  The Cauchy-Schwarz (CS) divergence was developed by Pr\'{i}ncipe et al. in
2000. In this paper, we extend the classic CS divergence to quantify the
closeness between two conditional distributions and show that the developed
conditional CS divergence can be simply estimated by a kernel density estimator
from given samples. We illustrate the advantages (e.g., rigorous faithfulness
guarantee, lower computational complexity, higher statistical power, and much
more flexibility in a wide range of applications) of our conditional CS
divergence over previous proposals, such as the conditional KL divergence and
the conditional maximum mean discrepancy. We also demonstrate the compelling
performance of conditional CS divergence in two machine learning tasks related
to time series data and sequential inference, namely time series clustering and
uncertainty-guided exploration for sequential decision making.
