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Generalized Entropy Power Inequalities and Monotonicity Properties of Information

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Generalized Entropy Power Inequalities and Monotonicity Properties of Information TO APPEAR IN IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY 2007 1 Generalized Entropy Power Inequalities and Monotonicity Properties of Information Mokshay Madiman, Member, IEEE, and Andrew Barron, Senior Member, IEEE Abstract— New families of Fisher information and entropy Our generalized entropy power inequality for subset sums power inequalities for sums of independent random variables is as follows: if is an arbitrary collection of subsets of are presented. These inequalities relate the information in the 1, 2,...,n , thenC sum of n independent random variables to the information { } contained in sums over subsets of the random variables, for an 2H(X +...+X ) 1 2H P s Xj e 1 n e j∈ , (3) arbitrary collection of subsets. As a consequence, a simple proof ≥ r of the monotonicity of information in central limit theorems is sX ∈C obtained, both in the setting of i.i.d. summands as well as in the where r is the maximum number of sets in in which any more general setting of independent summands with variance- C standardized sums. one index appears. In particular, note that 1) Choosing to be the class of all singletons yields Index Terms— Central limit theorem; entropy power; infor- C C1 mation inequalities. r =1 and hence (1). 2) Choosing to be the class n 1 of all sets of n 1 elements yieldsC r = n 1 andC hence− (2). − I. INTRODUCTION − 3) Choosing to be the class m of all sets of m elements C n 1 C ET X1,X2,...,Xn be independent random variables yields r = m− 1 and hence the inequality L with densities and finite variances. Let H denote the − (differential) entropy, i.e., if is the probability density f exp 2H X1 + . + Xn function of X, then H(X) = E[log f(X)]. The classical ≥ (4) entropy power inequality of Shannon− [36] and Stam [39] states 1 n 1 exp 2H Xi . n − m 1 sXm Xi s e2H(X1 +...+Xn) e2H(Xj ). (1) − ∈C ∈ ≥ 4) Choosing to be the class of all sets of consecutive Xj=1 k integers yieldsC r = min k,n +1 k and hence the In 2004, Artstein, Ball, Barthe and Naor [1] (hereafter denoted inequality { − } by ABBN [1]) proved a new entropy power inequality n exp 2H X1 + . + Xn 1 2H P X e2H(X1+...+Xn) e j6=i j , (2) ≥ ≥ n 1 Xi=1 1 − exp 2H X . min k,n +1 k i where each term involves the entropy of the sum of n 1 of the { − } sX Xi s variables excluding the i-th, and presented its implications− for ∈C ∈ In general, the inequality (3) clearly yields a whole family of arXiv:cs/0605047v2 [cs.IT] 2 Jul 2007 the monotonicity of entropy in the central limit theorem. It is entropy power inequalities, for arbitrary collections of subsets. not hard to see, by repeated application of (2) for a succession Furthermore, equality holds in any of these inequalities if and of values of n, that (2) in fact implies the inequality (4) and only if the X are normally distributed and the collection is hence (1). We will present below a generalized entropy power i “nice” in a sense that will be made precise later. C inequality for subset sums that subsumes both (2) and (1) and These inequalities are relevant for the examination of mono- also implies several other interesting inequalities. We provide tonicity in central limit theorems. Indeed, if X and X simple and easily interpretable proofs of all of these inequal- 1 2 are independent and identically distributed (i.i.d.), then (1) is ities. In particular, this provides a simplified understanding equivalent to of the monotonicity of entropy in central limit theorems. A similar independent and contemporaneous development of the X1 + X2 H H(X1), (5) monotonicity of entropy is given by Tulino and Verd´u[42]. √2 ≥ Manuscript received April 15, 2006; revised March 16, 2007. The material by using the scaling H(aX)= H(X)+log a . This fact im- n in this paper was presented in part at the Inaugural Workshop, Information | | Pi=1 Xi plies that the entropy of the standardized sums Yn = Theory and Applications Center, San Diego, CA, February 2006, and at the √n IEEE International Symposium on Information Theory, Seattle, WA, July for i.i.d. Xi increases along the powers-of-2 subsequence, i.e., 2006. H(Y2k ) is non-decreasing in k. Characterization of the change The authors are with the Department of Statistics, Yale Univer- in information quantities on doubling of sample size was used sity, 24 Hillhouse Avenue, New Haven, CT 06511, USA. Email: [email protected], [email protected] in proofs of central limit theorems by Shimizu [37], Brown Communicated by V. A. Vaishampayan, Associate Editor at Large. [8], Barron [4], Carlen and Soffer [10], Johnson and Barron TO APPEAR IN IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY 2007 2 2 [21], and Artstein, Ball, Barthe and Naor [2]. In particular, and the Fisher information I(X1 + . + Xn) = E[ρtot] has Barron [4] showed that the sequence H(Y ) converges to the bound { n } the entropy of the normal; this, incidentally, is equivalent to the 2 convergence to 0 of the relative entropy (Kullback divergence) E[ρ2 ] E wsρs . (7) tot ≤ from a normal distribution. ABBN [1] showed that H(Y ) is sX n ∈C in fact a non-decreasing sequence for every n, solving a long- For non-overlapping subsets, the independence and zero mean standing conjecture. In fact, (2) is equivalent in the i.i.d. case properties of the scores provide a direct means to express the to the monotonicity property right side in terms of the Fisher informations of the subset sums (yielding the traditional Blachman [7] proof of Stam’s X1 + . + Xn X1 + . + Xn 1 H H − . (6) inequality for Fisher information). In contrast, the case of over- √n ≥ √n 1 − lapping subsets requires fresh consideration. Whereas a naive application of Cauchy-Schwarz would yield a loose bound of Note that the presence of the factor n 1 (rather than n) in − s ws2 Eρs2 , instead a variance drop lemma yields that the denominator of (2) is crucial for this monotonicity. |C| the right∈C side of (7) is not more than r s ws2Eρs2 if each Likewise, for sums of independent random variables, the P i is in at most r subsets of . Consequently,P ∈C inequality (4) is equivalent to “monotonicity on average” C properties for certain standardizations; for instance, I(X + . + X ) r ws2I X (8) 1 n ≤ i sX Xi s X1 + . + Xn 1 i s Xi ∈C ∈ H H ∈ . s √n ≥ n P√m for any weights ws that add to 1 over all subsets m sXm ⊂ ∈C 1,...,n in . See Sections II and IV for details. Optimizing { } C A similar monotonicity also holds, as we shall show, for arbi- over w yields an inequality for inverse Fisher information trary collections and even when the sums are standardized that extends the Fisher information inequalities of Stam and by their variances.C Here again the factor r (rather than the ABBN: 1 1 1 cardinality of the collection) in the denominator of (3) for . (9) |C| I(X + . + X ) r I( X ) the unstandardized version is crucial. 1 n ≥ Xs i s i ∈C P ∈ Alternatively, using a scaling property of Fisher information Outline of our development. to re-express our core inequality (8), we see that the Fisher information of the total sum is bounded by a convex combi- We find that the main inequality (3) (and hence all of nation of Fisher informations of scaled subset sums: the above inequalities) as well as corresponding inequalities i s Xi for Fisher information can be be proved by simple tools. I(X1 + . + Xn) wsI ∈ . (10) P s ≤ Xs √w r Two of these tools, a convolution identity for score functions ∈C and the relationship between Fisher information and entropy This integrates to give an inequality for entropy that is an (discussed in Section II), are familiar in past work on infor- extension of the “linear form of the entropy power inequality” mation inequalities. An additional trick is needed to obtain the developed by Dembo et al [15]. Specifically we obtain denominator r in (3). This is a simple variance drop inequality i s Xi for statistics expressible via sums of functions of subsets of a H(X + . + X ) wsH ∈ . (11) 1 n P s collection of variables, particular cases of which are familiar ≤ Xs √w r in other statistical contexts (as we shall discuss). ∈C See Section V for details. Likewise using the scaling property We recall that for a random variable with differentiable X of entropy on (11) and optimizing over w yields our extension density f, the score function is ρ(x)= ∂ log f(x), the score ∂x (3) of the entropy power inequality, described in Section VI. is the random variable ρ(X), and its Fisher information is 2 To relate this chain of ideas to other recent work, we I(X)= E[ρ (X)]. point out that ABBN [1] was the first to show use of a Suppose for the consideration of Fisher information that variance drop lemma for information inequality development the independent random variables X1,...,Xn have absolutely (in the leave-one-out case of the collection n 1). For that continuous densities. To outline the heart of the matter, the first case what is new in our presentation is goingC − straight to step boils down to the geometry of projections (conditional n the projection inequality (7) followed by the variance drop expectations).
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