Uncertainty measurement with belief entropy on the interference effect in the quantum-like Bayesian networks. (English) Zbl 1429.91107

Summary: The Bayesian Network is a kind of probabilistic graphical models, having been applied to various fields for inference and learning. A quantum-like Bayesian Network has been proposed to model the prisoner’s dilemma, the famous example of social dilemma games. Recent findings reveal that people’s behaviors violate the Sure Thing Principle in such games. The quantum-like Bayesian network considers the people’s behaviors as wave functions and explains the violation as the interference effect. The determination of the interference effect is an essential part of modeling the Bayesian network.
In this paper, a belief entropy based uncertainty measurement is proposed to quantify the interference effect. Unlike most other methods, the measurement is a total unsupervised process, making the model predictive. On the other hand, the belief entropy based method provides a potential physical explanation of the interference effect. Tested with empirical data, the proposed method is proved to be precise to quantify the interference effect. The results further verify that the quantum-like Bayesian network is a delightful way to accommodate the paradoxical findings in social dilemma games.


91B06 Decision theory
81P15 Quantum measurement theory, state operations, state preparations
91B80 Applications of statistical and quantum mechanics to economics (econophysics)
81P45 Quantum information, communication, networks (quantum-theoretic aspects)
62B10 Statistical aspects of information-theoretic topics
Full Text: DOI arXiv


[1] Barros, J. A.D.; Suppes, P., Quantum mechanics, inference and the brain, J. Math. Psychol., 53, 5, 306-313 (2009) · Zbl 1178.81020
[2] Bhardwaj, K.; Marculescu, R., Non-stationary Bayesian learning for global sustainability, IEEE Trans. Sustain. Comput., 2, 3, 304-316 (2017)
[3] Bian, T.; Deng, Y., Identifying influential nodes in complex networks: a node information dimension approach, Chaos: Interdiscip. J. Nonlinear Sci., 28, 4, 043109 (2018) · Zbl 1390.90093
[4] Bian, T.; Zheng, H.; Yin, L.; Deng, Y., Failure mode and effects analysis based on D numbers and TOPSIS, Qual. Reliab. Eng. Int. (2018)
[5] Borunda, M.; Jaramillo, O.; Reyes, A.; Ibargüengoytia, P. H., Bayesian networks in renewable energy systems: a bibliographical survey, Renew. Sustain. Energy Rev., 62, 32-45 (2016)
[6] Bruza, P. D.; Wang, Z.; Busemeyer, J. R., Quantum cognition: a new theoretical approach to psychology, Trends Cognit. Sci., 19, 7, 383-393 (2015)
[7] Busemeyer, J. R.; Matthew, M. R.; Wang, Z., A quantum information processing explanation of disjunction effects, Proceedings of the 2006 Annual Meeting of the Cognitive Science Society, 28 (2006)
[8] Busemeyer, J. R.; Wang, Z.; Townsend, J. T., Quantum dynamics of human decision-making, J. Math. Psychol., 50, 3, 220-241 (2006) · Zbl 1186.91062
[9] de Campos, L. M.; Fernández-Luna, J. M.; Huete, J. F., Bayesian networks and information retrieval: an introduction to the special issue, Inf. Process. Manag., 40, 5, 727-733 (2004)
[10] Cinar, D.; Kayakutlu, G., Scenario analysis using Bayesian networks: a case study in energy sector, Knowl.-Based Syst., 23, 3, 267-276 (2010)
[11] Daim, T.; Kayakutlu, G.; Suharto, Y.; Bayram, Y., Clean energy investment scenarios using the Bayesian network, Int. J. Sustain. Energy, 33, 2, 400-415 (2014)
[12] D’Amato, D.; Droste, N.; Allen, B.; Kettunen, M.; Lähtinen, K.; Korhonen, J.; Leskinen, P.; Matthies, B.; Toppinen, A., Green, circular, bio economy: a comparative analysis of sustainability avenues, J. Clean. Prod., 168, 716-734 (2017)
[13] Deng, X.; Deng, Y., D-AHP method with different credibility of information, Soft Comput. (2018)
[14] Deng, X.; Jiang, W.; Zhang, J., Zero-sum matrix game with payoffs of Dempster-Shafer belief structures and its applications on sensors, Sensors, 17, 4 (2017)
[15] Deng, Y., Deng entropy, Chaos Solitons Fractals, 91, 549-553 (2016) · Zbl 1372.94368
[16] Dotzler, C.; Botzler, S.; Kierdorf, D.; Lang, W., Methods for optimising energy efficiency and renovation processes of complex public properties, Energy Build., 164, 254-265 (2018)
[17] Ellsberg, D.; Barro, R. J.; Helpman, E.; Katz, L. F.; Schleifer, A., Risk, ambiguity, and the savage axioms: reply, Q. J. Econ., 75, 4, 643-669 (1963)
[18] Friedman, N.; Linial, M.; Nachman, I.; Pe’er, D., Using Bayesian networks to analyze expression data, J. Comput. Biol., 7, 3-4, 601-620 (2000)
[19] Gambelli, D.; Alberti, F.; Solfanelli, F.; Vairo, D.; Zanoli, R., Third generation algae biofuels in italy by 2030: a scenario analysis using Bayesian networks, Energy Policy, 103, 165-178 (2017)
[20] He, Z.; Jiang, W., An evidential dynamical model to predict the interference effect of categorization on decision making results, Knowl.-Based Syst., 150, 139-149 (2018)
[21] Huang, Z.; Jiang, W.; Tang, Y., A new method to evaluate risk in failure mode and effects analysis under fuzzy information, Soft Comput., 22, 14, 4779-4787 (2017)
[22] Jiang, W.; Wei, B.; Tang, Y.; Zhou, D., Ordered visibility graph average aggregation operator: an application in produced water management, Chaos: Interdiscip. J. Nonlinear Sci., 27, 2 (2017)
[23] Jiang, W.; Wei, B.; Zhan, J.; Xie, C.; Zhou, D., A visibility graph power averaging aggregation operator: a methodology based on network analysis, Comput. Ind. Eng., 101, 260-268 (2016)
[24] Jiang, W.; Xie, C.; Luo, Y.; Tang, Y., Ranking z-numbers with an improved ranking method for generalized fuzzy numbers, J. Intell. Fuzzy Syst., 32, 3, 1931-1943 (2017) · Zbl 1375.03071
[25] Jiang, W.; Xie, C.; Zhuang, M.; Shou, Y.; Tang, Y., Sensor data fusion with Z-numbers and its application in fault diagnosis, Sensors, 16, 9 (2016)
[26] Jiang, W.; Xie, C.; Zhuang, M.; Tang, Y., Failure mode and effects analysis based on a novel fuzzy evidential method, Appl. Soft Comput., 57, 672-683 (2017)
[27] Jiang, W.; Zhan, J., A modified combination rule in generalized evidence theory, Appl. Intell., 46, 3, 630-640 (2017)
[28] Jiang, X.; Neapolitan, R. E.; Barmada, M. M.; Visweswaran, S., Learning genetic epistasis using Bayesian network scoring criteria, BMC Bioinform., 12, 1, 89 (2011)
[29] Kang, B.; Deng, Y.; Hewage, K.; Sadiq, R., Generating Z - number based on OWA weights using maximum entropy, Int. J. Intell. Syst., 33, 1745-1755 (2018)
[30] Khrennikov, A., Quantum-like model of cognitive decision making and information processing, Biosystems, 95, 3, 179-187 (2009)
[31] Khrennikov, A., Quantum Bayesianism as the basis of general theory of decision-making., Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 374, 2068 (2016) · Zbl 1353.94024
[32] Khrennikov, A. Y.; Haven, E., Quantum mechanics and violations of the sure-thing principle: the use of probability interference and other concepts, J. Math. Psychol., 53, 5, 378-388 (2009) · Zbl 1178.91043
[33] Koumenides, C. L.; Shadbolt, N. R., Combining link and content-based information in a Bayesian inference model for entity search, Proceedings of the First Joint International Workshop on Entity-Oriented and Semantic Search, 3 (2012), ACM
[34] Leicester, P. A.; Goodier, C. I.; Rowley, P., Using a Bayesian Network to Evaluate the Social, Economic and Environmental Impacts of Community Deployed Renewable Energy (2013), CISBAT
[35] Leifer, M. S.; Poulin, D., Quantum graphical models and belief propagation, Ann. Phys., 323, 8, 1899-1946 (2008) · Zbl 1146.81017
[36] Li, S.; Taplin, J., Examining whether there is a disjunction effect in Prisoner’s dilemma games, Chin. Psychol. Assoc. (2002)
[37] Liu, T.; Deng, Y.; Chan, F., Evidential supplier selection based on DEMATEL and game theory, Int. J. Fuzzy Syst., 20, 4, 1321-1333 (2018)
[38] Moreira, C.; Wichert, A., Quantum-like Bayesian networks for modeling decision making, Front. Psychol., 7 (2016)
[39] Moreira, C.; Wichert, A., Exploring the relations between quantum-like Bayesian networks and decision-making tasks with regard to face stimuli, J. Math. Psychol., 78, 86-95 (2017) · Zbl 1382.91034
[40] Moreira, C.; Wichert, A., Are quantum-like Bayesian networks more powerful than classical Bayesian networks?, J. Math. Psychol., 82, 73-83 (2018) · Zbl 1402.91112
[41] Myung, J. I.; Karabatsos, G.; Iverson, G. J., A Bayesian approach to testing decision making axioms., J. Math. Psychol., 49, 3, 205-225 (2005) · Zbl 1104.91016
[42] Russel, S. J.; Norvig, P., Artificial intelligence - a modern approach, Appl. Mech. Mater., 263, 5, 2829-2833 (2002)
[43] Shaikh, P. H.; Nor, N. B.M.; Nallagownden, P.; Elamvazuthi, I.; Ibrahim, T., A review on optimized control systems for building energy and comfort management of smart sustainable buildings, Renew. Sustain. Energy Rev., 34, 409-429 (2014)
[44] Song, T.-T.; Qin, S.-J.; Wen, Q.-Y.; Wang, Y.-K.; Jia, H.-Y., Finite-key security analyses on passive decoy-state QKD protocols with different unstable sources, Sci. Rep., 5, 15276 (2015)
[45] Takada, H. H.; Stern, J. M.; Costa, O. L.; Ribeiro, C. O., Classical-equivalent Bayesian portfolio optimization for electricity generation planning, Entropy, 20, 1, 42 (2018)
[46] Tanimoto, J.; Sagara, H., Relationship between dilemma occurrence and the existence of a weakly dominant strategy in a two-player symmetric game, Biosystems, 90, 1, 105 (2007)
[47] Tucci, R. R., Quantum Bayesian nets, Int. J. Mod. Phys. B, 9, 3, 295-337 (1997) · Zbl 1264.81026
[48] Tversky, A.; Kahneman, D., Judgment under uncertainty: heuristics and biases, Science, 185, 4157, 1124-1131 (1974)
[49] Tversky, A.; Koehler, D. J., Support theory: a nonextensional representation of subjective probability., Psychol. Rev., 101, 4, 547 (1994)
[50] Tversky, A.; Shafir, E., The disjunction effect in choice under uncertainty., Psychol. Sci., 3, 5, 305-309 (1992)
[51] Vourdas, A., Lower and upper probabilities in the distributive lattice of subsystems, J. Phys. A Math. Theor., 47, 34, 249-278 (2014) · Zbl 1309.81016
[52] Vourdas, A., Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces, J. Math. Phys., 55, 8, 823 (2014) · Zbl 1303.81041
[53] Vourdas, A., Mobius operators and non-additive quantum probabilities in the Birkhoff-von Neumann lattice, Physics, 101, 38-51 (2015) · Zbl 1331.60009
[54] Wang, Z.; Bauch, C. T.; Bhattacharyya, S.; dOnofrio, A.; Manfredi, P.; Perc, M.; Perra, N.; Salathé, M.; Zhao, D., Statistical physics of vaccination, Phys. Rep., 664, 1-113 (2016) · Zbl 1359.92111
[55] Wang, Z.; Jusup, M.; Wang, R. W.; Shi, L.; Iwasa, Y.; Moreno, Y.; Kurths, J., Onymity promotes cooperation in social dilemma experiments, Sci. Adv., 3, 3, e1601444 (2017)
[56] Wang, Z.; Kokubo, S.; Jusup, M.; Tanimoto, J., Universal scaling for the dilemma strength in evolutionary games, Phys. Life Rev., 14, 47-48 (2015)
[57] Xu, H.; Deng, Y., Dependent evidence combination based on Shearman coefficient and Pearson coefficient, IEEE Access, 6, 1, 11634-11640 (2018)
[58] Yin, L.; Deng, Y., Measuring transferring similarity via local information, Phys. A Stat. Mech. Appl., 498, 102-115 (2018)
[59] Yu, F.; Qiu, D.; Situ, H.; Wang, X.; Long, S., Enhancing user privacy in SARG04-based private database query protocols, Quant. Inf. Process., 14, 11, 4201-4210 (2015) · Zbl 1327.81177
[60] Yu, F.; Zhou, Q.; Lu, X.; Zhao, S., A first-order logic framework of major choosing decision making with an uncertain reasoning function, IEEE Trans. Syst. Man Cybern. Syst. (2018)
[61] Yu, Y.; Xiao, G.; Zhou, J.; Wang, Y.; Wang, Z.; Kurths, J.; Schellnhuber, H. J., System crash as dynamics of complex networks, Proc. Natl. Acad. Sci., 113, 42, 11726-11731 (2016) · Zbl 1355.92101
[62] Yukalov, V. I.; Sornette, D., Entanglement production in quantum decision making, Phys. Atom. Nuclei, 73, 3, 559-562 (2010)
[63] Yukalov, V. I.; Sornette, D., Mathematical Structure of Quantum Decision Theory, 13 (2010), Social Science Electronic Publishing · Zbl 1201.91040
[64] Yukalov, V. I.; Sornette, D., Decision theory with prospect interference and entanglement, Theory Decis., 70, 3, 283-328 (2011) · Zbl 1209.91063
[65] Yukalov, V. I.; Sornette, D., Quantum decision theory as quantum theory of measurement, Phys. Lett. A, 372, 46, 6867-6871 (2015) · Zbl 1227.81109
[66] Zachow, E. W., Positive-difference structures and bilinear utility functions, J. Math. Psychol., 17, 2, 152-164 (1978) · Zbl 0375.90004
[67] Zhang, Q.; Li, M.; Deng, Y., Measure the structure similarity of nodes in complex networks based on relative entropy, Phys. A Stat. Mech. Appl., 491, 749-763 (2018)
[68] Zhang, R.; Ashuri, B.; Deng, Y., A novel method for forecasting time series based on fuzzy logic and visibility graph, Adv. Data Anal. Classif., 11, 4, 759-783 (2017) · Zbl 1414.40004
[69] Zheng, H.; Deng, Y., Evaluation method based on fuzzy relations between Dempster-Shafer belief structure, Int. J. Intell. Syst. (2017)
[70] Zhou, X.; Hu, Y.; Deng, Y.; Chan, F. T.S.; Ishizaka, A., A DEMATEL-based completion method for incomplete pairwise comparison matrix in AHP, Ann. Oper. Res. (2018) · Zbl 1434.90075
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