Normal cones corresponding to credal sets of lower probabilities. (English) Zbl 07611305
Summary: Credal sets are one of the most important models for describing probabilistic uncertainty. They usually arise as convex sets of probabilistic models compatible with judgments provided in terms of coherent lower previsions or more specific models such as coherent lower probabilities or probability intervals. In finite spaces, credal sets usually take the form of convex polytopes. Many properties of convex polytopes can be derived from their normal cones, which form polyhedral complexes called normal fans. We analyze the properties of normal cones corresponding to credal sets of coherent lower probabilities. For two important classes of coherent lower probabilities, 2-monotone lower probabilities and probability intervals, we provide a detailed description of the normal fan structure. These structures are related to the structure of the extreme points of the credal sets. To arrive at our main results, we provide some general results on triangulated normal fans of convex polyhedra and their adjacency structure.
MSC:
| 68T37 | Reasoning under uncertainty in the context of artificial intelligence |
Keywords:
normal cone; credal set; convex polyhedron; extreme point; imprecise probability; coherent lower probabilityReferences:
| [1] | Abellan, J.; Mantas, C. J.; Castellano, J. G.; Moral-Garcia, S., Increasing diversity in random forest learning algorithm via imprecise probabilities, Expert Syst. Appl., 97, 228-243 (2018) |
| [2] | Antonucci, A.; Cuzzolin, F., Credal sets approximation by lower probabilities: application to credal networks, (International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (2010), Springer), 716-725 |
| [3] | Augustin, T.; Coolen, F. P.; de Cooman, G.; Troffaes, M. C., Introduction to Imprecise Probabilities (2014), John Wiley & Sons · Zbl 1290.62003 |
| [4] | Bradley, S., Imprecise probabilities, (Computer Simulation Validation (2019), Springer), 525-540 |
| [5] | Bronevich, A.; Augustin, T., Approximation of coherent lower probabilities by 2-monotone measures, (Augustin, T.; Coolen, F. P.A.; Moral, S.; Troffaes, M. C.M., ISIPTA’09: Proceedings of the Sixth International Symposium on Imprecise Probability: Theories and Applications. ISIPTA’09: Proceedings of the Sixth International Symposium on Imprecise Probability: Theories and Applications, SIPTA, Durham, UK (July 2009)), 61-70 |
| [6] | Bruns, W.; Gubeladze, J., Polytopes, Rings, and K-Theory (2009), Springer · Zbl 1168.13001 |
| [7] | Campos, L. D.; Huete, J.; Moral, S., Probability intervals: a tool for uncertain reasoning, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 2, 2 (1994) · Zbl 1232.68153 |
| [8] | Chateauneuf, A.; Jaffray, J.-Y., Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion, Math. Soc. Sci., 17, 3, 263-283 (1989) · Zbl 0669.90003 |
| [9] | Coolen, F., On the use of imprecise probabilities in reliability, Qual. Reliab. Eng. Int., 20, 3, 193-202 (2004) |
| [10] | Cuzzolin, F., The geometry of consonant belief functions: simplicial complexes of necessity measures, Fuzzy Sets Syst., 161, 10, 1459-1479 (2010) · Zbl 1207.68385 |
| [11] | Cuzzolin, F., The Geometry of Uncertainty: The Geometry of Imprecise Probabilities (2020), Springer Nature |
| [12] | De Bock, J., The limit behaviour of imprecise continuous-time Markov chains, J. Nonlinear Sci., 27, 1, 159-196 (2017) · Zbl 1373.60131 |
| [13] | de Cooman, G.; Hermans, F.; Quaeghebeur, E., Imprecise Markov chains and their limit behavior, Probab. Eng. Inf. Sci., 23, 4, 597-635 (2009) · Zbl 1183.60026 |
| [14] | De Loera, J.; Rambau, J.; Santos, F., Triangulations: Structures for Algorithms and Applications, vol. 25 (2010), Springer Science & Business Media · Zbl 1207.52002 |
| [15] | Dempster, A. P., Upper and lower probabilities induced by a multivalued mapping, (Classic Works of the Dempster-Shafer Theory of Belief Functions (2008), Springer), 57-72 |
| [16] | Denneberg, D., Non-additive Measure and Integral, vol. 27 (1994), Springer Science & Business Media · Zbl 0826.28002 |
| [17] | Dolžan, D.; Bukovšek, D. K.; Omladič, M.; Škulj, D., Some multivariate imprecise shock model copulas, Fuzzy Sets Syst., 428, 34-57 (2022) · Zbl 1522.62029 |
| [18] | Erreygers, A.; Rottondi, C.; Verticale, G.; De Bock, J., Imprecise Markov models for scalable and robust performance evaluation of flexi-grid spectrum allocation policies, IEEE Trans. Commun., 66, 11, 5401-5414 (2018) |
| [19] | Gruber, P., Convex and Discrete Geometry (2007), Springer-Verlag Berlin Heidelberg · Zbl 1139.52001 |
| [20] | Grünbaum, B.; Kaibel, V.; Klee, V.; Ziegler, G., Convex Polytopes, Graduate Texts in Mathematics (2003), Springer · Zbl 1024.52001 |
| [21] | Jansen, C.; Schollmeyer, G.; Augustin, T., Concepts for decision making under severe uncertainty with partial ordinal and partial cardinal preferences, Int. J. Approx. Reason., 98, 112-131 (2018) · Zbl 1451.91040 |
| [22] | Jensen, A. N., A non-regular grobner fan, Discrete Comput. Geom., 37, 3, 443-453 (2007) · Zbl 1109.13023 |
| [23] | Khakzad, N., System safety assessment under epistemic uncertainty: using imprecise probabilities in Bayesian network, Saf. Sci., 116, 149-160 (2019) |
| [24] | Krak, T.; De Bock, J.; Siebes, A., Imprecise continuous-time Markov chains, Int. J. Approx. Reason., 88, 452-528 (2017) · Zbl 1427.60164 |
| [25] | Lu, S.; Robinson, S. M., Normal fans of polyhedral convex sets, Set-Valued Anal., 16, 2, 281-305 (2008) · Zbl 1144.52008 |
| [26] | Miranda, E., A survey of the theory of coherent lower previsions, Int. J. Approx. Reason., 48, 2, 628-658 (2008), In Memory of Philippe Smets (1938-2005) · Zbl 1184.68519 |
| [27] | Miranda, E.; Couso, I.; Gil, P., Extreme points of credal sets generated by 2-alternating capacities, Int. J. Approx. Reason., 33, 1, 95-115 (2003) · Zbl 1029.68135 |
| [28] | Miranda, E.; Destercke, S., Extreme points of the credal sets generated by comparative probabilities, J. Math. Psychol., 64-65, 44-57 (2015) · Zbl 1318.60060 |
| [29] | Miranda, E.; Montes, I., Shapley and Banzhaf values as probability transformations, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 26, 06, 917-947 (2018) · Zbl 1470.91024 |
| [30] | Montes, I.; Miranda, E.; Montes, S., Decision making with imprecise probabilities and utilities by means of statistical preference and stochastic dominance, Eur. J. Oper. Res., 234, 1, 209-220 (2014) · Zbl 1305.91106 |
| [31] | Montes, I.; Miranda, E.; Vicig, P., 2-monotone outer approximations of coherent lower probabilities, Int. J. Approx. Reason., 101, 181-205 (2018) · Zbl 1448.68422 |
| [32] | Nau, R., Imprecise probabilities in non-cooperative games, (Proceedings of ISIPTA (2011), Citeseer), 297-306 |
| [33] | Oberguggenberger, M.; King, J.; Schmelzer, B., Classical and imprecise probability methods for sensitivity analysis in engineering: a case study, Int. J. Approx. Reason., 50, 4, 680-693 (2009) |
| [34] | Omladič, M.; Škulj, D., Constructing copulas from shock models with imprecise distributions, Int. J. Approx. Reason., 118, 27-46 (2020) · Zbl 1471.62360 |
| [35] | Omladič, M.; Stopar, N., Final solution to the problem of relating a true copula to an imprecise copula, Fuzzy Sets Syst., 393, 96-112 (2020) · Zbl 1452.62353 |
| [36] | Omladič, M.; Stopar, N., A full scale Sklar’s theorem in the imprecise setting, Fuzzy Sets Syst., 393, 113-125 (2020) · Zbl 1452.62354 |
| [37] | Pearl, J., On probability intervals, Int. J. Approx. Reason., 2, 3, 211-216 (1988) |
| [38] | Pelessoni, R.; Vicig, P., Convex imprecise previsions, Reliab. Comput., 9, 6, 465-485 (2003) · Zbl 1037.60003 |
| [39] | Quaeghebeur, E., Introduction to the theory of imprecise probability, (Uncertainty in Engineering (2022), Springer: Springer Cham), 37-50 |
| [40] | Quost, B.; Destercke, S., Classification by pairwise coupling of imprecise probabilities, Pattern Recognit., 77, 412-425 (2018) |
| [41] | Rockafellar, R., Convex Analysis, Princeton Landmarks in Mathematics and Physics (1970), Princeton University Press · Zbl 0193.18401 |
| [42] | Schmeidler, D., Integral representation without additivity, Proc. Am. Math. Soc., 97, 255-261 (1986) · Zbl 0687.28008 |
| [43] | Shafer, G., Belief functions and possibility measures, (Analysis of Fuzzy Information (1987), CRC Press Inc.), 51-84 · Zbl 0655.94025 |
| [44] | Shapley, L. S., Cores of convex games, Int. J. Game Theory, 1, 1, 11-26 (1971) · Zbl 0222.90054 |
| [45] | Škulj, D., Perturbation bounds and degree of imprecision for uniquely convergent imprecise Markov chains, Linear Algebra Appl., 533, 336-356 (2017) · Zbl 1386.60254 |
| [46] | Škulj, D., Errors bounds for finite approximations of coherent lower previsions on finite probability spaces, Int. J. Approx. Reason., 105, 98-111 (2019) · Zbl 1452.68226 |
| [47] | Škulj, D., Computing bounds for imprecise continuous-time Markov chains using normal cones (2020), preprint |
| [48] | Smets, P., Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem, Int. J. Approx. Reason., 9, 1, 1-35 (1993) · Zbl 0796.68177 |
| [49] | Sundberg, C.; Wagner, C., Characterizations of monotone and 2-monotone capacities, J. Theor. Probab., 5, 1, 159-167 (1992) · Zbl 0746.60003 |
| [50] | Troffaes, M. C., Decision making under uncertainty using imprecise probabilities, Int. J. Approx. Reason., 45, 1, 17-29 (2007) · Zbl 1119.91028 |
| [51] | Troffaes, M. C.; De Cooman, G., Lower Previsions (2014), John Wiley & Sons · Zbl 1305.60007 |
| [52] | Utkin, L. V., An imprecise extension of SVM-based machine learning models, Neurocomputing, 331, 18-32 (2019) |
| [53] | Utkin, L. V.; Coolen, F. P., Imprecise reliability: an introductory overview, (Computational Intelligence in Reliability Engineering (2007)), 261-306 · Zbl 1158.62069 |
| [54] | Utkin, L. V.; Kovalev, M. S.; Coolen, F. P., Imprecise weighted extensions of random forests for classification and regression, Appl. Soft Comput., 92, Article 106324 pp. (2020) |
| [55] | Vicig, P., Financial risk measurement with imprecise probabilities, Int. J. Approx. Reason., 49, 1, 159-174 (2008) · Zbl 1185.91201 |
| [56] | Škulj, D., Discrete time Markov chains with interval probabilities, Int. J. Approx. Reason., 50, 8, 1314-1329 (Sep 2009) · Zbl 1195.60100 |
| [57] | Škulj, D., Efficient computation of the bounds of continuous time imprecise Markov chains, Appl. Math. Comput., 250, 165-180 (Jan 2015) · Zbl 1328.60180 |
| [58] | Škulj, D., Computing bounds for imprecise continuous-time Markov chains using normal cones, (Vasile, M.; Quagliarella, D., Advances in Uncertainty Quantification and Optimization Under Uncertainty with Aerospace Applications: Proceedings of the 2020 UQOP International Conf. (2020)) |
| [59] | Walley, P., Statistical Reasoning with Imprecise Probabilities (1991), Chapman and Hall: Chapman and Hall London, New York · Zbl 0732.62004 |
| [60] | Wallner, A., Extreme points of coherent probabilities in finite spaces, Int. J. Approx. Reason., 44, 3, 339-357 (2007) · Zbl 1114.60010 |
| [61] | Weichselberger, K., Elementare Grundbegriffe einer allgemeineren Wahrscheinlichkeitsrechnung. I: Intervallwahrscheinlichkeit als umfassendes Konzept (2001), Physica-Verlag: Physica-Verlag Heidelberg · Zbl 0979.60001 |
| [62] | Yu, L.; Destercke, S.; Sallak, M.; Schon, W., Comparing system reliabilities with ill-known probabilities, (International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (2016), Springer), 619-629 · Zbl 1458.93074 |
| [63] | Zhang, J.; Shields, M., On the quantification and efficient propagation of imprecise probabilities with copula dependence, Int. J. Approx. Reason., 122, 24-46 (2020) · Zbl 1445.68225 |
| [64] | Ziegler, G., Lectures on Polytopes, Graduate Texts in Mathematics (2012), Springer: Springer New York |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
