  In broad applications, it is routinely of interest to assess whether there is
evidence in the data to refute the assumption of conditional independence of
$Y$ and $X$ conditionally on $Z$. Such tests are well developed in parametric
models but are not straightforward in the nonparametric case. We propose a
general Bayesian approach, which relies on an encompassing nonparametric Bayes
model for the joint distribution of $Y$, $X$ and $Z$. The framework allows $Y$,
$X$ and $Z$ to be random variables on arbitrary spaces, and can accommodate
different dimensional vectors having a mixture of discrete and continuous
measurement scales. Using conditional mutual information as a scalar summary of
the strength of the conditional dependence relationship, we construct null and
alternative hypotheses. We provide conditions under which the correct
hypothesis will be consistently selected. Computational methods are developed,
which can be incorporated within MCMC algorithms for the encompassing model.
The methods are applied to variable selection and assessed through simulations
and criminology applications.
