Causal inference grows increasingly complex as the number of confounders increases. Given treatments $X$, confounders $Z$ and outcomes $Y$, we develop a non-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 the Hilbert Schmidt Independence Criterion (HSIC) for marginal independence testing, we propose backdoor-HSIC (bd-HSIC) and demonstrate that it is calibrated and has power for both binary and continuous treatments under a large number of confounders. Additionally, we establish convergence properties of the estimators of covariance operators used in bd-HSIC. We investigate the advantages and disadvantages of bd-HSIC against parametric tests as well as the importance of using the do-null testing in contrast to marginal independence testing or conditional independence testing. A complete implementation can be found at \hyperlink{https://github.com/MrHuff/kgformula}{\texttt{https://github.com/MrHuff/kgformula}}.

翻译：随着混杂变量数量的增加，因果推断变得越来越复杂。给定处理变量$X$、混杂变量$Z$和结果变量$Y$，我们开发了一种非参数方法来检验\textit{干预零假设}$H_0:\; p(y|\text{\it do}(X=x))=p(y)$以对抗一般备择假设。基于边际独立性检验的希尔伯特-施密特独立性准则（HSIC），我们提出了后门HSIC（bd-HSIC），并证明其在大量混杂变量下对于二元和连续处理变量均具有校准能力和检验功效。此外，我们建立了bd-HSIC中使用的协方差算子估计量的收敛性质。我们研究了bd-HSIC相对于参数检验的优缺点，以及使用干预零假设检验相较于边际独立性检验或条件独立性检验的重要性。完整实现可在\hyperlink{https://github.com/MrHuff/kgformula}{\texttt{https://github.com/MrHuff/kgformula}}找到。