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 the De-Bruijn identity, where it measures the variation of the differential entropy under a Gaussian perturbation, and via the convolution inequality J(X + Y) ^-1 ≥ 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 consequence of a "data-processing inequality" for the Cramer-Rao lower bound on parameter estimation.