nach oben

Erschienen in:

2018 | OriginalPaper | Buchkapitel

Lower and Upper Probability Bounds for Some Conjunctions of Two Conditional Events

verfasst von : Giuseppe Sanfilippo

Erschienen in: Scalable Uncertainty Management

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In this paper we consider, in the framework of coherence, four different definitions of conjunction among conditional events. In each of these definitions the conjunction is still a conditional event. We first recall the different definitions of conjunction; then, given a coherent probability assessment (x, y) on a family of two conditional events \(\{A|H,B|K\}\), for each conjunction \((A|H) \wedge (B|K)\) we determine the (best) lower and upper bounds for the extension \(z=P[(A|H) \wedge (B|K)]\). We show that, in general, these lower and upper bounds differ from the classical Fréchet-Hoeffding bounds. Moreover, we recall a notion of conjunction studied in recent papers, such that the result of conjunction of two conditional events A|H and B|K is (not a conditional event, but) a suitable conditional random quantity, with values in the interval [0, 1]. Then, we remark that for this conjunction, among other properties, the Fréchet-Hoeffding bounds are preserved.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel On Instantiating Generalised Properties of Gradual Argumentation Frameworks

Nächstes Kapitel Qualitative Probabilistic Relational Models

Adams, E.W.: The Logic of Conditionals. Reidel, Dordrecht (1975)CrossRef

Benferhat, S., Dubois, D., Prade, H.: Nonmonotonic reasoning, conditional objects and possibility theory. Artif. Intell. 92, 259–276 (1997)MathSciNetCrossRef

Bochvar, D., Bergmann, M.: On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus. Hist. Philos. Log. 2(1–2), 87–112 (1981)MathSciNetCrossRef

Ciucci, D., Dubois, D.: Relationships between connectives in three-valued logics. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012. CCIS, vol. 297, pp. 633–642. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31709-5_64CrossRef

Ciucci, D., Dubois, D.: A map of dependencies among three-valued logics. Inf. Sci. 250, 162–177 (2013)MathSciNetCrossRef

Coletti, G., Gervasi, O., Tasso, S., Vantaggi, B.: Generalized bayesian inference in a fuzzy context: from theory to a virtual reality application. Comput. Stat. Data Anal. 56(4), 967–980 (2012)MathSciNetCrossRef

Coletti, G., Scozzafava, R.: Conditional probability, fuzzy sets, and possibility: A unifying view. Fuzzy Sets Syst. 144, 227–249 (2004)MathSciNetCrossRef

Coletti, G., Scozzafava, R., Vantaggi, B.: Coherent conditional probability, fuzzy inclusion and default rules. In: Yager, R., Abbasov, A., Reformat, M., Shahbazova, S. (eds.) Soft Computing: State of the Art Theory and Novel Applications. Studies in Fuzziness and Soft Computing, vol. 291, pp. 193–208. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-34922-5_14CrossRefMATH

Coletti, G., Scozzafava, R., Vantaggi, B.: Possibilistic and probabilistic logic under coherence: Default reasoning and System P. Math. Slovaca 65(4), 863–890 (2015)MathSciNetCrossRef

10.

Dubois, D., Prade, H.: Conditional objects as nonmonotonic consequence relationships. IEEE Trans. Syst. Man Cybern. 24(12), 1724–1740 (1994)MathSciNetCrossRef

11.

de Finetti, B.: La Logique de la Probabilité. In: Actes du Congrès International de Philosophie Scientifique, Paris, 1935, pp. IV 1–IV 9. Hermann et C.ie, Paris (1936)

12.

de Finetti, B.: Teoria delle probabilità. In: Einaudi (ed.) 2 vol., Torino (1970). English version: Theory of Probability 1 (2). Chichester, Wiley, 1974 (1975)

13.

Gale, D.: The Theory of Linear Economic Models. McGraw-Hill, New York (1960)MATH

14.

Gilio, A.: Criterio di penalizzazione e condizioni di coerenza nella valutazione soggettiva della probabilità. Boll. Un. Mat. Ital. 4–B(3, Serie 7), 645–660 (1990)MathSciNet

15.

Gilio, A.: \(C_0\)-Coherence and extension of conditional probabilities. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (eds.) Bayesian Statistics 4, pp. 633–640. Oxford University Press (1992)

16.

Gilio, A.: Probabilistic consistency of knowledge bases in inference systems. In: Clarke, M., Kruse, R., Moral, S. (eds.) ECSQARU 1993. LNCS, vol. 747, pp. 160–167. Springer, Heidelberg (1993). https://doi.org/10.1007/BFb0028196CrossRef

17.

Gilio, A.: Algorithms for conditional probability assessments. In: Berry, D.A., Chaloner, K.M., Geweke, J.K. (eds.) Bayesian Analysis in Statistics and Econometrics: Essays in Honor of Arnold Zellner, pp. 29–39. Wiley, New York (1996)

18.

Gilio, A.: Generalizing inference rules in a coherence-based probabilistic default reasoning. Int. J. Approx. Reason. 53(3), 413–434 (2012)MathSciNetCrossRef

19.

Gilio, A., Sanfilippo, G.: Quasi conjunction and p-entailment in nonmonotonic reasoning. In: Borgelt, C. (ed.) Combining Soft Computing and Statistical Methods in Data Analysis. AISC, vol. 77, pp. 321–328. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14746-3_40CrossRef

20.

Gilio, A., Sanfilippo, G.: Coherent conditional probabilities and proper scoring rules. In: Proceedings of ISIPTA 2011, Innsbruck, pp. 189–198 (2011)

21.

Gilio, A., Sanfilippo, G.: Quasi conjunction, quasi disjunction, t-norms and t-conorms: probabilistic aspects. Inf. Sci. 245, 146–167 (2013)MathSciNetCrossRef

22.

Gilio, A., Sanfilippo, G.: Conditional random quantities and compounds of conditionals. Stud. Log. 102(4), 709–729 (2014)MathSciNetCrossRef

23.

Gilio, A., Sanfilippo, G.: Conjunction and disjunction among conditional events. In: Benferhat, S., Tabia, K., Ali, M. (eds.) IEA/AIE 2017. LNCS (LNAI), vol. 10351, pp. 85–96. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-60045-1_11CrossRef

24.

Gilio, A., Sanfilippo, G.: Generalized logical operations among conditional events. Applied Intelligence (in press). https://doi.org/10.1007/s10489-018-1229-8

25.

Goodman, I.R., Nguyen, H.T.: Conditional objects and the modeling of uncertainties. In: Gupta, M.M., Yamakawa, T. (eds.) Fuzzy Computing, pp. 119–138. North-Holland, Amsterdam (1988)

26.

Goodman, I.R., Nguyen, H.T., Walker, E.A.: Conditional Inference and Logic for Intelligent Systems: A Theory of Measure-Free Conditioning. North-Holland, Amsterdam (1991)

27.

Milne, P.: Bruno de Finetti and the logic of conditional events. Br. J. Philos. Sci. 48(2), 195–232 (1997)MathSciNetCrossRef

28.

Pelessoni, R., Vicig, P.: The Goodman-Nguyen relation within imprecise probability theory. Int. J. Approx. Reason. 55(8), 1694–1707 (2014)MathSciNetCrossRef

29.

Sanfilippo, G., Pfeifer, N., Over, D.E., Gilio, A.: Probabilistic inferences from conjoined to iterated conditionals. Int. J. Approx. Reason. 93, 103–118 (2018)MathSciNetCrossRef

Titel: Lower and Upper Probability Bounds for Some Conjunctions of Two Conditional Events
verfasst von: Giuseppe Sanfilippo
Verlag: Springer International Publishing
Buch: Scalable Uncertainty Management
Print ISBN: 978-3-030-00460-6

Electronic ISBN: 978-3-030-00461-3

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-030-00461-3_18

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"