On Distance and Kernel Measures of Conditional Dependence
On Distance and Kernel Measures of Conditional Dependence
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发表时间：
2023
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通讯作者：
T. Sheng;Bharath K. Sriperumbudur
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作者：
T. Sheng;Bharath K. Sriperumbudur
测量条件依赖是统计推断中的重要任务之一，并且是因果发现、特征选择、降维、贝叶斯网络学习等的基础。在这项工作中，我们探讨了度量空间上的距离和再生核与再生核希尔伯特空间（RKHS）的条件依赖措施之间的连接。对于一定的距离和内核对，我们显示的距离为基础的条件依赖措施是等价的，基于内核的措施。另一方面，我们还表明，一些流行的核条件依赖措施的基础上的Hilbert-Schmidt范数的某一交叉条件协方差算子，没有一个简单的距离表示，除了在某些限制情况下。
Measuring conditional dependence is one of the important tasks in statistical inference and is fundamental in causal discovery, feature selection, dimensionality reduction, Bayesian network learning, and others. In this work, we explore the connection between conditional dependence measures induced by distances on a metric space and reproducing kernels associated with a reproducing kernel Hilbert space (RKHS). For certain distance and kernel pairs , we show the distance-based conditional dependence measures to be equivalent to that of kernel-based measures. On the other hand, we also show that some popular kernel conditional dependence measures based on the Hilbert-Schmidt norm of a certain cross-conditional covariance operator, do not have a simple distance representation, except in some limiting cases.