  The conditional mutual information I(X;Y|Z) measures the average information
that X and Y contain about each other given Z. This is an important primitive
in many learning problems including conditional independence testing, graphical
model inference, causal strength estimation and time-series problems. In
several applications, it is desirable to have a functional purely of the
conditional distribution p_{Y|X,Z} rather than of the joint distribution
p_{X,Y,Z}. We define the potential conditional mutual information as the
conditional mutual information calculated with a modified joint distribution
p_{Y|X,Z} q_{X,Z}, where q_{X,Z} is a potential distribution, fixed airport. We
develop K nearest neighbor based estimators for this functional, employing
importance sampling, and a coupling trick, and prove the finite k consistency
of such an estimator. We demonstrate that the estimator has excellent practical
performance and show an application in dynamical system inference.
