Minimum mean squared error estimation. prediction, and smoothing are usually considered to be different measures of performance in relation to mutual information, except in the special case of Gaussian distributions. However, recent results have established a relationship between entropy power and minimum mean squared estimation error quantities. In this chapter, we developed these relationships and show that the change in mutual information obtained by minimum mean squared error filtering, smoothing, and prediction can be expressed as the log ratio of entropy powers. The key result is that the ratios of entropy powers is the ratio of mean squared errors for an important set of probability densities whose differential entropy has a simple form.