  Measuring conditional independence 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 independence 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 independence measures to be
equivalent to that of kernel-based measures. On the other hand, we also show
that some popular---in machine learning---kernel conditional independence
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. This paper, therefore, shows the distance and kernel
measures of conditional independence to be not quite equivalent unlike in the
case of joint independence as shown by Sejdinovic et al. (2013).
