Counterexamples to "The Blessings of Multiple Causes" by Wang and Blei

This note has been updated (April, 2020) to respond to "Towards Clarifying the Theory of the Deconfounder" by Yixin Wang, David M. Blei (arXiv:2003.04948). This original note, posted in January, 2020, is meant to complement our previous comment on "The Blessings of Multiple Causes" by Wang and Blei (2019). We provide a more succinct and transparent explanation of the fact that the deconfounder does not control for multi-cause confounding. The argument given in Wang and Blei (2019) makes two mistakes: (1) attempting to infer independence conditional on one variable from independence conditional on a different, unrelated variable, and (2) attempting to infer joint independence from pairwise independence. We give two simple counterexamples to the deconfounder claim.