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2024 | OriginalPaper | Buchkapitel

3. Information and Entropy

verfasst von : Eduardo Souza de Cursi

Erschienen in: Uncertainty Quantification with R

Verlag: Springer Nature Switzerland

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Abstract

This chapter presents the notions connected to Shannon’s entropy and information, namely the joint, conditional, relative (Kullback–Leibler) entropies, and the mutual information, with their implementations in R. Applications to surprise quantification are shown, with their implementation in R. Examples show the use of the programs presented.

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Vorheriges Kapitel Beliefs

Nächstes Kapitel Maximum Entropy

Ash, R. (1965, reedited 2003). Information Theory. New York, NY, USA: Dover.

Attiaoui, D., Doré, P., Martin, A., & Ben Yaghlane, B. (2012). A Distance between Continuous Belief Functions. In E. Hüllermeier, S. Link, T. Fober, & B. Seeger (Ed.), Scalable Uncertainty Management: SUM 2012 (pp. 194–205). Marburg, Germany: Springer. doi:https://doi.org/10.1007/978-3-642-33362-0_15CrossRef

Aurnhammer, C., & Frank, S. L. (2019). Evaluating information-theoretic measures of word prediction in naturalistic sentence reading. Neuropsychologia, 134. doi:https://doi.org/10.1016/j.neuropsychologia.2019.107198

Baldi, P., & Itti, L. (2010). Of bits and wows: A Bayesian theory of surprise with applications to attention. Neural Networks, 23(5), 649–666. doi:https://doi.org/10.1016/j.neunet.2009.12.007CrossRef

Bayarria, M. J., & Morales, J. (2003). Bayesian measures of surprise for outlier detection. Journal of Statistical Planning and Inference, 111(1–2), 3–22. doi:https://doi.org/10.1016/S0378-3758(02)00282-3CrossRef

Benavoli, A. (2014). Belief function and multivalued mapping robustness in statistical estimation. International Journal of Approximate Reasoning, 55, 311–329. doi:https://doi.org/10.1016/j.ijar.2013.04.014CrossRef

Beretta, G. P. (2008). Axiomatic Definition of Entropy for Nonequilibrium States. International Journal of Thermodynamics, 11(2), 39–48. doi:https://doi.org/10.5541/ijot.211CrossRef

Boivin, C. (2022a, December 25). https://cran.r-project.org/web/packages/dst/vignettes/Captain_Example.html. Retrieved from Captain’s Example: https://cran.r-project.org/web/packages/dst/vignettes/Captain_Example.html

Boivin, C. (2022b, 12 24). Introduction to Belief Functions. Retrieved from https://cran.r-project.org/web/packages/dst/vignettes/: https://cran.r-project.org/web/packages/dst/vignettes/Introduction_to_Belief_Functions.html#fn2

Boivin, C. (2022c, December 25). Introduction to Belief Functions: The Monty Hall Game. Retrieved 2022, from https://cran.microsoft.com/snapshot/2018-08-11/web/packages/dst/vignettes/Monty-hall-Example.html: https://cran.microsoft.com/snapshot/2018-08-11/web/packages/dst/vignettes/Monty-hall-Example.html

Boltzmann, L. E. (1866). Über die mechanische Bedeutung des zweiten Hauptsatzes der Wärmetheorie. Wiener Berichte, 53, 195–220. Retrieved from http://opacplus.bsb-muenchen.de/title/BV020135572/ft/bsb10133426?page=3

Boltzmann, L. E. (1877 traduction 2002). On the relationship between the second main theorem of mechanical heat theory and the probability calculation with respect to the results about the heat equilibrium. Akademie der Wissenschaften in Wien Mathematisch-naturwissenschaftliche Klasse Sitzungsberichte, 2(76), 373–435. Retrieved February 14, 2023, from http://users.polytech.unice.fr/~leroux/boltztrad.pdf

Boltzmann, L. E. (1896). Vorlesungen ̈uber Gastheorie (Vol. 1). Leipzig: Barth.

Boltzmann, L. E. (1995). Lectures on Gas Theory. (S. G. Brush, Trans.) New York, NY, USA: Dover.

Boyle, G. H. (n.d.). modelingcommons.org/file/download/6101?file_id=3384. Retrieved from http://modelingcommons.org/file/download/6101?file_id=3384

Brillouin, L. (1956). Science and Information Theory. New York, NY, USA: Dover.CrossRef

Campagner, A., Ciucci, D., & Denœux, T. (2022). Belief Functions and Rough Sets: Suvey and New Insights. International Journal of Approximate Reasoning, 143, 92–215. doi:https://doi.org/10.1016/j.ijar.2022.01.011CrossRef

Chakrabarti, C. G., & Indranil, C. (2005). Shannon entropy: axiomatic characterization and application. International Journal of Mathematics and Mathematical Sciences. doi:https://doi.org/10.1155/IJMMS.2005.2847

Cheung, V. K., M.C., P., Meyer, L., Pearce, M. T., Haynes, J.-D., & Koelsch, S. (2019). Uncertainty and Surprise Jointly Predict Musical Pleasure and Amygdala, Hippocampus, and Auditory Cortex Activity. Current Biology, 29(23), 4084–4092.e4. doi:https://doi.org/10.1016/j.cub.2019.09.067

Clausius, R. J. (1850). Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen. Annalen der Physik, 368–397 , 500–524.CrossRef

Clausius, R. J. (1851 republished in 2009). On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(8), 1–21 , 102–119. doi:https://doi.org/10.1080/14786445108646819 , https://doi.org/10.1080/14786445108646840

Cobb, B. R., & Shenoy, P. (2006, April). On the plausibility transformation method for translating belief function models to probability models. Journal of Approximate Reasoning, 41(3), pp. 314–330. doi:https://doi.org/10.1016/j.ijar.2005.06.008

Couso, I., Dubois, D., & Sanchez, L. (2014). Random Sets and Random Fuzzy Sets as Ill-Perceived Random Variables. Springer. doi:https://doi.org/10.1007/978-3-319-08611-8

Csiszár, I. (2008). Axiomatic Characterizations of Information Measures. Entropy, 10(3), 261–273. doi:https://doi.org/10.3390/e10030261CrossRef

Dale, A. I. (1982). Bayes or Laplace? An Examination of the Origin and Early Applications of Bayes’ Theorem. Archive for History of Exact Sciences, 27(1), pp. 23–47.CrossRef

Dale, A. I. (1999). A History of Inverse Probability—From Thomas Bayes to Karl Pearson (2nd. ed.). New York: Springer.CrossRef

De Finetti, B. (2017). Theory of Probability—A Critical Introductory Treatment. (A. M. Smith, Trans.) UK: John Wiley and Sons.

DeGroot, M. H. (2004). Optimal Statistical Decisions. New Jersey: John Wiley & Sons.CrossRef

Dempster, A. (1967, April). Upper and Lower Probabilities Induced by a Multivalued Mapping. Ann. Math. Statist., 38(2), pp. 325–339,. doi:https://doi.org/10.1214/aoms/1177698950

Denoeux, T. (2019, June). Decision-making with belief functions: A review. International Journal of Approximate Reasoning, 109, 87–110. doi:https://doi.org/10.1016/j.ijar.2019.03.009

Denoeux, T. (2021). Uncertainty Analysis using Belief Functions: Applications to Statistical Inference and Pattern Recognition. Université de technologie de Compiègne, Department of Computer Science, Compiègne, France. Retrieved December 20, 2022, from https://www.hds.utc.fr/~tdenoeux/dokuwiki/_media/en/book_bf.pdf

Denœux, T., Younes, Z., & Abdallah, F. (2010). Representing uncertainty on set-valued variables using belief functions. Artificial Intelligence, 174(7–8), 479–499. doi:https://doi.org/10.1016/j.artint.2010.02.002CrossRef

Dezert, J., & Smarandache, F. (2008). A new probabilistic transformation of belief mass assignment. Fusion 2008: International Conference on Information Fusion, (pp. 1410–1417). Retrieved December 19, 2022, from https://hal.archives-ouvertes.fr/hal-00304319/document

Dezert, J., & Smarandache, F. (2009). Transformations of belief masses into subjective probabilities. In J. Dezert, & F. Smarandache (Eds.), Advances and Applications of DSmT for Information Fusion (pp. 85–136). Rehoboth: American Research Press . Retrieved December 19, 2022, from https://www.researchgate.net/publication/306413841_Transformations_of_belief_masses_into_subjective_probabilities

Dezert, J., Smarandache, F., & Daniel, M. (2004). A Generalized Pignistic Transformation. In J. Dezert, & F. Smarandache (Eds.), Advances and Applications of DSmT for Information Fusion (pp. 143–153). Rehoboth: American Research Press. Retrieved December 19, 2022, from https://www.onera.fr/sites/default/files/297/C023-Dezert-Fusion2004Stockholm.pdf

Doré, P. E., Fiche, A., & Martin, A. (2010). Models of belief functions—Impacts for patterns recognitions. 13th International Conference on Information Fusion. Edinburgh, UK. doi:https://doi.org/10.1109/ICIF.2010.5711936

Doré, P., Martin, A., Abi-Zeid, I., Jousselme, A., & Maupin, P. (2011a, January). Belief functions induced by multimodalprobability density functions, an application to the search and rescue problem. RAIRO—Operations Research, 44(4), 323–343. doi:https://doi.org/10.1051/ro/2011001

Doré, P., Osswald, C., Martin, A., Jousselme, A., & Maupin, P. (2011b). Continuous belief functions to qualify sensors performances. In W. Liu (Ed.), Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2011 (pp. 350–361). Belfast, Ireland: Springer. doi:https://doi.org/10.1007/978-3-642-22152-1_30CrossRef

Dragulescu, A., & Yakovenko, V. (2000). Statistical mechanics of money. Eur. Phys. J. B, 17, 723–729. doi:https://doi.org/10.1007/s100510070114CrossRef

Dubois, D., & Prade, H. (1986). A Set-Theoretic View of Belief Functions. International Journal of General Systems, 12(3), 193–226. doi:https://doi.org/10.1080/03081078608934937CrossRef

Fetz, T., & Oberguggenberger, M. (2016, November). Imprecise random variables, random sets, and Monte Carlo simulation. 78, 252–264. doi:https://doi.org/10.1016/j.ijar.2016.06.012

Fiche, A., Martin, A., Cexus, J.-C., & Khenchaf, A. (2010). Continuous belief functions and α-stable distributions. 13th Conference on Information Fusion. Edinburgh. doi:https://doi.org/10.1109/ICIF.2010.5711934

Fort, H. (2022). Forecasting with Maximum Entropy. IOP Publishing Ltd.CrossRef

Fox, C. R., & See, K. E. (2006). Belief and Preference in Decision Under Uncertainty. In D. Hardman, & L. Macchi (Eds.), Thinking: Psychological Perspectives on Reasoning, Judgment and Decision Making. John Wiley & Sons. doi:https://doi.org/10.1002/047001332X.ch14CrossRef

Gibbs, J. W. (1901 reprinted 1960). Elementary Principles of Statistical Mechanics. Dover.

Grassi, P. R., & Bartels, A. (2021). Magic, Bayes and wows: A Bayesian account of magic tricks. Neuroscience & Biobehavioral Reviews, 126, 515–527. doi:https://doi.org/10.1016/j.neubiorev.2021.04.001CrossRef

Guan, J., & Bell, D. (1993). Discounting and Combination Operations in Evidential Reasoning. In D. Heckerman, & A. Mamdani (Ed.), Ninth Conference on Uncertainty in Artificial Intelligence (pp. 477–484). Washington, USA: Elsevier. doi:https://doi.org/10.1016/B978-1-4832-1451-1.50062-7CrossRef

Halmos, P. (1978). Measure theory. Berlin, Heidelberg, New York: : Springer Verlag.

Halpern, J., & Fagin, R. (1992). Two views of belief: belief as generalized probability and belief as evidence. Artificial Intelligence,, 54, pp. 275–317.CrossRef

Hartley, R. (1928, July). Transmission of Information. Bell System Technical Journal, 7(3), 535–563.

He, Y. (2013). Uncertainty Quantification and Data Fusion based on Dempster-Shafer Theory. Florida State University, Mathematics. Florida State University Libraries.

Howson, C., & Urbach, P. (2006). Scientific reasoning : the Bayesian approach . Chicago: Open Court.

Hüllermeier, E., Kruse, R., & Hoffmann, F. (Eds.). (2010). Consonant continuous belief functions conflicts calculation. IPMU’10: Proceedings of the Computational intelligence for knowledge-based systems design, and 13th international conference on Information processing and management of uncertainty (pp. 706–715). Dortmund, Germany: Springer.

Hulse, A., Schumacher, B., & Westmoreland, M. D. (2018). Axiomatic Information Thermodynamics. Entropy, 20(4). doi:https://doi.org/10.3390/e20040237

Itti, L., & Baldi, P. (2009). Bayesian surprise attracts human attention. Vision Research, 49(10), 1295–1306. doi:https://doi.org/10.1016/j.visres.2008.09.007CrossRef

Jaffray, J.-Y., & Wakker, P. (1993). Decision making with belief functions: Compatibility and incompatibility with the sure-thing principle. Journal of Risk and Uncertainty, 7(3), 255–271. doi:https://doi.org/10.1007/BF01079626CrossRef

Jaynes, E. (1989). Clearing up Mysteries—The Original Goal. (J. Skilling, Ed.) Dordrecht: Springer. doi:https://doi.org/10.1007/978-94-015-7860-8_1

Jaynes, E. (2003). Probability Theory: The Logic of Science. Cambridge: Cambridge University Press.CrossRef

Jaynes, E. T. (1965). Gibbs vs. Boltzmann Entropies. American Journal of Physics, 33(5), 391–398.

Jeffreys, H. (1939). Theory of probability. Oxford: University Press.

Khinchin, A. Y. (1957). Mathematical Foundations of Information Theory. New York, NY, USA: Dover.

Klopotek, M. A., & Wierzchon, S. T. (1998). A New Qualitative Rough-Set Approach to Modeling Belief Functions. In L. Polkowski, & A. Skowron (Ed.), Rough Sets and Current Trends in Computing, First International Conference, RSCTC’98 (pp. 346–354). Warzsaw, Poland: Springer. doi:https://doi.org/10.1007/3-540-69115-4_47CrossRef

Kojadinovic, I., Marichal, J.-L., & Roubens, M. (2005). An axiomatic approach to the definition of the entropy of a discrete Choquet capacity. Information Sciences, 172(1–2), 131–153. doi:https://doi.org/10.1016/j.ins.2004.05.011CrossRef

Kolossa, A., Kopp, B., & Fingscheidt, T. (2015). A computational analysis of the neural bases of Bayesian inference. NeuroImage, 106, 222–237. doi:https://doi.org/10.1016/j.neuroimage.2014.11.00CrossRef

Kullback, S. (1951 reed. 1969 reprint 1979). Information Theory and Statistics. New York: Wiley, reed. Dover.

Kullback, S., & Leibler, R. A. (1951, March). On Information and Sufficiency. The Annals of Mathematical Statistics, 22(1), 79–86. doi:https://doi.org/10.1214/aoms/1177729694

Laghmara, H., Laurain, T., Cudel, C., & Lauffenburger, J. P. (2020). Heterogeneous sensor data fusion for multiple object association using belief functions. Information Fusion, 57, 44–58. doi:https://doi.org/10.1016/j.inffus.2019.11.002CrossRef

Laplace, P.-S. (1774). Memoire sur la probabilité des causes par les événements. Memoires de Mathématique et de Physique, Presentés à l’Académie Royale des Sciences par divers Savans & lus dans ses Assemblées, pp. 621–656.

Laplace, P.-S. (1891). Oeuvres complètes (Vol. 8). (A. d. Paris, Ed.) Paris: Gauthiers-Villars.

Laplace, P.-S. (1986). Memoir on the Probability of the Causes of Events. Statistical Science, 1(3), pp. 364–78. Retrieved from http://www.jstor.org/stable/2245476

Lazo, A. V., & Rathie, P. (1978). On the entropy of continuous probability distributions (Corresp.). IEEE Transactions on Information Theory, 24(1), 120–122. doi:https://doi.org/10.1109/tit.1978.1055832CrossRef

Lee, J., Fan, Y., & Sisson, S. (2015). Bayesian threshold selection for extremal models using measures of surprise. Computational Statistics and Data Analysi, 85, 84–99. doi:https://doi.org/10.1016/j.csda.2014.12.004CrossRef

Lian, C. (2017). Information Fusion and Decision Making using Belief Functions. Compiègne, France: Université de Technologie de Compiègne.

Liboff, R. L. (1974). Gibbs vs. Shannon entropies. J Stat Phys, 11, 343–357. doi:https://doi.org/10.1007/BF01009793

Liu, W. (2006, August). Analyzing the degree of conflict among belief functions. Artificial Intelligence, 170 (11), 909–924. doi:https://doi.org/10.1016/j.artint.2006.05.002

Martin, A. (2019). Conflict management in information fusion with belief functions. In E. Bossé, & G. Rogova (Eds.), Information quality in information fusion and decision making (pp. 79–97). Springer. doi:https://doi.org/10.1007/978-3-030-03643-0_4CrossRef

Martin, R., Zhang, J., & Liu, C. (2010, June). Dempster–Shafer Theory and Statistical Inference with Weak Beliefs. Statistical Science, 25(1), 72–87. doi:https://doi.org/10.1214/10-STS322

Mercier, D., Quost, B., & Denœux, T. (2005). Contextual Discounting of Belief Functions. In L. Godo (Ed.), ECSQARU 2005: Symbolic and Quantitative Approaches to Reasoning with Uncertainty (pp. 552–562). Barcelona, Spain: Springer. doi:https://doi.org/10.1007/11518655_47CrossRef

Miranda, E., Couso, I., & Gil, P. (2005, July). Random sets as imprecise random variables. Journal of Mathematical Analysis and Applications, 307(1), 32–47. doi:https://doi.org/10.1016/j.jmaa.2004.10.022

Modirshanechi, A., Brea, J., & Gerstner, W. (2022). A taxonomy of surprise definitions. Journal of Mathematical Psychology, 110. doi:https://doi.org/10.1016/j.jmp.2022.102712

Nambiar, K. K., Varma, P. K., & Saroch, V. (1992). An axiomatic definition of Shannon’s entropy. Appl. Math. Lett., 5(4), 45–46. doi:https://doi.org/10.1016/0893-9659(92)90084-MCrossRef

Nguyen, H. T. (1977). On Random sets and Belief Functions. University of California, Berkeley, EECS Department. Retrieved from http://www2.eecs.berkeley.edu/Pubs/TechRpts/1977/28879.html

Nguyen, H. T. (1978). On Random Sets and Belief Functions. Journal of Mathematical Analysis and Applications, 65, 531–542. doi:https://doi.org/10.1016/0022-247X(78)90161-0CrossRef

Nguyen, H., & Wang, T. (1997). Belief Functions and Random Sets. In J. Goutsias, R. Mahler, & H. Nguyen (Eds.), Random Sets (pp. 243–255). Springer. doi:https://doi.org/10.1007/978-1-4612-1942-2_11CrossRef

Ni, S., Lei, Y., & Tang, Y. (2020). Improved Base Belief Function-Based Conflict Data Fusion Approach Considering Belief Entropy in the Evidence Theory. Entropy, 22(8). doi:https://doi.org/10.3390/e22080801

Ostwald, D., Spitzer, B., Guggenmos, M., Schmidt, T. T., Kiebel, S. J., & Blankenburg, F. (2012). Evidence for neural encoding of Bayesian surprise in human somatosensation. NeuroImage, 62(1), 177–188. doi:https://doi.org/10.1016/j.neuroimage.2012.04.050CrossRef

Palm, G. (2023). Novelty, Information and Surprise. Germany: Springer-Verlag. doi:https://doi.org/10.1007/978-3-662-65875-8CrossRef

Planck, M. K. (1901). Über das Gesetz der Energieverteilung im Normalspektrum. Annalen der Physik, 309(3), 553–563. doi:https://doi.org/10.1002/andp.19013090310CrossRef

Planck, M. K. (1914). The Theory of Heat Radiation. (M. Masius, Trans.) Philadelphia: P. Blakiston’s Son & Co.

Press, S. J., & Tanur, J. M. (2001). The Subjectivity of Scientists and the Bayesian Approach. New York: John Wiley & Sons.CrossRef

Quiroga-Martinez, D., Hansen, N., Højlund, A., Pearce, M., Brattico, E., & Vuust, P. (2020). Decomposing neural responses to melodic surprise in musicians and non-musicians: Evidence for a hierarchy of predictions in the auditory system. NeuroImage. doi:https://doi.org/10.1016/j.neuroimage.2020.116816

Rényi, A. (1961). On Measures of Entropy and Information. In J. Neyman (Ed.), 4th Berkeley Symposium on Mathematics, Statistics and Probability,. 1, pp. 547–561. University of California Press.

Risti, B., & Smets, P. (2006). Belief function theory on the continuous space with an application to model based classification. In B. Bouchon-Meunier, G. Coletti, & R. R. Yager (Ed.), Modern Information Processing: From Theory to Applications. IPMU’04 (pp. 11–24). Perugia, Italy.: Elsevier. doi:https://doi.org/10.1016/B978-044452075-3/50002-9CrossRef

Saravanan, R., & Levine, R. (2022). Surprisal analysis of diffusion processes. Chemical Physics, 556. doi:https://doi.org/10.1016/j.chemphys.2022.111450

Savchuk, V. P., & Tsokos, C. P. (2011). Bayesian Theory and Methods with Applications. Paris: Atlantis Press.CrossRef

Shafer, G. (1976). A Mathematical Theory of Evidence . New Jersey: Princeton University Press.CrossRef

Shafer, G. (1990). Perspectives on the Theory and Practice of Belief Functions. International Journal of Approximate Reasoning, 4(5–6), 323–362. doi:https://doi.org/10.1016/0888-613X(90)90012-QCrossRef

Shannon, C. E. (1948a). A Mathematical Theory of Communication. The Bell System Technical Journal, 27(3), 379–423. doi:https://doi.org/10.1002/j.1538-7305.1948.tb01338.xCrossRef

Shannon, C. E. (1948b). A Mathematical Theory of Communication. Bell System Technical Journal, 27(4), 623–666. doi:https://doi.org/10.1002/j.1538-7305.1948.tb00917.xCrossRef

Shannon, C. E. (n.d.). A mathematical theory of comunication—Nokia Bell Labs. Retrieved 2 13, 2023, from https://www.bell-labs.com/claude-shannon/assets/images/discoveries: https://www.bell-labs.com/claude-shannon/assets/images/discoveries/1948-04-21-a-mathematical-theory-of-communication-parts-I-and-carousel-01.pdf

Smets, P. (1990). Constructing the Pignistic Probability Function in a Context of Uncertainty. Machine Intelligence and Pattern Recognition, 10, 29–39. doi:https://doi.org/10.1016/B978-0-444-88738-2.50010-5

Smets, P. (2000). Data fusion in the transferable belief model. Proceedings of the Third International Conference on Information Fusion. 1, pp. 21–33. Paris, France: IEEE. doi:https://doi.org/10.1109/IFIC.2000.862713.

Smets, P. (2005). Belief functions on real numbers. International Journal of Approximate Reasoning, 40, 181–223. doi:https://doi.org/10.1016/j.ijar.2005.04.001CrossRef

Sohrab, S. H. (2014). Boltzmann entropy of thermodynamics versus Shannon entropy of information theory. International Journal of Mechanics, 8, 73–84. Retrieved February 15, 2023, from https://www.naun.org/main/NAUN/mechanics/2014/a182003-086.pdf

Souza de Cursi, E. (2023). Uncertainty Quantification using R. Springer Cham.CrossRef

Stigler, S. M. (1982). Thomas Bayes’s Bayesian Inference. Journal of the Royal Statistical Society. Series A (General), 145(2), pp. .250–258.

Strat, T. (1984). Continuous belief functions for evidential reasoning. Proceedings of the 4th National Conference on Artificial Intelligence. Austin, Texas. Retrieved December 18, 2022, from https://www.aaai.org/Papers/AAAI/1984/AAAI84-035.pdf

Strat, T. (1987). The Generation of Explanations within Evidential Reasoning Systems. In J. P. McDermott (Ed.), 10th. International Joint Conference on Artificial Intelligence (IJCAI), (pp. 1097–1104). Milan, Italy. Retrieved December 20, 2022, from https://www.ijcai.org/Proceedings/87-2/Papers/104.pdf

Strat, T. (1990). Decision analysis using belief functions. International Journal of Approximate Reasoning, 4(5–6), 391–417. doi:https://doi.org/10.1016/0888-613X(90)90014-SCrossRef

Taillandier, P., & Therond, O. (2011). Use of the Belief Theory to formalize Agent DecisionMaking Processes : Application to cropping Plan Decision Making. European Simulation and Modelling Conference, (pp. 138–142). Guimaraes, Portugal. Retrieved 12 17, 2022, from https://hal.archives-ouvertes.fr/hal-00688405

Turing, A. M. (1941). The Applications of Probability to Cryptography. Bletchley Park. Available at Archive.org and https://www.nationalarchives.gov.uk/. Retrieved from https://archive.org/details/hw-25-37

Turing, A. M. (2015, May 26). The Applications of Probability to Cryptography. doi:https://doi.org/10.48550/arXiv.1505.04714

Wasserman, L. A. (1990, September). Belief functions and statistical inference. Canadian Journal of Statistics, 18(3), 183–196. doi:https://doi.org/10.2307/3315449

Wu, W.-Z., & Mi, J.-S. (2008). An Interpretation of Belief Functions on Infinite Universes in the Theory of Rough Sets. In C. Chan, J. Grzymala-Busse, & W. P. Ziarko (Ed.), Rough Sets and Current Trends in Computing. RSCTC 2008 (pp. 71–80). Akron, OH, USA: Springer. doi:https://doi.org/10.1007/978-3-540-88425-5_8CrossRef

Yakovenko, V. M. (2010). Statistical Mechanics of Money, Debt, and Energy Comsumption. Science and Culture, 76(9–10), 430–436. doi:https://doi.org/10.48550/arXiv.1008.2179CrossRef

Yakovenko, V. M., & Rosser, J. B. (2009, December 2). Statistical mechanics of money, wealth, and income. Rev. Mod. Phys., 81(4), 1703–1725. doi:https://doi.org/10.1103/RevModPhys.81.1703

Yao, Y. Y., & Lingras, P. J. (1998). Interpretations of Belief Functions in the Theory of Rough Sets. Information Sciences, 104(1–2), 81–106. doi:https://doi.org/10.1016/S0020-0255(97)00076-5CrossRef

Zanchini, E., & Beretta, G. P. (2008). Rigorous Axiomatic Definition of Entropy Valid Also for Non-Equilibrium States. In G. P. Beretta, A. Ghoniem, & G. Hatsopoulos (Ed.), MEETING THE ENTROPY CHALLENGE: An International Thermodynamics Symposium in Honor and Memory of Professor Joseph H. Keenan. 1033. Cambridge, MA, USA: AIP Conference Proceedings. doi:https://doi.org/10.1063/1.2979048

Zhang, H., & Deng, Y. (2020). Weighted belief function of sensor data fusion in engine fault diagnosis. Soft Comput, 24, 2329–2339. doi:https://doi.org/10.1007/s00500-019-04063-7CrossRef

Zhou, K., Martin, A., & Pan, Q. (2018). A belief combination rule for a large number of sources. .Journal of Advances in Information Fusion, 13(2). Retrieved December 2022, 20, from https://www.researchgate.net/deref/https%3A%2F%2Fhal.archives-ouvertes.fr%2Fhal-01883239

Titel: Information and Entropy
verfasst von: Eduardo Souza de Cursi
Verlag: Springer Nature Switzerland
Buch: Uncertainty Quantification with R
Print ISBN: 978-3-031-48207-6

Electronic ISBN: 978-3-031-48208-3

Copyright-Jahr: 2024
DOI: https://doi.org/10.1007/978-3-031-48208-3_3

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