A PROOF OF THE FISHER INFORMATION INEQUALITY VIA A DATA-PROCESSING ARGUMENT

The Fisher information J(X) of a random variable X under a translation parameter appears in information theory in the classical proof of the Entropy-Power Inequality (EPI). It enters the proof of the EPI via th e De-Bruijn identity, where it measures the variation of the different ial entropy under a Gaussian perturbation, and via the convolution ine quality J(X + Y)(-1) greater than or equal to J(X)(-1) + J(Y)(-1) (for independent X and Y), known as the Fisher Information Inequality (FII ). The FII, is proved in the literature directly, in a rather involved way. We give an alternative derivation of the FII, as a simple conseq uence of a ''data-processing inequality'' for the Cramer-Rao lower bou nd on parameter estimation.