{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T11:26:55Z","timestamp":1760441215248,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2014,2,24]],"date-time":"2014-02-24T00:00:00Z","timestamp":1393200000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>There are two entropy-based methods to deal with linear inverse problems, which we shall call the ordinary method of maximum entropy (OME) and the method of maximum entropy in the mean (MEM). Not only doesMEM use OME as a stepping stone, it also allows for greater generality. First, because it allows to include convex constraints in a natural way, and second, because it allows to incorporate and to estimate (additive) measurement errors from the data. Here we shall see both methods in action in a specific example. We shall solve the discretized version of the problem by two variants of MEM and directly with OME. We shall see that OME is actually a particular instance of MEM, when the reference measure is a Poisson Measure.<\/jats:p>","DOI":"10.3390\/e16021123","type":"journal-article","created":{"date-parts":[[2014,2,25]],"date-time":"2014-02-25T03:23:38Z","timestamp":1393298618000},"page":"1123-1133","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Relationship between the Ordinary Maximum Entropy Method and the Method of Maximum Entropy in the Mean"],"prefix":"10.3390","volume":"16","author":[{"given":"Henryk","family":"Gzyl","sequence":"first","affiliation":[{"name":"Centro de Finanzas, IESA, Ave. Iesa, San Bernardino, Caracas 1010, Venezuela"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Enrique","family":"Ter Horst","sequence":"additional","affiliation":[{"name":"Centro de Finanzas, IESA, Ave. Iesa, San Bernardino, Caracas 1010, Venezuela"},{"name":"Institute 2, CESA, Calle 35 #6-16, Bogot\u00e1, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2014,2,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1103\/PhysRev.108.171","article-title":"Information theory and statistical physics","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys. Rev"},{"key":"ref_2","unstructured":"Kapur, J (1989). Maximum Entropy Models in Science and Engineering;, Wiley Eastern Ltd."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Jaynes, E (2003). Probability Theory: The Logic of Science;, Cambridge University Press.","DOI":"10.1017\/CBO9780511790423"},{"key":"ref_4","first-page":"175","article-title":"On the probability function in the collective theory of risk","volume":"15","author":"Escher","year":"1932","journal-title":"Scan. Aktuarietidskr"},{"key":"ref_5","unstructured":"Kullback, S (1968). Information Theory and Statistics;, Dover Pubs."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1137\/0802010","article-title":"Dual Methods in Entropy Maximization. Application to Some Problems in Crystallography","volume":"2","author":"Decarreau","year":"1992","journal-title":"SIAM J. Optim"},{"key":"ref_7","first-page":"567","article-title":"Maximum d\u2019entropie et probl\u00e8me des moments","volume":"26","author":"Gamboa","year":"1990","journal-title":"Ann. Inst. Henri Poincar\u00e9"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Gzyl, H., and Vel\u00e1squez, Y (2011). Linear Inverse Problems: The Maximum entropy Connection;, World Scientific Pubs.","DOI":"10.1142\/9789814338783"},{"key":"ref_9","first-page":"117","article-title":"Maxentropic reconstruction of Fourier and Laplace transforms under non-linear constraints","volume":"25","author":"Gzyl","year":"1995","journal-title":"Appl. Math. Comput"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/S0895-7177(97)00046-0","article-title":"Maxentropic solutions of linear Fredholm equations","volume":"25","author":"Gamboa","year":"1997","journal-title":"Math. Comput. Model"},{"key":"ref_11","unstructured":"Gallon, S., Gamboa, F., and Loubes, M (2013). Functional Calibration estimation by the maximum entropy in the mean principle., arXiv: 1302.1158[math.ST]."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1214\/aop\/1176996454","article-title":"I-divergence geometry of probability distributions and minimization problems","volume":"3","author":"Csiszar","year":"1975","journal-title":"Ann. Probab"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"768","DOI":"10.1214\/aop\/1176993227","article-title":"Generalized I-projection and a conditional limit theorem","volume":"12","author":"Csiszar","year":"1984","journal-title":"Ann. Probab"},{"key":"ref_14","first-page":"447","article-title":"On minimization and maximization of entropy functionals in various disciplines","volume":"17","author":"Cherny","year":"2003","journal-title":"Theory Probab. Appl"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v032.i04","article-title":"An R package for solving large system of nonlinear equations and for optimizing a high dimensional nonlinear objective function","volume":"32","author":"Varadhan","year":"2009","journal-title":"J. Stat. Softw"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/2\/1123\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:08:30Z","timestamp":1760216910000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/2\/1123"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,24]]},"references-count":15,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2014,2]]}},"alternative-id":["e16021123"],"URL":"https:\/\/doi.org\/10.3390\/e16021123","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2014,2,24]]}}}