{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T08:02:03Z","timestamp":1780992123925,"version":"3.54.1"},"reference-count":46,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,3]],"date-time":"2021-01-03T00:00:00Z","timestamp":1609632000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Uncertainty is at the heart of decision-making processes in most real-world applications. Uncertainty can be broadly categorized into two types: aleatory and epistemic. Aleatory uncertainty describes the variability in the physical system where sensors provide information (hard) of a probabilistic type. Epistemic uncertainty appears when the information is incomplete or vague such as judgments or human expert appreciations in linguistic form. Linguistic information (soft) typically introduces a possibilistic type of uncertainty. This paper is concerned with the problem of classification where the available information, concerning the observed features, may be of a probabilistic nature for some features, and of a possibilistic nature for some others. In this configuration, most encountered studies transform one of the two information types into the other form, and then apply either classical Bayesian-based or possibilistic-based decision-making criteria. In this paper, a new hybrid decision-making scheme is proposed for classification when hard and soft information sources are present. A new Possibilistic Maximum Likelihood (PML) criterion is introduced to improve classification rates compared to a classical approach using only information from hard sources. The proposed PML allows to jointly exploit both probabilistic and possibilistic sources within the same probabilistic decision-making framework, without imposing to convert the possibilistic sources into probabilistic ones, and vice versa.<\/jats:p>","DOI":"10.3390\/e23010067","type":"journal-article","created":{"date-parts":[[2021,1,3]],"date-time":"2021-01-03T19:54:46Z","timestamp":1609703686000},"page":"67","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A New Hybrid Possibilistic-Probabilistic Decision-Making Scheme for Classification"],"prefix":"10.3390","volume":"23","author":[{"given":"Basel","family":"Solaiman","sequence":"first","affiliation":[{"name":"Image &amp; Information Processing Department (iTi), IMT-Atlantique, Technop\u00f4le Brest Iroise CS 83818, 29238 Brest, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3320-1234","authenticated-orcid":false,"given":"Didier","family":"Gu\u00e9riot","sequence":"additional","affiliation":[{"name":"Image &amp; Information Processing Department (iTi), IMT-Atlantique, Technop\u00f4le Brest Iroise CS 83818, 29238 Brest, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shaban","family":"Almouahed","sequence":"additional","affiliation":[{"name":"Image &amp; Information Processing Department (iTi), IMT-Atlantique, Technop\u00f4le Brest Iroise CS 83818, 29238 Brest, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bassem","family":"Alsahwa","sequence":"additional","affiliation":[{"name":"Image &amp; Information Processing Department (iTi), IMT-Atlantique, Technop\u00f4le Brest Iroise CS 83818, 29238 Brest, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9658-7595","authenticated-orcid":false,"given":"\u00c9loi","family":"Boss\u00e9","sequence":"additional","affiliation":[{"name":"Image &amp; Information Processing Department (iTi), IMT-Atlantique, Technop\u00f4le Brest Iroise CS 83818, 29238 Brest, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,3]]},"reference":[{"key":"ref_1","unstructured":"Dubois, D. (2008). Uncertainty theories: A unified view. SIPTA School 08\u2014UEE 08, SIPTA."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Klir, G.J., and Wierman, M.J. (1999). Uncertainty-Based Information: Elements of Generalized Information Theory, Physica-Verlag HD.","DOI":"10.1007\/978-3-7908-1869-7"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ijar.2016.07.010","article-title":"40 years of Dempster-Shafer theory","volume":"79","author":"Denoeux","year":"2016","journal-title":"Int. J. Approx. Reason."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Yager, R.R., and Liu, L. (2008). Classic Works of the Dempster-Shafer Theory of Belief Functions, Springer.","DOI":"10.1007\/978-3-540-44792-4"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Shafer, G. (1976). A Mathematical Theory of Evidence, Princeton University Press.","DOI":"10.1515\/9780691214696"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/0165-0114(78)90029-5","article-title":"Fuzzy Sets as the Basis for a Theory of Possibility","volume":"1","author":"Zadeh","year":"1978","journal-title":"Fuzzy Sets Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.csda.2006.04.029","article-title":"Generalized theory of uncertainty (GTU)\u2014principal concepts and ideas","volume":"51","author":"Zadeh","year":"2006","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.fss.2015.09.004","article-title":"The legacy of 50 years of fuzzy sets: A discussion","volume":"281","author":"Dubois","year":"2015","journal-title":"Fuzzy Sets Syst."},{"key":"ref_10","unstructured":"Duda, R.O., Hart, P.E., and Stork, D.G. (2012). Pattern Classification, John Wiley & Sons."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Solaiman, B., and Boss\u00e9, \u00c9. (2019). Possibility Theory for the Design of Information Fusion Systems, Springer.","DOI":"10.1007\/978-3-030-32853-5"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Fr\u00e9licot, C. (1998). On unifying probabilistic\/fuzzy and possibilistic rejection-based classifiers. Advances in Pattern Recognition, Springer.","DOI":"10.1007\/BFb0033298"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.inffus.2013.08.004","article-title":"A conceptual definition of a holonic processing framework to support the design of information fusion systems","volume":"21","author":"Solaiman","year":"2015","journal-title":"Inf. Fusion"},{"key":"ref_14","unstructured":"Tou, J.T., and Gonzalez, R.C. (1974). Pattern Recognition Principles, Addison-Wesley."},{"key":"ref_15","unstructured":"Dubois, D., and Prade, H. (1988). Possibility Theory: An Approach to Computerized Processing of Uncertainty, Plenum Press."},{"key":"ref_16","unstructured":"Dubois, D.J. (1980). Fuzzy Sets and Systems: Theory and Applications, Academic Press."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/0165-0114(92)90110-P","article-title":"When upper probabilities are possibility measures","volume":"49","author":"Dubois","year":"1992","journal-title":"Fuzzy Sets Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.ins.2011.11.027","article-title":"Fuzzy logic-based generalized decision theory with imperfect information","volume":"189","author":"Aliev","year":"2012","journal-title":"Inf. Sci."},{"key":"ref_19","first-page":"1","article-title":"How to Combine Probabilistic and Possibilistic (Expert) Knowledge: Uniqueness of Reconstruction in Yager\u2019s (Product) Approach","volume":"2","author":"Buntao","year":"2011","journal-title":"Int. J. Innov. Manag. Inf. Prod. (IJIMIP)"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.fss.2013.09.010","article-title":"Possibilistic and probabilistic likelihood functions and their extensions: Common features and specific characteristics","volume":"250","author":"Coletti","year":"2014","journal-title":"Fuzzy Sets Syst."},{"key":"ref_21","unstructured":"Fargier, H., Amor, N.B., and Guezguez, W. (2012). On the complexity of decision making in possibilistic decision trees. arXiv."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Guo, P. (2007). Possibilistic Decision-Making Approaches. The 2007 International Conference on Intelligent Systems and Knowledge Engineering, Atlantis Press.","DOI":"10.2991\/iske.2007.88"},{"key":"ref_23","unstructured":"Weng, P. (2012). Qualitative decision making under possibilistic uncertainty: Toward more discriminating criteria. arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/s10700-011-9098-1","article-title":"A measure based approach to the fusion of possibilistic and probabilistic uncertainty","volume":"10","author":"Yager","year":"2011","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Yager, R.R. (2011, January 11\u201315). On the fusion of possibilistic and probabilistic information in biometric decision-making. Proceedings of the 2011 IEEE Workshop on Computational Intelligence in Biometrics and Identity Management (CIBIM), Paris, France.","DOI":"10.1109\/CIBIM.2011.5949205"},{"key":"ref_26","unstructured":"Bouyssou, D., Dubois, D., Prade, H., and Pirlot, M. (2013). Decision Making Process: Concepts and Methods, John Wiley & Sons."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1023\/B:REOM.0000032115.22510.b5","article-title":"Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities","volume":"10","author":"Dubois","year":"2004","journal-title":"Reliab. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Dubois, D., and Prade, H. (1998). Possibility theory: Qualitative and quantitative aspects. Quantified Representation of Uncertainty and Imprecision, Springer.","DOI":"10.1007\/978-94-017-1735-9_6"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/S0165-0114(97)00080-8","article-title":"The three semantics of fuzzy sets","volume":"90","author":"Dubois","year":"1997","journal-title":"Fuzzy Sets Syst."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Dubois, D., Prade, H., and Sandri, S. (1993). On possibility\/probability transformations. Fuzzy Logic, Springer.","DOI":"10.1007\/978-94-011-2014-2_10"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/S0165-0114(98)00405-9","article-title":"Modeling vague beliefs using fuzzy-valued belief structures","volume":"116","year":"2000","journal-title":"Fuzzy Sets Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.fss.2011.05.022","article-title":"Maximum likelihood estimation from fuzzy data using the EM algorithm","volume":"183","year":"2011","journal-title":"Fuzzy Sets Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1109\/TKDE.2011.201","article-title":"Maximum likelihood estimation from uncertain data in the belief function framework","volume":"25","author":"Denoeux","year":"2013","journal-title":"Knowl. Data Eng."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Walley, P. (1991). Statistical Reasoning With Imprecise Probabilities, Chapman and Hall.","DOI":"10.1007\/978-1-4899-3472-7"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/S0888-613X(00)00031-1","article-title":"Towards a unified theory of imprecise probability","volume":"24","author":"Walley","year":"2000","journal-title":"Int. J. Approx. Reason."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Linda, O., Manic, M., Alves-Foss, J., and Vollmer, T. (2011, January 9\u201311). Towards resilient critical infrastructures: Application of Type-2 Fuzzy Logic in embedded network security cyber sensor. Proceedings of the 2011 4th International Symposium on Resilient Control Systems (ISRCS), Boise, ID, USA.","DOI":"10.1109\/ISRCS.2011.6016083"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1109\/91.995115","article-title":"Type-2 fuzzy sets made simple","volume":"10","author":"Mendel","year":"2002","journal-title":"Fuzzy Syst."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Ozen, T., and Garibaldi, J.M. (2004, January 25\u201329). Effect of type-2 fuzzy membership function shape on modelling variation in human decision making. Proceedings of the IEEE International Conference on Fuzzy Systems, Budapest, Hungary.","DOI":"10.1109\/FUZZY.2004.1375539"},{"key":"ref_39","unstructured":"Luce, R.D., and Raiffa, H. (1957). Games and Decisions, Wiley."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"3224","DOI":"10.1016\/j.fss.2009.01.009","article-title":"Na\u00efve possibilistic network classifiers","volume":"160","author":"Haouari","year":"2009","journal-title":"Fuzzy Sets Syst."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Benferhat, S., and Tabia, K. (2008). An efficient algorithm for naive possibilistic classifiers with uncertain inputs. Scalable Uncertainty Management, Springer.","DOI":"10.1007\/978-3-540-87993-0_7"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.fss.2013.07.012","article-title":"Naive possibilistic classifiers for imprecise or uncertain numerical data","volume":"239","author":"Bounhas","year":"2014","journal-title":"Fuzzy Sets Syst."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"733","DOI":"10.1007\/s00500-012-0947-9","article-title":"Possibilistic classifiers for numerical data","volume":"17","author":"Bounhas","year":"2013","journal-title":"Soft Comput."},{"key":"ref_44","unstructured":"Kikuchi, S., and Perincherry, V. (2020, December 30). Handling Uncertainty in Large Scale Systems with Certainty and Integrity. Available online: http:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.132.7069."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1109\/TFUZZ.2011.2159725","article-title":"Set measure directed multi-source information fusion","volume":"19","author":"Yager","year":"2011","journal-title":"Fuzzy Syst."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Yager, R.R. (2010, January 15\u201316). Hard and soft information fusion using measures. Proceedings of the 2010 IEEE International Conference on Intelligent Systems and Knowledge Engineering, Hangzhou, China.","DOI":"10.1109\/ISKE.2010.5680785"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/1\/67\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:06:31Z","timestamp":1760159191000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/1\/67"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,3]]},"references-count":46,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["e23010067"],"URL":"https:\/\/doi.org\/10.3390\/e23010067","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,3]]}}}