Comment: Reflections on the Deconfounder

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Acknowledgments

I would like to thank Avi Feller, Alex Franks, Elizabeth Ogburn, James Atwood, and D. Sculley for helpful comments. Thanks to Yixin Wang and David Blei for open discussions about their work.

Notes

1 The “no unobserved single-cause confounders” assumption does not uniquely identify the factor model by itself. Some structure also needs to be put on the latent variable, and even then, the factor model may not be identified. See D’Amour (2019) for an example where the factor model P(A, Z) is itself not identified.

2 To see this, note that the independence copula $c(Y^{\text{obs}},Z|A=a)=1$ implies that $P(Y(a))=P(Y^{\text{obs}}|A=a)$. Thus, because $P(Y(a))\ne P(Y^{\text{obs}}|A=a)$, this copula and the true copula yield different conclusions about $P(Y(a))$.

3 This is not how the theorem is stated, but this function restriction is implied by the subsequent overlap condition.