We propose a framework for hypothesis testing on conditional probability distributions, which we then use to construct statistical tests of functionals of conditional distributions. These tests identify the inputs where the functionals differ with high probability, and include tests of conditional moments or two-sample tests. Our key idea is to transform confidence bounds of a learning method into a test of conditional expectations. We instantiate this principle for kernel ridge regression (KRR) with subgaussian noise. An intermediate data embedding then enables more general tests -- including conditional two-sample tests -- via kernel mean embeddings of distributions. To have guarantees in this setting, we generalize existing pointwise-in-time or time-uniform confidence bounds for KRR to previously-inaccessible yet essential cases such as infinite-dimensional outputs with non-trace-class kernels. These bounds also circumvent the need for independent data, allowing for instance online sampling. To make our tests readily applicable in practice, we introduce bootstrapping schemes leveraging the parametric form of testing thresholds identified in theory to avoid tuning inaccessible parameters. We illustrate the tests on examples, including one in process monitoring and comparison of dynamical systems. Overall, our results establish a comprehensive foundation for conditional testing on functionals, from theoretical guarantees to an algorithmic implementation, and advance the state of the art on confidence bounds for vector-valued least squares estimation.

翻译：我们提出了一种针对条件概率分布进行假设检验的框架，并利用该框架构建了条件分布泛函的统计检验方法。这些检验能够以高概率识别出泛函存在差异的输入点，其应用包括条件矩检验或双样本检验。我们的核心思想是将学习方法的置信边界转化为条件期望的检验。我们将这一原理具体应用于具有次高斯噪声的核岭回归（KRR）中。通过中间数据嵌入，借助分布的核均值嵌入技术，可实现更一般的检验——包括条件双样本检验。为了在此设定下获得理论保证，我们将现有针对KRR的逐点或时间一致置信边界推广至先前难以处理但至关重要的情形，例如具有非迹类核的无限维输出场景。这些边界还绕过了对独立数据的需求，允许例如在线采样等应用。为使我们的检验在实践中易于应用，我们引入了自助法方案，该方案利用理论中识别出的检验阈值参数化形式，避免了难以调参的问题。我们通过实例展示了这些检验的应用，包括过程监控和动态系统比较的案例。总体而言，我们的研究为泛函的条件检验建立了从理论保证到算法实现的完整基础，并推动了向量值最小二乘估计置信边界的前沿进展。