Lower Bounds on Multivariate Higher Order Derivatives of Differential Entropy.

Abstract

This paper studies the properties of the derivatives of differential entropy in Costa’s entropy power inequality. For real-valued random variables, Cheng and Geng conjectured that for , , while McKean conjectured a stronger statement, whereby . Here, we study the higher dimensional analogues of these conjectures. In particular, we study the veracity of the following two statements: , where n denotes that is a random vector taking values in , and similarly, . In this paper, we prove some new multivariate cases: . Motivated by our results, we further propose a weaker version of McKean’s conjecture , which is implied by and implies . We prove some multivariate cases of this conjecture under the log-concave condition: and . A systematic procedure to prove is proposed based on symbolic computation and semidefinite programming, and all the new results mentioned above are explicitly and strictly proved using this procedure.