Weighting methods in causal inference have been widely used to achieve a
desirable level of covariate balancing. However, the existing weighting methods
have desirable theoretical properties only when a certain model, either the
propensity score or outcome regression model, is correctly specified. In
addition, the corresponding estimators do not behave well for finite samples
due to large variance even when the model is correctly specified. In this
paper, we consider to use the integral probability metric (IPM), which is a
metric between two probability measures, for covariate balancing. Optimal
weights are determined so that weighted empirical distributions for the treated
and control groups have the smallest IPM value for a given set of
discriminators. We prove that the corresponding estimator can be consistent
without correctly specifying any model (neither the propensity score nor the
outcome regression model). In addition, we empirically show that our proposed
method outperforms existing weighting methods with large margins for finite
samples.

本文提出了一种使用积分概率度量进行协变量平衡的加权方法，并证明该方法可以在不正确指定模型的情况下保持一致性，同时在有限样本情况下的表现也比现有方法更好。

利用积分概率度量进行协变量平衡的因果推断

Covariate balancing using the integral probability metric for causal  inference

Weighting methods are widely used to adjust for covariates in observational
studies, sample surveys, and regression settings. In this paper, we study a
class of recently proposed weighting methods which find the weights of minimum
dispersion that approximately balance the covariates. We call these weights
"minimal weights" and study them under a common optimization framework. The key
observation is the connection between approximate covariate balance and
shrinkage estimation of the propensity score. This connection leads to both
theoretical and practical developments. From a theoretical standpoint, we
characterize the asymptotic properties of minimal weights and show that, under
standard smoothness conditions on the propensity score function, minimal
weights are consistent estimates of the true inverse probability weights. Also,
we show that the resulting weighting estimator is consistent, asymptotically
normal, and semiparametrically efficient. From a practical standpoint, we
present a finite sample oracle inequality that bounds the loss incurred by
balancing more functions of the covariates than strictly needed. This
inequality shows that minimal weights implicitly bound the number of active
covariate balance constraints. We finally provide a tuning algorithm for
choosing the degree of approximate balance in minimal weights. We conclude the
paper with four empirical studies that suggest approximate balance is
preferable to exact balance, especially when there is limited overlap in
covariate distributions. In these studies, we show that the root mean squared
error of the weighting estimator can be reduced by as much as a half with
approximate balance.

本文研究一种最小化权重的方法，它可以近似平衡协变量，提出了一种调整权重的算法并应用于实证研究，结果表明这种方法可以有效降低加权估计器的均方误差