{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T22:33:26Z","timestamp":1777761206981,"version":"3.51.4"},"reference-count":101,"publisher":"Emerald","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,12,9]]},"abstract":"<jats:p>The usual answer to the question \u201cWhat probability distribution maximizes entropy or differential entropy of a random variable X subject to the constraint that the expected value of a real-valued function g applied to X has a specified value \u00b5?\u201d is an exponential distribution (probability mass or probability density function), with g(x) in the exponent multiplied by a parameter \u03bb, and with the parameter chosen so the exponential distribution causes the expected value of g(X) to equal \u00b5. The latter is called moment matching. While it is well-known that, when there are multiple expected value constraints, there are functions and expected value specifications for which moment matching is not possible, it is not well-known that this can happen when there is a single expected value constraint and a single parameter.<\/jats:p>\n                  <jats:p>This motivates the present monograph, whose goal is to reexammine the question posed above, and to derive its answer in an accessible, self-contained and complete fashion. It also derives the maximum entropy\/differential entropy when there is a constraint on the support of the probability distributions, when there is only a bound on expected value and when there is a variance constraint. Properties of the resulting maximum entropy\/differential entropy as a function of \u00b5 are derived, such as its convexity and its monotonicities. Example functions are presented, including many for which moment matching is possible for all relevant values of \u00b5, and some for which it is not. Indeed, there can be only subtle differences between the two kinds of functions.<\/jats:p>\n                  <jats:p>As one-parameter exponential probability distributions play a dominant role, one section of this monograph provides a self-contained discussion and derivation of their properties, such as the finiteness and continuity of the exponential normalizing constant (sometimes called the partition function) as \u03bb varies, the finiteness, continuity, monotonicity and limits of the expected value of g(X) under the exponential distribution as \u03bb varies, and similar issues for entropy and differential entropy. Most of these are needed in deriving the maximum entropy\/differential entropy or the properties of the resulting function of \u00b5.<\/jats:p>\n                  <jats:p>Aside from addressing the question posed initially, this monograph can be viewed as a warmup for discussions of maximizing entropy\/differential entropy with multiple expected value constraints and of multiparameter exponential families. It also provides a small taste of information geometry.<\/jats:p>","DOI":"10.1561\/0100000132","type":"journal-article","created":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T08:09:16Z","timestamp":1733731756000},"page":"589-846","source":"Crossref","is-referenced-by-count":0,"title":["Maximizing Entropy with an Expectation Constraint and One-Parameter Exponential Families of Distributions: A Reexamination"],"prefix":"10.1108","volume":"21","author":[{"given":"David L.","family":"Neuhoff","sequence":"first","affiliation":[{"name":"University of Michigan Department of Electrical Engineering and Computer Science, ,","place":["USA"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"140","published-online":{"date-parts":[[2024,12,9]]},"reference":[{"key":"2026032712174465800_ref001","article-title":"Methods of information geometry","volume":"191","author":"Amari","year":"2000","journal-title":"Oxford: Am. Math. Soc."},{"key":"2026032712174465800_ref002","volume-title":"Information and Exponential Families","author":"Barndorff-Nielsen","year":"1978"},{"issue":"1","key":"2026032712174465800_ref003","first-page":"39","article-title":"A maximum entropy approach to natural language processing","volume":"22","author":"Berger","year":"1996","journal-title":"Computational Linguistics"},{"key":"2026032712174465800_ref004","volume-title":"Maximum Entropy Solutions to Scientific Problems","author":"Bevensee","year":"1993"},{"key":"2026032712174465800_ref005","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.sysconle.2014.08.001","article-title":"Explicit solution of relative entropy weighted control","volume":"72","author":"Bierkens","year":"2014","journal-title":"Syst. & Control Lett."},{"key":"2026032712174465800_ref006","first-page":"397","article-title":"\u00dcber das W\u00e4rmegleichgewicht zwischen mehratomigen Gasmolek\u00fclen (On the thermal equilibrium between polyatomic gas molecules)","volume":"63","author":"Boltzmann","year":"1871","journal-title":"Wiener Berichte"},{"key":"2026032712174465800_ref007","volume-title":"Fundamentals of Statistical Exponential Families","author":"Brown","year":"1986"},{"key":"2026032712174465800_ref008","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198539414.001.0001","volume-title":"Maximum Entropy in Action: A Collection of Expository Essays","author":"Buck","year":"1991"},{"issue":"3","key":"2026032712174465800_ref009","doi-asserted-by":"crossref","first-page":"1119","DOI":"10.1109\/18.135651","article-title":"Balanced codes and nonequiprobable signaling","volume":"38","author":"Calderbank","year":"1992","journal-title":"IEEE Tr. 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Thy."},{"key":"2026032712174465800_ref015","doi-asserted-by":"crossref","DOI":"10.1016\/j.heliyon.2018.e00596","article-title":"An introduction to the maximum entropy approach and its application to inference problems in biology","volume":"4","author":"De Martino","year":"2018","journal-title":"Heliyon"},{"issue":"1","key":"2026032712174465800_ref016","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1007\/BF00773592","article-title":"The singly truncated normal distribution: A none-steep exponential family","volume":"46","author":"Del Castillo","year":"1994","journal-title":"Ann. Inst. Statist. Math."},{"issue":"4","key":"2026032712174465800_ref017","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1109\/34.588021","article-title":"Inducing features of random fields","volume":"19","author":"Della Pietra","year":"1997","journal-title":"IEEE Tr. Patt. Anal. and Mach. Intel."},{"issue":"2","key":"2026032712174465800_ref018","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1214\/aos\/1176344130","article-title":"The geometry of exponential families","volume":"6","author":"Efron","year":"1978","journal-title":"Ann. Statist."},{"issue":"2","key":"2026032712174465800_ref019","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1109\/18.119688","article-title":"Trellis precoding: Combined coding, precoding and shaping for intersymbol interference channels","volume":"38","author":"Eyuboglu","year":"1992","journal-title":"IEEE Tr. Inform. Thy."},{"key":"2026032712174465800_ref020","doi-asserted-by":"crossref","DOI":"10.1002\/0471439002","volume-title":"Precoding and Signal Shaping for Digital Transmission","author":"Fischer","year":"2002"},{"key":"2026032712174465800_ref021","doi-asserted-by":"crossref","first-page":"941","DOI":"10.1109\/49.29616","article-title":"Multidimensional constellations\u2014part II: Voronoi constellations","volume":"7","author":"Forney","year":"1989","journal-title":"IEEE J. Select. Areas Commun"},{"issue":"1","key":"2026032712174465800_ref022","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1109\/18.119687","article-title":"Trellis shaping","volume":"38","author":"Forney","year":"1992","journal-title":"IEEE Tr. Inform. Theory"},{"key":"2026032712174465800_ref023","volume-title":"Information Theory. 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J."},{"key":"2026032712174465800_ref029","volume-title":"Maximum Entropy Econometrics: Robust Estimation with Limited Data","author":"Golan","year":"1996"},{"issue":"7435","key":"2026032712174465800_ref030","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1038\/nature11875","article-title":"Towards practical, high-capacity, low maintenance information storage in synthesized DNA","volume":"494","author":"Goldman","year":"2013","journal-title":"Nature"},{"key":"2026032712174465800_ref031","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.cviu.2004.02.002","article-title":"Maximum entropy model-based baseball highlight detection and classification","volume":"96","author":"Gong","year":"2004","journal-title":"Int. J. Computer Vision and Image Understanding"},{"key":"2026032712174465800_ref032","volume-title":"Machine Learning in Multimedia Content Analysis","author":"Gong","year":"2007"},{"issue":"3","key":"2026032712174465800_ref033","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.1109\/18.256522","article-title":"A generalized Gilbert-Varshamov bound derived via analysis of a code-search algorithm","volume":"39","author":"Gu","year":"1993","journal-title":"IEEE Tr. Inform. Thy."},{"key":"2026032712174465800_ref034","volume-title":"Information Theory with Applications","author":"Guiasu","year":"1977"},{"key":"2026032712174465800_ref035","doi-asserted-by":"crossref","DOI":"10.1515\/9783110516074","volume-title":"Loss Data Analysis: The Maximum Entropy Approach","author":"Gzyl","year":"2018"},{"key":"2026032712174465800_ref036","doi-asserted-by":"crossref","DOI":"10.1093\/acprof:oso\/9780199593415.001.0001","volume-title":"Maximum Entropy and Ecology: A Theory of Abundance, Distribution, and Energetics","author":"Harte","year":"2011"},{"issue":"7","key":"2026032712174465800_ref037","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.tree.2014.04.009","article-title":"Maximum information entropy: A foundation for ecological theory","volume":"29","author":"Harte","year":"2014","journal-title":"Trends in Ecology and Evolution"},{"issue":"3","key":"2026032712174465800_ref038","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1109\/18.79946","article-title":"Variable-length state splitting with applications to average runlength-constrained (arc) codes","volume":"37","author":"Heegard","year":"1991","journal-title":"IEEE Tr. 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Inform. Thy."},{"key":"2026032712174465800_ref047","volume-title":"Maximum-Entropy Models in Science and Engineering","author":"Kapur","year":"1989"},{"key":"2026032712174465800_ref048","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-011-2430-0_1","volume-title":"Entropy Optimization Principles with Applications","author":"Kapur","year":"1992"},{"key":"2026032712174465800_ref049","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-36212-8","volume-title":"Entropy Measures, Maximum Entropy Principle and Emerging Applications","author":"Karmeshu","year":"2003"},{"key":"2026032712174465800_ref050","first-page":"1348","article-title":"Evaluating the Gilbert-Varshamov bound for constrained systems","author":"Keshav","year":"2022"},{"issue":"6","key":"2026032712174465800_ref051","doi-asserted-by":"crossref","first-page":"1799","DOI":"10.1109\/18.265491","article-title":"Shaping multidimensional signal spaces\u2014part I: Optimum shaping, shell mapping,","volume":"39","author":"Khandani","year":"1993","journal-title":"IEEE Tr. Inform. Thy."},{"issue":"6","key":"2026032712174465800_ref052","doi-asserted-by":"crossref","first-page":"1809","DOI":"10.1109\/18.265493","article-title":"Shaping multidimensional signal space\u2014part II: Shell-addressed constellations,","volume":"39","author":"Khandani","year":"1993","journal-title":"IEEE Tr. Inform. Thy."},{"key":"2026032712174465800_ref053","volume-title":"Coding Communications and Broadcasting","author":"Khandekar","year":"2000"},{"key":"2026032712174465800_ref054","article-title":"The capacity of costly noiseless channels","author":"Khayrallah","year":"1988","journal-title":"IBM Res."},{"issue":"3","key":"2026032712174465800_ref055","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1109\/18.490550","article-title":"Coding for channels with cost constraints","volume":"42","author":"Khayrallah","year":"1996","journal-title":"IEEE Tr. Inform. Thy."},{"issue":"4","key":"2026032712174465800_ref056","doi-asserted-by":"crossref","first-page":"2018","DOI":"10.1109\/TAC.2022.3168168","article-title":"Maximum entropy optimal control of continuous-time dynamical systems","volume":"68","author":"Kim","year":"2023","journal-title":"IEEE Tr. Auto. Control"},{"issue":"11","key":"2026032712174465800_ref057","doi-asserted-by":"crossref","first-page":"778","DOI":"10.1109\/18.79947","article-title":"Generating functions and lower bounds on rates for limiting error-correcting codes","volume":"37","author":"Kolesnik","year":"1991","journal-title":"IEEE Tr. Inform Thy."},{"key":"2026032712174465800_ref058","first-page":"421","article-title":"On modulation coding for channels with cost constraints","author":"Krachkovsky","year":"2014"},{"issue":"3","key":"2026032712174465800_ref059","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1109\/18.256499","article-title":"Optimal nonuniform signaling for Gaussian channels","volume":"39","author":"Kschischang","year":"1993","journal-title":"IEEE Tr. Inform. Thy."},{"issue":"3","key":"2026032712174465800_ref060","doi-asserted-by":"crossref","first-page":"892","DOI":"10.1109\/18.335900","article-title":"Optimal shaping properties of the truncated polydisc","volume":"40","author":"Kschischang","year":"1994","journal-title":"IEEE Tr. Inform. Thy."},{"key":"2026032712174465800_ref061","volume-title":"Information Theory and Statistics","author":"Kullback","year":"1959"},{"issue":"1","key":"2026032712174465800_ref062","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On information and sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Stat."},{"issue":"4","key":"2026032712174465800_ref063","doi-asserted-by":"crossref","first-page":"1044","DOI":"10.1109\/18.335969","article-title":"On optimal shaping of multidimen-sional constellations","volume":"40","author":"Laroia","year":"1994","journal-title":"IEEE Tr. Inform. Thy."},{"key":"2026032712174465800_ref064","first-page":"1080","article-title":"Adaptive language modeling using the maximum entropy principle","author":"Lau","year":"1993"},{"key":"2026032712174465800_ref065","first-page":"2885","article-title":"Coding for efficient DNA synthesis","author":"Lenz","year":"2020"},{"key":"2026032712174465800_ref066","first-page":"2494","article-title":"Exact asymptotics for discrete noiseless channels","author":"Lenz","year":"2023","journal-title":"IEEE Int\u2019l Symp. Inform. Thy."},{"issue":"8","key":"2026032712174465800_ref067","doi-asserted-by":"crossref","first-page":"5261","DOI":"10.1109\/TIT.2020.2986318","article-title":"Rateconstrained shaping codes for structured sources","volume":"66","author":"Liu","year":"2020","journal-title":"IEEE Tr. Inform. Thy."},{"key":"2026032712174465800_ref068","first-page":"1003","article-title":"Performance of optimal data shaping codes","author":"Liu","year":"2017"},{"key":"2026032712174465800_ref069","first-page":"1354","article-title":"Rate-constrained shaping codes for finite-state channels with cost","author":"Liu","year":"2022"},{"key":"2026032712174465800_ref070","first-page":"1","volume-title":"IEEE Global Commun. Conf. (GLOBECOM)","author":"Liu","year":"2016"},{"issue":"4","key":"2026032712174465800_ref071","doi-asserted-by":"crossref","first-page":"1347","DOI":"10.1109\/18.144714","article-title":"Shaping using variable-size regions","volume":"38","author":"Livingston","year":"1992","journal-title":"IEEE Tr. Inform. Thy."},{"key":"2026032712174465800_ref072","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4939-6795-7","volume-title":"Amazing and Aesthetic Aspects of Analysis","author":"Loya","year":"2017"},{"issue":"4","key":"2026032712174465800_ref073","doi-asserted-by":"crossref","first-page":"1213","DOI":"10.1109\/18.144702","article-title":"Improved Gilbert-Varshamov bound for constrained systems","volume":"38","author":"Marcus","year":"1992","journal-title":"IEEE Tr. Inform. Thy."},{"key":"2026032712174465800_ref074","article-title":"An introduction to coding for constrained systems","author":"Marcus","year":"2001","journal-title":"Lecture Notes"},{"key":"2026032712174465800_ref075","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1080\/14786446008642818","article-title":"Illustrations of the dynamical theory of gases","volume":"19","author":"Maxwell","year":"1860","journal-title":"Philosophical Magazine, Series 4"},{"key":"2026032712174465800_ref076","volume-title":"The Theory of Information and Coding","author":"McEliece","year":"1977","edition":"2"},{"key":"2026032712174465800_ref077","first-page":"245","article-title":"A maximum entropy Markov chain","author":"McEliece","year":"1983","journal-title":"Proc. 17th Annual Conf. on Inform. Sci. and Sys."},{"key":"2026032712174465800_ref078","volume-title":"Information Theory, Inference and Learning Algorithms","author":"McKay","year":"2003"},{"issue":"2","key":"2026032712174465800_ref079","doi-asserted-by":"crossref","first-page":"692","DOI":"10.1109\/18.556124","article-title":"Shaping codes constructed from cost constrained graphs","volume":"43","author":"McLaughlin","year":"1997","journal-title":"IEEE Tr. Inf. Thy."},{"issue":"10","key":"2026032712174465800_ref080","doi-asserted-by":"publisher","first-page":"1100","DOI":"10.3390\/e22101100","article-title":"An elementary introduction to information geometry","volume":"22","author":"Nielsen","year":"2020","journal-title":"Entropy"},{"key":"2026032712174465800_ref081","first-page":"61","article-title":"Using maximum entropy for text classification","author":"Nigam","year":"1999"},{"key":"2026032712174465800_ref082","doi-asserted-by":"crossref","DOI":"10.1017\/9781108966351","volume-title":"Information Theory: From Coding to Learning","author":"Polyanskiy","year":"2024"},{"issue":"4","key":"2026032712174465800_ref083","doi-asserted-by":"crossref","first-page":"1000","DOI":"10.1214\/aoms\/1177698768","article-title":"Epsilon entropy of stochastic processes","volume":"38","author":"Posner","year":"1967","journal-title":"Annals of Math. Stat."},{"issue":"3","key":"2026032712174465800_ref084","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1103\/RevModPhys.85.1115","article-title":"Principles of maximum entropy and maximum caliber in statistical physics","volume":"85","author":"Press\u00e9","year":"2013","journal-title":"Reviews of Modern Physics"},{"key":"2026032712174465800_ref085","first-page":"133","volume-title":"Conf. on Empirical Methods in Natural Lang. Proc.","author":"Ratnaparkhi","year":"1996"},{"key":"2026032712174465800_ref086","article-title":"A simple introduction to maximum entropy models for natural language processing","volume":"81","author":"Ratnaparkhi","year":"1997","journal-title":"IRCS Technical Reports Series"},{"key":"2026032712174465800_ref087","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1023\/A:1007502103375","article-title":"Learning to parse natural language with maximum entropy models","volume":"34","author":"Ratnaparkhi","year":"1999","journal-title":"Machine Learning"},{"key":"2026032712174465800_ref088","doi-asserted-by":"crossref","DOI":"10.1515\/9781400873173","volume-title":"Convex Analysis","author":"Rockafellar","year":"1970"},{"key":"2026032712174465800_ref089","author":"Rosenfeld","year":"1994","journal-title":"Adaptive Statistical Languarge Modeling: A Maximum Entropy Approach"},{"key":"2026032712174465800_ref090","volume-title":"A First Course in Probability","author":"Ross","year":"2002","edition":"6"},{"key":"2026032712174465800_ref091","volume-title":"Real Analysis","author":"Royden","year":"1988","edition":"3"},{"issue":"10","key":"2026032712174465800_ref092","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1109\/MCOM.002.2200249","article-title":"Innovation in constrained codes","volume":"60","author":"Schouhamer Immink","year":"2022","journal-title":"IEEE Comm. Mag."},{"key":"2026032712174465800_ref093","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"issue":"1","key":"2026032712174465800_ref094","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1109\/TIT.1980.1056144","article-title":"Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy","volume":"26","author":"Shore","year":"1980","journal-title":"IEEE Trans. Inform. Thy."},{"issue":"4","key":"2026032712174465800_ref095","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1109\/TIT.1981.1056373","article-title":"Properties of cross-entropy minimization","volume":"27","author":"Shore","year":"1981","journal-title":"IEEE Trans. Inform. Thy."},{"key":"2026032712174465800_ref096","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-69438-2","volume-title":"Maximum-Entropy Networks: Pattern Detection, Network Reconstruction and Graph Combinatorics","author":"Squartini","year":"2017"},{"key":"2026032712174465800_ref097","volume-title":"Exploiting Continuity: Maximum Entropy Estimation of Continuous Distributions","author":"Theil","year":"1984"},{"key":"2026032712174465800_ref098","first-page":"739","article-title":"Estimate of the number of signals in error correcting codes","volume":"117","author":"Varshamov","year":"1957","journal-title":"Dokl. Akad. 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