Potential Conditional Mutual Information: Estimators, Properties and Applications
This work addresses a need in learning problems like conditional independence testing and causal estimation, but it is incremental as it builds on existing mutual information concepts with a modified distribution.
The authors tackled the problem of estimating conditional mutual information using only the conditional distribution p_{Y|X,Z} by defining potential conditional mutual information and developing k-nearest neighbor estimators with importance sampling and a coupling trick, achieving excellent practical performance as demonstrated in dynamical system inference.
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.