Mutual information and MMSE in gaussian channels

Consider arbitrarily distributed input signals observed in additive Gaussian noise. A new fundamental relationship is found between the input-output mutual information and the minimum mean-square error (MMSE) of an estimate of the input given the output: The derivative of the mutual information (nats) with respect to the signal-to-noise ratio (SNR) is equal to half the MMSE. This identity holds for both scalar and vector signals, as well as for discrete- and continuous-time noncausal MMSE estimation (smoothing). A consequence of the result is a new relationship in continuous-time nonlinear filtering: Regardless of the input statistics, the causal MMSE achieved at snr is equal to the expected value of the noncausal MMSE achieved with a channel whose SNR is chosen uniformly distributed between 0 and snr