Several corrections are necessary for the above-named work are presented.

This paper proposes a novel variational approach for proving the extremal entropy inequality (EEI) [1]. Unlike previous proofs [1], [2], the proposed variational approach is simpler and it does not require neither the classical entropy power inequality (EPI) [1], [2] nor the channel enhancement technique [1]. The proposed approach is versatile and can be easily adapted to numerous other applications such as proving or extending other fundamental information theoretic inequalities such as the EPI, worst additive noise lemma, and Cramér-Rao inequality.

This paper proposes a unifying variational approach for proving and extending some fundamental information theoretic inequalities. Fundamental information theory results such as maximization of differential entropy, minimization of Fisher information (Cramér-Rao inequality), worst additive noise lemma, entropy power inequality, and extremal entropy inequality are interpreted as functional problems and proved within the framework of calculus of variations. Several applications and possible extensions of the proposed results are briefly mentioned.

This paper illustrates the equivalence between two fundamental results: Stein identity, originally proposed in the statistical estimation realm, and de Bruijn identity, considered for the first time in the information theory field. Two distinctive extensions of de Bruijn identity are presented as well. For arbitrary but fixed input and noise distributions, the first-order derivative of differential entropy is expressed by means of a function of the posterior mean, while the second-order derivative of differential entropy is manifested in terms of a function of Fisher information. Several applications exemplify the utility of the proposed results.

This paper focuses on developing an alternative proof for an extremal entropy inequality, originally presented in [1]. The proposed alternative proof is simply based on the classical entropy power inequality and the data processing inequality. Compared with the proofs in [1], the proposed alternative proof is simpler, more direct, and information theoretic, and presents the advantage of providing the structure of the optimal solution covariance matrix. Also, the proposed proof might also be used as a novel method to address applications such as calculation of the vector Gaussian broadcast channel capacity, establishing a lower bound for the achievable rate of distributed source coding with a single quadratic distortion constraint, and the secrecy capacity of the Gaussian wire-tap channel.

This paper focuses on illustrating 1) the equivalence between Stein's identity and De Bruijn's identity, and 2) two extensions of De Bruijn's identity. First, it is shown that Stein's identity is equivalent to De Bruijn's identity under additive noise channels with specific conditions. Second, for arbitrary but fixed input and noise distributions under additive noise channels, the first derivative of the differential entropy is expressed by a function of the posterior mean, and the second derivative of the differential entropy is expressed in terms of a function of Fisher information. Several applications over a number of fields, such as signal processing and information theory, are presented to support the usefulness of the developed results in this paper.