Potential conditional mutual information: Estimators and properties

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.