On counterfactual inference with unobserved confounding

TL;DR

This paper develops a method for estimating counterfactual distributions in observational studies with unobserved confounding, using a joint learning approach for heterogeneous exponential family models from limited data.

Contribution

It introduces a convex pooling objective for jointly learning heterogeneous exponential family parameters with one sample per unit, handling unobserved confounding and heterogeneity.

Findings

Error bound scales linearly with parameter space complexity

Method enables consistent imputation of missing covariates

Applicable to distributions satisfying the logarithmic Sobolev inequality

Abstract

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 that introduces statistical biases between interventions and outcomes as well as exacerbates the heterogeneity across units. Modeling the conditional distribution of the outcomes 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 $n$ parameter vectors, 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 $O(s\log k/p)$. En route, we derive sufficient conditions for compactly supported distributions to satisfy the logarithmic Sobolev inequality. As an application of the framework, our results enable consistent imputation of sparsely missing covariates.