Cauchy-Schwarz bounded trade-off weighting for causal inference with small sample sizes

The difficulty of causal inference for small-sample-size data lies in the issue of inefficiency that the variance of the estimators may be large. Some existing weighting methods adopt the idea of bias-variance trade-off, but they require manual specification of the trade-off parameters. To overcome this drawback, in this article, we propose a Cauchy-Schwarz Bounded Trade-off Weighting (CBTW) method, in which the trade-off parameter is theoretically derived to guarantee a small Mean Square Error (MSE) in estimation. We theoretically prove that optimizing the objective function of CBTW, which is the Cauchy-Schwarz upper-bound of the MSE for causal effect estimators, contributes to minimizing the MSE. Moreover, since the upper-bound consists of the variance and the squared ${\ell }_2$-norm of covariate differences, CBTW can not only estimate the causal effects efficiently, but also keep the covariates balanced. Experimental results on both simulation data and real-world data show that the CBTW outperforms most existing methods especially under small sample size scenarios.