  For a continuous random variable $Z$, testing conditional independence $X
\perp\!\!\!\perp Y |Z$ is known to be a particularly hard problem. It
constitutes a key ingredient of many constraint-based causal discovery
algorithms. These algorithms are often applied to datasets containing binary
variables, which indicate the 'context' of the observations, e.g. a control or
treatment group within an experiment. In these settings, conditional
independence testing with $X$ or $Y$ binary (and the other continuous) is
paramount to the performance of the causal discovery algorithm. To our
knowledge no nonparametric 'mixed' conditional independence test currently
exists, and in practice tests that assume all variables to be continuous are
used instead. In this paper we aim to fill this gap, as we combine elements of
Holmes et al. (2015) and Teymur and Filippi (2020) to propose a novel Bayesian
nonparametric conditional two-sample test. Applied to the Local Causal
Discovery algorithm, we investigate its performance on both synthetic and
real-world data, and compare with state-of-the-art conditional independence
tests.
