发表机构
University of Copenhagen（哥本哈根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究条件独立性检验中用嵌入替代原始变量导致的偏差问题，提出基于残差相关检验的稳健方法，在合成数据、文本嵌入及语言模型生成文本上验证了该方法的有效性。
AI 中文摘要
为检验给定文本或图像Z时X与Y的条件独立性，人们会用嵌入ψ(Z)替代Z进行条件检验。若给定ψ(Z)时Z与X或Y独立，则嵌入检验有效，但这一点无法从数据中确认；当该条件不成立时，原假设下的拒绝概率会趋近于1。我们研究了这种失效情况，表明聚焦于特定形式的依赖关系可降低嵌入需保留的信息要求。对于受广义协方差度量启发的残差相关检验，其有效性仅要求E[X|Z]和E[Y|Z]中未被E[X|ψ(Z)]和E[Y|ψ(Z)]捕捉到的部分互不相关，否则我们将被丢弃的信息视为遗漏变量。在原假设下，偏差等于未被捕捉部分的绝对相关系数乘以两个偏R²值的几何均值。该恒等式可得到一个在几何均值的声明容差下有效的稳健检验，该容差类似敏感性参数，无法从数据中识别。在合成数据和文本嵌入上，该稳健检验的水平近似符合要求；在语言模型生成的文本上，当原假设完全成立时，即使是生成器自身的状态，也会使嵌入检验产生偏差。
英文摘要
To test conditional independence of $X$ and $Y$ given a text or an image $Z$, one conditions on an embedding $ψ(Z)$ in place of $Z$. The embedded test is valid if $Z$ is independent of $X$ or of $Y$ given $ψ(Z)$, which cannot be confirmed from data, and when this fails, the rejection probability under the null hypothesis can tend to one. We study this failure, and show that focusing on a specific form of dependence relaxes what the embedding must retain. For a residual correlation test inspired by the Generalised Covariance Measure, validity only requires that the parts of $\mathbb{E}[X \mid Z]$ and $\mathbb{E}[Y \mid Z]$ missed by $\mathbb{E}[X \mid ψ(Z)]$ and $\mathbb{E}[Y \mid ψ(Z)]$ are uncorrelated. Otherwise, we treat the discarded information as an omitted variable. Under the null hypothesis, the bias equals the absolute correlation of the missed parts times the geometric mean of two partial $R^2$ values. This identity yields a robust test valid under a declared tolerance for the geometric mean, which, like a sensitivity parameter, is not identified from the data. On synthetic data and text embeddings, the robust test holds its level approximately. On text generated by a language model, under an exact null hypothesis, every embedding, even the generator's own states, biases the embedded test.
