive estimator that assumes unconfoundedness. It would be useful to discuss this point.  4) In the introduction you write: “We discuss below why this [parametric correction] assumption is possibly reasonable […]”. I think that the insight from your paper — pushing the burden of extrapolation to the confounding term — is a very good one. You discuss this briefly on lines 156-159, but I think that you should emphasize this more.   Clarity:  I would usually include these remarks as minor comments, but in this particular case they are not minor. I am taking the time to be specific because I really believe that the paper could be improved dramatically if the writing was tighter.  1) The language you use is very imprecise:  - l.68: what do you mean by “scale” and “scope”?  - l.69: you talk about unconfoundedness, but you only define it on l.91. The sentence on l. 90: “the key assumption we make about the unconfounded data is its unconfoundedness” should alert you to the fact that something is wrong.  - The example in l.110-119 is confusing. What you assume is that dutiful patients are more likely to seek treatment. You should be explicit about this. As an aside, it is a matter of personal preference but you could consider wrapping this into an “example” environment to set it apart from the main text.  - l.131: \omega is not an estimator, and unbiased is not the right word for it.   - l 132: “[…] which interpolates between \hat{omega} to the RCT outcomes” what does this sentence mean?  - l. 238-242 is confusing. Why do you superscript Y by CATE?  - l. 248: you mean that you are sampling a *fraction* q’ of the rural…  etc…. Please go over your paper carefully eliminating these kinds of things.  2) l.139-144 should be a lemma. The proof does not add to the comprehension.   3) Please consider not using superscripts like “Conf” and “Unc”. Single letters “C” and “U” would do. Also, using the letter E is usually not a good idea when you take expectations (even if you use a slightly different symbol for expectations).  4) Typos:  - l.40: “this observational sample which might suffer […]” -> remove the word “which”.  - l.131 and l.132: its \omega not omega  - l.138: “doing so would be difficuly” -> difficult  - l 139: “[…] relies on a simply identity” -> simple etc…