{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T16:28:30Z","timestamp":1776356910053,"version":"3.51.2"},"reference-count":81,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,12]],"date-time":"2024-07-12T00:00:00Z","timestamp":1720742400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fondecyt 1210315 ANID-Chile","award":["RGPIN-2024-05730"],"award-info":[{"award-number":["RGPIN-2024-05730"]}]},{"name":"Advanced Center for Electrical and Electronic Engineering, Basal Project FB0008","award":["RGPIN-2024-05730"],"award-info":[{"award-number":["RGPIN-2024-05730"]}]},{"name":"NID Basal Project FB0008 and the NSERC Discovery","award":["RGPIN-2024-05730"],"award-info":[{"award-number":["RGPIN-2024-05730"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A central challenge in hypothesis testing (HT) lies in determining the optimal balance between Type I (false positive) and Type II (non-detection or false negative) error probabilities. Analyzing these errors\u2019 exponential rate of convergence, known as error exponents, provides crucial insights into system performance. Error exponents offer a lens through which we can understand how operational restrictions, such as resource constraints and impairments in communications, affect the accuracy of distributed inference in networked systems. This survey presents a comprehensive review of key results in HT, from the foundational Stein\u2019s Lemma to recent advancements in distributed HT, all unified through the framework of error exponents. We explore asymptotic and non-asymptotic results, highlighting their implications for designing robust and efficient networked systems, such as event detection through lossy wireless sensor monitoring networks, collective perception-based object detection in vehicular environments, and clock synchronization in distributed environments, among others. We show that understanding the role of error exponents provides a valuable tool for optimizing decision-making and improving the reliability of networked systems.<\/jats:p>","DOI":"10.3390\/e26070596","type":"journal-article","created":{"date-parts":[[2024,7,12]],"date-time":"2024-07-12T14:04:35Z","timestamp":1720793075000},"page":"596","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Survey on Error Exponents in Distributed Hypothesis Testing: Connections with Information Theory, Interpretations, and Applications"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7836-2541","authenticated-orcid":false,"given":"Sebasti\u00e1n","family":"Espinosa","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, Universidad de Chile, Santiago 9170022, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0256-282X","authenticated-orcid":false,"given":"Jorge F.","family":"Silva","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Universidad de Chile, Santiago 9170022, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1686-2644","authenticated-orcid":false,"given":"Sandra","family":"C\u00e9spedes","sequence":"additional","affiliation":[{"name":"Department of Computer Science & Software Engineering, Concordia University, Montreal, QC H3G 1M8, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,12]]},"reference":[{"key":"ref_1","unstructured":"Cover, T.M., and Thomas, J.A. (2012). Elements of Information Theory, John Wiley & Sons."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Guzdial, M. (2013, January 11\u201314). Exploring hypotheses about media computation. Proceedings of the 9th Annual International ACM Conference on International Computing Education Research, San Diego, CA, USA.","DOI":"10.1145\/2493394.2493397"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1177\/0962280206079046","article-title":"A review of modern multiple hypothesis testing, with particular attention to the false discovery proportion","volume":"17","author":"Farcomeni","year":"2008","journal-title":"Stat. Methods Med. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"548","DOI":"10.2307\/1240503","article-title":"Hypotheses testing in economic science","volume":"65","author":"Tweeten","year":"1983","journal-title":"Am. J. Agric. Econ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1792","DOI":"10.1002\/2016WR020116","article-title":"Debates\u2014Hypothesis testing in hydrology: Theory and practice","volume":"53","author":"Pfister","year":"2017","journal-title":"Water Resour. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1561\/0100000086","article-title":"Asymptotic estimates in information theory with non-vanishing error probabilities","volume":"11","author":"Tan","year":"2014","journal-title":"Found. Trends Commun. Inf. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Csiszar, I., and K\u00f6rner, J. (2011). Information Theory: Coding Theorems for Discrete Memoryless Systems, Cambridge University Press.","DOI":"10.1017\/CBO9780511921889"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"El Gamal, A., and Kim, Y.H. (2011). Network Information Theory, Cambridge University Press.","DOI":"10.1017\/CBO9781139030687"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1647","DOI":"10.1109\/JSAC.2006.879411","article-title":"Error exponents for channel coding with application to signal constellation design","volume":"24","author":"Huang","year":"2006","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"7019","DOI":"10.1109\/TIT.2021.3111541","article-title":"On error exponents of encoder-assisted communication systems","volume":"67","author":"Merhav","year":"2021","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5095","DOI":"10.1109\/TIT.2014.2331952","article-title":"Error exponent for multiple-access channels: Lower bounds","volume":"60","author":"Nazari","year":"2014","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_12","unstructured":"Moon, T.K., and Stirling, W.C. (2000). Mathematical Methods and Algorithms for Signal Processing, Prentice Hall."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Pichler, G., Piantanida, P., and Matz, G. (October, January 29). Distributed information-theoretic biclustering of two memoryless sources. Proceedings of the 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton), Monticello, IL, USA.","DOI":"10.1109\/ALLERTON.2015.7447035"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1109\/TIT.1986.1057194","article-title":"Hypothesis testing with communication constraints","volume":"32","author":"Ahlswede","year":"1986","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_15","first-page":"777","article-title":"On the Exponential Approximation of Type II Error Probability of Distributed Test of Independence","volume":"7","author":"Espinosa","year":"2021","journal-title":"IEEE Trans. Signal Inf. Process. Netw."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1109\/MSP.2007.361598","article-title":"Wireless Sensors in Distributed Detection Applications","volume":"24","author":"Chamberland","year":"2007","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1109\/79.985674","article-title":"Detection, classification, and tracking of targets","volume":"19","author":"Li","year":"2002","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1109\/79.985685","article-title":"Information-driven dynamic sensor collaboration","volume":"19","author":"Zhao","year":"2002","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Mhanna, M., Piantanida, P., and Duhamel, P. (2017, January 21\u201325). Privacy-preserving quantization learning with applications to smart meters. Proceedings of the IEEE International Conference on Communications, ICC 2017, Paris, France.","DOI":"10.1109\/ICC.2017.7997332"},{"key":"ref_20","unstructured":"Chen, M., Liu, W., Chen, B., and Matyjas, J. (2010, January 26\u201329). Quantization for distributed testing of independence. Proceedings of the 2010 13th International Conference on Information Fusion, Edinburgh, UK."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1109\/7.489500","article-title":"Censoring sensors: A low-communication-rate scheme for distributed detection","volume":"32","author":"Rago","year":"1996","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Mhanna, M., and Piantanida, P. (2015, January 14\u201319). On secure distributed hypothesis testing. Proceedings of the 2015 IEEE International Symposium on Information Theory (ISIT), Hong Kong, China.","DOI":"10.1109\/ISIT.2015.7282727"},{"key":"ref_23","unstructured":"Mahler, R. (2007). Statistical Multisources-Multitarget Information Fusion, Artech House."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2022","DOI":"10.1109\/TSP.2019.2899291","article-title":"Algorithm and Fundamental Limits for Unlabeled Detection Using Types","volume":"67","author":"Marano","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4291","DOI":"10.1109\/TSP.2018.2849704","article-title":"Signal Amplitude estimation and detection from unlabeled binary quantized samples","volume":"66","author":"Wang","year":"2018","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2130","DOI":"10.1109\/LCOMM.2017.2695998","article-title":"Parameter Estimation via unlabelled sensing using distributed sensors","volume":"21","author":"Zhu","year":"2017","journal-title":"IEEE Commun. Letter"},{"key":"ref_27","first-page":"306","article-title":"The importance of being earnest: Social Network with unknown agent quality","volume":"2","author":"Marano","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3237","DOI":"10.1109\/TIT.2018.2809002","article-title":"Unlabeled sensing with random linear measurements","volume":"64","author":"Unnikrishnam","year":"2018","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2431","DOI":"10.1109\/LCOMM.2018.2875454","article-title":"Signal Detection from unlabeled ordered samples","volume":"22","author":"Liu","year":"2018","journal-title":"IEEE Commun. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1242","DOI":"10.1109\/TSP.2017.2786276","article-title":"Signal Recovery from unlabeled samples","volume":"66","author":"Haghighasthoar","year":"2018","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Varshney, P.K. (1996). Distributed Detection and Data Fusion, Springer. [1st ed.].","DOI":"10.21236\/ADA301116"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1109\/TAES.1981.309178","article-title":"Detection with Distributed Sensors","volume":"4","author":"Tenney","year":"1981","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1150","DOI":"10.1109\/7.395226","article-title":"Optimal m-ary data fusion with distributed sensors","volume":"31","author":"Baek","year":"1995","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2875","DOI":"10.1109\/TSP.2011.2121065","article-title":"Distributed Detection Over Gaussian Multiple Access Channels With Constant Modulus Signaling","volume":"59","author":"Tepedelenlioglu","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2636","DOI":"10.1109\/TSP.2016.2529583","article-title":"Design of Binary Quantizers for Distributed Detection Under Secrecy Constraints","volume":"64","author":"Nadendla","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_36","unstructured":"Chandramouli, R., and Ranganathan, N. (June, January 31). Quantization for robust sequential M-ary signal detection. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems, ISCAS \u201998 (Cat. No.98CH36187), Monterey, CA, USA."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Wigger, M., and Timo, R. (2016, January 12\u201315). Testing against independence with multiple decision centers. Proceedings of the 2016 International Conference on Signal Processing and Communications (SPCOM), Bangalore, India.","DOI":"10.1109\/SPCOM.2016.7746613"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Xiang, Y., and Kim, Y.H. (2012, January 1\u20135). Interactive hypothesis testing with communication constraints. Proceedings of the 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton), Monticello, IL, USA.","DOI":"10.1109\/Allerton.2012.6483336"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2409","DOI":"10.1109\/TCOMM.2018.2798659","article-title":"On hypothesis testing against conditional independence with multiple decision centers","volume":"66","author":"Salehkalaibar","year":"2018","journal-title":"IEEE Trans. Commun."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1109\/LSP.2021.3050381","article-title":"Finite-Length Bounds on Hypothesis Testing Subject to Vanishing Type I Error Restrictions","volume":"28","author":"Espinosa","year":"2021","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Espinosa, S., Silva, J.F., and Piantanida, P. (2019, January 11\u201314). New Results on Testing Against Independence with Rate-Limited Constraints. Proceedings of the 2019 IEEE Global Conference on Signal and Information Processing (GlobalSIP), Ottawa, ON, Canada.","DOI":"10.1109\/GlobalSIP45357.2019.8969535"},{"key":"ref_42","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."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1098\/rsta.1933.0009","article-title":"On the problem of the most efficient tests of statistical hypotheses","volume":"231","author":"Neyman","year":"1933","journal-title":"Phil. Trans. R. Soc. Lond. A"},{"key":"ref_44","unstructured":"Chung, K.L. (2001). A Course in Probability Theory, Academic Press."},{"key":"ref_45","unstructured":"Han, T.S. (2013). Information-Spectrum Methods in Information Theory, Springer Science & Business Media."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Gray, R.M. (2011). Entropy and Information Theory, Springer Science & Business Media.","DOI":"10.1007\/978-1-4419-7970-4"},{"key":"ref_47","unstructured":"Gallager, R.G. (1968). Information Theory and Reliable Communication, Springer."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1214\/aoms\/1177729330","article-title":"A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations","volume":"23","author":"Chernoff","year":"1952","journal-title":"Ann. Math. Stat."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1109\/18.42171","article-title":"Exponential-type error probabilities for multiterminal hypothesis testing","volume":"35","author":"Han","year":"1989","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1109\/18.212293","article-title":"On the converse theorem in statistical hypothesis testing","volume":"39","author":"Nakagawa","year":"1993","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1109\/TIT.1974.1055254","article-title":"Hypothesis testing and information theory","volume":"20","author":"Blahut","year":"1974","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_52","unstructured":"Strassen, V. (2024, June 06). Asymptotic Estimates in Shannon\u2019s Information Theory. Available online: https:\/\/www.google.com\/url?sa=t&source=web&rct=j&opi=89978449&url=https:\/\/pi.math.cornell.edu\/~pmlut\/strassen.pdf&ved=2ahUKEwj38ayXjZ6HAxXtiq8BHQk6BHoQFnoECBAQAQ&usg=AOvVaw34btneOU208HU2LXaxWDci."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"2307","DOI":"10.1109\/TIT.2010.2043769","article-title":"Channel coding rate in the finite blocklength regime","volume":"56","author":"Polyanskiy","year":"2010","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_54","unstructured":"Feller, W. (1991). An Introduction to Probability Theory and Its Applications, Volume 2, John Wiley & Sons."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1214\/aoms\/1177700150","article-title":"Asymptotically optimal tests for multinomial distributions","volume":"36","author":"Hoeffding","year":"1965","journal-title":"Ann. Math. Stat."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Huang, D., and Meyn, S. (2012, January 25\u201330). Error exponents for composite hypothesis testing with small samples. Proceedings of the 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Kyoto, Japan.","DOI":"10.1109\/ICASSP.2012.6288611"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1032","DOI":"10.1109\/TIT.2021.3125683","article-title":"On the error exponents of binary state discrimination with composite hypotheses","volume":"68","author":"Mosonyi","year":"2021","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"4923","DOI":"10.1109\/TIT.2017.2778252","article-title":"Neyman\u2013Pearson Test for Zero-Rate Multiterminal Hypothesis Testing","volume":"64","author":"Watanabe","year":"2018","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_59","unstructured":"Kendall, M., Stuart, A., Ord, K.J., and Arnold, S. (1999). Kendall\u2019s Advanced Theory of Statistics: Volume 2A\u2014Classical Inference and the Linear Model, Hodder Education Publishers."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1109\/TIT.1987.1057383","article-title":"Hypothesis testing with multiterminal data compression","volume":"33","author":"Han","year":"1987","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_61","first-page":"585","article-title":"A kernel statistical test of independence","volume":"20","author":"Gretton","year":"2007","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_62","first-page":"29","article-title":"A non parametric test for independence","volume":"Volume 26","author":"Deheuvels","year":"1981","journal-title":"Annales de l\u2019ISUP"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1086\/171931","article-title":"A simple test of independence for truncated data with applications to redshift surveys","volume":"399","author":"Efron","year":"1992","journal-title":"Astrophys. J."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1080\/07350015.1993.10509931","article-title":"Tests of independence in parametric models with applications and illustrations","volume":"11","author":"Cameron","year":"1993","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"2337","DOI":"10.1214\/08-AAP527","article-title":"The asymptotic distribution and Berry\u2013Esseen bound of a new test for independence in high dimension with an application to stochastic optimization","volume":"18","author":"Liu","year":"2008","journal-title":"Ann. Appl. Probab."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1214\/11-EJS610","article-title":"Asymptotic independence of correlation coefficients with application to testing hypothesis of independence","volume":"5","author":"Zhang","year":"2011","journal-title":"Electron. J. Statist."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1109\/TSP.2021.3135689","article-title":"Data-driven representations for testing independence: Modeling, analysis and connection with mutual information estimation","volume":"70","author":"Gonzalez","year":"2021","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_68","unstructured":"Tishby, N., Pereira, F.C., and Bialek, W. (2000). The information bottleneck method. arXiv."},{"key":"ref_69","doi-asserted-by":"crossref","unstructured":"Shkel, Y., Raginsky, M., and Verd\u00fa, S. (2017, January 25\u201330). Universal lossy compression under logarithmic loss. Proceedings of the 2017 IEEE International Symposium on Information Theory (ISIT), Aachen, Germany.","DOI":"10.1109\/ISIT.2017.8006710"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1109\/MCOM.2004.1341264","article-title":"Cooperative communication in wireless networks","volume":"42","author":"Nosratinia","year":"2004","journal-title":"IEEE Commun. Mag."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"878","DOI":"10.1109\/JPROC.2009.2015716","article-title":"Cooperative communications for cognitive radio networks","volume":"97","author":"Letaief","year":"2009","journal-title":"Proc. IEEE"},{"key":"ref_72","unstructured":"Tishby, N., Pereira, F., and Biale, W. (1999, January 22\u201324). The Information Botleneck Method. Proceedings of the 37th Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, USA."},{"key":"ref_73","unstructured":"Berger, T. (1971). Rate-Distortion Theory: A Mathematical Basis for Data Compression, Prentice Hall."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"740","DOI":"10.1109\/TIT.2013.2288257","article-title":"Multiterminal source coding under logarithmic loss","volume":"60","author":"Courtade","year":"2014","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_75","unstructured":"Tishby, N., and Zaslavsky, N. (May, January 26). Deep learning and the information bottleneck principle. Proceedings of the IEEE Information Theory Workshop, Jerusalem, Israel."},{"key":"ref_76","unstructured":"Alemi, A., Fischer, I., Dillon, J., and Murphy, K. (2017, January 24\u201326). Deep variational information bottleneck. Proceedings of the International Conference on Learning Representations (ICLR), Tulon, France."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"1611","DOI":"10.1162\/NECO_a_00961","article-title":"The deterministic Information Bottleneck","volume":"29","author":"Strouse","year":"2017","journal-title":"Neural Comput."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"2225","DOI":"10.1109\/TPAMI.2019.2909031","article-title":"Learning representations for Neural Network-Based Classification using the Information Bottleneck Principle","volume":"42","author":"Amjad","year":"2019","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_79","doi-asserted-by":"crossref","unstructured":"Espinosa, S., Silva, J.F., and Piantanida, P. (2022, January 22\u201327). A Data-Driven Quantization Design for Distributed Testing Against Independence with Communication Constraints. Proceedings of the 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2022), Singapore.","DOI":"10.1109\/ICASSP43922.2022.9746197"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"10261","DOI":"10.1109\/JIOT.2019.2937315","article-title":"Location verification for emerging wireless vehicular networks","volume":"6","author":"Ihsan","year":"2019","journal-title":"IEEE Internet Things J."},{"key":"ref_81","doi-asserted-by":"crossref","unstructured":"Espinosa, S., Silva, J.F., and Piantanida, P. (2019). Finite Sample-Size Regime of Testing Against Independence with Communication Constraints. arXiv.","DOI":"10.1109\/GlobalSIP45357.2019.8969535"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/7\/596\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:15:58Z","timestamp":1760109358000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/7\/596"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,12]]},"references-count":81,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["e26070596"],"URL":"https:\/\/doi.org\/10.3390\/e26070596","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7,12]]}}}