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Causal inference grows increasingly complex as the number of confoundersincreases. Given treatments $X$, confounders $Z$ and outcomes $Y$, we develop anon-parametric method to test the \textit{do-null} hypothesis $H_0:\;p(y|\text{\it do}(X=x))=p(y)$ against the general alternative. Building on theHilbert Schmidt Independence Criterion (HSIC) for marginal independencetesting, we propose backdoor-HSIC (bd-HSIC) and demonstrate that it iscalibrated and has power for both binary and continuous treatments under alarge number of confounders. Additionally, we establish convergence propertiesof the estimators of covariance operators used in bd-HSIC. We investigate theadvantages and disadvantages of bd-HSIC against parametric tests as well as theimportance of using the do-null testing in contrast to marginal independencetesting or conditional independence testing. A complete implementation can befound at\hyperlink{https://github.com/MrHuff/kgformula}{\texttt{https://github.com/MrHuff/kgformula}}.

A Kernel Test for Causal Association via Noise Contrastive Backdoor Adjustment