Motivated by conditional independence testing, an essential step in constraint-based causal discovery algorithms, we study the nonparametric Von Mises estimator for the entropy of multivariate distributions built on a kernel density estimator. We establish an exponential concentration inequality for this estimator. We design a test for conditional independence (CI) based on our estimator, called VM-CI, which achieves optimal parametric rates under smoothness assumptions. Leveraging the exponential concentration, we prove a tight upper bound for the overall error of VM-CI. This, in turn, allows us to characterize the sample complexity of any constraint-based causal discovery algorithm that uses VM-CI for CI tests. To the best of our knowledge, this is the first sample complexity guarantee for causal discovery for continuous variables. Furthermore, we empirically show that VM-CI outperforms other popular CI tests in terms of either time or sample complexity (or both), which translates to a better performance in structure learning as well.

基于我们的估计器建立的多元分布的熵的非参数Von Mises估计器，在条件独立性测试这一关键步骤上受到启发，我们设计了一种基于估计器的条件独立性测试（VM-CI），在光滑性假设下达到了最优的参数速率。利用指数集中不等式，我们证明了VM-CI的总体误差的紧密上限。反过来，这使我们能够表征使用VM-CI进行条件独立性测试的任何基于约束的因果发现算法的样本复杂度。据我们所知，这是连续变量因果发现的首个样本复杂度保证。此外，我们经验证明，无论是时间复杂度还是样本复杂度（或两者兼有），VM-CI在性能上优于其他常见的条件独立性测试，这也反映在结构学习中表现出更好的性能。

关于利用冯·米塞斯估计器进行条件独立性检验的样本复杂性及其应用于因果推断