Conditional local independence is an asymmetric independence relation among continuous time stochastic processes. It describes whether the evolution of one process is directly influenced by another process given the histories of additional processes, and it is important for the description and learning of causal relations among processes. We develop a model-free framework for testing the hypothesis that a counting process is conditionally locally independent of another process. To this end, we introduce a new functional parameter called the Local Covariance Measure (LCM), which quantifies deviations from the hypothesis. Following the principles of double machine learning, we propose an estimator of the LCM and a test of the hypothesis using nonparametric estimators and sample splitting or cross-fitting. We call this test the (cross-fitted) Local Covariance Test ((X)-LCT), and we show that its level and power can be controlled uniformly, provided that the nonparametric estimators are consistent with modest rates. We illustrate the theory by an example based on a marginalized Cox model with time-dependent covariates, and we show in simulations that when double machine learning is used in combination with cross-fitting, then the test works well without restrictive parametric assumptions.

翻译：条件局部独立性是连续时间随机过程之间的一种非对称独立性关系。它描述了在给定其他过程历史信息的条件下，一个过程的演进是否直接受到另一个过程的影响，这对于描述和学习过程之间的因果关系至关重要。我们开发了一个无模型框架，用于检验计数过程在条件上局部独立于另一个过程的假设。为此，我们引入了一个新的泛函参数——局部协方差度量（LCM），该参数量化了相对于假设的偏差。遵循双机器学习原理，我们提出了一种基于非参数估计器和样本分割或交叉拟合的LCM估计量及假设检验方法。我们将该检验称为（交叉拟合）局部协方差检验（(X)-LCT），并证明只要非参数估计量以适度速率一致，该检验的水平和功效就能得到均匀控制。我们通过一个基于含时依协变量的边际化Cox模型的示例来说明该理论，并在模拟中表明，当双机器学习与交叉拟合结合使用时，该检验无需严格参数假设即可表现良好。