Solution to problem 87-6* : The entropy of a Poisson distribution Boersma, J.

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1 Solution to problem 87-6* : The entropy of a Poisson distribution Boersma, J. Published in: SIAM Review DOI: /10359 Published: 01/01/1988 Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publication after peer review. The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication Citation for published version (APA): Boersma, J. (1988). Solution to problem 87-6* : The entropy of a Poisson distribution. SIAM Review, 30(2), DOI: /10359 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 28. Nov. 2018

2 314 PROBLEMS AND SOLUTIONS Finally, substituting for x from (11), we end up considering the simple unconstrained scalar optimization problem.vt T7~~~~~lFz~~2( ViT7 ~ l)1 (15) max ( 2 2 z z L 412(l + hz)? V4 - Z4-2(12/h2)Z2 + 8/hz} + Also recall that z lies in the range (14) and the optimal z must satisfy (12), namely (16) z _ Z4-2(1 + 2gh2)z2 + 8ghz _ hz -Z2. Finally, the optimal fluid level x is found by substituting the optimal z in (13). The problem parameters are A = and h = For these parameters the quartic equation associated with (14) has a unique real positive solution z = and therefore (14) yields the range 0 < z _ z = Furthermore, maximum in (15) is then attained at the boundary z of the z-range, i.e., z * = z = and thus x* = l/z = Finally, note that (12) is then satisfied since < < The insight gained above affords us the possibility of ascertaining the range of validity (in parameter space) of our solution. In particular, if A is sufficiently small, then z > 1, x * = l/z < 1 and the left-hand part of ( 12) is violated. Hence, the smallest possible value of the parameter A for our analysis to be valid is that which makes = 1 the unique real, positive solution of the quartic solution z4+ 2(1 + 2gh2)z2-8ghz - 4 = 0. In other words,u_ = (I/4h(h - 2)). Furthermore, it is required that h> h 2. Hence, the parameter space (A, h) is partitioned by the curve 8(h) = (1/4h(h- 2)), h >)h= 2. The Entropy of a Poisson Distribution Problem 87-6*, by C. ROBERT APPLEDORN (Indiana University). The following problem arose during a study of data compression schemes for digitally encoded radiographic images. Specifically, Huffman optimum (minimum) codes are employed to reduce the storage requirements for high resolution gray-level images, for example, 2048 x 2048 pixels by 10 bits per pixel for a single image. Entropy estimates provide a lower bound for the average storage in terms of bits per pixel that can be achieved using these coding methods. Due to the nature of the problem, the Poisson probability density function with mean value parameter m is of particular interest: mke -m p(k) = k- The entropy H associated with this probability density function is given by H=- i p(k) logp(k). The problem that develops during the entropy calculation is to determine a closedform solution to the following summation: ke-rn E log (k!) k!

3 PROBLEMS AND SOLUTIONS 315 It is conjectured by the author that for large m (say, m > 10), the entropy H can be approximated by H=2 log (2-rem). Prove or disprove. Solution by RONALD J. EVANS (University of California, San Diego). Let SP(x) = log F(x + 1). By definition of the entropy H, where H=, p(k)(qp(k) + m - k log m) = m(1 - log m) + (P., sm=e p (k)(p(k). By [2, Chap. 3, Entry 10], [3, Entry 10], as (real) m -- oo, ( ) + m'p(2)(m)+ mp3) (m)+ m 2P4) (m) + O(m-2) (Several minor errors in [2, Chap. 3,? 10] and [3,? 10] are corrected in [4].) Substituting in the well-known asymptotic expansions for the derivatives (P()(m) [1, (6.1.41), (6.3.18),(6.4.11)], we deduce that, as m -> oo, H= log (27rem) m REFERENCES [1] M. ABRAMOWITZ AND I. STEGUN, eds., Handbook of Mathematical Functions, Dover, New York, [2] B. C. BERNDT, Ramanujan's Notebooks, Part I, Springer-Verlag, New York, [3] B. C. BERNDT, R. J. EVANS, AND B. M. WILSON, Chapter 3 of Ramanujan's Second Notebook, Adv. in Math., 49 (1983), pp [4] R. J. EVANS, Ramanujan's Second Notebook: Asymptotic expansions for hypergeometric series and relatedfunctions, in Proc. Ramanujan Centenary Conference, Academic Press, New York, Solution by J. BOERSMA (Eindhoven University of Technology, Eindhoven, the Netherlands). It is easily seen that the entropy H associated with the Poisson probability density function is given by ~~~~~~~mk,e-m H=- p(k) log p(k) = m-m logm+ log (k!)"k In the latter series replace log (k!) by Malmsten's representation [1, Formula 1.9(1)] then it is found that log (k!)=j k- 1_ -t1 dt; H=m-mlogm+ {m- 1ex[m(- e )} tdt by the substitution 1 - e-t = s. [ I e-ms ds = m- mloggm- (m- 1 s)

4 316 PROBLEMS AND SOLUTIONS The final integral in (*) is decomposed into -mf, [ ds + m (_-e ds +, - ds s j[og (I -s) s 2d- (s -log(i S) s -]d whereby the successive integrals are shortly denoted by I1, I2, substitution s = 1 - e-t it is found that J0 (-it - e-t) d= I5. By back where -y denotes Euler's constant (cf.[1, Formula 1.7(19)]). Through an integration by parts the second integral reduces to 1e-ms M ei- 1 edt I2 =+mf dsl-em-m+m d s 0 o s o t = 1-e-m-m+m[El(m)+log m+,y], in which El (m) = f e-tt-u dt stands for the exponential integral (cf.[1, Formulae 9.7(1), (5)]). In the same manner we have I3 = 1(1 e dt= -[E1 (m) + log m +-y] In the fourth integral we substitute s = 1 - e-t, which yields next, I4 is decomposed into 1 I4= JI4-0 ( - et ')2 ) e-t ]t; dt; The latter integrals can be evaluated directly and by use of [1, Formulae 1.9(3), 1.7(19)], viz. J I - t t2 J 2 ] t o 22t ie-t t 2 t dt 21 og(2 ), f ( - + -tlt- 1-1 d t

5 PROBLEMS AND SOLUTIONS 317 Thus we find I4 =2+2log (21r) -2 By inserting the previous results into (*), we obtain the representation H= 2log(27rem)+ m+ E(m)-e-m)- f I)+- --] ds. It does not seem possible to evaluate the latter integral in closed form; however, an evaluation by numerical integration is certainly feasible. The asymptotics of the integral, as m -> oo, is readily established by use of Watson's lemma. To that end we inserthe Taylor series 1 1~ ~~~~~~0 0 1 ~~~~ aksk, IsI<1 log (1 -s) s 2 k=1 with coefficients al =, a2 = 1, a3 = 79, etc., and integrate term by term. Then the complete asymptotic expansion of the entropy H is found as I H~- log (2irem)- E ak(k- )!m-k 2 k=1 apart from exponentially small terms of order O(e-m). (m >oo) REFERENCE [1] A. ERDtLYI, W. MAGNUS, F. OBERHETTINGER, AND F. G. TRICOMI, Higher Transcendental Functions, Vols. I, II, McGraw-Hill, New York, A similar solution was obtained by PETER SMITH (Keele University, U.K.). N. M. BLACHMAN (GTE Government Systems Corp., Mountain View, California) notes that since the Poisson distribution has variance m and becomes approximately normal for large m, its entropy becomes that part of the N(m, n) distribution, viz., log v2e. A. A. JAGERS (Universiteit Twente, Enschede, the Netherlands) shows that log (27rm)<H < log (27rm)+ 1. Also solved by W. B. JORDAN (Scotia, New York), J. VAN KAN, P. SONNENVELD (Delft University of Technology), T. KAUFFMAN (Houston, Texas), S. L. PAVERI- FONTANA (Virginia Polytechnic Institute and State University, Blacksburg, Virginia), T. J. OTT (Bell Communications Research, Morristown, New Jersey), P. R. PUDAITE (student, University of Illinois, Urbana), M. RENARDY (Virginia Polytechnic Institute and State University, Blacksburg, Virginia), and P. WAGNER (University of Innsbruck, Austria). An Identity Problem 87-8, by JOHN W. MOON (University of Alberta). Show that X 56n 2+33n-8f21.nS. (n +, 2)(n J, 1)