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Fundamental Inequalities and Identities Involving Mutual and Directed Informations in Closed-Loop Systems

We present several novel identities and inequalities relating the mutual information and the directed information in systems with feedback. The internal blocks within such systems are restricted only to be causal mappings, but are allowed to be non-linear, stochastic and time varying. Moreover, the involved signals can be arbitrarily distributed. We bound the directed information between signals inside the feedback loop by the mutual information between signals inside and outside the feedback loop. This fundamental result has an interesting interpretation as a law of conservation of information flow. Building upon it, we derive several novel identities and inequalities, which allow us to prove some existing information inequalities under less restrictive assumptions. Finally, we establish new relationships between nested directed informations inside a feedback loop. This yields a new and general data-processing inequality for systems with feedback.