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We present several novel identities and inequalities relating the mutualinformation and the directed information in systems with feedback. The internalblocks within such systems are restricted only to be causal mappings, but areallowed to be non-linear, stochastic and time varying. Moreover, the involvedsignals can be arbitrarily distributed. We bound the directed informationbetween signals inside the feedback loop by the mutual information betweensignals inside and outside the feedback loop. This fundamental result has aninteresting interpretation as a law of conservation of information flow.Building upon it, we derive several novel identities and inequalities, whichallow us to prove some existing information inequalities under less restrictiveassumptions. Finally, we establish new relationships between nested directedinformations inside a feedback loop. This yields a new and generaldata-processing inequality for systems with feedback.

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