Potential Conditional Mutual Information: Estimators, Properties and Applications
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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.
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