On the equivalence between Stein and de Bruijn identities

TL;DR

This paper proves the equivalence between Stein's and de Bruijn's identities under certain conditions and explores their extensions, relating entropy derivatives to Fisher information and posterior means.

Contribution

It establishes the theoretical equivalence between Stein's and de Bruijn's identities and extends de Bruijn's identity to broader distributions.

Findings

Stein's identity is equivalent to de Bruijn's identity under specific conditions.

Extensions of de Bruijn's identity relate entropy derivatives to Fisher information.

Applications demonstrate the practical usefulness of these theoretical results.

Abstract

This paper focuses on proving the equivalence between Stein's identity and de Bruijn's identity. Given some conditions, we prove that Stein's identity is equivalent to de Bruijn's identity. In addition, some extensions of de Bruijn's identity are presented. For arbitrary but fixed input and noise distributions, there exist relations between the first derivative of the differential entropy and the posterior mean. Moreover, the second derivative of the differential entropy is related to the Fisher information for arbitrary input and noise distributions. Several applications are presented to support the usefulness of the developed results in this paper.