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The Blessings of Multiple Causes: Rejoinder

We thank all the discussants for taking the time and energy to build on this work; and we thank the editors for putting together an engaging and thought-provoking collection of discussions. After reading these contributions, we were struck that these are not mere discussions—indeed, each is an article in itself. This collection pushes forward “The Blessings of Multiple Causes” in significant ways, offering new theory, new criticism, and new application. After highlighting some of the themes of these articles, we will turn to each individually. “The Blessings of Multiple Causes” provide assumptions, theory, and algorithms for multiple causal inference. The deconfounder algorithm involves modeling the causes, using the model to infer a substitute confounder, and then using the substitute confounder in a downstream causal inference. The deconfounder is not a black-box solution to causal inference. Rather, it is a way to use careful domain-specific modeling in the service of causal inference. Causal inference with the deconfounder involves a number of assumptions and trade-offs, and many of the discussants highlighted these. Among them are the following. (1) There can be no unobserved single-cause confounders. (2) When we apply the deconfounder, we trade an increase in estimation variance for a reduction in confounding bias; there is no free lunch. (3) We do not recommend using the deconfounder with causally dependent causes, such as a time series; finding a substitute confounder may be too difficult in these scenarios. There are many directions for further research, and the discussants have pointed out several of the most important ones. We need a more complete picture of identification; D’Amour (2019) and the discussions here make good progress (see Table1). We need to understand the finite-sample properties of the deconfounder, and how to estimate uncertainty about causal inferences when using a substitute multi-cause confounder. We need rigorous methods of model criticism for assessing the validity of the substitute confounder. Deconfounder-like methods have already been used for genome-wide association studies (e.g., Pritchard et al. 2000) and estimating peer effects in networks (Shalizi and McFowland III 2016). More broadly, the deconfounder strategy points to many applications, including in genetics, psychology, education, and marketing, where factor models are routinely fit to largescale data. We hope that statisticians and machine learners will