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Primary Submission Category: Heterogeneous Treatment Effects

On counterfactual inference with unobserved confounding

Authors: Abhin Shah, Raaz Dwivedi, Devavrat Shah, Gregory Wornell,

Presenting Author: Abhin Shah*

Given an observational study with $n$ independent but heterogeneous units, our goal is to learn the counterfactual distribution for each unit using only one $p$-dimensional sample per unit containing covariates, interventions, and outcomes. Specifically, we allow for unobserved confounding which introduces statistical biases between interventions and outcomes as well as exacerbates the heterogeneity across units. Modeling the underlying joint distribution as an exponential family, we reduce learning the unit-level counterfactual distributions to learning $n$ exponential family distributions with heterogeneous parameters and only one sample per distribution. We introduce a convex objective that pools all $n$ samples to jointly learn all $n$ parameters and provide a unit-wise mean squared error bound that scales linearly with the metric entropy of the parameter space. For example, when the parameters are $s$-sparse linear combination of $k$ known vectors, the error is $O(slog k/p)$. En route, we derive sufficient conditions for compactly supported distributions to satisfy the logarithmic Sobolev inequality. As an application, our results enable consistent imputation of missing covariates when they are sparse.