{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T16:03:09Z","timestamp":1761580989249,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2012,5,21]],"date-time":"2012-05-21T00:00:00Z","timestamp":1337558400000},"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>The minimum error entropy (MEE) criterion has been receiving increasing attention due to its promising perspectives for applications in signal processing and machine learning. In the context of Bayesian estimation, the MEE criterion is concerned with the estimation of a certain random variable based on another random variable, so that the error\u2019s entropy is minimized. Several theoretical results on this topic have been reported. In this work, we present some further results on the MEE estimation. The contributions are twofold: (1) we extend a recent result on the minimum entropy of a mixture of unimodal and symmetric distributions to a more general case, and prove that if the conditional distributions are generalized uniformly dominated (GUD), the dominant alignment will be the MEE estimator; (2) we show by examples that the MEE estimator (not limited to singular cases) may be non-unique even if the error distribution is restricted to zero-mean (unbiased).<\/jats:p>","DOI":"10.3390\/e14050966","type":"journal-article","created":{"date-parts":[[2012,5,21]],"date-time":"2012-05-21T11:24:56Z","timestamp":1337599496000},"page":"966-977","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Some Further Results on the Minimum Error Entropy Estimation"],"prefix":"10.3390","volume":"14","author":[{"given":"Badong","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Precision Instruments and Mechanology, Tsinghua University, Beijing, 100084, China"},{"name":"Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jose C.","family":"Principe","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,5,21]]},"reference":[{"key":"ref_1","unstructured":"Cover, T.M., and Thomas, J.A. (1991). Element of Information Theory, Wiley & Son, Inc."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kapur, J.N., and Kesavan, H.K. (1992). Entropy Optimization Principles with Applications, Academic Press, Inc.","DOI":"10.1007\/978-94-011-2430-0_1"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1109\/TIT.1970.1054444","article-title":"Entropy analysis of estimating systems","volume":"16","author":"Weidemann","year":"1970","journal-title":"IEEE Trans. Inform. Theor."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/S0019-9958(76)90140-6","article-title":"An application of the information theory to estimation problems","volume":"32","author":"Tomita","year":"1976","journal-title":"Inf. Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1016\/j.sigpro.2004.11.028","article-title":"Minimum-entropy estimation in semiparametric models","volume":"85","author":"Wolsztynski","year":"2005","journal-title":"Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"690","DOI":"10.1080\/00207170701701031","article-title":"An extended orthogonal forward regression algorithm for system identification using entropy","volume":"81","author":"Guo","year":"2008","journal-title":"Int. J. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1384","DOI":"10.1016\/j.ins.2010.11.037","article-title":"\u25b3-entropy: Definition, properties and applications in system identification with quantized data","volume":"181","author":"Chen","year":"2011","journal-title":"Inf. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0020-0255(79)90039-2","article-title":"Linear prediction, filtering and smoothing: an information theoretic approach","volume":"17","author":"Kalata","year":"1979","journal-title":"Inf. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1109\/9.587329","article-title":"Optimal state estimation for stochastic systems: An information theoretic approach","volume":"42","author":"Feng","year":"1997","journal-title":"IEEE Trans. Automat. Contr."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1109\/TAC.2006.872771","article-title":"Minimum entropy filtering for multivariate stochastic systems with non-Gaussian Noises","volume":"51","author":"Guo","year":"2006","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1109\/9.983388","article-title":"Minimum entropy control of non-Gaussian dynamic stochastic systems","volume":"47","author":"Wang","year":"2002","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1109\/TAC.2002.806663","article-title":"Minimum entropy control of closed-loop tracking errors for dynamic stochastic systems","volume":"48","author":"Yue","year":"2003","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1780","DOI":"10.1109\/TSP.2002.1011217","article-title":"An error-entropy minimization algorithm for supervised training of nonlinear adaptive systems","volume":"50","author":"Erdogmus","year":"2002","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1109\/SP-M.2006.248709","article-title":"From linear adaptive filtering to nonlinear information processing\u2014The design and analysis of information processing systems","volume":"23","author":"Erdogmus","year":"2006","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Principe, J.C. (2010). Information Theoretic Learning: Renyi's Entropy and Kernel Perspectives, Springer.","DOI":"10.1007\/978-1-4419-1570-2"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1109\/9.989082","article-title":"Robustness and risk-sensitive filtering","volume":"47","author":"Boel","year":"2002","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/0020-0255(94)90043-4","article-title":"Minimum entropy of error principle in estimation","volume":"79","author":"Janzura","year":"1994","journal-title":"Inf. Sci."},{"key":"ref_18","first-page":"331","article-title":"Minimum entropy of error estimate for multi-dimensional parameter and finite-state-space observations","volume":"31","author":"Otahal","year":"1995","journal-title":"Kybernetika"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3166","DOI":"10.1109\/TIT.2008.924686","article-title":"On the minimum entropy of a mixture of unimodal and symmetric distributions","volume":"54","author":"Chen","year":"2008","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1016\/j.jfranklin.2009.11.009","article-title":"On optimal estimations with minimum error entropy criterion","volume":"347","author":"Chen","year":"2010","journal-title":"J. Frankl. Inst.-Eng. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3313","DOI":"10.1016\/j.sigpro.2010.05.028","article-title":"A new interpretation on the MMSE as a robust MEE criterion","volume":"90","author":"Chen","year":"2010","journal-title":"Signal Process."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/5\/966\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:50:23Z","timestamp":1760219423000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/5\/966"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,5,21]]},"references-count":21,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2012,5]]}},"alternative-id":["e14050966"],"URL":"https:\/\/doi.org\/10.3390\/e14050966","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2012,5,21]]}}}