Lower Bounds on Multivariate Higher Order Derivatives of Differential Entropy

Abstract

This paper studies the properties of the derivatives of differential entropy H(Xt) in Costa’s entropy power inequality. For real-valued random variables, Cheng and Geng conjectured that for m≥1, (−1)m+1(dm/dtm)H(Xt)≥0, while McKean conjectured a stronger statement, whereby (−1)m+1(dm/dtm)H(Xt)≥(−1)m+1(dm/dtm)H(XGt). Here, we study the higher dimensional analogues of these conjectures. In particular, we study the veracity of the following two statements: C1(m,n):(−1)m+1(dm/dtm)H(Xt)≥0, where n denotes that Xt is a random vector taking values in Rn, and similarly, C2(m,n):(−1)m+1(dm/dtm)H(Xt)≥(−1)m+1(dm/dtm)H(XGt)≥0. In this paper, we prove some new multivariate cases: C1(3,i),i=2,3,4. Motivated by our results, we further propose a weaker version of McKean’s conjecture C3(m,n):(−1)m+1(dm/dtm)H(Xt)≥(−1)m+11n(dm/dtm)H(XGt), which is implied by C2(m,n) and implies C1(m,n). We prove some multivariate cases of this conjecture under the log-concave condition: C3(3,i),i=2,3,4 and C3(4,2). A systematic procedure to prove Cl(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