Independence testing is a fundamental and classical statistical problem that has been extensively studied in the batch setting when one fixes the sample size before collecting data. However, practitioners often prefer procedures that adapt to the complexity of a problem at hand instead of setting sample size in advance. Ideally, such procedures should (a) allow stopping earlier on easy tasks (and later on harder tasks), hence making better use of available resources, and (b) continuously monitor the data and efficiently incorporate statistical evidence after collecting new data, while controlling the false alarm rate. It is well known that classical batch tests are not tailored for streaming data settings: valid inference after data peeking requires correcting for multiple testing but such corrections generally result in low power. Following the principle of testing by betting, we design sequential kernelized independence tests (SKITs) that overcome such shortcomings. We exemplify our broad framework using bets inspired by kernelized dependence measures, e.g, the Hilbert-Schmidt independence criterion. Our test is valid under non-i.i.d. time-varying settings, for which there exist no batch tests. We demonstrate the power of our approaches on both simulated and real data.

翻译：独立性检验是一个基础且经典的统计问题，在固定样本量的批处理设定下已有广泛研究。然而，实践者通常倾向于采用能够根据问题复杂度自适应调整的方法，而非预先设定样本量。理想情况下，此类方法应具备以下特点：(a) 允许在简单任务上提前停止（复杂任务则延迟停止），从而更高效地利用可用资源；(b) 持续监测数据并在收集新数据后有效整合统计证据，同时控制虚警率。众所周知，经典批处理检验不适用于流数据设定：数据窥探后的有效推断需修正多重检验问题，但此类修正通常会导致检验效能降低。遵循"通过赌注进行检验"原则，我们设计了克服上述缺陷的序贯核化独立性检验（SKIT）。我们以受核化相关性度量（如希尔伯特-施密特独立性准则）启发的赌注为例，展示了该通用框架的适用性。本检验在非独立同分布时变设定下仍保持有效性——而此类设定尚不存在对应的批处理检验方法。通过模拟与实际数据实验，我们证明了所提方法的检验效能。