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

TL;DR

This paper presents counterexamples demonstrating that the deconfounder method does not effectively control for multi-cause confounding, clarifying misconceptions in prior theoretical claims.

Contribution

It provides clear counterexamples and critiques to the claims made by Wang and Blei regarding the deconfounder's effectiveness.

Findings

Counterexamples show the deconfounder fails in certain confounding scenarios

Clarification of errors in the original deconfounder theory

Demonstrates the deconfounder does not guarantee control of multi-cause confounding

Abstract

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.