论文标题
最小值最佳条件独立性测试
Minimax Optimal Conditional Independence Testing
论文作者
论文摘要
我们认为，给定$ x，y $和$ z $的$ x $和$ y $有条件独立测试的问题是三个真正的随机变量，$ z $是连续的。我们专注于两个主要案例 - 当$ x $和$ y $都是离散的时候，当$ x $和$ y $都是连续的时。鉴于有条件独立测试的最新结果（Shah和Peters，2018年），人们无法设计非平凡的测试，这些测试控制着所有绝对连续有条件独立的分布的I型错误，同时仍确保对有趣的替代方案进行权力。因此，我们确定了$ x，y | z = z $的各种自然平稳性假设在$ z $的支持方面有所不同，并研究了这些平滑度假设下有条件独立测试的硬度。我们在零变化度量中的null和替代假设之间的分离临界半径上得出匹配的下限和上限。我们考虑的测试很容易实现，并依靠汇总连续变量$ z $的支持。为了补充这些结果，我们提供了Shah和Peters的硬度结果的新证明。
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.
