On the Relationship between Mutual Information and Minimum Mean-Square Errors in Stochastic Dynamical Systems

摘要

We consider a general stochastic input-output dynamical system with outputevolving in time as the solution to a functional coefficients, It\^{o}'sstochastic differential equation, excited by an input process. This generalclass of stochastic systems encompasses not only the classical communicationchannel models, but also a wide variety of engineering systems appearingthrough a whole range of applications. For this general setting we findanalogous of known relationships linking input-output mutual information andminimum mean causal and non-causal square errors, previously established in thecontext of additive Gaussian noise communication channels. Relationships arenot only established in terms of time-averaged quantities, but also theirtime-instantaneous, dynamical counterparts are presented. The problem ofappropriately introducing in this general framework a signal-to-noise rationotion expressed through a signal-to-noise ratio parameter is also taken intoaccount, identifying conditions for a proper and meaningful interpretation.