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Lower Bounds on Multivariate Higher Order Derivatives of Differential Entropy

Authors:	Laigang Guo Chun-Ming Yuan Xiao-Shan Gao

Affiliation:	1.Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China;2.KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;3.University of Chinese Academy of Sciences, Beijing 100049, China

Abstract:	This paper studies the properties of the derivatives of differential entropy $H (X_{t})$ in Costa’s entropy power inequality. For real-valued random variables, Cheng and Geng conjectured that for $m \geq 1$ , ${(- 1)}^{m + 1} (d^{m} / d t^{m}) H (X_{t}) \geq 0$ , while McKean conjectured a stronger statement, whereby ${(- 1)}^{m + 1} (d^{m} / d t^{m}) H (X_{t}) \geq {(- 1)}^{m + 1} (d^{m} / d t^{m}) H (X_{G t})$ . Here, we study the higher dimensional analogues of these conjectures. In particular, we study the veracity of the following two statements: $C_{1} (m, n) : {(- 1)}^{m + 1} (d^{m} / d t^{m}) H (X_{t}) \geq 0$ , where n denotes that $X_{t}$ is a random vector taking values in $R^{n}$ , and similarly, $C_{2} (m, n) : {(- 1)}^{m + 1} (d^{m} / d t^{m}) H (X_{t}) \geq {(- 1)}^{m + 1} (d^{m} / d t^{m}) H (X_{G t}) \geq 0$ . In this paper, we prove some new multivariate cases: $C_{1} (3, i), i = 2, 3, 4$ . Motivated by our results, we further propose a weaker version of McKean’s conjecture $C_{3} (m, n) : {(- 1)}^{m + 1} (d^{m} / d t^{m}) H (X_{t}) \geq {(- 1)}^{m + 1} \frac{1}{n} (d^{m} / d t^{m}) H (X_{G t})$ , which is implied by $C_{2} (m, n)$ and implies $C_{1} (m, n)$ . We prove some multivariate cases of this conjecture under the log-concave condition: $C_{3} (3, i), i = 2, 3, 4$ and $C_{3} (4, 2)$ . A systematic procedure to prove $C_{l} (m, n)$ is proposed based on symbolic computation and semidefinite programming, and all the new results mentioned above are explicitly and strictly proved using this procedure.

Keywords:	differential entropy, completely monotone, Mckean’ s conjecture, log-concavity, Gaussian optimality

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