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 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 $n$ 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.

翻译：给定一项包含$n$个独立但异质性单元（单位）的观察性研究，我们的目标是仅使用每个单元的一个$p$维样本（包含协变量、干预和结果）来学习每个单元的反事实分布。具体而言，我们允许未观测混杂的存在，这种混杂会在干预与结果之间引入统计偏差，并加剧单元间的异质性。将联合分布建模为指数族后，我们将单元级反事实分布的学习问题简化为学习$n$个具有异质性参数且每个分布仅对应一个样本的指数族分布。我们引入一个汇集所有$n$个样本的凸优化目标函数，以联合学习所有$n$个参数向量，并给出一个单元均方误差界，该误差界随参数空间的度量熵线性增长。例如，当参数为$k$个已知向量的$s$稀疏线性组合时，误差为$O(s\log k/p)$。在此过程中，我们推导了紧支撑分布满足对数Sobolev不等式的充分条件。作为该框架的应用，我们的结果能够实现稀疏缺失协变量的一致插补。