Mutual information and conditional mean estimation in Poisson channels

Following the recent discovery of new connections between information and estimation in Gaussian channels, this paper reports parallel results in the Poisson regime. Both scalar and continuous-time Poisson channels are considered. It is found that, regardless of the statistics of the input, the derivative of the input-output mutual information with respect to the dark current can be expressed in the expected difference between the logarithm of the input and the logarithm of its conditional mean estimate (noncausal in case of continuous-time). The same is true for the derivative with respect to input scaling, but with the logarithmic function replaced by x log x.