Minimax optimal conditional independence testing
Minimax optimal conditional independence testing
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DOI：
10.1214/20-aos2030
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发表时间：
2020-01
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影响因子：
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通讯作者：
Matey Neykov;Sivaraman Balakrishnan;L. Wasserman
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文献类型：
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作者：
Matey Neykov;Sivaraman Balakrishnan;L. Wasserman
考虑给定Z的X，Y和Z的条件独立性检验问题，其中X，Y和Z是三个真实的随机变量，Z是连续的.我们主要关注两种情况--当$X$和$Y$都是离散的，以及当$X$和$Y$都是连续的。鉴于最近关于条件独立性检验的结果（Shah和Peters 2018），人们不能希望设计非平凡的检验，它控制所有绝对连续条件独立分布的I型错误，同时仍然确保对有趣的替代品的功效。因此，我们确定了各种自然的光滑假设的条件分布的$X，Y| Z=z$ as $z$在$Z$的支持下变化，并研究了在这些光滑性假设下条件独立性检验的难度。我们得出匹配的下限和上限的临界半径之间的分离的零和备择假设的总变差度量。我们考虑的测试很容易实现，并且依赖于连续变量$Z$的支持。为了补充这些结果，我们提供了一个新的证明的硬度结果的沙阿和彼得斯，并表明，在没有平滑假设条件独立性测试仍然很难，即使当$X，Y$是离散变量的有限（而不是缩放与样本大小）的支持。
We consider the problem of conditional independence testing of $X$ and $Y$ given $Z$ where $X,Y$ and $Z$ are three real random variables and $Z$ is continuous. We focus on two main cases -- when $X$ and $Y$ are both discrete, and when $X$ and $Y$ are both continuous. In view of recent results on conditional independence testing (Shah and Peters 2018), one cannot hope to design non-trivial tests, which control the type I error for all absolutely continuous conditionally independent distributions, while still ensuring power against interesting alternatives. Consequently, we identify various, natural smoothness assumptions on the conditional distributions of $X,Y|Z=z$ as $z$ varies in the support of $Z$, and study the hardness of conditional independence testing under these smoothness assumptions. We derive matching lower and upper bounds on the critical radius of separation between the null and alternative hypotheses in the total variation metric. The tests we consider are easily implementable and rely on binning the support of the continuous variable $Z$. To complement these results, we provide a new proof of the hardness result of Shah and Peters and show that in the absence of smoothness assumptions conditional independence testing remains difficult even when $X,Y$ are discrete variables of finite (and not scaling with the sample-size) support.